Dispersion theory and sum rules in linear and nonlinear optics

RIVISTA DEL NUOVO CIMENTO DOI 10.1393/ncr/i2003-10006-x

Vol. 26, N. 12

2003

Dispersion theory and sum rules in linear and nonlinear optics V. Lucarini(1 )(∗ ), F. Bassani(2 )(3 ), K.-E. Peiponen(1 ) and J. J. Saarinen(1 ) (1 ) Department of Physics, University of Joensuu P.O. Box 111, FIN-80101 Joensuu, Finland (2 ) Scuola Normale Superiore - Piazza dei Cavalieri 7, 56126 Pisa, Italy (3 ) Istituto Nazionale di Fisica della Materia - Corso Perrone 22, 16152 Genova, Italy (ricevuto il 10 Dicembre 2003)

3 6 6 6 10 12 12 15 16 17 19

1. 2.

20 21 21 23 23 25 26 27 28 28 28 29 30 31 32 33 34 34 36 39

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Introduction Linear optical response . 2 1. Introduction . 2 2. Maxwell’s equations and optical functions . 2 3. Fresnel’s equations . 2 4. Microscopic definition of the susceptibility . 2 4.1. Lagrangian and Hamiltonian functions of the system . 2 4.2. Quantum description of the susceptibility . 2 4.3. Linear Green function and susceptibility . 2 4.4. The asymptotic behaviour of the linear susceptibility . 2 4.5. An equivalent derivation of the asymptotic behaviour of the linear susceptibility . 2 4.6. Corrections due to relativistic and spatial dispersion effects . 2 5. Local electric field and effective-medium approximation . 2 5.1. Homogeneous media . 2 5.2. Two-phase media . 2 5.3. The Maxwell Garnett system . 2 5.4. The Bruggeman effective-medium theory . 2 5.5. Layered nanostructures . 2 5.6. Asymptotic behaviour of the linear susceptibility for nanostructures Dispersion relations and sum rules in linear optical spectroscopy . 3 1. The principle of causality . 3 2. Titchmarsch’s theorem and Kramers-Kronig relations . 3 2.1. Kramers-Kronig relations for conductors . 3 2.2. Kramers-Kronig relations for nanostructures . 3 3. Superconvergence theorem and sum rules . 3 4. Sum rules for conductors . 3 5. Sum rules for nanostructures . 3 6. Integral properties of the optical constants . 3 6.1. Integral properties of the index of refraction . 3 6.2. Linear reflectance spectroscopy . 3 7. Generalization of the integral properties for more effective data analysis

(∗ ) E-mail: [email protected] c Societ` a Italiana di Fisica

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2 39 41 42 45 45 46 49 51 53 55 55 56 57 58 59 60 62 65 67 70 74 76 76 77 77 80 82 82 84 85 86 88 88 88 88 91 96 100 103 105 106 107 111

V. LUCARINI, F. BASSANI, K.-E. PEIPONEN

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. 3 7.1. Generalized Kramers-Kronig relations . 3 7.2. Subtractive Kramers-Kronig relations . 3 8. The harmonic oscillator model 4. General properties of the nonlinear optical response . 4 1. Historical and introductory remarks on nonlinear optics . 4 2. Nonlinear Green functions and nonlinear optical functions . 4 3. Pump and probe susceptibility . 4 4. Quantum-mechanical expression of the nonlinear Green functions . 4 5. Local field effects . 4 5.1. Relevance of local field effects in nanostructured media . 4 5.2. The Maxwell Garnett system . 4 5.3. The Bruggeman effective-medium theory . 4 5.4. Layered nanostructures 5. Dispersion relations and sum rules for nonlinear optical processes . 5 1. Kramers-Kronig relations for nonlinear susceptibility: independent frequency variables . 5 2. Scandolo’s theorem and nonlinear Kramers-Kronig relations . 5 3. Nonlinear Kramers-Kronig relations and sum rules: the case of pump and probe . 5 4. Additional Kramers-Kronig relations and sum rules for the case of pump and probe susceptibility . 5 5. Nonlinear oscillator model for pump and probe systems . 5 6. Generalized Kramers-Kronig–type relations and sum rules . 5 7. Modified Kramers-Kronig relations and sum rules for meromorphic total index of refraction 6. General properties of the harmonic generation susceptibility . 6 1. Introduction . 6 2. Scandolo’s theorem and harmonic generation susceptibility . 6 3. Asymptotic behaviour of the harmonic generation susceptibility . 6 4. Kramers-Kronig relations and sum rules . 6 4.1. Case of nonlinear conductors . 6 5. Nonlinear oscillator model for harmonic generation processes . 6 5.1. Generalized Miller’s rule . 6 5.2. Kramers-Kronig relations and sum rules . 6 6. Subtractive Kramers-Kronig relations for harmonic generation susceptibility 7. Kramers-Kronig relations and sum rules for data analysis: examples . 7 1. Introduction . 7 2. Efficacy of generalized Kramers-Kronig relations for data inversion . 7 2.1. Kramers-Kronig inversion of the susceptibility . 7 2.2. Kramers-Kronig inversion of the second power of the susceptibility . 7 3. Verification of the sum rules . 7 4. Application of singly subtractive Kramers-Kronig relations . 7 5. Application of sum rules in second harmonic generation processes 8. Conclusions Appendix A Derivation of the asymptotic behaviour of the pump and probe susceptibility Appendix B Derivation of the asymptotic behaviour of the n-th–order harmonic generation susceptibility Appendix C Convergence of Kramers-Kronig–type dispersion relations

DISPERSION THEORY AND SUM RULES IN LINEAR AND NONLINEAR OPTICS

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1. – Introduction There is an ever-growing demand in optical-material research to measure more accurately linear and nonlinear optical spectra of various bulk and composite materials. On the one hand, this is made possible by the continuous development of novel optoelectronic devices which are used, for instance, as light sources and detectors. On the other hand, in the field of life science, such as bio-optical medicine, interaction of light with different species can be exploited, for instance, in novel drug development. Recording of the wavelength-dependent optical spectrum, in the field of linear optics, is probably the best and most important tool in the analysis of different constituents of any medium. In many cases it is impossible to gain information on all optical properties of media just by recording spectra and, therefore, other means are needed. They are obtained by using various mathematical methods, which provide important steps in the assessment of the optical properties of the medium. In the case of nonlinear optical spectroscopy, which has greatly developed during the last decades thanks to tunable lasers, the requirement of reliable spectra analysis of nonlinear optical properties has crucial importance in basic optical-material research, proper device development and in the field of life science. The requirement of space-time causality in the optical response provides the optical functions with general properties such as Kramers-Kronig dispersion relations and sum rules. The Kramers-Kronig (K-K) dispersion relations [1, 2] and the sum rules [3-6] have long constituted fundamental tools in the investigation of the linear optical properties of condensed matter, gases, molecules, and liquids, besides having had a key role in the initial development of quantum mechanics. Furthermore, K-K relations have played a very relevant role in different fields, such as high-energy physics, acoustics, statistical physics, and signal processing. The K-K relations describe a fundamental connection between the real and imaginary part of the complex optical functions descriptive of the light-matter interaction phenomena, such as the susceptibility, the dielectric function or the index of refraction. The real and imaginary parts are not wholly independent, but are connected by a special form of the Hilbert transforms, which are termed K-K relations. The sum rules are universal constraints that determine the results of the integration over the infinite spectral range of the functions descriptive of relevant optical properties of the medium under investigation. Hence, these integral properties provide constraints for checking the self-consistency of experimental or model-generated data. Furthermore, by applying the K-K relations it is possible to perform the so-called inversion of optical data, i.e. to acquire knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole spectrum or vice versa [7-13]. The K-K relations have been traditionally exploited in linear optical spectroscopy for data inversion and phase retrieval from measured spectra of insulating, metallic and semiconducting media. Actually much of our present knowledge about the linear optical properties of these media, such as their real refractive index and extinction coefficient, is based on the exploitation of the K-K relations. The conceptual foundations of such general properties lie upon the principle of causality [14] in the light-matter interaction, and the connection between the mathematical properties of the functions describing the physics in the time and frequency domains is provided by the Titchmarsch theorem [10]. Under the quite general hypotheses of validity of the perturbative approach [15, 16], the n-th–order nonlinear optical properties of any material can be completely described by the corresponding n-th–order nonlinear susceptibilities [16-18]. These are the multivariable Fourier transforms of the nonlinear Green functions, which describe the higher-

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order dynamics of the system. The Scandolo theorem [19, 20] is a recent extension of the Titchmarsch theorem which permits to frame in very general terms the possibility of establishing dispersion relations for the nonlinear optical phenomena [19-25]. Scandolo’s theorem permits the determination of the class of nonlinear susceptibility functions that obey K-K relations up to any order. In spite of the ever-increasing scientific and technological relevance of nonlinear optical phenomena, relatively little attention has been paid for a long time to the experimental investigation of the K-K relations and the sum rules of the corresponding nonlinear susceptibilities [26-29]. The research has usually focused on achieving high resolution in both experimental data and theoretical calculations, while these integral properties are especially relevant for experimental investigations of frequency-dependent nonlinear optical properties. In the context of this class of experiments, K-K relations and sum rules could provide information on whether or not a coherent, common picture of the nonlinear properties of the material under investigation is available [30]. In particular, these integral properties permit the framing of peculiar phenomena related to mattermatter or light-matter couplings that are very relevant at given frequencies, such as the excitonic or polaritonic effects in solids [31], in the context of the interaction on the whole spectral range, showing that their dispersive and absorptive contributions are connected to all the other contributions in the entire spectral range. The technical problem of measuring the nonlinear excitation spectrum on a relatively wide frequency range is probably the single most important reason why experimental research in this field has been subdued for a long time. In this paper we present a detailed analysis of the general integral properties of the susceptibility functions descriptive of linear and nonlinear optical processes. We deduce rigorously, using a general approach, a complete set of independent K-K dispersion relations and sum rules, which provide constraints determining the self-consistency of experimental or model-generated data. Furthermore, we show that by applying K-K relations it is possible to get information on the real part of the susceptibility by measurements over all the spectrum of the imaginary part or vice versa. We then describe the applications of K-K relations and sum rules on experimental data. We also present the general expressions and describe the applications of singly and multiply subtractive K-K relations, which have a great relevance in experimental data analysis since the consideration of one or more anchor points in data inversion relaxes the error due to the inevitably finite spectral range. In terms of microscopic physics, in this paper we focus our attention on the nonrelativistic domain, also neglecting spatial dispersion effects. Consideration of relativistic and dispersive effects give much smaller contributions [32]. We study the response of the system to the electromagnetic perturbation through the formalism developed by Kubo [33], since, in spite of its rather abstract formulation, it is well suited for universal treatment of all materials, it can be easily generalized to treat nonlinear phenomena, and allows to obtain in a very natural way the information relevant to the establishment of the integral properties of the susceptibility. We treat the case of a general quantum system by using the density matrix formalism [34]. In this paper we also emphasize the relevant result that some general integral properties for the linear and nonlinear susceptibility can be obtained from simple classical oscillator models [13, 21-23, 35, 36], the Drude-Lorentz model for linear processes and its generalization with the introduction of a generic non-harmonic potential for nonlinear processes [16, 37, 38]. We wish to underline that a quite comprehensive treatment of the integral properties


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of nonlinear susceptibilities descriptive of the transients, which can be experimentally obtained by using very short pulse lasers, characterized by pulse durations of the order of picoseconds or less, has been presented in [25, 39]. We will not present these results again, focusing instead on the continuous-wave approximation. The paper is organized as follows: • In sect. 2 we define the general microscopic system under examination and introduce the susceptibility functions, obtained as the Fourier transform of the linear Green function of the system. We also present a treatment of local field effects with specific applications to homogenous and nanostructured media. • In sect. 3 we present the general properties of the linear susceptibility; we derive in particular the K-K relations and the sum rules for the susceptibility and other optical constants such as the index of refraction and reflectivity. We also review some extensions of dispersion theory for linear optics, which permit a much more efficient treatment of experimental data. We mention the effect of spatial dispersion and relativistic corrections. • In sect. 4 we describe the higher-order dynamics of the system and consequently introduce the nonlinear Green function, showing its relation to the general nonlinear susceptibility. We also present a discussion of nonlinear local field effects and show how they can be described with the effective medium approximation for both homogeneous and nanostructured media. • In sect. 5 we discuss the holomorphic properties of the general nonlinear susceptibility with respect to the frequency variables and we present the K-K relations and the sum rules for the experimentally and theoretically relevant case of pump and probe systems. We then introduce the nonlinear oscillator model and show the consistency of its results with the quantum theory findings. • In sect. 6 we study the general integral properties of the arbitrary-order harmonic generation susceptibility and establish a full set of independent K-K relations and sum rules. We compare the results obtained using a full quantum treatment with those derived using the classical nonlinear oscillator model. We also introduce the theory of singly and multiply subtractive K-K relations with the purpose of proposing a technique suited for retrieving the full optical constants from a few experimental data. • In sect. 7 we present applications of the theoretical results to the experimental data of third harmonic generation susceptibility on polymers, and we show how stringent are the constraints imposed by the K-K relations and how the consideration of anchor points improves the optical data inversion process. We also discuss the role of sum rules in the analysis of second harmonic generation processes for different materials. • In sect. 8 we present our conclusions and perspective for future work. • In appendices A, B, C, and we present some detailed mathematical derivations, which are useful but a bit cumbersome to be included in the main text.

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2. – Linear optical response . 2 1. Introduction. – Linear optical spectroscopy is probably the most thoroughly exploited tool in optical-material research. In this subsection we first introduce the linear optical response of a medium to an external electromagnetic perturbation by using very basic macroscopic electromagnetic theory. In this context we provide the definition of susceptibility and of the most relevant optical constants that can be derived from it. We then provide a microscopic picture of the above quantities by introducing a general quantum framework able to describe thoroughly the optical response at microscopic level. The linear susceptibility results to be the Fourier transform of the linear Green function of the quantum microscopic system. We also present a treatment of local field effects and detailed results regarding homogenous and nanostructured media. . 2 2. Maxwell’s equations and optical functions. – Maxwell’s equations relate the space and time derivatives of the electric and magnetic fields to each other throughout the continuous medium. In this subsection we only present the main definitions and results which are needed for the scopes of this work; for a detailed description of the electromagnetic theory see, e.g., [7] and [40]. We adopt c.g.s.-Gauss units, and express Maxwell equations as follows [40]: (1)

r, t) = 4πρ(r, t), ∇ · D(

(2)

r, t) = 0, ∇ · B(

(3)

(4)

r, t) = − ∇ × E(

r, t) = ∇ × H(

1 ∂ B(r, t), c ∂t

1 ∂ 4π J(r, t) + D(r, t), c c ∂t

r, t) denotes the electric field, where r is the 3-dimensional coordinate vector. Here E( r, t) the magnetic inducH(r, t) the magnetic field, D(r, t) the electric displacement, B( tion, J(r, t) the current density, and ρ(r, t) the charge density. The electric displacement and the magnetic induction are connected to electric field and magnetic field, respectively, by the constitutive equations: (5)

r, t) = E( r, t) + 4π P (r, t), D(

(6)

r, t) = H( r, t) + 4π M (r, t), B(

(r, t) are the polarization and magnetization of the medium, respecwhere P (r, t) and M tively. These quantities describe the response of the medium to the applied electromagnetic field. If we apply the Fourier transform [41] for both space and time to the Maxwell equations, we obtain the following set of equations: (7)

k · D( k, ω) = 4πρ(k, ω) ,


(8)

k · B( k, ω) = 0 .

(9)

k × E( k, ω) , k, ω) = ω B( c

(10)

k × H( k, ω) − ω D( k, ω) , k, ω) = − 4iπ J( c c

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In the equations above, k and ω are the wave vector and the angular frequency of the electromagnetic fields under consideration. At optical frequencies the materials are usually non-magnetic and the magnetization k, ω) = B( k, ω). Under this fairly good approximation, the can be omitted, so that H( optical response of a medium to an electromagnetic perturbation is completely described by the constitutive relation between the polarization and the electric field. The linear polarization P (1) (r, t) provides an extensive description of the light-matter interaction when low radiation intensities are considered. When we consider more intense light sources, new effects come into play and the picture of the phenomena becomes more and more complex, since entirely new classes of processes can be experimentally observed. The linear polarization can be expressed in general with the following convolution product [12]:

(11)

(1) Pi (r, t)

∞ =

(1) Gij (r , t )Ejloc (r − r , t − t )dt dr ,

R3 −∞

loc (r, t) is the local electric field induced by the external perturbation acting in where E (1) the location r at time t , while the tensor Gij (r, t) is the linear Green function accounting for the behaviour of the system caused by the presence of an additional electric field. We emphasize that the polarization is a macroscopic quantity, which derives from a suitably defined small-scale spatial average of the corresponding microscopic quantity, the polarizability [7, 42-44], which describes the distortion of the dipole field on an atomic scale induced by the interaction with the external oscillating field. Simple geometric arguments suggest that the ratio between the polarization and the polarizability is just the number of elementary components of the medium per unit volume. Effectively, this is not precisely the case, since in the definition of the averaged macroscopic quantity not only the external applied field but also the local dipole fields induced by the single elementary components of the medium yield contributions [43-47]. Therefore, the electric field inducing the polarization is not the macroscopic external field of the incoming loc (r, t) = E( r, t), where the latter is the external electric radiation. This implies that E field of the incoming radiation. In the derivation of the main properties of the linear loc (r, t) ∼ optical response we will ignore the local field corrections and assume that E . r, t). In subsect. 2 5 we will deal with the local field effects and show how their E( inclusion modify our results. We compute the Fourier transform in the time and space domains of both members of expression (11), and considering that the Fourier transform of a convolution product of two functions is the conventional product of the Fourier transforms of the two

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functions [41], we obtain (12)

(1) (1) Pi (k, ω) = Φ F Pi (r, t) = (1) (1) = Φ F Gij (r, t) Φ {F [E(r, t)]} = χij (k, ω)Ej (k, ω),

where Φ and F indicate, respectively, the application of the Fourier transform in the space and time domains. We have indicated the linear susceptibility of the system as (13)

(1) (1) χij (k, ω) = Φ F Gij (r, t) .

If we apply the Fourier transform to the first constitutive equation presented in (5), we obtain (14)

(1) (k, ω) = E( k, ω) + 4π P (1) (k, ω). D

By inserting into eq. (14) the result obtained in (12) for the linear polarization, we derive the following expression for each component of the electric-displacement vector: (15)

(1) (1) Di (k, ω) = δij + 4πχij (k, ω) Ej (k, ω),

where δij is the Kronecker delta, whose value is 1 if i = j and 0 if i = j. We can then express the constitutive relation between the electric displacement and the electric field as follows: (16)

(1) (1) Di (k, ω) = ij (k, ω)Ej (k, ω),

where ij (k, ω) is the linear dielectric tensor. If we consider an isotropic medium or a medium with cubic symmetry, the susceptibility (13) and the dielectric (16) tensors are diagonal in all coordinate systems [40], therefore they can be expressed in the following way: (1) (17) χij k , ω = δij χ(1) k , ω ,

(18)

(1)

ij

k , ω = δij (1) k , ω ,

where k is the length of the vector k. If the susceptibility tensor —and so also the dielectric tensor— is in the form (17), we can treat the linear susceptibility and the dielectric function as scalar quantities: k, ω), (19) P (1) (k, ω) = χ(1) k , ω E(

(20)

(1) (k, ω) = (1) k , ω E( k, ω). D


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In such a case, and if we assume that no sources are present (i.e., ρ(k, ω) = J(k, ω) = 0), it is possible to derive straightforwardly the electromagnetic wave equations [7, 42], thus obtaining for the electric field (21)

2 ω 2 k, ω . k E k, ω = (1) k , ω 2 E c

This implies that the following dispersion relation holds: 2 ω 2 k = (1) k , ω 2 . c

(22)

Hence, we have that each monochromatic component of the radiation wave can be written as    k c ω (r, t) = eÊ ω exp i ω  (23) E kˆ · r − ct + c.c. , c ω

(24)

ω (r, t) = c k × E ω (r, t) , B ω

where the unit vector eˆ gives the polarization of the light, kˆ is the unit vector in the ω is a constant vector. The electric and magnetic fields direction of propagation k, and E are orthogonal to k, as implied from the two Maxwell equations (7), (8) with no sources. The absolute value k can be expressed as a function of ω thanks to the dispersion relation (22), so that we obtain ω (1) ω (r, t) = eÊ ω exp i E + c.c. = k , ω kˆ · r − ct c ω exp i ω N k , ω kˆ · r − ct + c.c. = = eÊ c ω n k , ω kˆ · r − ct exp −κkˆ · r + c.c., = eÊω exp i c

(25)

where we have introduced the customary notation for the complex index of refraction [42]: (26) N k , ω = (1) k , ω = 1 + 4πχ(1) k , ω = n k , ω + iκ k , ω . The index of refraction, being defined as the square root of a complex function of the variable is in general a complex function. The real part of the index of refraction, ω, n k , ω , is responsible for the dispersive optical phenomena, since eq. (25) implies that the phase velocity of the travelling wave (23) is v k , ω = c/n k , ω . The imaginary part, κ k , ω , is related to the phenomena of light absorption, since it introduces a non-oscillatory real exponential which describes an extinction process. Using

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the definition (26), we can find the following equations relating the real and imaginary part of the index of refraction and of the susceptibility: (27)

2 2 = n k , ω (1) k , ω − κ k , ω ,

(28)

= 2n k , ω κ k , ω , (1) k , ω

where and indicate the real and imaginary part, respectively. The polarization of the electromagnetic wave is defined by its electric-field component. For the sake of completeness, we wish to underline that it is possible to present a similar treatment of the linear response of the medium to an external magnetic field. The linear response is fully described by the magnetic permittivity tensor, which allows the closure of the constitutive equation (6): (29)

(1) Bi (k, ω) = µij (k, ω)Hj (k, ω).

Analogously to the electric case, if we are dealing with a medium that is characterized by isotropy or cubic symmetry, the magnetic permittivity tensor can be treated as a scalar: (30)

k, ω). k, ω) = µ(1) k , ω H( B(

If we include the magnetic effects in the derivation of the wave equation for the electromagnetic field, the dispersion relation (22) is changed into the following: (31)

2 ω 2 k = (1) k , ω µ(1) k , ω 2 . c

As a consequence, the index of refraction has to be redefined as follows: (32)

N k , ω = (1) k , ω µ(1) k , ω .

We note that if we set µij (k, ω) = δij , we obtain the non-magnetic case. . 2 3. Fresnel’s equations. – Maxwell’s equations hold everywhere inside the continuous medium. At the boundary of materials 1 and 2 the boundary conditions are used to propagate the electromagnetic field. In order to simplify the notation, in this and in the next subsection we drop the space and time dependence when we refer to the electromagnetic quantities. The usual boundary conditions can be derived from the integral form of Maxwell’s equations and give continuity of the tangential components of the electric and magnetic field at the boundary and of the normal components of the electric displacement and magnetic induction. Suppose that a monochromatic electromagnetic plane wave with frequency ω of the form (25) arrives at the boundary of two media with complex refractive indices N1 (ω) and N2 (ω) at x = 0 as presented in fig. 1. The structure is assumed to be invariant in the y-direction and the components of the electromagnetic wave can be expressed in the form


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Fig. 1. – Geometry for the derivation of Fresnel’s equations.

= E(x, = H(x, E z, t) and H z, t). We consider two light beams, one with the electric field perpendicular to the plane of incidence (TE-polarization), and one with the electric field oscillating in the plane of the beams (TM-polarization). The angle of the reflected beam equals the angle of incidence, but the angle of the refracted beam is obtained from the law (33)

N1 (ω) sin ϕ1 = N2 (ω) sin ϕ2 ,

which can be derived from boundary conditions and constitutes the generalization of Snell’s law for the complex case. For TE-polarized light, the electric field, which oscillates along the y-direction, and the z-component of the magnetic field are continuous at the boundary. These conditions are given by (34)

Eωi + Eωr = Eωt ,

(35)

Eωi N1 (ω) cos ϕ1 + Eωr N1 (ω) cos ϕ1 = Eωt N2 (ω) cos ϕ2 ,

where eq. (35) has been obtained using eqs. (23)-(25). Using the fact that ϕi = ϕr , one obtains Fresnel’s amplitude reflection coefficient for TE-polarized light as follows: Eωr N1 (ω) cos ϕi − N2 (ω) cos ϕt (36) rs (ω) = = Eωi TE N1 (ω) cos ϕi + N2 (ω) cos ϕt and the amplitude transmission coefficient as Eωt 2N1 (ω) cos ϕi (37) ts (ω) = = . Eωi TE N1 (ω) cos ϕi + N2 (ω) cos ϕt For TM-polarized light the z-component of the electric field and the y-component of the magnetic field are continuous. Analogously with the previous case, we obtain (38)

Eωi cos ϕi − Eωr cos ϕr = Eωt cos ϕt ,

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(39)

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Ni (ω)Eωi + Nr (ω)Eωr = Nt (ω)Eωt .

The combination of equations results in an amplitude reflection coefficient for TMpolarized light that is (40)

rp (ω) =

Eωr Eωi

= TM

N2 (ω) cos ϕi − N1 (ω) cos ϕt . N2 (ω) cos ϕi + N1 (ω) cos ϕt

The amplitude transmission coefficient is of the form (41)

tp (ω) =

Eωt Eωi

= TM

2N1 (ω) cos ϕi . N1 (ω) cos ϕi + N2 (ω) cos ϕt

Fresnel’s equations are used in the calculations of the reflectance and transmittance through the boundaries. Furthermore, they are used to take into account the possible phase shifts occurring at the boundary. . 2 4. Microscopic definition of the susceptibility. – We present a microscopic treatment of the response of the system to the external perturbation and derive an expression for the linear susceptibility. We assume that the physics of the system does not depend appreciably on k, which is equivalent to considering that the wave vectors obey the following constraint: k d 1,

(42)

where d is of the order of the dimensions of the characteristic elementary constituent of the system, the atoms and the molecules for the gases and liquids, and the elementary crystalline cells for the solids. We then ignore in all calculations the spatial dispersion effects, so that we drop all the r dependences in eq. (11) and we assume the limit |k| → 0: (1) (1) χij (ω) ≡ lim χij (k, ω).

(43)

k→0

This is the usual form of the electric-dipole approximation [12], which is widely adopted for the description of the light-matter interaction when the optical frequencies are considered. . 2 4.1. Lagrangian and Hamiltonian functions of the system. We consider a volume V containing N electrons with charge −e that interact with a time-dependent electromagnetic field and are subjected to a static scalar potential and to mutual repulsion. The non-relativistic Lagrangian of the system can then be written as [7, 16] (44)

L=

N α=1

m

N N N e2 e ˙α α r˙ α · r˙ α − − V (r α ) − r · A(r , t), α β 2 |r − r | α=1 c α=1 α=β=1

is the vector potential of the where r is the vector denoting the spatial position, A external electromagnetic field, α is the index of the single particle, c is the speed of light in the vacuum, and the dot indicates the temporal derivative. We recall that using

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the Coulomb gauge we can establish the following relation between the external vector r, t) and the external electric field E( r, t) [7, 42]: potential A( r, t). r, t) = − 1 ∂t A( E( c

(45)

Using the conventional Legendre trasformation [48] we obtain 1 e α r˙ α = p α − A(r , t). m mc

(46)

The corresponding non-relativistic Hamiltonian H governing the dynamics of the system is given by the sum of two terms: (47)

H = H0 + HI ,

where the first term is the unperturbed, time-independent Hamiltonian:

H0 = T + V + Hee =

(48)

N N N p α · p α e2 + , V (r α ) + 2m |r α − r β | α=1 α=1 α=β=1

where T , V , and Hee are respectively the total kinetic energy, the total one-particle potential energy, and the electron-electron interaction potential energy. The second term is the Hamiltonian of interaction of the electrons with the electromagnetic field:

(49)

HI =

N α=1

2 α α e α α r α , t) · p α + e A(r , t) · A(r , t) p · A(r , t) + A( 2mc 2mc2

,

where we have symmetrized between the vector potential and the conju the dot products r α , t), p α = 0, with the usual definition of the commutator gate moments [16], since A( [a, b] = ab − ba. The dynamics of the system does not change if we add a total time derivative [48] to the original Lagrangian (44), so that a wholly equivalent Lagrangian is

(50)

˜ = L+ d L dt =

N α=1

m

N e ˙α α r · A(r , t) = c α=1

N N N e2 e α d α r˙ α · r˙ α − − r · A(r , t) . V (r α ) − α β 2 |r − r | α=1 c dt α=1 α=β=1

˜ which If we apply the Legendre transformation, we obtain a new Hamiltonian function H, is fully equivalent to H: (51)

˜ = H0 + H ˜ I, H

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where H0 is the same as (48), while we have a new form of the Hamiltonian describing the interaction between the external electromagnetic field and the electrons: (52)

˜I = − H

N e α d α r · A(r , t). c dt α=1

The two interaction Hamiltonians (49) and (52) are then totally equivalent and can be used interchangeably [49]. If we assume the dipolar approximation, we can neglect the so that it is only a function of time. Hence, spatial dependence of the vector potential A, for every α its commutator with the momentum operator gives zero: (53)

A(t), p α = 0.

We then obtain that in the dipolar approximation the interaction Hamiltonian (49) can be written as N 2 A(t) · A(t) e e + p α · A(t) . (54) HIt (v) = mc 2mc2 α=1 Expression (54) describes the light-matter interaction in the so-called gauge of velocity. We wish to emphasize that in the dipolar approximation we can find for the interaction Hamiltonian a very instructive expression that involves the conventional dipole–electricfield interaction. We first consider that in the dipolar approximation total and partial time derivative can be identified, so that (55)

d A(t) = ∂t A(t). dt

Considering that in the dipolar approximation the constitutive relation (45) can be written as (56)

1 E(t) = − ∂t A(t), c

the equivalent interaction Hamiltonian (54) can be eventually expressed in the following very instructive way: (57)

HIt = e

N

r α · E(t).

α=1

Expression (57) is often referred to as interaction Hamiltonian in the gauge of length. The notation on the temporal dependence of the interaction Hamiltonian functions (54) and (57) depends on the fact that we want to emphasize the choice of the dipolar approximation, which implies that the external fields are only time dependent. We emphasize that, even if we are guaranteed that the physics described with the two gauges is exactly the same, the two gauges have very different performances when adopted for calculations in real systems, especially in nonlinear cases. In fact, the equivalence is


15

respected only if a truly complete set of eigenstates of the unperturbed Hamiltonian is considered for the intermediate states in the calculations. In actual numerical evaluations of real systems it is impossible to use complete sets, therefore the issue of evaluating the relative efficiency of the length and velocity gauge for approximate calculations is of crucial importance [50, 51]. . 2 4.2. Quantum description of the susceptibility. All the statistical properties of a general quantum system of electrons can be described with the density operator ρ [34] defined as (58) ρ= ρab |ab|, a,b

in terms of a complete set of normalized (ai |aj = δij ) eigenstates of the unperturbed, time-independent Hamiltonian H0 in the Hilbert space of N indistinguishable fermionic particles, and the coefficients ρab describe the statistical mixture. The density is normalized to 1 by imposing that (59)

Tr{ρ} =

ρaa = 1.

a

We define the expectation value of the operator O as [34] (60)

Tr{Oρ} =

Oab ρba ,

a,b

where we have considered the usual definition Oab = a|O|b. The electric polarization per unit volume can then be defined as the expectation value of the dipole moment per unit volume [17, 18, 34, 42]: (61)

1 P (t) = Tr −e r α ρ(t) , V a

where ρ(t) is the evolution at the time t of the density matrix of the system, with the initial condition given by the Boltzmann equilibrium distribution: (62)

ρ(0) =

a

e−Ea /KT |aa| −E /KT . e a a

We assume that the material under examination has no permanent electrical dipole at the thermodynamic equilibrium: (63)

Tr −e

α

r ρ(0)

= 0.

α

In this study we ignore the effects of natural radiative decay and noise-induced quantum dephasing [16]. Hence, we consider a purely semiclassical treatment of the light-matter

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interaction so that ρ(t) obeys the Liouville differential equation [17] where the relaxation terms are ignored: (64)

i∂t ρ(t) = [H, ρ(t)] = [H0 , ρ(t)] + HIt , ρ(t) ,

with boundary condition (62) at the time t = 0. We can solve eq. (64) using a perturbative approach and we can write ρ(t) as a sum consisting of terms of decreasing magnitude: (65)

ρ(t) = ρ(0) + ρ(1) (t) + . . . + ρ(n) (t) + . . . .

Substituting (65) in eq. (64) and equating the terms of the same order we obtain the concatenated system of coupled differential equations: i∂t ρ(1) (t) = H0 , ρ(1) (t) + HIt , ρ(0) (66)

... = ... i∂t ρ(n) (t) = H0 , ρ(n) (t) + HIt , ρ(n−1) (t) ... = ... .

The validity of the perturbative approach is related to the assumption that the absolute value of the oscillating electric field affecting the electrons is much smaller than the characteristic static electric field, due to the interactions with the nucleus and with the other electrons. Inasmuch as the initial condition of eq. (64) is given by eq. (63), we have that ρ(n) (0) = 0 for each n > 0. In the case of the linear response we will limit ourselves to the first order and therefore consider the solution of the first differential equation from the top of the system (66). . 2 4.3. Linear Green function and susceptibility. We follow [17] and use the length gauge formulation for the interaction Hamiltonian (57) so that the differential equation for ρ(1) (t) can be written as

(67)

(1)

i∂t ρ

(1)

(t) = H0 , ρ

N α (t) + e r · E(t), ρ(0) . α=1

This is a first-order differential equation that can be solved with the method of the variation of the arbitrary constants. By considering the first-order terms in expression (61), we obtain the following definition for the linear polarization:

(68)

1 (1) α (1) P (t) = Tr −e r ρ (t) . V a

Inserting the expression of ρ(1) (t) obtained by solving eq. (67), we derive the explicit

17


expression for the linear polarization:

(69)

(1) Pi (t)

∞

(1)

Gij (τ )Ej (t − τ )dτ =

= −∞

e2 =− iV

∞ Ej (t − τ )θ(τ )Tr

N

rjα (−τ ),

α=1

−∞

N

riα ρ(0) dτ,

α=1

where we have introduced the usual Heaviside function θ(t), and obtained the following expression for the linear Green function of the system: (1) Gij (t)

(70)

e2 θ(t)Tr =− iV

N α=1

rjα (−t),

N

riα

ρ(0) ,

α=1

which is null if t < 0. We have used the interaction representation for the evolution of the position operators: rjα (−t) = exp

(71)

iH0 (−t) α iH0 (−t) rj exp − .

The polarization is the convolution product of two functions depending only on the time variables, and therefore its Fourier transform is the ordinary product of the temporal Fourier transforms of the two functions: (1) (1) (1) (72) Pi (ω) = F Pi (t) = χij (ω)Ej (ω). Expression (72) corresponds to the limit k → 0 of expression (12), which is consistent with our decision to use the dipolar approximation. . 2 4.4. The asymptotic behaviour of the linear susceptibility. We emphasize that the semiclassical approach we have adopted does not permit a detailed representation of the absorption phenomena because in our treatment the evolution is driven by purely Hermitian operators. As well known [12, 17, 18, 31], this approximation implies that absorption peaks are essentially Dirac δ-functions centered on the transition frequencies. In a more precise picture the δ-functions are substituted by Lorentzian functions when a finite lifetime is introduced. The asymptotic behaviour of the total susceptibility turns out to be related only to the real part. From the standard Fourier transform theory we see that

(73)

k

(−t) F θ (t) k!

∞ = −∞

(−t)k ik dk θ(t)eiωt dt = k! k! dω k

1 i℘ + πδ(ω) , ω

where ℘ indicates the pricipal part. If we consider the asymptotic behaviour for large values of ω, we can drop the δ(ω) and its derivatives as well as the principal part, because

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they are relevant only for ω = 0, thus obtaining 1 (−t)k F θ(t) ≈ −(−i)k+1 k+1 . k! ω

(74)

This implies that in order to deduce information on the behaviour of the susceptibility we must analyze the short-time behaviour of the response function. Considering that by definition the relaxation processes are not important on the shortest time scales of the system, we have a relevant conceptual argument supporting the fact that the assumption of a purely Hermitian Hamiltonian operator does not affect the leading asymptotic behaviour of the linear susceptibility. From the Heisenberg equations we can deduce that for short times the following Taylor expansion holds: ∞

rjα (−t) =

(75)

α ak,j

k=0

(−t)k , k!

where (76)

α ak,j

=−

1 i

α H0 , ak−1,j ,

α a0,j = rjα .

Substituting expression (75) in the linear Green function (70) and adding on the index α, we obtain the following infinite Taylor expansion: (1) Gij

(77)

(t) =

∞

(k)

Aij θ(t)

k=0

(−t)k , k!

with the following definition for the tensorial coefficients: (k) Aij

(78)

e2 Tr =− iV

N

α ak,j ,

α=1

N

riα , ρ(0) .

α=1

Applying the Fourier transform to the Taylor expansion of the Green function (77) and considering the result (74), we obtain that for large values of ω the linear susceptibility can be written as (79)

(1) χij (ω)

∞ e2 ≈ Tr iV

=

∞ k=0

k=0

N

α ak,j ,

α=1

Akij (−i)k+1

N α=1

riα

ρ0

(−i)k+1

1 = ω k+1

1 . ω k+1

Therefore, the leading term in the asymptotic expansion of the linear susceptibility can be found by determining which is the lowest value of k at which the tensorial coefficient (78) does not vanish.


19

When considering k = 0 in expression (78), we see that (80)

(0) Aij

e2 Tr =− iV

N α=1

rjα

N

riα , ρ0

= 0,

α=1

since it involves the commutator of two purely spatial variables. The value of the k = 1 coefficient in expression (78) is (81)

(1) Aij

N N 1 H0 , rjα riα , ρ0 = i α=1 α=1 N N pjα e2 α Tr − , r ρ0 = = iV m α=1 i α=1

e2 Tr = iV

=

ωp2 e2 N e2 N Tr {δij ρ0 } = δij = δij , Vm mV 4π

where we have considered that the trace of the density matrix is 1 and introduced the plasma frequency ωp . Therefore, we have obtained that asymptotically the susceptibility tensor is diagonal and decreases as ω −2 : (1)

χij (ω) ≈ −

(82)

ωp2 1 δij 2 + o(ω −2 ), 4π ω

where o(ω −2 ) indicates terms that have an asymptotic decrease that is faster than ω −2 . We can observe that the first asymptotic term is isotropic and does not contain physical parameters other than the electron charge, mass and density, thus excluding the quantum parameter . This implies that the material, if interacting with a very high-frequency radiation, behaves universally like a classical free-electron gas with the same density. . 2 4.5. An equivalent derivation of the asymptotic behaviour of the linear susceptibility. It is possible to derive a more conventional expression of the linear susceptibility by applying the Fourier transform to the Green function (70) and expanding the time-dependent position operators r α (t) using expression (71). We then obtain (83)

(1)

χij (ω) =

N e2 2ωba a|xi |bb|xj |a ρaa , 2 V α=1 ω 2 − ωba a,b

where ωba = (Eb − Ea )/ = −ωab , and where we have used definition (58) for the density operator and taken a| and b| as eigenstates of the Hamiltonian H0 . For the sake of completeness of our study, we wish to show that from this expression it is possible to obtain the same asymptotic behaviour presented in (82), provided we use the commutation properties, as shown by Heisenberg. If external magnetic fields can be neglected, it is always possible to select the eigenstates of a Hamiltonian system in such a way that they are described in the space representation by real functions [52]. Since xi is a real operator, this implies that (84)

a|xi |b = b|xi |a.

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Using this result in eq. (83) we obtain that (1)

χij (ω) =

(85)

N e2 ωba (a|xi |bb|xj |a + a|xj |bb|xi |a) ρaa . 2 V α=1 ω 2 − ωba a,b

The asymptotic behaviour of expression (83) can then be obtained by assuming ω ωba , ∀a|, b|: (86)

(1)

χij (ω) =

N e2 ωba (a|xi |bb|xj |a + a|xj |bb|xi |a) ρaa + o(ω −2 ). ω 2 V α=1 a,b

Considering that the following relation holds for any operator O: ωba a|O|b = 1/a|OH0 − H0 O|b = 1/a| [O, H0 ] |b,

(87)

we then obtain that ωba a|xi |b = 1/a|[xi , H0 ]|b = i/ma|pi |b = −ωba b|xi |a.

(88)

By applying the result (88) into expression (86), we eventually obtain (89)

(1)

χij (ω) =

N ωp2 1 e2 i −2 δij 2 + o(ω −2 ), a|[p , x ]|aρ + o(ω ) = − i j aa 2 ω V m α=1 a 4π ω

which corresponds to the asymptotic behaviour obtained in (82) using the short-time expansion of the Green function (70). . 2 4.6. Corrections due to relativistic and spatial dispersion effects. The problem of the relativistic corrections to the linear susceptibility asymptotic behaviour was first dealt with in the 1960s [53] and was more recently revived in the 1990s [54]. Recent theoretical findings [32] have shown that if first-order relativistic and spatial dispersion effects are simultaneously and consistently taken into account and the wave vector k is expressed as a function of the angular frequency ω [55], the asymptotic behaviour of the linear susceptibility of a homogeneous medium is (90)

ωp2 2 T 0 1 χ(1) k(ω) , ω ≈ − + o(ω −2 ), 1− 4π 3 mc2 ω 2

where the quantity T 0 is defined as the ground-state expectation value of the total kinetic energy of the system introduced in eq. (48): N p α · p α ρ(0) . T 0 = Tr 2m α=1

(91)

Therefore the correction is related to a measure of the ratio between the non-relativistic total energy (when the Virial Theorem is considered [48]) and the relativistic mass energy.


21

Such ratio increases with the atomic number of the atoms under consideration, being of the order of 10−1 for Ag. It is reasonable to expect that very high frequency radiation sources may permit the observation of this correction to the conventional asymptotic behaviour (82). In principle, we expect that the relativistic effects become relevant when ω αmc2 ,

(92)

where α = e2 /c is the fine-structure constant. Substituting the appropriate numerical values in eq. (92), we obtain that the relativistic effects become relevant when ω 4 keV. Moreover, by reversing (42), we can estimate that the spatial dispersion effects come into play when k d 1 ,

(93)

where d is the characteristic length for the elementary component of the medium. We observe that if we choose as characteristic length the hydrogen atom Bohr radius a0 = 2 /me2 —which nevertheless gives a very general length scale for many systems— the two estimates (92) and (93) coincide. . 2 5. Local electric field and effective-medium approximation. – The response of a medium to an external electric field cannot be described exactly by means of macroscopic electric fields. The external field drives the bound charges of the medium apart and induces a collection of dipole moments [56]. In an optically dense medium, the interaction of the induced dipoles is taken into account by a local field factor, which relates the macroscopic fields to the local ones. Conventionally, the local field is derived from the macroscopic properties of the medium (see, for instance, ref. [43]). Unfortunately, such an approach lacks generality and does not provide any insight into the physical principles responsible for the dielectric properties of the medium. Alternatively, one can arrive at macroscopic properties by averaging the microscopic response over the volume [45]. Recently, rigorous derivation of local field corrections was provided with the aid of particle correlation techniques [57]. For our purposes, the macroscopic approach is suitable. In order to avoid a too cumbersome presentation, we have chosen to develop a scalar theory, which corresponds to treating isotropic matter. Nevertheless, it can be immediately seen that the very same conclusions apply for a full tensorial theory. . 2 5.1. Homogeneous media. For materials with linear optical response, the local field is connected to the microscopic polarization p, which is related to the polarizability α(ω) of one microscopic constituent charge, according to the relation p(ω) = α(ω)E(ω).

(94)

The macroscopic polarization of the medium is derived by averaging eq. (94) over the investigated volume V , as follows [45]: (95)

1 P (ω) = V

V

loc (ω), p(ω)dv = ℵα(ω)E

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where ℵ is the density of microscopic constituents. We emphasize that if we consider a monophase medium, we have that ne ℵ =

(96)

N , V

where N/V is the number of electrons per unit volume and ne is the number of electrons per elementary constituent of the medium. On the other hand, the polarization can be expressed in terms of the external electromagnetic field as follows: (97)

(1) P (ω) = χeff (ω)E(ω).

Hence, in order to express the effective susceptibility in terms of the microscopic polarizability, we have to express the local electric field in terms of the external electric field. We have that the local electric field can be expressed as follows [43]: (98)

P (ω) loc (ω) = E(ω) E . + 4π 3

By inserting the definition of the macroscopic polarization (95) into eq. (98) we obtain (ω)+2 4πℵα(ω) loc 1 loc (ω) = E(ω)+ (99) E 3E(ω)+[(ω)−1]E(ω) = E (ω) = E(ω), 3 3 3 The combination of eqs. (14)-(16) and (99) and the use of the definition of the local field yields the result (100)

4π (ω) − 1 ℵα(ω) = , 3 (ω) + 2

which is known as the classical Clausius-Mossotti equation [7, 42]. The local field factor corrects the values (ω) − 1 that are calculated without the presence of a local field, e.g., in dilute gases [16]. We can redefine the constitutive relation between the induced macroscopic polarization P (1) (ω) and the applied external field E(ω) presented in eq. (72) into the following: (101)

(1)

P (1) (ω) = χeff (ω)E(ω) . (1)

Equivalently, the linear macroscopic susceptibility χeff (ω) presented in expression (97) is related to the linear microscopic polarizability α(ω) by the following equation: (102)

(1)

χeff (ω) =

ℵα(ω) . 1 − 4π 3 ℵα(ω)

(1)

If the density ℵ is small, we have that χeff (ω) ∼ ℵα(ω). The linear susceptibility χ(1) (ω), presented in (79) precisely obeys this approximation, i.e. (103)

χ(1) (ω) = ℵα(ω).


23

Therefore we can establish the following functional dependence between the effective macroscopic susceptibility and the susceptibility we computed in (79): (1)

(104)

χeff (ω) =

χ(1) (ω) . (1) (ω) 1 − 4π 3 χ

Inserting the asymptotic behaviour of the susceptibility (82) into eq. (104), we deduce (1) that χeff (ω) and χ(1) (ω) are asymptotically equivalent: (105)

(1)

χeff (ω) ≈ χ(1) (ω) ≈ −

ωp2 1 + o(ω −2 ) . 4π ω 2

. 2 5.2. Two-phase media. The classical Clausius-Mossotti formula (100) can be extended to the first-order effective-medium approximation by considering a mixture of two constituents. For a two-component system, where the constituents have different polarizabilities αa (ω) and αb (ω), the effective-medium approximation gives (106)

4π eff (ω) − 1 = (ℵa αa (ω) + ℵb αb (ω)) , eff (ω) + 2 3

where ℵi is the number of constituent dipoles per unit of volume with i = a or b. Equation (106) can be expressed with the aid of eq. (100): (107)

a (ω) − 1 b (ω) − 1 eff (ω) − 1 = fa + fb , eff (ω) + 2 a (ω) + 2 b (ω) + 2

where fi = ℵi /(ℵa + ℵb ) denotes the volume fraction of the constituent i and fa + fb = 1. Equation (107) is the Lorentz-Lorenz effective-medium approximation, which can be used in the derivation of the optical properties of some nanostructures. Nanostructures have attracted much interest during the past two decades as a new class of optical materials. The history of the research on optical properties of nanostructures dates back to the studies of Maxwell Garnett [58, 59], who explained the colors induced by minute metal spheres by the effective-medium theory. For example, gold nanospheres embedded into glass display a ruby-red colour due to an anomalous absorption band [60]. The classical electromagnetism related to randomly oriented nanostructures is concerned with the effective-medium approximation originally provided by Bruggeman [61] (see also Landauer [62]). The size of the constituents is assumed to be much smaller than the wavelength of the incident light, so that there is no light scattering, but sufficiently large to possess macroscopic dielectric constants. . 2 5.3. The Maxwell Garnett system. Perhaps the simplest case of an effective medium is a two-phase Maxwell Garnett (MG) nanosphere system, as presented in fig. 2(a). In such a system the solid nanospheres, whose characteristic size is much smaller than the wavelength of the incident light, are embedded into a host material; in general the spheres can be insulating, semiconducting, or conducting. We assume that the nanospheres are dielectric. If the volume fraction of the nanospheres fi is much smaller than the volume fraction of the host material fh = 1 − fi , the

24


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Fig. 2. – Schematic diagrams of two-phased media: (a) Maxwell Garnett, (b) Bruggeman, and (c) layered nanostructures.

linear effective dielectric function eff (ω) obeys the following equation: (108)

eff (ω) − h (ω) i (ω) − h (ω) = fi , eff + 2h (ω) i (ω) + 2h (ω)

where i (ω) is the complex linear dielectric function of the nanospheres and h (ω) the corresponding quantity of the host material. Equation (108) treats the constituent materials asymmetrically, i.e. the nanospheres are embedded in the host material. The MG approximation of an effective medium [58, 59] can be reformulated as follows: (109)

eff (ω) = h (ω)

1 + 2υ(ω)fi , 1 − υ(ω)fi

where υ(ω) is given by (110)

υ(ω) =

i (ω) − h (ω) . i (ω) + 2h (ω)


25

The reformulated version (109) is useful in the derivation of the nonlinear optical properties of an MG nanosphere system [63]. Unfortunately, in an MG approximation it must usually be assumed that there is no interaction between the nanospheres and that the scattering of light can be neglected. Furthermore, the MG model takes into account neither the size nor the the distance between the nanospheres, i.e. agglomeration and polydispersion are neglected. In the literature, the extended versions of the MG approximation, which take into account the size of the nanospheres, have been discussed [64,65]. However, in the frame of the conventional MG approximation the volume fraction of the nanospheres must be relatively low fi 1, although Boyd et al. [66] used a value fi = 0.5 as an upper limit in their theoretical study. In experimental measurements [67], the upper limit of the volume fraction, which gives good agreement between the measured and the predicted values, is rather small (fi ≈ 0.1). The real and imaginary parts of the effective dielectric function of the MG nanosphere system in the case of non-absorbing host material ( {h (ω)} = 0) were given by Ward [68]. We present here the expressions for the real and imaginary parts of the MG nanosphere system in the case of {h (ω)} = 0. After lengthy algebra, the real and imaginary parts of the complex effective dielectric function are explicitly given in their general form as follows [69]: (1 + 2fi )({i }{h }− {i } {h }) + 2fh ({h }2 − {h }2 ) − Ψ2 + Φ2 (1 + 2fi )({i } {h } + {i }{h }) + 4fh {i } {h } −Φ , Ψ2 + Φ2

(111) {eff } = Ψ

(112)

{eff } =

Φ{eff } + (1 + 2fi )( {i } {h } + {i } {h }) + 4fh {h } {h } , Ψ

where (113)

Ψ = fh {i } + (2 + fi ){h },

(114)

Φ = −fh {i } − (2 + fi ) {h }.

The general expression can be used to estimate the maximum of the imaginary part of the effective dielectric function. The maximum is obtained when {i } ≈ −{h }(2+fi )/(fh ) and the denominator approaches zero. The increase in the volume fraction shifts the resonance angular frequency of the nanosphere system towards higher energies, which has been demonstrated by, e.g., the papers of Dalacu and Martinu [70] and Peiponen et al. [69]. Unfortunately, a direct measurement of the complex effective dielectric function of an MG nanosphere system is not possible at optical frequencies. However, the reflectance of a liquid-phase MG system can be obtained in order to calculate the complex effective dielectric function of the liquid. . 2 5.4. The Bruggeman effective-medium theory. In the framework of the MG approximation the volume fraction of the nanospheres is restricted to relatively low values. This restriction can be removed by using the Bruggeman effective-medium model. In this model, presented in fig. 2(b), host and guest materials are indistinguishable and the

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system is completely random. Mathematically the low volume fraction limit arises from the asymmetric equation (108) for the effective dielectric function. For a two-phase MG nanosphere system with comparable volume fractions, the effective dielectric function is dependent on the choice of the host material. Bruggeman [61] solved this problem with the assumption that the inclusions are embedded in the effective medium itself and the dielectric function of the host material is replaced by the effective dielectric function in eq. (108). Such a procedure gives the expression [61] (115)

fa

a (ω) − eff (ω) b (ω) − eff (ω) + fb = 0, a (ω) + 2eff (ω) b (ω) + 2eff (ω)

where a and b denote the two components having different complex dielectric functions (a , b ) and volume fractions (fa , fb ), respectively. Now the effective dielectric function is invariant if the constituents are interchanged. Zeng et al. [46] generalized the results of Bruggeman for different shapes of inclusions, obtaining (116)

fa

a (ω) − eff (ω) b (ω) − eff (ω) + fb = 0. eff (ω) + g[a (ω) − eff (ω)] eff (ω) + g[b (ω) − eff (ω)]

Here, g is a geometric factor, which depends on the shape of the inclusions. For spherical inclusions, g = 1/3 and for two-dimensional circular inclusions, g = 1/2. If we substitute g = 1/3, eq. (116) reduces to the form (115). The quadratic equation (116) has two explicit solutions for the complex effective dielectric function (117)

eff (ω) =

−c(ω) ±

c(ω)2 + 4g(1 − g)a (ω)b (ω) , 4(1 − g)

where (118)

c(ω) = (g − fa )a (ω) + (g − fb )b (ω).

In eq. (117) only the positive branch is physically reasonable. This can be justified by inserting g = 1/3, fa = fb = 0.5 and a = b = 1. Then the effective dielectric function has to take the value of vacuum, which is possible only by allowing the positive sign of the square root. . 2 5.5. Layered nanostructures. In fig. 2(c), a two-phase layered nanostructure is presented, which is built by alternating layers of sub-wavelength thicknesses. Layered nanostructures are assumed to have a wide range of optoelectronic applications from all-optical switching, modulating and computing devices to quantum well lasers. The analysis of the linear optical properties of such a composite can be based on Maxwell’s equations and boundary conditions presented in this section. For TE-polarized light (the electric field is parallel to the layers), the tangential component of the electric field ETE is continuous at the boundary between the layers. The electric field is distributed uniformly between the constituents a and b with volume fractions fa and fb . The average electric displacement inside the nanostructure can be expressed as follows: (119)

(a)

(b)

DTE (ω) = fa DTE (ω) + fb DTE (ω) = [fa a (ω) + fb b (ω)]ETE (ω),


27

since ETE is continuous over the structure. On the other hand, the average electric displacement is related to the effective dielectric function by the following relation: (120)

DTE (ω) = eff (ω)ETE (ω).

By combining eqs. (119) and (120), one obtains for TE-polarized light the expression for the effective dielectric function (121)

eff (ω) = fa a (ω) + fb b (ω).

For TM-polarized light (the electric field is perpendicular to the layers), the normal component of the electric displacement DTM is continuous at the boundary. The average electric field is fa fb (a) (b) (122) ETM (ω) = fa ETM (ω) + fb ETM (ω) = + DTM (ω), a (ω) b (ω) where DTM is continuous. The effective dielectric function for TM-polarized light is then given by fa fb 1 = + . eff (ω) a (ω) b (ω)

(123)

It is worth mentioning that the linear optical properties are independent of the thicknesses of individual layers. Hence, only the volume fraction of different constituents matters. The linear optical response of a stack of dielectric layers, whose dimensionality is much smaller than the incident wavelength, is anisotropic, yielding a form birefringence, i.e. the refractive index of a medium is different for TE- and TM-polarized light [71]. . 2 5.6. Asymptotic behaviour of the linear susceptibility for nanostructures. We have shown in eq. (105) that the asymptotic behaviour of the effective linear susceptibility relative to each material constituting the nanostructure can be expressed as follows: (1)

χeff i (ω) = −

(124)

2 ωpi 1 + o(ω −2 ), 4π ω 2

where the index i defines the material we are referring to, host and inclusion in the MG and Bruggemann cases, and layer 1 and layer 2 in the layered nanostructure case, and ωp is the corresponding plasma frequency. Considering for both phases as well as for the effective medium the usual correspondence between the dielectric function and the susceptibility presented in eq. (16), the asymptotic behaviour of the effective susceptibilities for the systems considered can be found by direct substitution of the asymptotic behaviour (124) into the effective dielectric functions. We obtain that in all cases analyzed —MG, Bruggemann, layered nanostructures with TE- and TM-polarized light— the asymptotic fall-off of the effective susceptibility is (125)

(1)

χeff (ω) = −

2 2 + fb ωpb ] 1 [fa ωpa + o(ω −2 ) . 4π ω −2

Note that the asymptotic behaviour (125) is, regardless of the topology of the system, the volume average of the asymptotic behaviours of the two materials.

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3. – Dispersion relations and sum rules in linear optical spectroscopy . 3 1. The principle of causality. – Kramers-Kronig (K-K) dispersion relations [1, 2] and sum rules [3-6] have long constituted fundamental tools in the investigation of lightmatter interaction phenomena in condensed matter, gases, molecules, and liquids. They provide constraints for checking the self-consistency of experimental or model-generated data. Moreover, by applying K-K relations it is possible to invert optical data, i.e. to acquire knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole spectrum or vice versa [7-9,11-13]. In particular, these general properties permit the framing of peculiar phenomena that are very relevant at given frequencies, showing that their dispersive and absorptive contributions are connected to all the other contributions in the rest of the spectrum [11]. The conceptual foundations of such general properties can be found in the principle of causality [14] in the light-matter interaction. The connection between the mathematical properties of the functions describing the physics in the domains of time and frequency is provided by the Titchmarsch theorem [10]. In this section we present a detailed and general derivation of K-K relations and related sum rules in both cases of conducting and non-conducting materials. The concept of dispersion relations and sum rules is then generalized to hold also for the linear susceptibility of the most typical nanostructures, which tend to have more and more importance for optical materials in various engineering fields. . 3 2. Titchmarsch’s theorem and Kramers-Kronig relations. – Titchmarch’s Theorem connects, within fairly loose hypotheses, the causality of a function a(t) to the analytical properties of its Fourier transform a(ω) = F [a(t)]: Theorem 1.: The three statements 1., 2. and 3. are mathematically equivalent: 1. a(t) = 0 if t ≤ 0; a(t) ∈ L2 ; 2. a(ω) = F [a(t)] ∈ L2 if ω ∈ R ; a(ω) = limω →0 a(ω + iω ), a(ω + iω ) is holomorphic if ω > 0 ; 3. Hilbert transforms [41] connect the real and imaginary part of a(ω): {a(ω)} =

1 ℘ π

∞

−∞

1 {a(ω)} = − ℘ π

{a(ω )} dω , ω − ω

∞

−∞

{a(ω )} dω . ω − ω

Thus, the causality of a(t), together with its property of being a function belonging to the space of the square-integrable functions L2 , implies that its Fourier transform a(ω) is analytic in the upper complex ω-plane and that the real and imaginary parts of a(ω) are not independent, but connected by non-local, integral relations termed dispersion relations. Under the physically reasonable assumption that all the tensorial components of the linear Green function belong to the L2 space we deduce that the Hilbert relations connect the real and imaginary parts of the tensorial components of the linear susceptibility.


29

Considering that the components of the polarization P (1) (t) are real functions, we can deduce that ∗ (1) (1) (126) χij (−ω) = χij (ω) , where x∗ denotes the complex conjugate of x. Equation (126) implies that for every ω ∈ R the following relations hold: (1) (1) χij (ω) = χij (−ω) , (1) (1) χij (ω) = − χij (−ω) .

(127) (128)

This allows to write the K-K relations [1, 2] as follows: ∞ ω χ(1) (ω ) ij 2 (1) dω , χij (ω) = ℘ π ω 2 − ω 2 0 ∞ χ(1) (ω ) ij 2ω (1) dω . χij (ω) = − ℘ 2 π ω − ω2

(129)

(130)

0

The relevance of the K-K relations in physics goes beyond the purely conceptual sphere. Since every function that models the linear susceptibility of a material, as well as any set of experimental data of the real and imaginary parts of the linear susceptibility, must respect the K-K relations, they also constitute a fundamental test of self-consistency. Furthermore, the real and imaginary parts of the susceptibility are related to qualitatively different phenomena, light dispersion and light absorption respectively, whose measurement requires different experimental set-ups and instruments [7-9, 11-13]. The causality of the physical system under consideration, which, when local effects are considered, also includes contributions in the Hamiltonian related to the local dipole (1) fields induced by the single elements, ensures that χeff (ω) is holomorphic in the upper complex ω-plane. This property and the asymptotic equivalence between χ(1) (ω) and (1) χeff (ω) imply that the latter function obeys the very same K-K relations and sum rules of the former function. Therefore the results obtained in sect. 2 are still valid in the more general and realistic context where the local field effects are considered. . 3 2.1. Kramers-Kronig relations for conductors. In the special case of conductors the above dispersion relations have to be modified [72-74] because of the presence of a pole in the linear susceptibility for ω = 0. It is possible in general to express the susceptibility as a function of the linear conductivity [7, 42]:

(131)

(1) (1) (1) σij (ω) σij (ω) (ω) σ ij (1) χij (ω) = i =i − , ω ω ω

where the linear-conductivity tensor describes the physically indistinguishable effects of polarization and of current. The conductors, by definition, have a non-vanishing real

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conductivity for static electric fields, so that the imaginary part of the linear susceptibility has a pole for ω = 0, i.e. the following relation holds for low values of ω: σ (1) (0) ij (1) . χij (ω) ≈ ω

(132)

As a consequence, when computing the Hilbert transform of the linear susceptibility there is a boundary contribution from the pole at the origin. We eliminate this pole by (1) considering the function χij (ω)−σij (0)/ω and obtain that the second dispersion relation presented in (130) is changed as follows: ∞ χ(1) (ω ) σ (1) (0) ij 2ω ij (1) =− ℘ dω . χij (ω) − ω π ω 2 − ω 2

(133)

0

. 3 2.2. Kramers-Kronig relations for nanostructures. The linear optical properties of nanostructures can be described with the effective linear dielectric function of the structure. The expressions for the effective dielectric function of different models including Maxwell Garnett, Bruggeman and layered structures have been presented in the previous section. The causality of the physical system under consideration, which, when local effects are considered, also includes contributions in the Hamiltonian that are related to local (1) dipole fields induced by the single elements, ensures that χeff (ω) is holomorphic in the upper complex ω-plane. This property and the asymptotic equivalence between χ(1) (ω) (1) and χeff (ω) imply that the latter function obeys the very same K-K relations and sum rules of the former function. Therefore the results obtained in this section are still valid in the more general and realistic context where the local field effects are considered. It (1) has been directly proved that the effective dielectric function χeff (ω) is holomorphic and fulfills the symmetry property (126) in the cases of MG [69], Bruggeman [75] and layered nanostructures [76]. Since the fall-off the effective susceptibility function is in general strictly faster than ω −1 , as has been shown in eq. (125), it is possible to obtain the following K-K relations relating the real and imaginary parts of the effective susceptibility:

(134)

(135)

(1) {χeff (ω)}

(1) {χeff (ω)}

2 = ℘ π

∞ 0

2ω =− ℘ π

(1)

ω {χeff (ω )} dω , ω 2 − ω 2 ∞ 0

(1)

{χeff (ω )} dω . ω 2 − ω 2

From eq. (134), one obtains the DC susceptibility function simply by setting ω = 0. Following the work by Stroud [77], we can define the average linear susceptibility function of the nanostructure in the form (136)

(1) (1) χ(1) av (ω) = fa χa (ω) + fb χa (ω).


31

It is easy to derive that the dominant term in the asymptotic behaviour of the function (1) (1) χav (ω) is the same as for the function χeff (ω). Considering the parity properties of the real part of the susceptibility functions, we can then derive that the asymptotic fall-off of the function difference between the average and the effective susceptibility is ψ + o(ω −4 ). ω4 (1) (1) (1) (1) The functions χeff (ω)−χav (ω) and ω 2α χeff (ω) − χav (ω) are holomorphic in the upper (137)

(1)

χeff (ω) − χ(1) av (ω) ∼

complex place of the variable ω and have an asymptotic decrease strictly faster than ω −1 , therefore the following independent K-K relations connect the real and imaginary parts: ∞ ω 2α+1 χ(1) (ω ) − χ(1) (ω ) av eff 2 (1) ω 2α χeff (ω) − χ(1) ℘ (138) dω , av (ω) = 2 2 π ω −ω 0 ∞ ω 2α χ(1) (ω ) − χ(1) (ω ) av eff 2 (1) (139) dω , ω 2α−1 χeff (ω) − χ(1) av (ω) = − ℘ π ω 2 − ω 2 0

with α = 0, 1. We note that when we consider suitable functions with a faster asymptotic decrease, it is generally possible to derive a larger set of independent K-K relations. . 3 3. Superconvergence theorem and sum rules. – From the K-K relations and the knowledge of the asymptotic behaviour of the susceptibility function it is possible to deduce the value of the zero-degree moment of the real part and the value of the first moment of the imaginary part of the susceptibility. These results can be obtained by taking advantage of the superconvergence theorem [6, 78]: Theorem 2.: If ∞ g (y) = ℘ 0

f (x) dx, y 2 − x2

where 1. f (x) is continuously differentiable; −1 2. f (x) = o (x ln x) ; ⇒ for y x the following asymptotic expansion holds: 1 g (y) = 2 y

∞

" ! f (x) dx + o y −2 .

0

The thesis of the theorem can also be written as ∞ (140) 0

! " f (x) dx = lim y 2 g (y) . y→∞

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We then consider the first of the K-K relations in eq. (129), we replace x = ω , y = ω, and we set f (x) and g(y) equal to the general tensorial component of the imaginary and real parts of the linear susceptibility, respectively. The discussion presented in this section shows that the asymptotic decrease for all tensorial components of the imaginary part is strictly faster than ω −2 . Therefore, by applying the superconvergence theorem [6, 78] to the first K-K relation (129), we conclude that ∞ (141) 0

π ω 2 p (1) (1) δij , ω χij (ω ) dω = lim − ω 2 χij (ω) = ω→∞ 2 8

where we have considered the general result for the asymptotic behaviour of the real part of the susceptibility presented in eq. (82). This law is commonly referred to as Thomas-Reiche-Kuhn (TRK) or f -sum rule [3-5]. Considering that the quantity under the integral is proportional to the absorption of the material under examination, we may see that the total absorption over the entire spectrum is proportional to the total electronic density. If we take into account the relativistic and spatial dispersion corrections presented in expression (90) for a homogeneous material, we then find that the TRK sum rule is accordingly modified in the following way: ∞ (142) 0

ω χ(1) k(ω ) , ω dω = ω 2 π 2 T 0 p 2 (1) = = lim − ω χ . 1− k(ω) , ω ω→∞ 2 8 3 mc2

It is possible to obtain an additional sum rule by applying the superconvergence theorem to the second dispersion relation presented in eq. (129):

(143)

∞ π (1) (1) ω χij (ω) χij (ω ) dω = lim = 0. ω→∞ 2 0

This sum rule implies that the average value of the real part of the permittivity over all the spectrum is 1. We would point out that the verification of linear sum rules from experimental data is usually hard to obtain because of the critical contributions made by the out-of-range asymptotic part of the real or imaginary parts of the susceptibility under examination [11, 79, 80]. However, in the case of linear optics, information about the response of the material to very high frequency radiation can be obtained using synchrotron radiation [11]. . 3 4. Sum rules for conductors. – Applying the superconvergence theorem in the usual way to the K-K relations for the conductors presented in eq. (133), we find that while the TRK sum rule is still obeyed, the boundary term contained in the second K-K relation


33

changes the sum rule for the real part of the susceptibility:

(144)

∞ σ (1) (0) π (1) π ij (1) (1) ω χij (ω) − = − σij (0) . χij (ω ) dω = lim ω→∞ 2 ω 2 0

Therefore, in the case of conductors the average value of the linear permittivity is not equal to 1 [72]. . 3 5. Sum rules for nanostructures. – Consider next the effective linear susceptibility functions of nanostructures. We have previously derived the K-K relations (134), (135) for the effective susceptibility of nanostructures for all investigated topologies (Maxwell Garnett [69], Bruggeman [75], and layered structure [76]). Therefore, by applying the superconvergence theorem to the two K-K relations and taking into account the general asymptotic behaviour presented in eq. (125), it is possible to derive sum rules for the real and imaginary parts of the effective susceptibility. The corresponding TRK sum rule for the effective susceptibility is of the form ∞ (145) 0

2 2 fi ωpi + fh ωph (1) . ω χeff (ω ) dω = 8

It is worth noting that the high-frequency behaviour of the composite transforms the TRK sum rule but the average sum rule is valid for all topologies: ∞ (1) χeff (ω ) dω = 0 ,

(146)

0

which is totally analogous to that presented in eq. (143). Moreover, if we consider the set of K-K relations for the real and imaginary parts for (1) (1) the function χeff (ω) − χav (ω) presented in eqs. (138), (139) by applying the superconvergence theorem to the equations, we obtain the following sum rules: ∞ (147) 0 ∞

(148) 0

(1) ω 2α χeff (ω ) − χ(1) av (ω ) dω = 0,

α = 0, 1 ,

 0, (1) ω 2α+1 χeff (ω ) − χ(1) av (ω ) dω = − π ψ, 2

α = 0, α = 1.

The last sum rule was given by Stroud [77] in the case of MG composite, for which we have that (149)

ψ=

2 1 2 fa fb ωpb . − ωpa 3

We wish to emphasize that these results can be easily generalized for multi-phase nanostructures.

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Fig. 3. – Real part of the linear permittivity (1) (ω) obtained from K-K analysis of absorbance of La1.87 Sr0.13 CuO4 . Adapted from [82].

. 3 6. Integral properties of the optical constants. – The linear susceptibility function describes at a fundamental level the connection between the microscopic dynamics of the system under consideration and its linear optical properties. Nevertheless, it is experimentally much easier to measure other quantities that are more directly related to the behaviour of the light as influenced by its interaction with the matter. The most commonly used optical constants are the complex index of refraction N (ω) and the reflectivity at normal incidence rˆ(ω). It has been shown that relevant integral relations can be established also for these quantities. Hence, the K-K relations and sum rules find wider experimental application in the analysis of the data of these optical constants. In this subsection we consider the simplifying assumption of isotropic matter and we briefly present the integral properties for the index of refraction N (ω). We show that they are analogous to those presented for the linear susceptibility. An excellent example of application of the integral properties to the analysis of experimental data of various optical constants can be found in ref. [72]. . 3 6.1. Integral properties of the index of refraction. We observe that the index of refraction N (ω) presented in eq. (26) is the square root of a function that is is holomorphic in the upper complex ω-plane. Following the procedure proposed in ref. [34] it is possible to derive that N (ω) itself is holomorphic and has no branching points in the upper complex ω-plane, and that the usual crossing relation N (ω) = [N (−ω)]∗ holds. Moreover it is simple to prove that the function N (ω) − 1 decreases asymptotically as (150)

N (ω) − 1 ∼ −

ωp2 . 2ω 2

Therefore, we conclude that we can write the following K-K relations connecting the real and imaginary parts of the refractive index:

(151)

2 n(ω) − 1 = ℘ π

∞ 0

ω κ(ω ) dω , ω 2 − ω 2


35

Fig. 4. – Imaginary part of the linear permittivity (1) (ω) obtained from K-K analysis of absorbance of La1.87 Sr0.13 CuO4 . Adapted from [82].

2ω κ(ω) = − ℘ π

(152)

∞ 0

n(ω ) − 1 dω . ω 2 − ω 2

We wish to emphasize that this is the original formulation of the K-K relations proposed by Kramers [1]. Moreover, it should be noted that the above dispersion relations are valid both in the case of conductors and non-conductors; we refer to [81] for a thorough discussion of this issue. In figs. 3, 4 we show the result of the application of the K-K relations to the experimental data of k(ω). From the measurement of the absorbance only it has been possible to deduce all the information needed to reconstruct the real and imaginary parts of the linear permittivity [82]. If we apply the superconvergence theorem to eq. (151) and consider the asymptotic behaviour presented in (150), we obtain ∞ (153)

π πω 2 p − ω 2 (n (ω) − 1) = , ω→∞ 2 4

ω κ (ω ) dω = lim

0

which is equivalent to the TRK sum rule [3-5] presented in eq. (141). Following the same procedure for the second dispersion relation (152) we obtain the Altarelli-DexterNussenzveig-Smith (ADNS) sum rule [6]:

(154)

∞ π ωκ (ω) = 0. (n (ω ) − 1)dω = lim ω→∞ 2 0

We underline that this sum rule is not wholly equivalent to the sum rule (143) obtained for the real part of the susceptibility. The reason is that while if we compute the sum rule for the imaginary part of the susceptibility function in the case of conductors we obtain a measure of the static conductance as expressed in (144), the result (154) is valid in both cases of conducting and non-conducting matter. We can guess that the reason

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Fig. 5. – Computation of the sum rule (143) using optical data for molybdenum. Note that, instead of {χ(1) (ω)}, the equivalent quantity 1 − {(1) (ω)} = −4π{χ(1) (ω)} is considered in the integration. Adapted from [83].

why the consideration of the susceptibility permits us to discern between conductors and non-conductors is that the susceptibility is a more fundamental physical quantity, since it is directly related to the dynamics of the system. In figs. 5 and 6 we show the results obtained for the integration of the real part of the susceptibility presented in expression (143) and the sum rule for [n(ω) − 1] presented in eq. (154) from optical data for molybdenum [83], respectively. While in the former case the integral converges asymptotically to a positive quantity, which corresponds to the actual conductivity as predicted by (143), in the latter case the integral converges to zero, as predicted by the ADNS sum rule (154). . 3 6.2. Linear reflectance spectroscopy. A common approach to measure the optical constants of an investigated sample is to measure the reflectance at normal incidence.

Fig. 6. – Computation of the ADNS sum rule (154) using optical data for molybdenum. Note that a factor appears in this figure. Adapted from [83].


37

Then dispersion relations are used in the phase retrieval procedure to determine the optical constants [84, 85]. The measurement of the normal reflectance is especially well suited for the determination of optical constants in the regions of high absorbance [86], where no radiation goes through the sample and absorption cannot be measured directly. At normal incidence, the complex reflectivity rˆ(ω) from the boundary of the sample and the vacuum is related to the complex refractive index of the sample, according to eqs. (36) and (40): (155)

rˆ(ω) =

N (ω) − 1 n(ω) − 1 + iκ(ω) = . N (ω) + 1 n(ω) + 1 + iκ(ω)

There is an analytic continuation of function rˆ(ω) from the real axis into the upper half of the complex (ω)-plane [86]. In the case of an insulating medium, the real axis is in the region of holomorphicity, but for metals there is a branch point at ω = 0. The effect of the branch point on a dispersion relation for a phase spectrum has been discussed by Nash et al. [87] and Lee and Sindoni [88]. One obtains a symmetry relation for the complex reflectivity with linearly polarized light (156)

rˆ(−ω ∗ ) = [ˆ r(ω)]∗ .

For circularly and elliptically polarized light the reflectivity has a lower symmetry after the mixing of left- and right-hand modes [89]. The quantity obtained from the reflectance measurement is usually the amplitude of the reflected beam, which is the square of the modulus of the complex reflectivity (157)

R(ω) = |ˆ r(ω)|2 = rˆ(ω)[ˆ r(ω)]∗ .

The complex reflectivity can be expressed in polar coordinates rˆ(ω) = |ˆ r(ω)| exp[iθ(ω)] and taking a natural logarithm leads to (158)

ln[ˆ r(ω)] = ln |ˆ r(ω)| + iθ(ω).

Unfortunately, the function diverges logarithmically at the limit |ω| → ∞ and is not square-integrable [90]. The divergence of the integrals eliminates the possibility for deriving analogous dispersion relations for the complex reflectivity as for optical constants [91]. Another important requirement for the complex reflectivity is the lack of zeros [86], which are branch points of a logarithm. However, there are several possibilities to avoid the divergence of the integral. On the one hand, the function [92] (159)

F (ω) =

ln[ˆ r(ω)] 2 ω − ω 2

gives the well-known relation for the phase of the complex reflectivity, as follows:

(160)

2ω θ(ω) = − ℘ π

∞ 0

ln |ˆ r(ω )| dω . ω 2 − ω 2

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Fig. 7. – (a) Reflectance spectrum of KI and real (b) and imaginary (c) parts of the susceptibility obtained with Kramers-Kronig analysis. Adapted from [84].

On the other hand, the function [92]

(161)

1 1 G(ω) = ln rˆ(ω) − ω − ω2 ω − ω2

allows to derive a dispersion relation for the difference between the amplitudes of the complex reflectivity evaluated at two frequencies ω1 and ω2 :

(162)

2 ln |ˆ r(ω1 )| − ln |ˆ r(ω2 )| = ℘ π

∞ 0

1 1 ω θ(ω ) 2 − 2 dω . ω − ω12 ω − ω22


39

We see that the knowledge of the phase does not uniquely determine the amplitude |ˆ r(ω)|. This is caused by the possible existence of a so-called Blaschke product, which takes into account the zeros of the complex reflectivity [93, 94]. The Blaschke product has been exploited, e.g., in phase recovery of a Raman excitation profile [95] and in the phase retrieval procedure of the transfer function for the interferometers [96]. The first significant experimental application of the normal reflectivity measurements for the determination of the optical constants of semiconductors have been carried out in 1962 by Ehrenreich and Philipp [97, 98]. Since then, the method has become a standard spectroscopic technique [9]. For purpose of exemplification, we present in fig. 7 the analysis so performed in KI by Rossler [84].

. 3 7. Generalization of the integral properties for more effective data analysis. – A serious practical problem in applying the K-K relations effectively is that they require information over the whole spectrum, while experimentally the data range is unavoidably finite. One of the possible strategies to ease this problem is to improve the approximate truncated integration by extending the integration to infinity by an a priori choice of the behaviour of the susceptibility outside the data range. Nevertheless, the choice of the asymptotic behaviour cannot be totally arbitrary. Peiponen and Vartiainen [99] used Gaussian line shape for the extinction coefficient in the data extrapolation. They observed that calculated values of the real refractive index obtained from K-K analysis were erroneous. The errors were caused by the failure of the symmetry property of the Gaussian line shape, which is not an odd function. In the next subsections we present a more detailed description of two promising approaches that, by exploiting some refinements to the conventional Kramers-Kronig relations, provide more efficient and practical tools for data analysis.

. 3 7.1. Generalized Kramers-Kronig relations. In order to overcome the problem of the relatively low convergence of the K-K relations, Altarelli and Smith [81] introduced generalized K-K relations by considering the functions [N (ω) − 1]m and ω m [N (ω)]m . All the functions belonging to both classes have the same holomorphic properties of N (ω), are square-integrable functions, and for a given m the convergence is proportional to ω −2m and ω −m , respectively. The symmetry relation of the latter function depends on the parity of the power ω m . If we consider odd numbers m = 2α + 1, α ≥ 0, the generalized K-K relations can be written as follows:

(163)

(164)

ω

ω

2α+1

2α+1

2α+1

[n(ω) − 1]

2ω ℘ = π

2α+1

{[N (ω) − 1]

∞ 0

ω 2α+1 {[N (ω ) − 1]2α+1 } dω , ω 2 − ω 2

2 }=− ℘ π

∞ 0

ω 2α+2 {[N (ω ) − 1]2α+1 } dω . ω 2 − ω 2

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When we consider even numbers m = 2α, α ≥ 0, the generalized K-K relations are of the form (165)

(166)

ω

2α

2 {[N (ω) − 1] } = ℘ π 2α

ω 2α {[N (ω) − 1]2α } = −

∞ 0

2ω ℘ π

ω 2α+1 {[N (ω ) − 1]2α } dω , ω 2 − ω 2 ∞ 0

ω 2α {[N (ω ) − 1]2α } dω . ω 2 − ω 2

For values m > 2 the convergence of the generalized K-K relations is strictly faster than that of conventional K-K relations, which is of crucial importance in experimental data analysis. If we apply the superconvergence theorem to eqs. (163)-(166) and consider the asymptotic behaviour (150) of N (ω) − 1, we derive a more general set of sum rules [81]: π

∞

m

ω {[N (ω ) − 1] }dω =

(167) 0 ∞

(168)

4 0,

ωp2 ,

0

(170) 0 ∞

(171)

α ≥ 0, π

ω 0 ∞

m > 1,

ω 2α {[N (ω ) − 1]2α+1 }dω = 0,

∞ (169)

m = 1,

2α+1

2α+1

{[N (ω ) − 1]

}dω =

4 0,

ω 2α−1 {[N (ω ) − 1]2α+1 }dω = 0,

ω 2α {[N (ω ) − 1]2α }dω = 0,

ωp2 ,

α = 0, α ≥ 1,

α ≥ 1,

α ≥ 1,

0

∞ (172)

ω 2α−2 {[N (ω ) − 1]2α }dω = 0,

α ≥ 2,

0

∞ (173)

ω m−1 {[N (ω ) − 1]m dω = 0,

0

∞ ω

(174) 0

2α+1

m ≥ 0,

π − ωp4 , 8 {[N (ω ) − 1] }dω = 0,

2α

α = 1, α ≥ 2.

Similar results could be obtained also for the higher powers of the linear susceptibility. Peiponen and Asakura [76] have presented the generalized K-K relations and sum rules for layered nanostructures by considering both even and odd powers of the function m (1) ω m χeff (ω) ; their results nevertheless apply to all of the nanostructural composites here considered.


41

Along the lines of Altarelli and Smith, Smith and Manogue [90] considered the powers of the complex reflectivity. They proved that the functions [ˆ r(ω)]m and ω m [ˆ r(ω)]m are square-integrable functions with proper asymptotic behaviour which allows the derivation of mixed K-K relations. These K-K relations mix both the amplitude and the phase of the complex reflectivity. By separating the real and imaginary parts one obtains for the function [ˆ r(ω)]m

(175)

(176)

2 |ˆ r(ω)| cos[mθ(ω)] = ℘ π m

|ˆ r(ω)|m sin[mθ(ω)] = −

∞ 0

ω|ˆ r(ω )|m sin[mθ(ω )] dω , ω 2 − ω 2 ∞

2ω ℘ π

0

|ˆ r(ω )|m cos[mθ(ω )] dω . ω 2 − ω 2

In a similar manner, Smith and Manogue [90] derived generalized mixed relations for even and odd powers of ω m [ˆ r(ω)]m , which are analogous to eqs. (163)-(166). Mixed K-K relations can be used in measurements where both the amplitude and the phase of the reflected beam are obtained. Moreover, they can be used in the derivation of various sum rules that provide tools for testing experimental data of measurements such as those found in ellipsometric studies. . 3 7.2. Subtractive Kramers-Kronig relations. Bachrach and Brown [100] introduced singly subtractive K-K relations (SKK) in order to reduce the errors caused by the finite spectrum as they calculated the real refractive index of a medium from a measured absorption spectrum. They measured independently one reference point (or anchor point), which was used to improve the accuracy of K-K analysis. A conventional K-K relation (151) can be written in the reference point, say at frequency ω1 , in the form

(177)

2 n(ω1 ) − 1 = ℘ π

∞ 0

ω κ(ω ) dω , ω 2 − ω12

where n(ω1 ) is a priori known. By subtracting eq. (177) from eq. (151) Bachrach and Brown [100] obtained a singly subtractive K-K relation as follows:

(178)

2 n(ω) − n(ω1 ) = ℘ π

∞ 0 2

=

ω κ(ω ) 2 dω − ℘ 2 2 ω −ω π

2(ω − π

ω12 )

∞ ℘ 0

∞ 0

ω κ(ω ) dω = ω 2 − ω12

ω κ(ω ) dω . (ω 2 − ω 2 )(ω 2 − ω12 )

The subtractive form of a K-K integral converges more rapidly than a conventional K-K relation. Singly subtractive K-K relations have been used in improving the convergence of the phase of measured reflectance spectra [101]. Recently, Palmer et al. [102] described multiply subtractive K-K relations (MSKK) in order to obtain the optical constants of the medium with a single reflectance measurement in a finite frequency range. Multiply

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subtractive K-K relations (MSKK) are derived in a manner similar to singly subtractive K-K relations: (179)

2 (ω 2 − ω22 )(ω 2 − ω32 ) · · · (ω 2 − ωQ ) n(ω) − 1 = 2 ) [n(ω1 ) − 1] + · · · (ω12 − ω22 )(ω12 − ω32 ) · · · (ω12 − ωQ 2 2 2 (ω 2 − ω12 ) · · · (ω 2 − ωj−1 )(ω 2 − ωj+1 ) · · · (ω 2 − ωQ ) + 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 ) [n(ωj ) − 1] + · · · (ωj2 − ω12 ) · · · (ωj2 − ωj−1 j j+1 j Q 2 2 2 2 2 2 (ω − ω1 )(ω − ω2 ) · · · (ω − ωQ−1 ) [n(ωQ ) − 1] + + 2 − ω 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 (ωQ 1 2 Q Q Q−1 ) 2 2 ) ℘× + (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ π ∞ ω κ(ω )dω × 2 ). (ω 2 − ω 2 )(ω 2 − ω12 ) · · · (ω 2 − ωQ 0

(180)

2 (ω 2 − ω22 )(ω 2 − ω32 ) · · · (ω 2 − ωQ ) κ(ω) = 2 ) κ(ω1 ) + · · · (ω12 − ω22 )(ω12 − ω32 ) · · · (ω12 − ωQ 2 2 2 (ω 2 − ω12 ) · · · (ω 2 − ωj−1 )(ω 2 − ωj+1 ) · · · (ω 2 − ωQ ) + 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 ) κ(ωj ) + · · · (ωj2 − ω12 ) · · · (ωj2 − ωj−1 j j+1 j Q 2 2 2 2 2 2 (ω − ω1 )(ω − ω2 ) · · · (ω − ωQ−1 ) κ(ωQ ) + + 2 − ω 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 (ωQ 1 2 Q Q Q−1 ) 2 2 + (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ ) ℘× π ∞ [n(ω) − 1]dω × 2 ). (ω 2 − ω 2 )(ω 2 − ω12 ) · · · (ω 2 − ωQ 0

Similar formulas can be written also for the linear susceptibility. Multiply subtractive K-K relations have very rapid convergence, which significantly reduces the errors caused by extrapolations. . 3 8. The harmonic oscillator model . – In this subsection we perform an analysis of the general integral properties of the linear susceptibility that can be defined by modelling the light-matter interaction with the classic harmonic oscillator Drude-Lorentz model [7,42]. Such a model is scalar and hence it is suitable for describing only the diagonal elements of the linear susceptibility tensor. We consider a material with N/V electrons per unit of volume. Every electron is subjected to a restoring elastic force towards the equilibrium position, to a friction proportional to the speed, and to an external periodic force due to an oscillating electric field. We assume that the motion of the electron remains within a reasonably limited


43

spatial range, so that we can safely consider the dipolar approximation for the electric field. The Hamiltonian for the single oscillator is (181)

H=

p2 + V (x) + eE(t)x, 2m

where x is the spatial coordinate, p = mx˙ is the momentum, E(t) is the applied electric field, and V (x) is the harmonic potential energy defined as (182)

V (x) =

mω02 x2 . 2

When damping is included, we obtain the equation of motion for each electron as x ¨ + γ x˙ + ω02 x = −

(183)

eE(t) . m

We define the polarization for unit of volume as P (t) = −e

(184)

N x(t). V

Applying the Fourier transform to the equation of motion, we obtain P (ω) = −e

(185)

N x(ω) = χ(1) (ω)E(ω), V

with (186)

1 e2 N = 2 m V ω0 − ω 2 − iγω ωp2 ωp2 ω02 − ω 2 γω = +i , 2 2 2 2 2 4π (ω0 − ω ) + γ ω 4π (ω02 − ω 2 )2 + γ 2 ω 2

χ(1) (ω) =

where we have used the previously presented definition of plasma frequency ωp . For large values of ω (ω ω0 ) the previous expression can be approximated as (187)

χ(1) (ω) ≈ −

ωp2 1 ωp2 γ + i , 4π ω 2 4π ω 3

which in the limit of γ → 0 reproduces the asymptotic behaviour obtained in eq. (82). We notice that the imaginary part has actually a faster asymptotic decrease as required. In the general case ω0 = 0, we are modelling a system where a restoring force acts on the electrons and permits them to wander around an equilibrium position. We can assume we are referring to a non-conducting material. The susceptibility (186) is a holomorphic function in the upper part of the complex ω-plane, because it is the Fourier transform of the Green function of a physical system that obeys causality. We can then

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write the K-K relations between the real and imaginary parts of the susceptibility, which correspond to the diagonal terms of the tensorial expressions in eqs. (129), (130):

χ

(188)

(1)

∞ & (1) ' ω χ (ω ) 2 (ω) = ℘ dω , π ω 2 − ω 2 0

(1)

χ

(189)

∞ & (1) ' χ (ω ) 2ω (ω) = − ℘ dω . π ω 2 − ω 2 0

From these dispersion relations we can deduce, using as usual the superconvergence theorem, the two sum rules for the first moment of the imaginary part and for the real part, obtaining the very same results as in eqs. (141) and (143), respectively: ∞ (190) 0

(191)

π ω 2 p , ω χ(1) (ω ) dω = lim − ω 2 χ(1) (ω) = ω→∞ 2 8

∞ π ω χ(1) (ω) χ(1) (ω ) dω = lim = 0. ω→∞ 2 0

Choosing ω0 = 0, there is no restoring force for the electrons, so that we may assume that we are modelling the free-electron gas of the metals (Drude model) [42]. In this case the linear susceptibility can be expressed as χ(1) (ω) = −

(192)

1 γ e2 N e2 N . + i mV ω 2 + γ 2 mV ω(ω 2 + γ 2 )

The imaginary part of the susceptibility turns out to be divergent as ω → 0, so that we can define a non-vanishing quantity descriptive of the static conductivity: (193)

σ(0) =

e2 N 1 . mV γ

Allowing the correspondence between the above-defined quantity and the conductivity of a quantum system, the holomorphic properties and the asymptotic behaviour of the susceptibility obtained with the Drude model correspond to the previously presented results for the conductors. Therefore, dispersion relations closely corresponding to those presented in (133) can be established:

(1)

χ

(194)

∞ & (1) ' ω χ (ω ) 2 (ω) = ℘ dω , π ω 2 − ω 2 0

∞ & (1) ' σ (1) (0) χ (ω ) 2ω (1) χ (ω) − =− ℘ dω . ω π ω 2 − ω 2

(195)

0


45

By applying the superconvergence theorem [6,78] to the second dispersion relation (195), we obtain, with considerable similarity to the quantum case (144), a correction to the ADNS sum rules resulting from the presence of a non-vanishing static conductivity:

(196)

∞ σ (1) (0) π π ω χ(1) (ω) − χ(1) (ω ) dω = lim = − σ (1) (0). ω→∞ 2 ω 2 0

The main reason why we have in all cases obtained the same integral properties for the susceptibility obtained with very simplified classic models and a full quantum treatment depends on the fact that the general properties we have considered derive directly from the causality of the response function [7, 10, 14, 42]. The detail of the interaction, and thus the choice of the model of the physical system, which determines the precise form of the Green function, is not relevant in this context. 4. – General properties of the nonlinear optical response . 4 1. Historical and introductory remarks on nonlinear optics. – Linear optics provides a complete description of light-matter interaction only in the limit of weak radiation sources. When we consider more powerful radiation sources, the phenomenology of the interaction is much more complex and interesting, since entirely new classes of processes can be experimentally observed, so that new theoretical tools have to be introduced in order to account for these phenomena. When matter interacts with one or more monochromatic intense light sources, a very notable effect of the nonlinear light-matter processes is the frequency mixing, i.e. the presence in the spectrum of the outgoing radiation of non-vanishing frequency components corresponding to the sum and differences of the frequency components of the incoming radiation. This process is greatly enhanced when either the incoming radiation or their combinations are in resonance with the permitted transitions in the material. Within the large family of the frequency mixing phenomena, a very relevant process is the harmonic generation. Harmonic generation is the production of outgoing radiation with relevant spectral components at frequencies that are the integer multiples of the frequencies of the incoming monochromatic light sources. Another very relevant family of nonlinear effects are the multi-photon absorption phenomena, i.e. processes where the electronic transitions are assisted by the simultaneous intervention of more than one photon. In recent years nonlinear optical investigations have gained fundamental importance in the study of the properties of gases, liquids and condensed matter, and they have assumed great relevance in technological applications as well as in the study of organic and biological materials. In 1931 Goppert-Mayer [49] proposed that two-photon assisted transitions between two states were theoretically possible, but only in the 1950s, when suitable monochromatic radiation sources became available in the regions of radiowaves and microwaves, were nonlinear processes observed. Hughes et al. [103] measured the two-photon transition between two hyperfine levels, while Battaglia et al. [104] observed the two-photon transition between the rotational levels of the OCS molecule. In the 1960s the advent of laser technology permitted the observation of nonlinear phenomena in the optical frequencies. In 1961 Kaiser et al. [105] observed the first twophoton transition in solid matter and in the same year Franken et al. [106] experimentally proved the possibility of the second harmonic generation. These experiments had a great

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impact on the scientific community and provided a strong impulse for the theoretical and experimental research in nonlinear optics. Woodbury and Ng [107] then observed the phenomenon of the stimulated Raman absorption [108, 109], Maker et al. [110] reported on the third harmonic generation, and Singh and Bradley [111] first observed a threephoton absorption process; the dynamic Stark effect [112] was later first observed by Schuda et al. [113]. The book by Bloembergen [16] provides a complete report of the scientific activity in this rapidly growing branch of physics, and more recently a very complete collection of the most relevant papers from the early days of nonlinear optics has been published [114]. Exploiting the phenomena of frequency mixing in suitable materials, and taking advantage of the transition resonances in order to obtain good power conversion, it was possible to obtain new coherent and monochromatic sources with peak frequencies different from those of the primary sources, and thus to increase the possibilities of experimental studies of nonlinear optical phenomena. In the 1970s tunable monochromatic light sources became available thanks to the development of solid-state and dye lasers [115, 116]. This permitted nonlinear spectroscopic studies that could span an extended frequency range, instead of being limited to given discrete frequencies. The passage from discrete frequencies to continuum greatly improved the investigation of the nonlinear optical properties of the materials. Thereafter, the development of the mode-locked laser permitted the generation of pulses shorter than 10−9 s with very high peak power. This permitted the experimental study of transient nonlinear phenomena and of nonlinear effects of a higher order. Since the 1980s, lasers with peak intensities of 1013 W/cm2 or more have become available so that very high-order nonlinear processes, such as production of harmonics of orders higher than 100, have also been possible [117-119]. The theory of nonlinear optics is mostly based on a perturbative approach introduced by Bloembergen [16], which is essentially a semiclassical analogue of Feynman graphs formalism [15], and which generally adopts the dipolar approximation. In the perturbative approach the nonlinear optical properties of the material are fully described by the nonlinear susceptibilities, which are obtained by applying the Fourier transform to the nonlinear Green function, which describes the higher-order dynamics of the system. The nonlinear Green function can be obtained using a generalization of Kubo’s linear response theory [33] used in deriving eq. (69). Results obtained using the perturbative theory are generally in good agreement with the experimental data unless we consider the interaction of matter with ultra-intense lasers, such as those responsible for very high-order harmonic generation. A detailed and more up-to-date presentation of the perturbative theory for nonlinear optics and of the comparison with experimental data can be found in recent books by Butcher [17] and Boyd [18]. In this section we will review the main results obtained in the past decade, leading to the establishment of general integral properties for nonlinear optical susceptibilities. We extend the results obtained in the previous section to higher orders of perturbation. We also discuss nonlinear local field effects and show how they can be described with the effective-medium approximation for both homogeneous and nanostructured media. . 4 2. Nonlinear Green functions and nonlinear optical functions. – In our analysis we adopt the dipolar approximation and generalize the definition of polarization presented in the previous section. The total polarization can then be expressed as the sum of the

47


linear and nonlinear contributions: P (t) = P (1) (t) + P NL (t) ,

(197)

where the total nonlinear polarization P NL (t) is defined as the sum over the various contributions at each order of nonlinearity: P NL (t) =

(198)

∞

P (n) (t) .

n=2

We define the n-th–order nonlinear polarization as a multiple convolution product:

(199)

(n) Pi (t)

∞

(n)

Gij1 ...jn (t1 . . . tn )Ej1 (t − t1 ) . . . Ejn (t − tn )dt1 . . . dtn ,

= −∞

where the nonlinear Green function is symmetric in the time variables and respects the temporal causality principle: (n)

Gij1 ...ji ...jk ...jn (t1 , . . . , ti , . . . , tk , . . . , tn ) =

(200)

(n)

= Gij1 ...jk ...ji ...jn (t1 , . . . , tk , . . . , ti , . . . , tn ) , Gij1 ...jn (t1 , . . . , tn ) = 0, ti < 0, 1 ≤ i ≤ n, Computing the Fourier transform of expression (199) we obtain

(201)

(n) Pi (ω)

∞ = −∞

n

(n) χij1 ...jn

×δ ω −

ωl ; ω1 , . . . , ωn

l=1 n

Ej1 (ω1 ) . . . Ejn (ωn ) ×

ωl

dω1 . . . dωn ,

l=1

where the Dirac δ-function guarantees that the sum of the arguments of the Fourier transforms of the electric field equals the argument of the Fourier transform of the polarization. This means that each ω component of the n-th–order nonlinear polarization results from the interaction of n photons mediated by the material under examination, which is described by the nonlinear susceptibility function, defined as

(202)

(n) χij1 ...jn

n l=1

ωl ; ω1 , . . . , ωn

∞ = −∞

(n) Gij1 ...jn

(t1 , . . . , tn ) exp i

n

ωl tl dt1 . . . dtn .

l=1

It is customary in the literature [17, 18] to set as the first argument of the susceptibility before the semicolon the sum of all the frequency arguments. From the definition (202) it is possible to derive the a priori properties of the nonlinear susceptibility functions by considering the properties of the nonlinear Green function described in expressions (200). The symmetry of the nonlinear Green function with respect to the time variable exchange

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implies that the nonlinear susceptibility functions obey a symmetric property for the frequency variables: (203)

(n) χij1 ...ji ...jk ...jn

=

n

ωl ; ω1 , . . . , ωi , . . . ωk , . . . , ωn

l=1

(n) χij1 ...jk ...ji ...jn

n

=

ωl ; ω1 , . . . , ωk , . . . ωi , . . . , ωn

, ∀i, j ∈ {1, . . . , n} .

l=1

Since P (n) (t) is real and since the nonlinear Green function obeys causality, we derive, along the lines of the previously analyzed linear case, that at every order k the following relation holds: n n ∗ (n) (n) (204) χij1 ...jn −ωl ; −ω1 , . . . , −ωn = χij1 ...jn ωl ; ω1 , . . . , ωn , l=1

l=1

so that, separating the real and imaginary parts, we can write the following generalizations of eqs. (127), (128):

(205)

(n) χij1 ...jn

(206) −

(n) χij1 ...jn

n

l=1 n

=

−ωl ; −ω1 , . . . , −ωn −ωl ; −ω1 , . . . , −ωn

(n) χij1 ...jn

(n) χij1 ...jn

=

l=1

n

l=1 n

,

ωl ; ω1 , . . . , ωn ωl ; ω1 , . . . , ωn

.

l=1

We next consider the realistic case of the interaction of the matter with m monochromatic plane waves with different amplitude, polarization and frequency: Ej (t) =

(207)

m i=1

(i)

Ej e−iωi t + c.c. , (i)

where c.c. indicates the complex conjugate, and Ej is the amplitude of the electric field of the i-th wave in the j-direction. We have (208)

Ej (ω) =

m i=1

(i)

(i)

Ej δ (ω − ωi ) + Ej δ (ω + ωi ).

From formulas (200) and (208) we deduce that the spectral components of the n-th–order polarization are all the possible sums of n among the 2m (m positive and m negative) frequencies characterizing the spectrum of the incoming radiation: (209)

(n) Pi

(ω) =

(n) χij1 ...jn

Ωi ∈{±ω1 ,...±ωm } (Ω ) ×Ej1 1

(Ω ) . . . Ejn n δ

ω−

n

Ωl ; Ω1 , . . . , Ωn

l=1 n i=1

Ωi

,

×


49

Fig. 8. – Scheme of a pump and probe experiment where the two laser beams have the same polarization unit vectors. Thick line: pump laser; thin line: probe laser.

(Ωi )

where Eji (210)

(i)

= Eji if Ωi = ±ωi . Separating each frequency component we have (n)

Pi

(ω) =

ωs

(n)

Pi

(ωs ) δ (ω − ωs ) ,

where we are summing over all the possible distinct values ωs of the sum of n among the (n) 2m frequencies in the electric-field spectrum. The coefficients P i (ωs ) are such that (211)

(n)

Pi

(t) =

ωs

(n)

Pi

(ωs ) e−iωs t .

The presence in the nonlinear polarization of all the possible combinations of the frequencies of the incoming radiation is commonly referred to as frequency mixing. . 4 3. Pump and probe susceptibility. – We first consider the relevant and instructive example of the pump and the probe processes. The ideal experimental setting is depicted in fig. 8, where two laser beams are present. We make the simplifying assumption that both light beams are linearly polarized along the x-direction: (212)

E(t) =x Ê (1) e−iω1 t + x Ê (2) e−iω2 t + c.c.

We assume that index (1) refers to the probe laser, whose frequency can be changed, and index (2) to the pump laser, whose frequency is fixed. We take (213)

E (2) E (1) .

We consider the third-order expression of the polarization at frequency ω1 , i.e. the first variation in the linear optical response at the frequency of the probe laser due to the

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Fig. 9. – Real and imaginary parts of the pump and probe susceptibility from a simplified model. The peaks corresponding to the dynamical Stark effect (D.S.E), two-photons absorption (T.P.A.) and stimulated Raman scattering (S.R.S.) are indicated. Adapted from [36]

light-matter coupling. Specifying in expression (210) n = 3 and ω = ω1 , and considering the form (212) of the incoming radiation, we obtain (214)

P

(3)

2 (3) (ω1 ) = 6E (1) E (2) χixxx (ω1 ; ω1 , ω2 , −ω2 ) + 3 (3) +3 E (1) χixxx (ω1 ; ω1 , ω1 , −ω1 ) ,

where the numeric coefficients can be deduced using combinatorial arguments, considering that the symmetry relation (203) holds. We underline that since the probe laser is much less intense than the pump laser, the first term is largely dominant in expression (214). We emphasize that, since expression (214) represents the first nonlinear correction to the linear response at the probe laser frequency, it is possible to derive a correspondence between the nonlinear change in the index of refraction and the pump and probe susceptibility: (215)

N NL (ω1 , ω2 , E1 , E2 ) = ∆N (ω1 ) = ∆n(ω1 ) + i∆κ(ω1 ) = "2 "2 ! ! 6 E (2) χ(3) (ω1 ; ω1 , ω2 , −ω2 ) + 3 E (1) χ(3) (ω1 ; ω1 , ω1 , −ω1 ) ∼ = 4π 2N (ω1 ) ! (2) "2 (3) E χ (ω1 ; ω1 , ω2 , −ω2 ) ∼ 12π = N NL (ω1 , ω2 , E1 ), N (ω1 )

where we have used a scalar notation and we have simplified the expression using the


51

conditions (213). We underline that in expression (215) N (ω1 ) is the linear index of refraction introduced in expression (26), and we have used the ∆-symbol to emphasize that N NL (ω1 , ω2 , E2 ) is the first correction to the linear refraction index. The dominant term of the quantity (214) —and, correspondingly, of the quantity (215)— is especially relevant in physical terms since the imaginary part is related to the first-order nonlinear induced absorption, which includes the dissipative phenomena of two-photon absorption [50, 51, 120-122], stimulated Raman scattering [123-126], and dynamic Stark effect [112, 113, 127-131], while the real part describes all the corresponding nonlinear dispersive effects [12]. Figure 9 shows the values of the real and imaginary parts of the first term of expression (214) obtained using a simple anharmonic oscillator model [36]. The second term of expression (214) describes all of the nonlinear phenomena resulting from the probe laser only and, when E (1) is factored out, is usually referred to as degenerate susceptibility: the imaginary part contributes to the absorption (dynamic Kerr effect), while the real part contributes to changing the linear refractive index (dynamic Kerr birefringence). . 4 4. Quantum-mechanical expression of the nonlinear Green functions. – In this subsection we provide the explicit expression of the general quantum nonlinear Green function, so that the nonlinear susceptibility can be derived. The nonlinear susceptibility contains all the information on the nonlinear optical properties of the system. We define the n-th–order nonlinear polarization as the expectation value [16-18] of the electric dipole moment per unit volume over the n-th–order perturbative term of the density operator (65) introduced in the previous section:

(n) Pi

(216)

N 1 α (n) (t) ≡ Tr −e ri ρ (t) . V α=1

We ignore the relaxation phenomena in describing the evolution of ρ(n) (t). We therefore consider the n-th differential equation in the concatenated system (66): i∂t ρ(n) (t) = H0 , ρ(n) (t) + HIt , ρ(n−1) (t) ,

(217)

with the boundary condition ρ(n) (0) = 0. The solution of this equation can be obtained recursively by considering each increasing perturbation order [17, 18], so that ρ(n) (t) can be expressed as a function of the dipole operators and of ρ(0) only:

(218)

(n)

ρ

t n−1 n t e (t) = ... Ej1 (t1 ) . . . Ejn (tn ) × −i −∞ −∞ N N α α rj1 (t1 − t) , . . . rjn (tn − t) , ρ (0) . . . dt1 . . . dtn . × α=1

α=1

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Considering that the following relation holds [17]: (219)

Tr

N

riα

α=1

N

rjα1 (−t1 ) , . . .

α=1

N

n

= (−1) Tr

α=1

rjαn

N

α=1

rjαn (−tn ) , ρ (0) . . .

N

(−tn ) , . . . ,

α=1

rjα1

(−t1 ) ,

N

= riα

. . . ρ (0) ,

α=1

we can express the n-th–order nonlinear polarization as a convolution product of a nonlinear Green function times n electric fields: (220)

(n) Pi

∞ (t) =

∞

−∞

(n)

Gij1 ...jn (t1 , . . . , tn ) Ej1 (t − t1 ) . . . Ejn (t − tn ) dt1 . . . dtn ,

... −∞

where the n-th–order nonlinear Green function is (221)

en+1 (n) Gij1 ...jn (t1 , . . . , tn ) = − n θ (t1 ) . . . θ (tn − tn−1 ) × V (−i) N N N rjαn (−tn ) , . . . , rjαi (−t1 ) , riα . . . ρ (0) . ×Tr α=1

α=1

α=1

Therefore, the n-th–order nonlinear polarization P (n) (t) can generally be expressed as the expectation value of a t-dependent function over the thermodynamic equilibrium unperturbed density operator of the physical system under analysis, as obtained in the previous section for the linear case. All the optical properties —linear and nonlinear— of a given physical system then depend only on its statistically relevant ground state. Using definition (202), the n-th–order nonlinear susceptibility of a general quantum system can then be expressed as n (n) χij1 ...jn ωj ; ω1 , . . . , ωn = (222) l=1

∞ =

(n) Gij1 ...jn

(t1 , . . . , tn ) exp i

n

ωl tl dt1 . . . dtn =

l=1

−∞

∞ en+1 θ (t1 ) . . . θ (tn − tn−1 ) × =− n V (−i) −∞ N N N rjαn (−tn ) , . . . , rjα1 (−t1 ) , riα . . . ρ (0) × ×Tr

α=1 n

× exp i

α=1

α=1

ωl tl dt1 . . . dtn .

l=1

In the linear case, the study of the asymptotic behaviour reveals that only the 0th moment of the susceptibility has an asymptotic behaviour fast enough for the dispersion


53

relation to converge. Hence, only one pair of independent dispersion relations —and of sum rules— can be established. We will show that the asymptotic decrease in the nonlinear susceptibilities for large values of each frequency variable is strictly faster than for the linear susceptibility. The conceptual reason is that the electrons interacting with very energetic photons behave indistinguishably from the components of a free-electron gas, since on the very short time-scale relevant for the light-matter interaction the other forces affecting the electrons are negligible, and the free-electron gas behaviour is already asymptotically saturated by considering the linear susceptibility alone, as seen in the previous section. We then expect that, if it is possible to frame a dispersion theory for nonlinear optical susceptibilities, the number of independent K-K relations and sum rules that can be established depends on how fast the asymptotic decrease is. . 4 5. Local field effects. – The nonlinear optical response of the elementary component of a physical system is thoroughly described by the suitable microscopic hyperpolarizability function β (n) which connects the microscopic nonlinear polarization p(n) of the elementary constituent of the medium to the electric field acting on it. Adopting a scalar notation, we have that

(223)

    n n p(n)  ωj  = β (n)  ωj ; ω1 , . . . , ωn  E(ω1 ) · E(ωn ). j=1

j=1

Along the lines of the linear case, the macroscopic nonlinear polarization results to be

(224)

    n n 1 ωj  = p(n)  ωj  dV = P (n)  V V j=1 j=1   n = ℵβ (n)  ωj ; ω1 , . . . , ωn  E loc (ω1 ) · E loc (ωn ), j=1

where eq. (96) connects the number of elementary constituents per unit volume with the number of electrons per unit volume. On the other hand, the nonlinear polarization can be expressed as the result of the nonlinear coupling of the external electric field with an effective nonlinear susceptibility:  (225)

P (n) 

n j=1

 ωj  =

(n) χeff

  n  ωj ; ω1 , . . . , ωn  E (ω1 ) · · · E (ωn ) . j=1

In order to obtain the expression of the effective susceptibility we have to equate expressions (224) and (225). Considering that the local field of a nonlinear medium takes the form (226)

E loc (ω) = E(ω) +

4π (1) 4π P (ω) + PNL (ω), 3 3

we obtain that the effective macroscopic nonlinear susceptibility, which takes into account the local field effects, is related to the n-th–order microscopic hyperpolarizability (223)

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in the following way:

(227)

  n (n) ωj ; ω1 , . . . , ωn  = χ  eff

j=1



= ℵL 

n





ωj  L(ω1 ) . . . L(ωn )β (n) 

j=1

n

 ωj ; ω1 , . . . , ωn  ,

j=1

where the factors L(ω) are defined as (1)

(228)

L(ω) =

eff (ω) + 2 4π (1) ℵα(1) (ω) 4π = 1+ χeff (ω) = 1 + = (1) (ω) 3 3 3 1 − 4π 3 ℵα = 1+

χ(1) (ω) 4π = (1) (ω) 3 1 − 4π 1− 3 χ

1

, 4π (1) (ω) 3 χ

where we have used eqs. (104), (105). Similarly to the linear case, if ℵ is small, we have that  (229)

(n) χeff



n





ωj ; ω1 , . . . , ωn  ∼ ℵβ (n) 

j=1

n

 ωj ; ω1 , . . . , ωn  .

j=1

The expression of the general nonlinear susceptibility presented in this work has been obtained with a strictly microscopic treatment through the nonlinear Green function formalism presented in eq. (221), so that  (230)

χ(n) 

n





ωj ; ω1 , . . . , ωn  = ℵβ (n) 

j=1

n

 ωj ; ω1 , . . . , ωn  .

j=1

Hence, we find that the following relation holds between the effective macroscopic nonlinear susceptibility and the nonlinear susceptibility proposed in expression (222):  (231)

(n) χeff



n

 ωj ; ω1 , . . . , ωn  =

j=1



= L

n j=1





ωj  L(ω1 ) . . . L(ωn )χ(n) 

n

 ωj ; ω1 , . . . , ωn  .

j=1

The experimental set-up for the measurement of linear and nonlinear local field factors was established by Maki et al. [132], who measured the reflectivity and surface phase conjugation of a dense atomic vapor. The former quantity was used to determine the linear microscopic polarizability and the latter to determine the second hyperpolarizability.


55

. 4 5.1. Relevance of local field effects in nanostructured media. In the field of optical physics, nanostructures have attracted much interest during the past two decades as a new class of optical materials. There is a great demand for materials with large nonlinear optical responses and suitable spectral properties because it is assumed that such materials will play an important role in future photonic devices and all-optical signal processing [133, 134]. Conventionally, the development of new devices has involved a search for materials with suitable nonlinear optical properties. However, a more advanced way to enhance the nonlinear properties would be to find suitable constituent materials and combine them into a nanostructure. The nonlinear optical properties of nanostructures including Maxwell Garnett [63, 69], Bruggeman [46, 75], metal-coated [135], and layered [47] nanostructures have been studied widely both theoretically and experimentally [136, 137]. The interest in such nanocomposites arises from the possibility to enhance their optical nonlinearity: the effective nonlinear optical susceptibility of the nanostructure material can in fact exceed those of its constituent components for a wide range of nonlinear processes such as four-wave mixing, third harmonic generation and nonlinear refractive index [138]. This enhancement is due to the collective enhancement of local electric fields in the medium. Recently, the enhancement of nonlinearity was verified experimentally [136]. Moreover, optical bistability, which has great potential for optical switching, has been demonstrated in weak nonlinear nanostructures [139] and even in a thin layer of optically nonlinear material [140], and has been explained by means of local fields. . 4 5.2. The Maxwell Garnett system. The investigation of nonlinear optical properties of nanospheres [63, 67, 141-143] and coated nanospheres [135, 144] has been the object of intensive study. The theory and mathematical formulas for the effective third-order degenerate susceptibility was given by Sipe and Boyd [63] for cases where the nanospheres are optically nonlinear while the host material acts in an optically linear manner and vice versa. They have also derived an expression for the general case where both the nanospheres and the host material are optically nonlinear. In this paper we concentrate on the properties of the degenerate third-order nonlinear susceptibility of nanostructures, since it is proportional to the nonlinear index of refraction. The effective degenerate MG third-order susceptibility for linearly polarized light of a single beam can be expressed as follows (for a more detailed derivation see ref. [63]): (232)

(3)

(3)

χeff (ω; ω, ω, −ω) = fi [Li (ω)]2 |Li (ω)|2 χi (ω; ω, ω, −ω) + (3)

+[Lh (ω)]2 |Lh (ω)|2 [fh + fi x(ω)]χh (ω; ω, ω, −ω), where the factor x(ω) is given by (233)

x(ω) =

' 8 6 2 18 & [υ(ω)]2 |υ(ω)|2 + υ(ω)|υ(ω)|2 + [υ(ω)]3 + |υ(ω)|2 + [υ(ω)]2 , 5 5 5 5

υ(ω) being defined by expression (110). The local field factors of the nanospheres (= inclusions) and the host material are, respectively, (234)

Li (ω) =

eff (ω) + 2h (ω) , i (ω) + 2h (ω)

56


(235)

Lh (ω) =

and

J. J. SAARINEN

eff (ω) + 2h (ω) . 3h (ω)

The general expression (232) can be simplified if the volume fraction of the inclusions is small, as assumed in the MG model. The effective dielectric function of such a composite approaches the value of the host material and the local field factor of the host approaches unity. Then eq. (232) can be written in the form (236)

(3)

(3)

(3)

χeff (ω; ω, ω, −ω) = fi [Li (ω)]2 |Li (ω)|2 χi (ω; ω, ω, −ω) + χh (ω; ω, ω, −ω).

The approximation of the effective third-order susceptibility is linearly dependent on the volume fraction fi of the inclusions at small values fi . The recent study of Prot et al. [67] dealt with gold nanospheres embedded in a silica host. A large enhancement was observed in the imaginary part of the degenerate third-order nonlinear susceptibility, which was plotted as a function of the volume fraction of the nanospheres. They observed that the linear dependence is lost when the volume fraction exceeds 10 percent, which is in agreement with the theory. Equation (236) predicts the possibility of cancellation of the nonlinear absorption in a Maxwell Garnett nanosphere system. Smith et al. [143] measured the nonlinear absorption from a metal colloid of gold nanospheres, that were embedded in the water, using the Z-scan technique. The cancellation is mainly due to an imaginary part of the local field factor of the inclusions. Materials with a large nonlinear refractive index and negligible nonlinear absorption are ideal, for instance, in optical switching. Unfortunately, in bulk materials a high nonlinear refractive index is usually related to high nonlinear absorption. Therefore, the aim in nonlinear optical nanostructure engineering is usually to find optimum constituent materials so that the nonlinear absorption of incident light is negligible. Conversely, in some cases it may even be desirable to have large nonlinear absorption such as a two-photon absorption induced fluorescence from microvolume, which has a great potential in drug discovery and in novel bioaffinity assays [145]. In conclusion, the nonlinear optical properties of an MG nanosphere system can be optimized according to the applications by an appropriate choice of the constituent materials and volume fraction. . 4 5.3. The Bruggeman effective-medium theory. The spectroscopic properties of liquids are usually measured by a reflectometer [146], which is based on the use of a prism as a probe. The information about the liquid is obtained from a thin layer at the prism-liquid interface in the region of an evanescent wave. For example, the islands of oil drops on the surface of water can be treated as a two-phase Bruggeman nanostructure. In the measurement of the effective reflectivity of a random nanostructure, the effective dielectric function at the probe-sample interface may differentiate from the sample average, which leads to an erroneous result. Therefore, Gehr et al. [147] suggested a measurement over the total volume based on the use of an interferometer and found good agreement between predicted values and measured data. For randomly intermixed constituents, the effective nonlinear susceptibilities should be considered with a statistical theory [66]. Fortunately, with the approximation of the uniformity of the electric field inside each constituent the effective degenerate third-order susceptibility for a single beam is given by [46] 1 ∂eff (ω) ∂eff (ω) (3) (3) (237) χeff (ω; ω, ω, −ω) = χi (ω; ω, ω, −ω), fi ∂i (ω) ∂i (ω) i


57

where i denotes the summation of the constituents. For a two-phase Bruggeman model, the explicit expressions for the derivatives have been presented in ref. [46]. Equation (237) predicts a small enhancement in the third-order nonlinear susceptibility, when the linear refractive index of the linear constituent is larger than that of the nonlinear constituent [66]. In addition, Lakhtakia [148] investigated the complex linear dielectric function and nonlinear susceptibility of the Bruggeman effective media using strong permittivity fluctuation theory. One of the few experimental results was presented by Gehr et al. [147] who investigated the nonlinear properties of a porous glass saturated with optically nonlinear liquids by the Z-scan method. The enhancement of the nonlinear susceptibility was verified and observed to be approximately 50%. Bruggeman formalism is applicable to nanosphere systems with shape distributions. Recently, it has been shown [149,150] that shape variation can cause the separation of the nonlinear absorption peak and the enhanced nonlinear susceptibility peak. This means that it is possible to optimize the nonlinear enhancement of the nanostructure by finding an optimum shape distribution. However, the spectral properties of the constituents must be known. . 4 5.4. Layered nanostructures. The mathematical expressions for the effective nonlinear susceptibility of different nonlinear optical processes for layered nanostructures were given by Boyd and Sipe [47] starting from the mesoscopic field inside each layer. The macroscopic field was obtained after an averaging process of the mesoscopic fields. For a two-phase structure with TE-polarized light, the effective degenerate third-order susceptibility of a single beam becomes a summation of the constituent susceptibilities multiplied by their volume fractions, as follows: (238)

(3)

(3)

χeff (ω; ω, ω, −ω) = fa χ(3) a (ω; ω, ω, −ω) + fb χb (ω; ω, ω, −ω).

Hence, their enhancement in a nonlinear optical process cannot exist. However, for TM-polarized light the situation changes drastically. Boyd and Sipe [47] formulated the mathematical expression for an effective degenerate third-order susceptibility of a single beam with TM-polarized light obtaining (239)

(3) χeff (ω; ω, ω, −ω)

(3)

eff (ω) 2 eff (ω) 2 (3) = fa χa (ω; ω, ω, −ω) + a (ω) a (ω) eff (ω) 2 eff (ω) 2 (3) +fb χb (ω; ω, ω, −ω). b (ω) b (ω) (3)

Therefore, if a > b and χa < χb , the effective third-order susceptibility exceeds those of its constituent components. The terms in brackets in eq. (239) contain the local field factors of the constituents. The analysis of the nonlinear properties of layered structures presented above is based on the effective-medium approximation. Furthermore, an analysis based on the nonlinear wave equation has been examined [151], yielding analogous results. Theoretically, there can be an enhancement up to a factor of 10 when the ratio of the linear refractive indices of the constituents is equal to 2. Nevertheless, in their experimental measurement, Fischer et al. [136] obtained an enhancement only of a factor of 1.35. This was caused by the small difference between the linear refractive indices of the investigated organic polymer and titanium dioxide, that is only 1.33. Therefore, the result is consistent with the predicted value.

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In the frame of nonlinear optics, layered nanostructures are assumed to play an important role. It has been demonstrated [152] that the electro-optic effect can be enhanced with layered nanostructures. The enhancement of nonlinearity in layered structures can be as large as a factor of 10, which makes them very attractive in all-optical signal processing. Kishida et al. [133, 134] have observed large nonlinearity in transition-metal oxides and halides. Such materials combined into a layered nanostructure may possess a huge third-order nonlinearity and may find applications in novel optoelectronic device development. 5. – Dispersion relations and sum rules for nonlinear optical processes In spite of the ever increasing scientific and technological relevance of nonlinear optical phenomena, relatively little attention has been paid for a long time to the experimental investigation of the K-K dispersion relations and sum rules of the corresponding nonlinear susceptibilities. The research has usually focused on achieving high resolution in both experimental data and theoretical calculations, while those properties are especially relevant for experimental investigations of frequency-dependent nonlinear optical properties. In the context of this class of experiments, the K-K relations and sum rules could provide information on whether or not a coherent, common picture of the nonlinear properties of the material under investigation is available. We remark that only a few data sets referring to nonlinear optical phenomena span a spectral range wide enough to permit the use of dispersion relations. The first heuristic applications of the K-K dispersion relations theory to nonlinear susceptibilities date back to the 1960s [153-155] and 1970s [156, 157], while a more systematic study began essentially within the last decade. Some authors have preferentially introduced the K-K relations in the context of ab initio or model calculations of materials properties [39, 158-161]; a very complete review of this approach can be found in [25]. Other authors have used a more general approach capable of providing the theoretical foundations of dispersion theory for nonlinear optics [19-24]. This approach permits a connection between the K-K relations and the establishment of the sum rules for the real and imaginary parts of the nonlinear susceptibility. The instruments of complex analysis permit the definition of the necessary and sufficient conditions for the applicability of the K-K relations, which require that the nonlinear susceptibility function descriptive of the nonlinear phenomena under examination be holomorphic in the upper complex plane of the relevant frequency variable. It has also been shown that the asymptotic behaviour of the nonlinear susceptibility determines the number of independent pairs of the K-K relations that hold simultaneously. Combining the K-K relations and the knowledge of the asymptotic behaviour of the nonlinear susceptibility, it is possible to derive, along the same lines as described for the linear case, sum rules for nonlinear optics. We underline that sum rules for nonlinear optics have been obtained by other authors by following different strategies [162]. We would emphasize that, as for the linear case, by generalizing the Drude-Lorentz . model presented in subsect. 3 8 with the introduction of a generic non-quadratic potential [16, 37, 38], nonlinear optics has also been provided with a qualitatively reliable simple model. The model is in essential agreement with experiments and with detailed theoretical calculations since the susceptibilities deduced using the toy model obey the same general integral properties of the susceptibilities obtained with a rigorous quantum treatment [21, 23, 35, 36]. In this section we study the holomorphic properties of the general nonlinear suscep-

59


tibility functions, and we deduce general criteria for establishing the cases in which the K-K relations connect the real and imaginary parts of the nonlinear susceptibility under consideration. We will consider in particular the analysis of the nonlinear integral properties of the pump and probe susceptibility as an experimentally and theoretically relevant example. We also briefly consider the case of the meromorphic total refractive index, when the optical constants have poles in the upper complex plane, and derive the related modified K-K relations and sum rules. We will analyze at the end of this section the anharmonic oscillator model in detail in the case of pump and probe processes and show how such a simple classical model permits nevertheless the derivation of some general properties. . 5 1.Kramers-Kronig relations for nonlinear susceptibility: independent frequency variables. – The nonlinear Green function obeys causality for each of its time variables, as shown in (200). Therefore, assuming as usual that the suitable integrability conditions are obeyed, we can apply Titchmarsch’s theorem [10] separately to each variable and deduce that the nonlinear susceptibility function (202) is holomorphic in the upper complex plane of each variable ωi , 1 ≤ i ≤ n. If we consider the first argument ω1 of the nonlinear susceptibility function (202), the following relation holds: ∞

(n)

χij1 ...jn

℘

(240)

−∞

=

n l=1 ω1

ωl ; ω1 , . . . , ωn − ω1

(n) iπχij1 ...jn

ω1 +

n

dω1 = ωl ; ω1 , ω2 , . . . , ωn

.

l=2

Repeating the same procedure for all the remaining n − 1 frequency variables, and applying the symmetry relation (205), we obtain (241)

n π n (n) − χij1 ...jn ωl ; ω1 , . . . , ωn = 2 l=1 ( n ) (n) ∞ ∞ ωl ; ω1 , . . . , ωn χij1 ...jn l=1 dω1 . . . dωn , = ω1 . . . ωn ℘ . . . ℘ (ω12 − ω12 ) . . . (ωn2 − ωn2 ) 0

(242)

π n 2

(n) χij1 ...jn

...℘ 0

n

0

ωl ; ω1 , . . . , ωn

l=1

∞ ω1

∞ =℘

0

(

. . . ωn

(n) χij1 ...jn

=

n l=1

) ωl ; ω1 , . . . , ωn

(ω12 − ω12 ) . . . (ωn2 − ωn2 )

dω1 . . . dωn ,

which is for nonlinear optics the equivalent of the conventional K-K relation previously . described for the linear case in subsect. 3 2. It is clear that these relations, in spite of their theoretical significance, are not interesting from an experimental point of view,

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since their verification as well as their utilization for optical data retrieving requires the possibility of independently changing n laser beams. Moreover, most of the physically relevant nonlinear phenomena are described by nonlinear susceptibilities where all or part of the frequency variables are mutually dependent. This occurs in any case when the number of the frequencies components of the incoming radiation is less than the order of the nonlinear process under examination, as occurs in the pump and probe case (214). The dispersion relations (240) are then a mere mathematical extension of the linear K-K relations. A more flexible theory is needed in order to provide the effectively relevant dispersion relations for nonlinear optical phenomena. . 5 2. Scandolo’s theorem and nonlinear Kramers-Kronig relations. – In nonlinear optics a relevant dispersion relation is a line integral in the space of the frequency variables, which entails the choice of a one-dimensional space embedded in an n-dimensional space, if we consider an n-th–order nonlinear process. This corresponds to the realistic experimental setting where only the frequency of one of the monochromatic beams described in eq. (207) is changed. Since in nonlinear optics we have frequency mixing, changing one frequency of the incoming radiation will change none, one or more than one arguments of the nonlinear susceptibility function considered, depending on how many photons of the tunable incoming radiation it takes into account. The choice of the parameterization then selects different susceptibilities and so refers to different nonlinear optical processes. Each component j of the straight line in Rn can be parameterized as ωj = vj s + wj , 1 ≤ j ≤ n,

(243)

where the parameter s ∈ (−∞, ∞), the vector v of its coefficients describes the direction of the straight line, and the vector w determines ω (0). We refer to the previously presented pump and probe susceptibility (214) to provide a well-suited example of the parameterizations (243). In this case, while the pump frequency ω2 is fixed, the probe frequency ω1 can be tuned. The correct parameterization of the straight line for the first term in expression (214) is then v = (1, 0, 0) and w = (0, ω2 , −ω2 ). The correct parameterization of the second term in (214) is instead v = (1, 1, −1) and w = (0, 0, 0), because all the arguments change simultaneously but one has the opposite sign of the other two. Scandolo’s theorem [19, 20] permits the determination in very general terms of the holomorphic properties of the nonlinear susceptibility tensor with respect to the varying parameter s. Substituting expression (243) in the definition (202), we obtain

(244)

(n)

χij1 ...jn

  n  (vj s + wj ) ; v1 s + w1 , . . . , vn s + wn  = j=1

∞ = −∞

(n)



Gij1 ...jn (t1 , . . . , tn ) exp is

n j=1





vj tj  exp i

n

 wj tj  dt1 . . . dtn .

j=1

When dealing with the s-parameterized form of the nonlinear susceptibility, from now

61


on we will use the simpler notation  (245)

(n)

(n)

χij1 ...jn (s) = χij1 ...jn 

n

 (vj s + wj ) ; v1 s + w1 , . . . , vn s + wn  .

j=1

We compute the inverse Fourier transform with respect to s of expression (244) and obtain the following parameterized one-variable nonlinear Green function: (n) Gij1 ...jn

(246)

1 (τ ) = 2π

∞

(n)

χij1 ...jn (s) e−isτ ds.

−∞

Under the condition vj ≥ 0 ,

(247)

1 ≤ j ≤ n,

we deduce that, since the nonlinear Green function obeys causality for each time variable, as remarked in condition (200), expression (246) obeys the following condition with respect to the variable τ : (n)

(248)

Gij1 ...jn (τ ) = 0 ,

τ ≤ 0.

With this result (248), and making the usual assumption that the function belongs to the space L2 , we can take advantage of the Titchmarsch theorem [10] and deduce that the nonlinear susceptibility (245) is holomorphic in the upper complex plane of the complex variable s. Hence, the Hilbert transforms connect the real and the imaginary parts of the nonlinear susceptibility (245): ∞ χ(n) (s ) ij ...j 1 1 n (n) χij1 ...jn (s) = ℘ ds , π s − s −∞ ∞ χ(n) (s ) ij1 ...jn 1 (n) χij1 ...jn (s) = − ℘ ds . π s − s

(249)

(250)

−∞

These general relations extend the results obtained in the 1960s [153-155] and 1970s [156, 157] for specific nonlinear phenomena, and the general conclusions drawn in the 1980s for specific models [21-23]. Condition (247) on the sign of the directional vectors of the straight line in Rn implies that only a particular class of nonlinear susceptibilities possess the holomorphic properties required to obey the dispersion relations (249). Hence, causality is not a sufficient condition for the existence of the K-K relations between the real and imaginary parts of a general nonlinear susceptibility function. The principle of causality [14] of the response function is reflected mathematically in the validity of the K-K relations for the nonlinear susceptibility in the form presented in eqs. (241), (242). Since sq is a holomorphic function for q ≥ 0 and integer, we expect that, in general, if the asymptotic behaviour of the nonlinear susceptibility (245) is strictly faster than that obtained in the linear case, it is possible to establish independent K-K relations,

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not only between the real and the imaginary parts of the nonlinear susceptibility, but also between the real and the imaginary parts of its q-th moments, for all the values of q such that the line integral in Rn of the dispersion relation converges. For the susceptibility functions that do not obey the conditions of the Scandolo theorem, i.e. if at least one component of the vector v in the parameterization (243) is negative, it is not possible to obtain the K-K relations. However, considering that these functions have poles in the upper complex s-plane, Cauchy’s integral theorem [41] permits us to write the following relation: ∞ (251)

℘ −∞

(3) χij1 ...jn (s) ds s − s

(n)

= iπχij1 ...jn (s) + 2πi

ps ,(ps )>0

(n) Res χij1 ...jn (s) ps − s

,

where in the second term on the right-hand side the sum is over the residua (Res) of the susceptibility in the poles ps in the upper complex s-plane. The property (204) implies that all the poles are symmetric with respect to the imaginary axis of the variable s. Therefore, for these susceptibility functions information on all the poles is required. A recent study of a simplified oscillator model [36] shows that the knowledge of the poles in the upper complex ω1 -plane, resulting from the contribution to the total pump and probe susceptibility relative to the dynamical Kerr effect, permits us to write explicitly a set of dispersion relations between the real and the imaginary parts, which differ from the usual K-K relations by a term that is proportional to the intensity of the probe beam. Generally, the inversion of optical data for this class of susceptibility functions can be performed with a high degree of accuracy using the algorithms based on the maximum entropy method [163], which in recent studies has shown its potentially great impact in applications [13, 75, 164-167]. Later in this paper we will briefly discuss the class of nonlinear susceptibilities that obey eq. (251). Such susceptibilities are termed meromorphic and describe relevant nonlinear optical processes, which are of interest especially in the case of nanostructured materials. . 5 3. Nonlinear Kramers-Kronig relations and sum rules: the case of pump and probe. – We consider the response of a physical system to a pump and probe experiment in the limit of very low intensity of the probe laser. We then consider the contribution to the nonlinear polarization at the probe frequency resulting from the interaction of photons of the pump laser corresponding to components of the fields of opposite frequency and one photon of the probe laser, corresponding to the first term in expression (214). Scandolo’s theorem implies that the two terms of the pump and probe susceptibility (214) have fundamentally different properties. The dominant term is holomorphic in the upper complex ω1 -plane, so that by considering the parameterization v = (1, 0, 0) and w = (0, ω2 , −ω2 ) for the line integral, the K-K relations can be established between the real and imaginary parts, obtaining in third order ∞ ω χ(3) (ω ; ω , ω , −ω ) 2 2 1 1 1 ij j j 2 1 2 3 (3) (252) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ℘ dω1 , π ω12 − ω12 0 ∞ χ(3) (ω ; ω , ω , −ω ) 2 2 1 1 ij1 j2 j3 2ω1 (3) ℘ (253) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = − dω1 , π ω12 − ω12

0


63

where we have considered the general tensorial component and have presented the usual semi-infinite integration by taking advantage of relation (203). Moreover, as shown in appendix A, the asymptotic behaviour for large values of ω1 of the dominant term in expression (214) can be written as follows [19, 20]: (3)

χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) ≈

(254)

cij1 j2 j3 (ω2 ) , ω14

where the explicit expression for cij1 j2 j3 (ω2 ) is (255)

cij1 j2 j3 (ω2 ) =

∞ 1 N e4 θ (t2 ) θ (t3 − t2 ) exp [−iω2 t2 + iω2 t3 ] × m2 3! (−i)2 V −∞ ∂ 2 V (r α ) α α ρ (0) dt2 dt3 . ×Tr rj3 (−t3 ) , rj2 (−t2 ) , ∂riα ∂rjα1

As with the linear case, the leading asymptotic term of the susceptibility for large values of ω1 is real. The numerical evaluation of the ω2 -dependent tensor appearing as the coefficient of the leading asymptotic order has been performed in the case of the hydrogen atom [20]. From the asymptotic behaviour (254) we deduce that the second moment of the nonlinear susceptibility considered here decreases asymptotically as ω1−2 and that it is holomorphic in the upper complex ω1 -plane because it is a product of two holomorphic functions. This allows us to deduce the following additional pair of independent K-K relations [19, 20]: (256)

(257)

(3) ω12 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ∞ ω 3 χ(3) 1 ij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) 2 = ℘ dω1 , π ω12 − ω12 0 (3) ω1 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = (3) ∞ ω12 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) 2 = − ℘ dω1 dω1 . π ω12 − ω12 0

This pair of K-K relations is peculiar to the nonlinear response at the probe frequency because they are independent of the previously obtained dispersion relations (252). This in turn implies that the experimental or model-based data of the considered susceptibility have to obey both dispersion relations pairs in order to be self-consistent. Similar dispersion relations can be established for the dominant term of the nonlinear change of the index of refraction presented in expression (215). Figure 10 shows the results of a study [26] where the K-K relations of the form (252), (253) for the nonlinear change in the index of refraction were used heuristically before they had been rigorously derived. By applying the superconvergence theorem [6] presented in the previous section to the K-K relations (252)-(257) and taking into account the asymptotic behaviour (254),

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Fig. 10. – Application of the K-K relations for the inversion of nonlinear optical data in pump and probe experiments; (a) absorption at probe frequency for different intensities of the pump laser; (b) dots: experimental data showing the change in the real part of the refractive index; solid lines: corresponding curves computed via the K-K relations from the absorption data in (a). Adapted from [26].

it is possible to obtain the following sum rules for the moments of the real and imaginary parts of the considered susceptibility [19, 20]:

(258)

∞ (3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = 0 , 0

∞ (259)

(3) ω 2 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = 0 ,

0

∞ (260)

(3) ω χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = 0 ,

0

∞ (261) 0

π (3) ω 3 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = − cij1 j2 j3 (ω2 ) . 2


65

Fig. 11. – Bottom graph: linear absorption. Middle graph: total absorption with pump laser switched on in EIT configuration. Top graph: nonlinear absorption. Adapted from [29].

These sum rules are a priori constraints that every set of experimental or modelgenerated data have to obey simultaneously. The first sum rule (258) implies that the average of the real part of the nonlinear susceptibility is zero, as for the real part of the linear susceptibility. This extends the linear sum rule (143) to include nonlinear effects. The second sum rule (259), which applies only in the nonlinear case, given that the second moment of the real part of the linear susceptibility does not converge, gives an additional constraint on the real part of the nonlinear susceptibility. The third sum rule (260) extends the TRK sum rule (141) to include nonlinear effects, and this implies that the imaginary part of the nonlinear susceptibility at the probe frequency has to be negative in some parts of the spectrum, so that nonlinear stimulated emissions compensate for nonlinear stimulated absorptions. In 1997 Cataliotti et al. [29] have experimentally verified this sum rule on cold cesium atoms (T < 10 µK). When the pump laser is opportunely tuned, there is a decrease in the main resonance absorption peak as a result from quantum interference effects, as prescribed by Electromagnetic Induced Transparency (EIT) theory [168], while new peaks of absorption outside the linear resonance are formed, as can be seen in fig. 11. Those peaks compensate for the reduced absorption resulting from the EIT effect so that the integral of the nonlinear absorption is zero, with good experimental precision. The fourth sum rule (261) is peculiar to the nonlinear susceptibility, and links the absorption spectrum to a function of the pump laser frequency and the specific material thermodynamic equilibrium density matrix only. Experimental analysis of the second and fourth sum rules (259)-(261) would be of particular relevance because these constraints hold only in the nonlinear case. Nevertheless, we can expect that the actual verification of the sum rules may be not easily achievable because, in general, sum rules are integral relations whose validification requires data over a very large spectral range [11, 79, 80]. . 5 4. Additional Kramers-Kronig relations and sum rules for the case of pump and probe susceptibility. – Along the lines of what has been presented in [81] for linear optics and in [169] in the case of nonlinear phenomena for the specific case of harmonic generation susceptibility, we wish to propose an extension of the previously derived K-K

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relations (252)-(257) and sum rules (258)-(261) for the holomorphic contribution to the pump and probe susceptibility by considering the higher powers of the considered susceptibility. The k-th (k ≥ 1) power of the pump and probe susceptibility is holomorphic in the upper complex ω-plane, since it derives from the product of functions that obey this property. From eq. (254) we also derive that for large values of ω: (262)

k k [cij1 j2 j3 (ω2 )] (3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) ≈ . ω14k

We thus derive that the following set of newly established general K-K relations hold simultaneously:

(263)

( k ) (3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ( k ) ∞ ω12α+1 χ(3) (ω ; ω , ω , −ω ) 2 2 1 1 ij1 j2 j3 2 ℘ dω1 , = πω12α ω12 − ω12 0

(264)

( k ) (3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ( k ) ∞ ω12α χ(3) (ω ; ω , ω , −ω ) 2 2 1 1 ij1 j2 j3 2 dω1 , = − 2α−1 ℘ 2 2 ω1 − ω1 πω1 0

where 0 ≤ α ≤ 2k − 1. From the generalized K-K relations (263), (264), using the superconvergence theorem and considering the asymptotic behaviour (262), it is possible to derive the following set of new generalized sum rules: ∞ (265)

ω12α

( k ) (3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = 0,

0 ≤ α ≤ 2k − 1 ,

0

∞ (266)

ω12α+1

(

(3) χij1 j2 j3

k )

(ω1 ; ω1 , ω2 , −ω2 )

dω1 = 0,

0 ≤ α ≤ 2k − 2 ,

0

∞ (267) 0

ω 4k−1

(

k ) π (3) k χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = − [cij1 j2 j3 (ω2 )] . 2

These sum rules reduce to those presented in (258)-(261) if we set k = 1. We also observe that from eq. (261) it is possible to derive a consistency relation between the non-vanishing sum rule for the k-th power of the pump and probe susceptibility and the


67

k-th power of the non-vanishing sum rule for the conventional susceptibility: (268)

2 − π

( ∞ k ) (3) 4k−1 ω1 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1 = 0

 = −

2 π

∞

k (3) ω13 χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) dω1  .

0

We propose these newly derived results (263)-(268) as tools of investigation of the pump and probe experimental data. We underline that it may be easier to obtain convergence in the integral properties if we consider the higher powers of the susceptibility, thanks to the realized faster asymptotic decrease. A fast asymptotic decrease relaxes the problems related to the unavoidable spectral range finiteness, as will be discussed in the harmonic generation case in sect. 7. . 5 5. Nonlinear oscillator model for pump and probe systems. – The nonlinear oscillator model [16, 37, 38] permits a basic understanding of all nonlinear processes and can be adopted for a qualitative analysis of the general properties of optical electronic transitions [13, 21-23]. In particular, it has been shown that the nonlinear oscillator model permits the derivation of general integral properties for pump and probe systems [36] and for harmonic generation processes to all orders [35]. In this subsection we restrict ourselves to the pump and probe case and follow the analysis presented in [36]. We gen. eralize the linear Drude-Lorentz scalar oscillator presented in subsect. 3 8 by considering a smooth general potential energy function V (x). Such a model allows a derivation of the most important features of pump and probe systems, considering all nonlinearities and including also the nonlinear effects arising from the probe beam. We assume that in our system we have N/V oscillators per unit volume, each forced by an external electric field E(t). In this case the Hamiltonian of the single forced oscillator is (269)

H=

p2 + V (x) + eE(t)x, 2m

with (270)

E(t) = E (1) e−iω1 t + E (2) e−iω2 t + c.c.

We then consider only two monochromatic beams such that condition (213) E (2) E (1) is respected, and ignore the spatial dispersion, thus adopting the conventional dipolar approximation. When we include damping, the equation of motion becomes (271)

x ¨ + γ x˙ + ω02 x +

∞ n ∂ V (x) 1 xn−1 eE(t) =− , n ∂x m 0 m (n − 1)! n=3

where we expand the potential energy function around the equilibrium position of the moving charge: (272)

V (x) =

∞ mω02 x2 ∂ n V (x) xn + . 2 ∂xn 0 n! n=3

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The solutions can be obtained by successive iterations [13, 16, 35, 36]. The procedure requires the expression of the solution as a sum of terms of decreasing magnitude: (273)

x(t) = x1 (t) + . . . + xn (t) + . . . ,

each term being responsible for nonlinear effects of increasingly high order, if we define the nonlinear polarization for unit volume as P (n) (t) = −e

(274)

N (n) x (t). V

We obtain for the linear polarization two spectral contributions: P (1) (ω) = χ(1) (ωj ) 2πE (j) δ(ω − ωj ),

(275)

j = 1, 2

with the usual definition (186) of the linear susceptibility: 1 e2 N , m V D (ωj )

j = 1, 2 ,

D (ωj ) = ω02 − ωj2 − iγωj ,

j = 1, 2.

χ(1) (ωj ) =

(276)

where (277)

It is possible to derive the expression for all spectral components of the arbitrary-order nonlinear susceptibility P (n) (t). It can be shown that the non-vanishing components can be expressed as (rω1 + sω2 ) with |r| + |s| ≤ n, along the lines of what presented for . the general case in subsect. 4 2 [36]. Here we concentrate on the third-order nonlinear response at the probe frequency ω1 . Following definition (211) of the spectral components of the nonlinear polarization, we obtain

P

(3)

2 2 (ω1 ) = χ(3) (ω1 ; ω1 , −ω2 , ω2 ) E (1) E (2) + χ(3) (ω1 ; ω1 , 0 [−ω2 , ω2 ]) E (1) E (2) + 2 2 + χ(3) (ω1 ; ω2 , ω1 − ω2 ) E (1) E (2) + χ(3) (ω1 ; −ω2 , ω1 + ω2 ) E (1) E (2) + 3 3 + χ(3) (ω1 ; ω1 , ω1 , −ω1 ) E (1) + χ(3) (ω1 ; ω1 , 0 [ω1 , −ω1 ]) E (1) + 3 + χ(3) (ω1 ; −ω1 , 2ω1 ) E (1) .


69

The various terms of the previous expression are responsible for distinct physical processes and are expressed as follows: (278) (279) (280) (281) (282) (283)

4 ∂ 4 V (x) 1 e N χ (ω1 ; ω1 , −ω2 , ω2 ) = − , 4 V D(ω )2 D(ω )D(−ω ) x4 m 1 2 2 0 3 2 4 ∂ V (x) 1 e N , χ(3) (ω1 ; ω1 , 0 [ω2 , −ω2 ]) = 6 V ω 2 D(ω )2 D(ω )D(−ω ) x3 m 1 2 2 0 0 3 2 4 ∂ V (x) 1 e N , χ(3) (ω1 ; ±ω2 , ω1 ∓ ω2 ) = 3 6 2 x 0 m V D(ω1 ) D(ω2 )D(−ω2 )D(ω1 ∓ ω2 ) 4 1 e N 1 ∂ 4 V (x) χ(3) (ω1 ; ω1 , −ω1 , ω1 ) = − , 4 V D(ω )3 D(−ω ) 2 x4 m 1 1 0 3 2 4 ∂ V (x) 1 e N (3) , χ (ω1 ; −ω1 , 0 [ω1 , −ω1 ]) = 2 3 6 3 x 0 m V ω0 D(ω1 ) D(−ω1 ) 2 4 1 e N 1 ∂ 3 V (x) (3) . χ (ω1 ; −ω1 , 2ω1 ) = 6 V D(ω )3 D(−ω )D(2ω ) 2 x3 m 1 1 1 0 (3)

We notice that two types of contributions appear, one proportional to ∂ 4 V (x) /∂ 4 x 0 , 3 2 the other to ∂ V (x) /∂ 3 x 0 . The first is due to the sum of three frequencies, the second comes from the sum of two frequencies, one of which is already a combination of two. The notation 0[ωj , −ωj ] indicates that we are considering the nonlinearity due to the static nonlinear field caused by the sum of two photons of opposite frequency ωj and −ωj . These are the only possible ways to contribute to the third-order susceptibility χ(3) (ω1 ). Obviously, if the system has particular symmetries, some terms can be equal to zero; for instance in the case of inversion symmetry all the odd derivatives of the potential are zero so that only terms (278) and (281) survive. We observe that both holomorphic contributions, which obey Scandolo’s theorem [19, 20], and meromorphic contributions are present. Among the holomorphic contributions, term (278) gives the usual correction to the linear resonance (dynamical Stark effect), term (279) originates from the optical rectification and is usually neglected, while contributions (280) include the two-photon absorption and the stimulated Raman scattering. The above-described contributions have been already discussed in the literature [13]. We observe that the meromorphic terms are due to the nonlinear effects of the probe beam only [13], (281) corresponding to the Kerr effect, (282) to the optical rectification, and (283) to the two-photon absorption of the probe beam. We remark that at every frequency the holomorphic contributions to the nonlinear polarization at the probe frequency ω1 described by terms (278)-(280) are dominant since they are proportional to the pump intensity; therefore, as a first approximation, (3) the meromorphic contributions can be neglected. We define χhol (ω1 ) as the sum of the holomorphic contributions (278)-(280). We notice that the asymptotic behaviour of the optical response for large values of the probe frequency ω1 is determined by the two holomorphic contributions (278) and (279):

(284)

(3) χhol (ω1 )

e4 N ∼− 4 m V

∂ 4 V (x) x4

3 2 ∂ V (x) 1 1 −4 − 2 2 4 + o(ω1 ). 3 m ω x ω 0 1 0 0

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and

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(3)

Considering the holomorphicity of χhol (ω1 ) in the upper complex ω1 -plane and the asymptotic behaviour (284), we can derive the following independent K-K relations:

(3) ω12α χhol (ω1 ) =

2 ℘ πω12

∞ ω 2α+1 χ(3) (ω ) 1 1 hol

dω1 , ω12 − ω12 0 ∞ ω 2α χ(3) (ω ) 1 1 hol 2 (3) ω12α−1 χhol (ω1 ) = − ℘ dω1 π ω12 − ω12

(285)

(286)

0

with α = 0, 1. These dispersion relations match perfectly with those derived with the general quantum treatment and presented in (252)-(257). If we apply the superconvergence theorem [6] to the K-K relations (285), (286) and consider the asymptotic behaviour established in eq. (284), we can derive the following set of sum rules:

(287)

∞ (3) χhol (ω1 ) dω1 = 0 , 0

∞ (288)

(3) ω 2 χhol (ω1 ) dω1 = 0 ,

0

∞ (289)

(3) ω χhol (ω1 ) dω1 = 0 ,

0

∞

3

ω

(290) 0

(3) χhol

(ω1 )

dω1

π e4 N = 2 m4 V

∂ 4 V (x) x4

3 2 ∂ V (x) 1 − 2 2 . m ω0 x3 0 0

If in our analysis we take into account also the meromorphic contributions (281)(283), it is necessary to explicitly consider their poles in the upper ω1 -plane in order to derive the K-K relations and sum rules for the total nonlinear response at the probe frequency. The poles of the upper complex plane appearing in terms (281)-(283) corre! "1/2 spond to the solutions of D (−ω1 ) = 0, i.e. ω1 = ± ω02 − γ 2 /4 + iγ/2. After lengthy calculations, it is possible to prove that the real and imaginary parts of the meromorphic contributions to the susceptibility obey non-vanishing sum rules for all the moments considered. Moreover, it is possible to show that there are only two independent sum rules, and that the residual terms that appear in the modified K-K relations for meromorphic susceptibilities (251) can also be expressed in terms of these two independent sum rules [36]. . 5 6. Generalized Kramers-Kronig–type relations and sum rules. – In this subsection we concentrate on the meromorphic third-order degenerate susceptibility, which in the case of the anharmonic oscillator corresponds to the sum of terms (281)-(283). For the sake of clarity, we denote χ(3) (ω; ω, ω, −ω) simply by χ(3) (ω). The asymptotic behaviour of the third-order degenerate susceptibility can be derived either quantum mechanically


71

or from the classical anharmonic oscillator model, which both yield

χ(3) (ω) = ψω −8 + o(ω −8 ),

(291)

where ψ is a high-frequency constant of the susceptibility and o(ω −8 ) denotes terms that converge strictly faster than ω −8 . Moreover we remind that nonlinear susceptibilities possess a symmetry property [36]

χ(3) (−ω ∗ ) = [χ(3) (ω)]∗ .

(292)

Hence, the real part of the nonlinear susceptibility is an even function and the imaginary part an odd function. We consider the moments and the powers of a third-order degenerate susceptibility, which have very rapid convergence. The relevance of considering higher powers of the susceptibility function lies on the fact that the errors in data inversion caused by the finite spectral range should diminish thanks to the realized faster asymptotic behaviour at infinity. We consider the function

f (ω ) =

(293)

ω m [χ(3) (ω )]k , ω − ω

where ω is a real-valued frequency. In the upper complex ω-plane we have one pole on the real axis, due to the denominator in (293), and several poles located symmetrically with respect to the imaginary axis, due to the meromorphic nature of the susceptibility function considered. According to Cauchy’s integral theorem and the theorem of residues, the complex integration around the closed circle C results to be * (294) C

f (ω ) dω = 2πi

poles

Res [f (ω )].

{ω>0}

Here, the summation of the residues is over the poles in the upper half-plane. On the other hand, the integration around the closed circle C can be split into the real and imaginary parts. The upper limits for the convergent moments are obtained by the investigation of counterclockwise integration along the semicircle in the high-frequency limit. Let us assume that m is an even integer, which allows the derivation of K-K–type dispersion relations. With the aid of the symmetry property of eq. (292) we obtain the

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and

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following K-K–type relations for even integers m with m ≤ 8k − 2: (295)

(296)

∞

ω m+1 {[χ(3) (ω )]k } dω − ω 2 − ω 2 0   ( m (3) k )  poles ω [χ (ω )] Res −2 ,   ω − ω {ω >0} ∞ m ω {[χ(3) (ω )]k } 2ω m (3) k dω − ω {[χ (ω)] } = − ℘ π ω 2 − ω 2 0   ( m (3) k )  poles ω [χ (ω )] Res −2 .   ω − ω m

ω {[χ

(3)

2 (ω)] } = ℘ π k

{ω >0}

In a similar way we obtain for an odd integer m ≤ 8k−1 the following K-K–type relations as follows: (297)

(298)

∞

ω m {[χ(3) (ω )]k } dω − ω 2 − ω 2 0   ( m (3) k )  poles ω [χ (ω )] Res −2 ,   ω − ω {ω >0} ∞ m+1 ω {[χ(3) (ω )]k } 2 dω − ω m {[χ(3) (ω)]k } = − ℘ 2 − ω 2 π ω 0  ( m (3) k )  poles ω [χ (ω )] Res −2 .   ω − ω

ω m {[χ(3) (ω)]k } =

2ω ℘ π

{ω >0}

For the value m = 8k, which is the highest convergent moment, the complex integration along the semicircle in the high-frequency limit does not converge to zero but yields information of the high-frequency value of the degenerate susceptibility. For higher values m the integral diverges. Therefore, the K-K–type relation (295) is transformed as follows: (299)

ω m {[χ(3) (ω)]k } =

∞

ω m+1 {[χ(3) (ω )]k } dω − 2 − ω 2 ω 0    poles ω m [χ(3) (ω )]k  −2 + ψk ,   ω − ω 2 ℘ π

{ω >0}

whereas the K-K–type relation (296) remains in the same form. The poles of the degenerate third-order susceptibility are of the first order. This was taken into account in [36] by considering explicitly the poles in the upper half-plane for pump and probe nonlinear processes in the frame of the anharmonic oscillator model, which provides exact results as concerns the residue terms. Here we generalize the previous results [36] and obtain better convergence with generalized K-K relations and sum rules by considering the powers of the degenerate nonlinear susceptibility. Unfortunately,

73


for the poles of the order ≥ 2 the derivatives of the complex function appear, and we obtain the residue terms with poles of order higher than 1 as follows: (300)

dk−1 [χ(3) (ω )]k 1 = lim Resω =ωj ω − ω ω →ωj (k − 1)! d ω k−1

(

(ω −

[χ(3) (ω )]k ωj )k ω − ω

) ,

where the index j refers to the various poles. For the first-order poles expression (300) gives the well-known result: (301)

Res

ω =ωj

( ) (3) [χ(3) (ω )]k (ω )] [χ = lim (ω − ωj ) . ω − ω ω →ωj ω −ω

Now we are able to move into the derivation of sum rules for the moments and the powers of the degenerate susceptibility. Let us now consider the function g(ω) = ω m [χ(3) (ω)]k .

(302)

The only poles that are located in the upper half-plane are caused by the degenerate susceptibility with the order k. Following the procedure above for the derivation of K-K–type relations we obtain sum rules by splitting the integrals into the real and imaginary parts. However, the upper limit of the convergent moment is now m = 8k − 1 due to the lack of pole in the real axis. For even integers m with m ≤ 8k − 2 we have   ∞  poles  (303) ω m {[χ(3) (ω )]k }dω = −πIm Res{ω m [χ(3) (ω )]k } ,   0 {ω >0}

(304)

  poles 

{ω >0}

  Res{ω m [χ(3) (ω )]k } = 0. 

For odd integers m with m ≤ 8k − 3 sum rules are of the form   ∞  poles  ω m {[χ(3) (ω )]k }dω = π Res{ω m [χ(3) (ω )]k } , (305)   0 {ω >0}

(306)

  poles 

{ω >0}

  Res{ω m [χ(3) (ω )]k } = 0. 

For odd m = 8k − 1 the sum rule given by eq. (306) remains in the same form, but the imaginary-part sum rule equation (305) is transformed according to the high-frequency value of susceptibility as follows:   ∞  poles  π (307) ω m {[χ(3) (ω )]k }dω = π Res{ω m [χ(3) (ω )]k } − ψ k .   2 0 {ω >0}

74


and

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Finally, we use the K-K–type dispersion relations given by eqs. (295)-(299) in the derivation of static sum rules for the degenerate third-order susceptibility by setting ω = 0. It is observed that this procedure gives exactly the same sum rules as those presented above in eqs. (303)-(307). Therefore, these sum rules yield information on the static limit. The meromorphicity of the degenerate susceptibility yields residue terms to sum rules, which are now presented for the generalized degenerate third-order susceptibility ω m [χ(3) (ω)]k . The developed K-K–type dispersion relations and sum rules have strictly faster convergence than the previously presented counterparts in the literature. Therefore, these dispersion relations and sum rules are assumed to have relevance in testing of theoretical models describing the degenerate third-order susceptibility. It is a straightforward procedure to extend the K-K–type dispersion relations and sum rules, which are now presented for the third-order degenerate susceptibility, to arbitrary-order meromorphic degenerate nonlinear susceptibilities. Unfortunately, the application of the developed theory to measured spectra remains yet cumbersome since at resonance points of the spectrum the knowledge of both the real and imaginary parts of the degenerate nonlinear susceptibility is required. . 5 7. Modified Kramers-Kronig relations and sum rules for meromorphic total index of refraction. – The meromorphic third-order degenerate susceptibility introduced in the previous subsection can be used to describe one of the most relevant nonlinear optical effects: the intensity-dependent change of the refractive index. Such a meromorphic nonlinear degenerate susceptibility appears in the context of a self-action process and involves only one input light beam. If we consider the case of isotropic or cubic-symmetric matter interacting with a monochromatic beam of frequency ω, we can obtain the following expression for the total index of refraction: N tot (ω) = ntot (ω) + iκtot (ω) = N (ω) + N NL (ω) = = n(ω) + iκ(ω) + ∆n(ω) + i∆κ(ω), where we have used the conventional notation for the real and imaginary parts of the index of refraction. The third-order nonlinear change of the index of refraction can be expressed as

(308)

2

N NL (ω) ∼ 6π (E)

χ(3) (ω; ω, ω, −ω) , N (ω)

which is strictly related to the pump and probe case presented in eq. (215). We note that the nonlinear change in the index of refraction is proportional to the intensity of the incoming field. The function N NL (ω) has the same number of poles and analytical properties as the function χ(3) (ω) in the upper complex ω-plane since the linear refractive index has no poles or zeros in that region [34]. Therefore, modified K-K relations can k be derived for the general function ω m N NL(ω) . For the sake of simplicity here we concentrate only on the case where m = 0 and k = 1. Modified K-K relations, which can be derived with the aid of complex contour integration, similar to eq. (251), were given purely for the total refractive index itself [170]. However, the following relations


75

involving the linear and nonlinear contributions are particularly instructive [171]:

(309)

tot

n

2 (ω) − 1 = ℘ π

∞ 0

2 + ℘ π (310)

κtot (ω) =

2ω ℘ π

ω κ(ω ) dω + ω 2 − ω 2

∞

{ω >0}

0

∞ 0

2ω + ℘ π

  poles NL ω ∆κ(ω ) (ω ) N , dω − 2  Res ω 2 − ω 2 ω − ω

[n(ω ) − 1]dω + ω 2 − ω 2

∞ 0

  poles NL ∆n(ω ) (ω ) N . dω − 2  Res ω 2 − ω 2 ω −ω {ω >0}

A problem with the modified dispersion relations is the calculation of the residue terms, which requires the knowledge of the nonlinear susceptibility at complex frequencies and the a priori knowledge of the resonance points of the system. Therefore, relations (309), (310), at the present stage, have little practical utility if compared with the procedure of numerical data inversion by K-K and MSKK relations. However, the dispersion relations (309), (310) provide a frame to test theoretical dispersion models of the electronic systems and incorporated nonlinear optical properties that may be suggested to hold for nonlinear media such as nanocomposites. Next we generalize our previous result [166] and give a sum rule for the powers of total meromorphic refractive index as follows: ∞ (311)

tot j n (ω ) − 1 dω = 2πi

poles

Res (ntot (ω ))j ,

{ω >0}

−∞

where j = 1, 2, . . . . Let us consider first the case j = 1. The real linear refractive index of a medium fulfills the well-known ADNS sum rule ∞ [n(ω ) − 1]dω = 0 .

(312)

0

Since n is an even function of the angular frequency variable, i.e. n(−ω) = n(ω), we can deduce from eqs. (311) and (312) that   ∞  poles  (313) ∆n(ω )dω = 2π N NL (ω ) ,   −∞

{ω >0}

and ∞ (314) −∞

[κ(ω ) + ∆κ(ω )] dω = −2π

  poles 

{ω >0}

  N NL (ω ) . 

76


and

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Equation (313) is the generalization of the ADNS sum rule to nonlinear optics with respect to the meromorphic optical constant. Sum rule (314) gives the integrated area of the extinction curve. In the case when j = 2 the nonlinear sum rules become rather complicated if the real and imaginary parts are resolved. Finally, we mention that modified K-K relations hold also for meromorphic total reflectivity [172]. 6. – General properties of the harmonic generation susceptibility . 6 1. Introduction. – The theoretical and experimental investigation of harmonic generation processes is one of the most important branches of nonlinear optics [16-18]. Since the pioneering study of the second harmonic generation by Franken et al. [106] and of the third harmonic generation by Maker et al. [110], the continuous development in this field has produced experimental and theoretical studies of harmonic generation in solids [173-176], molecules [177, 178], atoms [179, 180], and organic matter [181]. In atomic gases the third harmonics was first observed by New and Ward in 1967 [182], while only in the following decade was the fifth harmonics experimentally observed [179]. The first numerical calculations of harmonic generation susceptibility date back to the same period [183]. In the ’80s technological advances provided the scientists with lasers able to generate very short and very intense pulses, having peak intensity up to and beyond 1013 W cm−2 [117-119]. Ultra-intense lasers have been mostly used for the investigation of the highorder harmonic generation in noble gases (and their clusters [184]) up to the 135th order. Under extreme conditions other materials are usually subjected to structural damages [117-119, 185]. Nevertheless in the last years relatively high-order harmonics have been obtained also on solids by using intense lasers capable of femtosecond pulses: the very short duration of the pulses prevents damages to the crystalline structure [186]. The ultra-intense lasers generate electric fields having amplitude of the same order of magnitude or exceeding the typical amplitude of the static electric field effecting the electrons, so that perturbation theory appropriateness conditions break up. The results of the experiments of very high harmonic generation on various noble gases with ultraintense lasers show qualitatively similar properties that cannot be efficiently explained with the perturbative approach, since they cannot be framed in the context of nonlinear susceptibilities. Therefore we will not refer to these strongly driven systems in our work. In the case of harmonic generation, only recently has a complete formulation of general K-K relations and sum rules for n-th–order harmonic generation susceptibility in continuous wave approximation been obtained [35, 169, 187]. Unfortunately, even now there are relatively few studies that report on independent measurements of the real and imaginary parts of the harmonic generation susceptibilities [188, 189] and on the validity of the K-K relations in nonlinear experimental data inversion [27]. Most recently, very detailed studies of the integral properties of optical harmonic generation on experimental data on polymers have been presented [30, 190, 191]. In this section we review the most relevant theoretical achievements in the establishment of the general properties of the harmonic generation susceptibility. We also define the n-th–order harmonic generation susceptibility tensor. We derive its asymptotic behaviour for large values of the frequency and obtain its analytical properties by taking advantage of Scandolo’s theorem [19, 20]. Then we combine all the information gathered and, using the superconvergence theorem [6,78], we derive a set of independent K-K relations and sum rules for the moments of the harmonic generation susceptibility and of its powers. We also derive expressions for generalized MSKK relations for the moments of


77

the harmonic generation susceptibilities. Furthermore, we show that the results obtained display a strict analogy with those derived [35] using the classical anharmonic oscillator model [16,37,38], thus demonstrating its relevance to the theory of nonlinear susceptibilities. The general classical-quantum correspondence, as in the linear case, is also found for the nonlinear optical processes. This suggests that the general integral properties of the susceptibility functions do not depend on the model formulation [13, 21-23, 36], since they are a direct consequence of the principle of causality [14] in the light-matter interaction. A major problem in the effective experimental verification of the general properties descriptive of harmonic generation processes presented in this section is their integral formulation, which in principle requires data covering the whole of the infinite positive range. This will be discussed and exemplified in sect. 7. . 6 2. Scandolo’s theorem and harmonic generation susceptibility. – We focus on the typical experimental set-up for harmonic generation processes, where the incident radiation is given by a strictly monochromatic and linearly polarized field so that we can (t) as express E Ej (t) = Ej e−iωt + c.c.

(315)

Since we are interested in studying the n-th–order harmonic generation processes, we seek the nω frequency component of the induced nonlinear polarization P (n) (t) introduced in eqs. (209)-(211). Considering the term in eq. (209), where the product of only the positive frequency components ω is considered, we obtain that (316)

(n)

Pi

(n)

(nω) = χij1 ,...,jn (nω; ω, . . . , ω) Ej1 . . . Ejn .

We wish to emphasize that contributions to the nω frequency component of the nonlinear polarization also come from the higher nonlinear orders n + 2q, where in eq. (209) we select in the susceptibility function argument the positive ω frequency component n + q times and the negative −ω frequency component q times. We will ignore these much smaller higher-order contributions and concentrate on the n-th–order nonlinear process. When the frequency of the incoming radiation varies, all the arguments of the harmonic generation susceptibility in (316) change coherently, so that the correct straight line parameterization in Rn of the arguments of the susceptibility proposed in eq. (243) is v = (1, . . . , 1) and w = (0, . . . , 0). Since all the components of the vector v are positive and hence obey condition (247), Scandolo’s theorem [19, 20] permits us to deduce that the harmonic generation susceptibility is holomorphic in the upper complex ω-plane. We would also like to underline that the n-th–order harmonic generation susceptibility obeys the following symmetry relation: (317)

∗ (n) (n) χij1 ...jn (−nω; −ω, . . . , −ω) = χij1 ...jn (nω; ω, . . . , ω) .

. 6 3. Asymptotic behaviour of the harmonic generation susceptibility. – As thoroughly discussed in the previous section in connection with the general nonlinear case, the number of independent K-K relations and sum rules that simultaneously hold for the harmonic generation susceptibility and its moments is determined by the asymptotic behaviour of the susceptibility function. Therefore, the establishment of complete general integral

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properties for the considered susceptibility requires the microscopic treatment of the harmonic generation process, which we will approach using the conventional n-th–order perturbation theory presented in the previous subsections, which can be summarized with the concatenated system of eq. (66). Nevertheless, we have found that, while in the definition of the n-th–order nonlinear polarization and susceptibility functions the conventional length gauge for the light-matter interaction Hamiltonian presented in expression (57) is more efficient, the determination of the asymptotic behaviour of the harmonic generation susceptibility is much more straightforward if we adopt the physically equivalent —and theoretically more elegant [7, 192]— velocity gauge formulation presented in expression (54). Following the same procedure adopted in the derivation of expressions (216)-(221) in the length gauge, we obtain that, equivalently, the n-th–order nonlinear polarization P (n) (t) can be expressed as

(318)

(n) Pi

∞

(n)

Jij1 ...jn (t1 , . . . , tn ) Aj1 (t − t1 ) . . . Ajn (t − tn ) dt1 . . . dtn ,

(t) = −∞

where the velocity gauge n-th–order nonlinear Green function is (319)

en+1 (n) Jij1 ...jn (t1 , , . . . , tn ) = − n θ (t1 ) . . . θ (tn − tn−1 ) × V (−imc) N N N α α α ×Tr pjn (−tn ), . . . , pj1 (−t1 ), ri . . . ρ (0) . α=1

α=1

α=1

in the interaction HamilWe observe that the quadratic term of the vector potential A(t) tonian (54) is not present in the definition of the Green function (319) because it does not involve operators and so, its contribution is cancelled out in an iterative commutators structure as in (319). By applying the Fourier transform to both members of eq. (56) we obtain c A(ω) = E(ω), iω

(320)

so that we can express the polarization (318) in terms of the fields in the velocity gauge as c n (n) (n) (321) P i (nω) = ηij1 ,...,jn (nω; ω, . . . , ω) Ej1 . . . Ejn , iω where (322)

(n) ηij1 ,...,jn (nω; ω, . . . , ω)

∞ =

(n) Jij1 ...jn

(τ1 , . . . , τn ) exp iω

−∞

n

tl dt1 . . . dtn .

l=1

Because of gauge invariance we must have (323)

(n)

(n)

χij1 ,...,jn (nω; ω, . . . , ω) = ηij1 ,...,jn (nω; ω, . . . , ω)

c n . iω

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Therefore, the n-th–order harmonic generation susceptibility can be expressed in the velocity gauge as ∞ en+1 =− θ (t1 ) . . . θ (tn − tn−1 ) × n V (m) ω n −∞ N N N α α α pjn (−tn ), . . . , pj1 (−t1 ), ri . . . ρ (0) × ×Tr

(n) χij1 ,...,jn (nω, ω, . . . , ω)

(324)

α=1 n

× exp iω

α=1

α=1

tl dt1 . . . dtn .

l=1

We notice that the linear susceptibility presented in expression (79) is included in the above expression when we set n = 1. By applying the following variables change:

tj =

(325)

j

τi ,

1≤j≤n

i=1

we obtain for the harmonic generation susceptibility: en+1 (n) (326) χij1 ,...,jn (nω; ω, . . . , ω) = − × n V (m) ω n   ∞ n θ (τ1 ) . . . θ (τn ) exp iω (n + 1 − j) τj  × × j=1

−∞

×Tr

N α=1

pα jn (−τn −. . .

− τ1 ), . . . ,

N α=1

pα j1 (−τ1 ),

N

riα

. . . ρ(0) dτ1 . . . dτn .

α=1

After a calculation, which is fully reported in appendix B, we obtain the result that for large ω, the n-th–order harmonic generation susceptibility asymptotically decreases as ω −2n−2 with the following rule: n

(327)

(−1) en+1 N (n) × χij1 ,...,jn (nω; ω, . . . , ω) = 2 n n! mn+1 V " ! ∂ n+1 V (rα ) 1 ×Tr ρ (0) + o ω −2n−2 . α α α 2n+2 ∂rjn . . . ∂rji ∂ri ω

We observe that the fundamental quantum constant does not appear in the formula (327), thus suggesting that a detailed quantum treatment is not essential for capturing the asymptotic behaviour. The quantum aspect of the expression we have obtained appears only in the definition of the expectation value of the derivatives of the one-particle potential energy on the equilibrium density matrix of the system. We observe that the many-particle interactions in the Hamiltonian H0 are not directly represented in this result, apart from playing a relevant role in the definition of the ground state of the system. . We will show in subsect. 6 5 that it is possible to demonstrate a close correspondence

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between the above-described general properties of the quantum harmonic generation susceptibility and those obtained from a simple classical anharmonic oscillator model [35]. . 6 4. Kramers-Kronig relations and sum rules for the harmonic generation susceptibility. – The holomorphic properties and the asymptotic behaviour of the n-th–order harmonic generation susceptibility allow us to write the following set of independent K-K dispersion relations for the moments of the real and imaginary parts of the considered susceptibility for non-conducting materials [35, 187]: 2 ∞ ω 2α+1 χ(n) (nω ; ω , . . . , ω ) ij ...j 1 n (n) ω 2α χij1 ...jn (nω; ω, . . . , ω) = ℘ (328) dω , π ω 2 − ω 2 0 ∞ ω 2α χ(n) (nω ; ω , . . . , ω ) ij ...j 2 1 n (n) (329) ω 2α−1 χij1 ...jn (nω; ω, . . . , ω) = − ℘ dω , π ω 2 − ω 2 0

with 0 ≤ α ≤ n, where α is such that the α-th moment of the harmonic susceptibility under consideration decreases for large values of the frequency at least as fast as ω −2 . We observe that the number of independent K-K relations grows with the order of the harmonic generation process in question. Moreover, since when a function is holomorphic in a given domain, its positive integer powers share the same property, we find that the more general K-K relations stated below hold for the positive integer k-th powers of the susceptibility under examination [169]:

2α

(

k ) (nω; ω, . . . , ω) = ( k ) ∞ ω 2α+1 χ(n) (nω ; ω , . . . , ω ) ij1 ...jn

(n) χij1 ...jn

(330)

ω

(331)

2 ℘ dω , π ω 2 − ω 2 0 ( k ) (n) 2α−1 ω χij1 ...jn (nω; ω, . . . , ω) = ( k ) ∞ ω 2α χ(n) (nω ; ω , . . . , ω ) ij1 ...jn 2 dω , =− ℘ π ω 2 − ω 2 =

0

with 0 ≤ α ≤ k(n + 1) − 1. The dispersion relations presented in eqs. (328), (329) and (330), (331) extend to all orders the results previously obtained for the second- and third-order harmonic generation processes [153-157, 193, 194]. Comparing the asymptotic behaviours obtained from expression (B.16) with those obtained by applying the superconvergence theorem [6,78] to the general K-K relations (330), (331), we obtain the


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following set of sum rules: ∞ (332)

ω 2α

( k ) (n) χij1 ...jn (nω , ω , . . . , ω ) dω = 0,

0

∞ (333)

ω

2α+1

(

0 ≤ α ≤ k(n + 1) − 1 ,

k ) (nω , ω , . . . , ω ) dω = 0,

(n) χij1 ...jn

0 ≤ α ≤ k(n + 1) − 2 ,

0

∞

ω 2k(n+1)−1

(

0

(334)

π = 2

(n)

χij1 ...jn (nω , ω , . . . , ω )

n+1

(−1) n2 n!

en+1 N Tr mn+1 V

k )

dω =

k ∂ n+1 V (rα ) ρ (0) . ∂rjαn . . . ∂rjαi ∂riα

All the moments of the k-th power of the n-th–order harmonic generation susceptibility vanish except that of order 2k(n + 1) − 1 of the imaginary part. In the most fundamental case of k = 1, the non-vanishing sum rule creates a conceptual bridge between a measure of the (n + 1)-th–order nonlinearity of the potential energy of the system and the measurements of the imaginary part of the susceptibility in the entire spectrum, thus relating structural and optical properties of the material under consideration, i.e. the static and dynamic properties of the electronic system. We can also observe that, analogously to what presented in eq. (268), from the nonvanishing sum rule in eq. (334) it is possible to obtain a simple consistency relation between the non-vanishing sum rule for the k-th power of the harmonic generation susceptibility and the k-th power of the non-vanishing sum rule for the conventional susceptibility, where we select k = 1: 2 − π

∞ 0

ω 2k(n+1)−1

(

k ) (n) χij1 ...jn (nω , ω , . . . , ω ) dω =

 2 = − π

∞

k (n) ω 2n+1 χij1 ...jn (nω , ω , . . . , ω ) dω  .

0

The constraints presented here are, in principle, universal, since they descend from the principle of causality in the response of the matter to the external radiation, and are expected to hold for any material. They provide fundamental tests of self-consistency that any experimental or model-generated data have to obey. The verification of the K-K relations and sum rules constitutes an unavoidable benchmark for any investigation into the nonlinear response of matter to radiation over a wide spectral range. In appendix C we report an alternative derivation of the K-K relations and sum rules for harmonic generation susceptibilities of arbitrary order based on a more sophisticated use of complex analysis.

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. 6 4.1. Case of nonlinear conductors. Conducting materials are characterized by the presence of a non-vanishing static conductance, which changes the integral properties of their n-th–order harmonic generation susceptibilities as in linear optics [72-74, 81]. Remembering that at every order of nonlinearity the susceptibility can be expressed in terms of the conductivity: (n)

(n)

(335)

χij1 ...jn (nω; ω, . . . , ω) = i

σij1 ...jn (nω; ω, . . . , ω) , nω

in the case of conductors, by definition, for frequencies close to zero we can express the harmonic generation susceptibility in terms of the non-vanishing real tensor of the nonlinear static conductance of order n: (n) χij1 ...jn

(336)

(n)

σij ...j (0) . (nω; ω, . . . , ω) ≈ i 1 n nω

The presence of this pole at the origin of the ω-axis changes the K-K relation presented in (329) for α = 0 into σ (n) ij1 ...jn (0) (n) χij1 ...jn (nω; ω, . . . , ω) − = nω ∞ χ(n) (nω ; ω , . . . , ω ) ij1 ...jn 2ω =− ℘ dω . π ω 2 − ω 2

(337)

0

By applying the superconvergence theorem to the dispersion relation (337) and comparing with the asymptotic behaviour (327), we derive the following sum rule for the lowest moment of the susceptibilities: (338)

∞ π (n) (n) χij1 ...jn (nω ; ω , . . . , ω ) dω = − σij1 ...jn (0) . 2n 0

Equations (337) and (338) extend to the nonlinear case the results previously obtained . for linear optics in subsubsect. 3 2.1 and in eq. (144). All of the K-K relations (328), (329) and related sum rules (332)-(334) of the higher moments, corresponding to 1 ≤ α ≤ n, remain unchanged for conductors because the higher moments of the harmonic generation susceptibility do not have a pole at the origin. We suggest that the established general integral properties for the harmonic generation processes could be experimentally verified also on conductors. . 6 5. Nonlinear oscillator model for harmonic generation processes. – In this subsec. tion we use the nonlinear oscillator model previously presented in subsect. 5 5 in order to explore from another, non-microscopic, point of view the general properties of the harmonic generation susceptibilities. Also in this case we assume that in our system we have N/V generic nonlinear oscillators per unit of volume, each forced by an external monochromatic electric field E(t) of frequency ω: (339)

E(t) = Ee−iωt + c.c. ,

83


and we ignore spatial dispersion. We plug the above-defined electric field in the equation of motion (271) and obtain the solution by considering the perturbation theory with successive iterations, each iterative process describing dynamics of higher and higher order of nonlinearity. The procedure requires the expression of the solution as a sum of terms of decreasing magnitude: (340)

x(t) = x1 (t) + . . . + xn (t) + . . . .

Using definition (274) for the nonlinear polarization, we have that each term in eq. (340) is responsible for various optical effects, among which the production of the corresponding harmonic. Substituting (340) in (271) and separating the various orders terms, we obtain the solutions for harmonic generation. As shown in eq. (186), we find that the linear susceptibility is χ(1) (ω) =

(341)

e2 N 1 , m V D(ω)

with definition (277) for the quantity D(ω). The second-order susceptibility for the second harmonic, as suggested by some authors [13, 16], is

(342)

χ(2) (2ω; ω, ω) = −

1 e3 N 2! m3 V

∂ 3 V (x) ∂x3

1 0

D(2ω) [D(ω)]

2

.

The third-order susceptibility for the third harmonics is

(343)

1 1 e4 N ∂ 4 V (x) + 3! m4 V ∂x4 0 D(3ω) [D(ω)]3 2 1 e4 N ∂ 3 V (x) + 5 3. 3 m V ∂x 0 D(3ω)D(2ω) [D(ω)]

χ(3) (3ω; ω, ω, ω) = −

We can observe that the first term has a slower decreasing behaviour at infinity (∝ ω −8 ), while the second term goes to zero much faster (∝ ω −10 ). In previous studies [13, 16] the second term had not been considered. The higher-order susceptibilities for the harmonic generation can also be obtained. They can be expressed recursively order by order as 1 en+1 N n! mn+1 V

∂ n+1 V (x) ∂xn+1

1 n − D(nω) [D(ω)] 0   l   kj +1 ! (i ) "kj l 1 χ j (ω . . . ω)   ∂ j=1 1 V (x)  1     , − l l     D(nω) k ! k +1 j j kj −1 j=1 i,k,l y=1 j=1 ∂x (e(N/V )m) 0

(344) χ(n) (nω; ω, . . . , ω) = −

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with the constraints l

kj ij = n,

j=1

∀j, j |1 ≤ j < j ≤ l, ⇒ (1 ≤ ij < ij < n) ∧ (0 ≤ kj < n) . Note that the second condition in (345) is needed to separate the first term in (344) from the rest of the terms in the summation. We can observe that in (344) only the first term is relevant for the asymptotic behaviour, since all the other contributions go to zero much faster at infinity. Furthermore, when the system obeys inversion symmetry, only the odd-order terms of the susceptibilities are different from zero because in the evenorder susceptibilities the first term vanishes and the terms in the summation also vanish, because they always contain as a factor at least one odd-order derivative of the potential, which obviously vanishes. We then discover that the harmonic generation susceptibility has the following asymptotic behaviour: 1 (−1)n en+1 N ∂ n+1 V (x) (n) (345) χ (nω; ω, . . . , ω) = 2 + o(ω 2n+2 ). n+1 n+1 2n+2 n n! m V ∂x 0 ω We may observe that this expression (345) matches perfectly the result obtained with a full quantum treatment in (327), with the conventional rule of considering the classicalquantum correspondence between the derivatives at the equilibrium position of the external potential V (x) of the anharmonic oscillator, and the expectation values on the ground state of the corresponding derivatives of the one-particle electron potential. . 6 5.1. Generalized Miller’s rule. The Miller empirical rule [37] generalizes the observation that in many materials the second harmonic generation susceptibility in the transparency region can be expressed in terms of the product of suitable linear susceptibility functions: (346)

(2)

(1)

(1)

(1)

χijk (2ω; ω, ω) ∼ ∆ijk χii (2ω)χjj (ω)χkk (ω) ,

where ∆ijk is a coupling constant relating the nonlinear and the linear processes named Miller’s ∆. The constant ∆ijk was originally thought to be material independent, while later it has been found that this is not strictly correct. The anharmonic oscillator model above described allows to find a generalized scalar expression for Miller’s ∆ at every order n including n = 2 [35], which can be derived by considering the leading term in expression (344): (347)

n χ(n) (nω; ω, . . . , ω) ∼ ∆(n) χ(1) (nω) χ(1) (ω) ,

where (348)

∆(n) = −

1 1 n! en+1

N V

−n

∂ n+1 V (x) ∂xn+1

. 0

This expression shows that the coupling constant depends on the nonlinearity of the potential energy, and gives an explanation of why Miller’s ∆ at every order should be expected to be material dependent.


85

. 6 5.2. Kramers-Kronig relations and sum rules. The general harmonic generation susceptibility presented in (344) is for each order n a holomorphic function in the upper complex ω-plane, since it is proportional to the multiple products of factors 1/D(βω), with β > 0, which are holomorphic in that domain. The fundamental reason for this property is that time causality has been essentially introduced in the equation of motion. When the asymptotic behaviour is considered, we then obtain that for each order n several independent K-K relations connect the moments of the k-th power of the harmonic generation susceptibility: ( k ) ω 2α χ(n) (nω; ω, . . . , ω) (349) = ∞ ω 2α+1 χ(n) (nω ; ω , . . . , ω )k 2 = ℘ dω , π ω 2 − ω 2 0 ( k ) 2α−1 (n) ω χ (nω; ω, . . . , ω) (350) = ∞ ω 2α χ(n) (nω ; ω , . . . , ω )k 2 =− ℘ dω , π ω 2 − ω 2 0

with 0 ≤ α ≤ k(n + 1) − 1. With the above dispersion relations (349), comparing the related asymptotic behaviour presented in expression (345) to the asymptotic behaviour as obtained from the superconvergence theorem [6,78], we obtain the following sum rules: ∞ (351)

ω 2α

(

k ) χ(n) (nω ; ω , . . . , ω ) dω = 0,

0 ≤ α ≤ k(n + 1) − 1 ,

0

∞ (352)

ω

2α+1

( χ

(n)

k ) (nω ; ω , . . . , ω ) dω = 0,

0 ≤ α k(n + 1) − 2 ,

0

∞ (353)

ω

2k(n+1)−1

0

=−

( χ

(n)

k ) (nω ; ω , . . . , ω ) dω =

k π (−1)n en+1 N ∂ n+1 V (x) . 2 n2 n! mn+1 V ∂xn+1 0

Momenta of the susceptibility higher than those considered above do not give convergence. These integral properties show the very close correspondence with the results . obtained in subsect. 6 4 for the quantum harmonic generation susceptibility. The main reason for this correspondence, which extends to the nonlinear domain the results pre. sented in subsect. 3 8 for linear optics, is that the basic ingredients of the general integral relations are the analytical properties and asymptotic behaviour of the harmonic generation susceptibility. These properties do not essentially depend on the microscopic

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treatment of the interaction between light and matter, but are due to the validity of the causality principle in physical systems [7, 10, 14, 42]. We wish to emphasize that the classic oscillator (linear and nonlinear) we have adopted . . in this subsection as well as in subsects. 3 8 and 5 5 provides a simplified but conceptually complete picture of the general properties of the first-order and higher-orders transition probabilities of a generic system when we consider only one resonance or several, welldistinct resonances; on the contrary the behaviour of the general quantum oscillator is not efficiently captured, except in the linear case [24].

. 6 6. Subtractive Kramers-Kronig relations for harmonic generation susceptibility. – The characteristic integral structure of the K-K relations requires the knowledge of the spectrum on the entire angular-frequency range. Unfortunately, in practical spectroscopy the optical constants can be measured only on a limited spectral range. Moreover, technical difficulties in gathering information about nonlinear optical properties over a sufficiently wide spectral range make even the application of approximate K-K relations problematic. The extrapolations in K-K analysis, such as the estimation of the data beyond the measured spectral range, can be a serious source of errors [8, 75]. Recently, King [195] presented an efficient numerical approach to the application of the K-K relations. Nevertheless, the problem of data fitting is always present in regions outside the measured range. In the context of linear optics, singly [196] (SSKK) and multiply [102] subtractive Kramers-Kronig (MSKK) relations have been proposed in order to relax the limitations caused by finite-range data as previously described in sect. 3. Subtractive K-K relations have been proposed only recently in the context of nonlinear optics and especially for harmonic generation susceptibilities [190, 191]. The idea behind the subtractive Kramers-Kronig technique is that the inversion of the real (imaginary) part of the n-th–order harmonic generation susceptibility can be greatly improved if we have one or more anchor points, i.e. a single or multiple measurement of the imaginary (real) part at specific frequencies. Palmer et al. [102] have studied multiply subtractive K-K analysis in the case of phase retrieval problems related to linear reflection spectroscopy. Here, their results are generalized to apply for holomorphic nonlinear susceptibilities. Unfortunately, Palmer et al. [102] presented MSKK only for the phase angle (imaginary part of the logarithm of the linear reflectance). Here we extend their theory to be applicable both for the real and the imaginary parts of the arbitrary-order harmonic generation susceptibility. For simplicity, in this section we use the following simplified notation:

(354)

(n)

(n)

χij1 ...jn (nω; ω, . . . , ω) = χij1 ...jn (nω).

With the aid of mathematical induction (see appendix A in ref. [102]) we can derive the


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multiply subtractive K-K relation for the real and imaginary parts as follows: (355)

(356)

(n) ω 2α χij1 ...jn (nω) = 2 ) (ω 2 − ω22 )(ω 2 − ω32 ) · · · (ω 2 − ωQ (n) ω12α χij1 ...jn (nω1 ) + · · · = 2 2 2 2 2 2 (ω1 − ω2 )(ω1 − ω3 ) · · · (ω1 − ωQ ) 2 2 2 (ω 2 − ω12 ) · · · (ω 2 − ωj−1 )(ω 2 − ωj+1 ) · · · (ω 2 − ωQ ) + 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 ) × (ωj2 − ω12 ) · · · (ωj2 − ωj−1 j j+1 j Q 2α (n) ×ωj j χij1 ...jn (nωj ) + · · · 2 ) (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ−1 (n) 2α ω χ (nω ) + + Q Q ij1 ...jn 2 − ω 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 (ωQ 1 2 Q Q Q−1 ) 2 2 + (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ ) ℘× π (n) ∞ ω 2α+1 χij1 ...jn (nω ) × 2 ) dω , (ω 2 − ω 2 )(ω 2 − ω12 ) · · · (ω 2 − ωQ 0 (n) ω 2α−1 χij1 ...jn (nω) = 2 ) (ω 2 − ω22 )(ω 2 − ω32 ) · · · (ω 2 − ωQ (n) 2α−1 ω χ (nω ) + ··· = 1 1 ij1 ...jn 2) (ω12 − ω22 )(ω12 − ω32 ) · · · (ω12 − ωQ 2 2 2 (ω 2 − ω12 ) · · · (ω 2 − ωj−1 )(ω 2 − ωj+1 ) · · · (ω 2 − ωQ ) + 2 )(ω 2 − ω 2 ) · · · (ω 2 − ω 2 ) × (ωj2 − ω12 ) · · · (ωj2 − ωj−1 j j+1 j Q (n) ×ωj2α−1 χij1 ...jn (nωj ) + · · · 2 ) (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ−1 (n) 2α−1 χij1 ...jn (nωQ ) − + ωQ 2 2 2 2 2 2 (ωQ − ω1 )(ωQ − ω2 ) · · · (ωQ − ωQ−1 ) 2 2 ) ℘× − (ω 2 − ω12 )(ω 2 − ω22 ) · · · (ω 2 − ωQ π (n) ∞ ω 2α χij1 ...jn (nω ) × 2 ) dω , (ω 2 − ω 2 )(ω 2 − ω12 ) · · · (ω 2 − ωQ 0

with 0 ≤ α ≤ n; here ωj with j = 1, · · · , Q denote the anchor points. Note that the anchor points in eqs. (355) and (356) do not need to be the same. We observe that the integrands of eqs. (355) and (356) show a remarkably faster asymptotic decrease as functions of angular frequency than the conventional K-K relations given by eqs. (328), (329). In the case of Q-times subtracted K-K relations, the integrands asymptotically decrease as ω 2α−(2n+2+2Q) , whereas in the case of the conventional K-K relations the decrease is proportional to ω 2α−(2n+2) . Hence, it can be expected that the limitations related to the presence of an experimentally unavoidable finite frequency range are relaxed, and the precision of the integral inversions is then enhanced. The additional information for optical-data inversion derived from the knowledge of anchor points is mostly useful when they are located well outside the main spectral

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features in the considered range. The first reason is that if we already have direct access to information on the main features of the spectrum, the optical-data inversion may be unnecessary. Moreover, even small relative errors in the experimental value of anchor points located in the main features of the spectrum would “propagate” in eqs. (355) or (356) over all the spectral region under examination in such a way that the relative error in the out-of-resonance portions would be considerable, thus causing large error in the estimation of the complex phase. We note that according to Palmer et al. [102], under simplifying assumptions of the spectral shape, the choice of the anchor points is optimal when they are near the zeros of the Q-th order Chebyshev polynomial of the first kind. In linear optical spectroscopy it is usually easy to get information on the optical constants at various anchor points. However, in the field of nonlinear optics it is difficult to obtain the real and imaginary parts of the nonlinear susceptibility at various anchor points. Therefore, in the present study we wish to emphasize that even a single anchor point reduces the errors caused by the finiteness of the spectral range in the data inversion of nonlinear optical data. 7. – Kramers-Kronig relations and sum rules for data analysis: examples . 7 1. Introduction. – Only in a few investigations independent measurements of the real and imaginary parts of the harmonic generation susceptibilities have been performed across a relatively wide frequency range [27,28,188,189]. Consequently, verification of the coherence of measured data by checking the self-consistency of Kramers-Kronig relations is still of very limited use [27], while until very recently no studies at all have dealt with the experimental verification of the sum rules [30, 190, 191]. In this section we report the major results of the analysis of harmonic generation data where the full potential of the generalized nonlinear K-K relations and sum rules for harmonic generation susceptibilities, as well as of the subtractive K-K relations presented in the previous section, is exploited. We consider two published sets of wide spectral range data of the third harmonic generation susceptibility on two different polymers [27, 188]. We wish to underline that in our study, instead of considering the raw experimental data, we perform our analysis with data sets derived from the actual measurements by using theoretical best-fit and filtering. Nevertheless, in this section we will use the term experimental or measured data for the sake of clarity and simplicity. . 7 2. Efficacy of generalized Kramers-Kronig relations for data inversion. – We base our calculations on two published sets of experimental data on the third harmonic generation on polymers, where the real and imaginary parts of the susceptibility were independently measured. The first data set refers to measurements taken on polisylane [27] and spans a frequency range of 0.4–2.5 eV. The second data set refers to measurements taken on polythiophene [188] and spans a frequency range of 0.5–2.0 eV. We place ourselves in the worst scenario, which is nevertheless usually the most typical, namely data from a limited spectral range with no extrapolations beyond the measured range, and with no knowledge of anchor points [102, 190, 196]. We have no information about the tensorial component of the harmonic generation susceptibility that have been measured, so that we use a simplified scalar notation; we also denote χ(3) (3ω; ω, ω, ω) by χ(3) (3ω) to simplify the notation. . 7 2.1. Kramers-Kronig inversion of the susceptibility. We control the self-consistency of the two data sets by observing the efficacy of the K-K relations in inverting the optical


89

Fig. 12. – Efficacy of the K-K relations in retrieving χ(3) (3ω) on polysilane.

data for the functions ω 2α [χ(3) (3ω)]k , with k = 1, 2. We do not assume any asymptotic behaviour outside the data range, but use only the experimental data, since extrapolation is somewhat arbitrary and in K-K analysis can be quite problematic [8,13]. We effectively apply truncated K-K relations and use a self-consistent procedure. In the paper by Kishida et al. [27] a check of the validity of the K-K relations was performed by comparing the measured and retrieved χ(3) (3ω). Apart from the fact that our analysis also considers another data set, the consideration of the moments of the susceptibility is not a mere add-on to the work by Kishida et al. [27], but it represents a conceptual improvement. These additional independent relations are peculiar to the nonlinear phenomena, and provide independent double-checks of the experimental data that must be obeyed in addition to the more conventional K-K relations. In figs. 12 and 13 we present the results of the K-K inversion for, respectively, the real and imaginary parts of the third harmonic generation susceptibility data on polysilane. We observe that in both cases the retrieved data obtained with the choices α = 0, 1 are almost indistinguishable from the experimental data, while for α = 2 and α = 3, the agreement is quite poor in the lower part of the spectrum. The error induced by the presence of the cut-off at the high-frequency range becomes more critical in the data inversion for larger α, since a slower asymptotic decrease is realized. We expect that by inverting the data with the additional information given by anchor points located in the lower part of the data range these divergences can be cured. However, our object here is to deal with the worst case, i.e. when there is no a priori information about the phase of the complex nonlinear susceptibility at one or more fixed anchor points (fixed angular frequencies) [190]. From the theory, we expect that for α = 4 no convergence should occur. Actually we observe that, while the main features around 1.1 eV are represented, there is no convergence at all for the lower frequencies. The absence of a clear qualitative difference in the retrieving process performance between the α = 3 and α = 4 cases is due to the finiteness of the data range.

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Fig. 13. – Efficacy of the K-K relations in retrieving χ(3) (3ω) on polysilane.

Fig. 14. – Efficacy of the K-K relations in retrieving χ(3) (3ω) on polythiophene.


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Fig. 15. – Efficacy of the K-K relations in retrieving χ(3) (3ω) on polythiophene.

In figs. 14 and 15 we show the comparison between the retrieved and the experimental data of the third harmonic generation susceptibility on polythiophene for the real and the imaginary parts, respectively. The dependence on the quality of data inversion is similar to the previous case: for α = 0, 1 the agreement is virtually perfect, while for α = 2, 3 we have a progressively worse performance in the low-frequency range. Nonetheless, the peaks in the imaginary part are still well reproduced, while the dispersive structures in the real part are present but shifted towards the lower values. In this case the quality of the retrieved data for α = 3 and α = 4 is worse than in the previous data set. The inversion with α = 4 presents a notable disagreement in the whole lower half of the data range for both the real and the imaginary parts. In particular, we see that in fig. 14 the dispersive structure is absent, and the main peak in fig. 15 is essentially missing. Usually it may be expected that only the real or the imaginary part of the nonlinear susceptibility has been measured. The normal procedure is then to try the data inversion using K-K relations in order to calculate the missing part. The results in figs. 12-15 confirm that the best convergence is obtained when using the conventional K-K relations with α = 0. Hence, these should generally be used to obtain a first best guess for the inversion of optical data, and they should be used as seeds for any self-consistent retrieval procedure. Nevertheless, if there is good agreement with the inversions obtained with higher values of α, it is reasonable to conclude that the dispersion relations provide much more robust results. In this sense, the two data sets analyzed here are quite good in terms of self-consistency. . 7 2.2. Kramers-Kronig inversion of the second power of the susceptibility. We wish to emphasize that, while the consideration of a higher power of the susceptibility implicitly filters out noise and errors in the tails of the data, it greatly enhances experimental errors in the relevant features of the spectrum —resonances for the imaginary part and disper-

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Fig. 16. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polysilane —α = 0, 1, 2, 3, 4, 5.

sive structures for the real part. Therefore, the consistency between the K-K inversion of the different moments will improve for the powers of the susceptibility k > 1 with respect to the simple susceptibility (k = 1 case) only if the data are extremely accurate. In figs. 16-19 we show the results of the K-K inversion for the real and imaginary parts of the second power of the third harmonic generation susceptibility on polysilane [30]. We observe that for α = 0, 1, 2 the agreement between the experimental and retrieved data is almost perfect, while it becomes progressively worse for increasing α. Nevertheless, as long as α ≤ 6, the main features are reproduced well for both the real and the imaginary parts, and the retrieved data match well the experimental data when the photon energy is ≥ 1.0 eV. In figs. 20-23 we report the same analysis for the second power of the susceptibility data taken on the polythiophene. In this case the agreement is also very good if α = 0, 1, 2, but the narrower frequency range does not permit the data inversion if the very high moments are considered. If we consider the real part (fig. 20) for α = 3, 4, the K-K data inversion provides a good reproduction of the experimental data for photon energies ≤ 0.7 eV. For α ≥ 5 there is no convergence in the lower half of the spectral range. For the imaginary part (fig. 22) we can repeat the same observations, except that for α = 5 there is still a good reproduction of the main features of the curve. The theory predicts convergence for the K-K relations with α ≤ 7 and divergence for α = 8: in our analysis we have divergence for α = 7 in the case of polysilane and for α = 6, 7 in the case of polythiophene data. The disagreement between the theory and the experimental fact can be safely attributed to the truncation occurring in the high-frequency range. The poor representation of the far asymptotic behaviour mostly affects the convergence of the dispersion relations of the very high moments.


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Fig. 17. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polysilane —α = 6, 7, 8.

Fig. 18. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polysilane —α = 0, 1, 2, 3, 4, 5.

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Fig. 19. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polysilane —α = 6, 7, 8.

Fig. 20. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polythiophene —α = 0, 1, 2, 3, 4, 5.


95

Fig. 21. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polythiophene —α = 6, 7, 8.

Fig. 22. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polythiophene —α = 0, 1, 2, 3, 4, 5.

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Fig. 23. – Efficacy of the K-K relations in retrieving [χ(3) (3ω)]2 on polythiophene —α = 6, 7, 8.

We emphasize that if only one of the real or imaginary parts of the harmonic generation susceptibility has been measured experimentally, the use of the K-K relations relative to the higher powers of the susceptibility is not possible because the multiplication mixes the real and imaginary parts. Hence, in this case the K-K relations for k > 1 can be used as a test for the robustness of the results obtained with the dispersion relations applied to the conventional susceptibility. . 7 3. Verification of the sum rules. – In general a good accuracy in the verification of the sum rules from experimental data is more difficult to achieve. In consequence, a positive outcome of this test provides a very strong argument in support of the quality and coherence of the experimental data [79, 80]. In the case of harmonic nonlinear processes, the technical constraints for acquiring information about a very wide frequency range are very severe, and verification of the sum rules is critical, especially for those involving relatively large values of α which determine a slower asymptotic decrease. Nevertheless, if we consider increasingly large values of k, the integrands in eqs. (332) and (333) have a much faster asymptotic decrease, so that the missing high-frequency tails tend to become negligible. Therefore, we expect that for a given α the convergence of the sum rules should be more accurate for higher values of k, if we assume that the main features of the spectrum are well reproduced by the experimental data, as explained in the previous subsection. We focus first on the vanishing sum rules (332), (333). In order to have a measure of how precisely the vanishing sum rules are obeyed, we introduce the dimensionless


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Fig. 24. – Convergence to 0 of the vanishing sum rules for ω 2α [χ(3) (3ω)]k with 1 ≤ k ≤ 5; data on polysilane.

quantities Z and Z : (357)

3 ωmax 2α (3) k 3 ωmin ω {[χ (3ω)] }dω Z (α, k) = ωmax 2α , ω ω |{[χ(3) (3ω)]k }|dω min

(358)

3 ωmax 2α+1 {[χ(3) (3ω)]k }dω 3 ωmin ω Z (α, k) = ωmax 2α+1 . ω ω | {[χ(3) (3ω)]k }|dω min

Low values of Z (α, k) and Z (α, k) imply that the negative and positive contributions to the corresponding sum rule cancel out quite precisely compared to their total absolute value. The two data sets of the polymers have quite different performances in the verification of these sum rules. In figs. 24 and 25 we present the results obtained with the data taken with polysilane by considering 1 ≤ k ≤ 5 for, respectively, the sum rules of the real and the imaginary parts. In both cases, we see that for every given α, we have a better convergence when a higher k is considered, with a remarkable increase in the accuracy of the sum rules for k ≥ 3. Consistent with the argument that the speed of the asymptotic behaviour is critical in determining the accuracy of the sum rule, we generally also have a decrease in the quality of the convergence to zero when, for a given k, we consider higher moments, i.e. larger values of α. Particularly impressive is the increase in the performance in the convergence of the sum rules of χ(3) (3ω) for both the real part (2α = 0, 2, 4, 6) and the

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Fig. 25. – Convergence to 0 of the vanishing sum rules for ω 2α+1 [χ(3) (3ω)]k with 1 ≤ k ≤ 5; data on polysilane.

imaginary part (2α + 1 = 1, 3, 5) when we consider k = 4, 5 instead of k = 1: the values of Z and Z decrease by more than three orders of magnitude in all cases considered. In figs. 26 and 27 we present the corresponding results for the experimental data on polythiophene. Most of the sum rules computed with these data show a rather poor convergence to zero, since the corresponding Z and Z are above 10−1 . Nevertheless, we can draw conclusions similar to those of the previous case in terms of change in the accuracy of the convergence for different values of k and α. Consistent with the relevance of the asymptotic behaviour, the precision increases with increasing k and with decreasing α. But in this case, for a given α, the improvement in the convergence of the sum rules obtained by considering a high value of k instead of k = 1 is generally small, in most cases consisting of the decrease of Z and Z of about an order of magnitude. The two data sets for polymers differ greatly in the precision achieved in the verification of the sum rules. In many corresponding cases the polysilane data provide results that are better by orders of magnitude. The main reason for this discrepancy is the much stronger dependence of the sum rules precision on the position of the high-frequency experimental cut-off relative to the saturation of the electronic transitions of the material. It is likely that the data on polythiophene, apart from being extended on a narrower range, do not completely capture the electronic properties of the material. This result is consistent with the previously presented slightly worse performance of this data set in the K-K inversion of the second power of χ(3) (3ω), where the relevance of the out-of-range data is also quite prominent. For both of the data sets considered in this study there is no consistency between the non-vanishing sum rules referring to the various powers 1 ≤ k ≤ 5 of the susceptibility under examination, since the previously established consistency relation (335) is essen-

99


0

Value of Z ℜ

10

10

1

k=1 k=2 k=3 k=4 k=5

2

10

0

2

6

10

14

18 22 Value of 2α

26

30

34

38

Fig. 26. – Convergence to 0 of the vanishing sum rules for ω 2α [χ(3) (3ω)]k with 1 ≤ k ≤ 5; data on polythiophene.

0

Value of Z

ℑ

10

10

-1

k=1 k=2 k=3 k=4 k=5

-2

10

1

5

9

13

17 21 25 Value of 2α+1

29

33

37

Fig. 27. – Convergence to 0 of the vanishing sum rules for ω 2α+1 [χ(3) (3ω)]k with 1 ≤ k ≤ 5; data on polythiophene.

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tially not obeyed. Hence, the structural constant determined by the value resulting from the integration of the highest moment of the imaginary part of the harmonic generation susceptibility presented in (334) cannot be reliably evaluated with the data sets we are considering. The poor performance of the experimental data in reproducing this theoretical result is essentially due to the fact that we are dealing with the slowest converging sum rules for each k. These depend on the asymptotic behaviour of the data, which, as already noted, is poorly represented in the experiments, given the narrow frequency range. We conclude that better data covering a much wider spectral range are required to deal effectively with the non-vanishing sum rules. . 7 4. Application of singly subtractive Kramers-Kronig relations to experimental data. – For one anchor point, say at frequency ω1 , we obtain from eqs. (355) and (356) the following singly subtractive K-K relations: (359)

(n) (n) ω 2α χij1 ...jn (nω) = ω12α χij1 ...jn (nω1 ) + 2(ω 2 − ω12 ) ℘ + π

(360)

∞

(n) ω 2α+1 χij1 ...jn (nω ) (ω 2 − ω 2 )(ω 2 − ω12 )

0

dω ,

(n) ω 2α−1 χij1 ...jn (nω) = ω12α−1 χ(n) (nω1 ) − −

2

2(ω − π

ω12 )

℘ 0

∞

(n) ω 2α χij1 ...jn (nω ) (ω 2 − ω 2 )(ω 2 − ω12 )

dω ,

with 0 ≤ α ≤ n. Here we apply the singly subtractive K-K relations (359), (360) to the analysis of the experimental values of the real and imaginary part of the nonlinear susceptibility of the previously presented third-order harmonic wave generation on polysilane, obtained by Kishida et al. [27]. First, we consider only data ranging from 0.9 to 1.4 eV, in order to simulate a low data availability scenario, and then we compare the quality of the data inversion obtained with the α = 0 conventional K-K and SSKK relations within this energy range. This interval constitutes a good test since it contains the most relevant features of both parts of the susceptibility. However, a lot of the spectral structure is left outside the interval and the asymptotic behaviour is not established for either parts. Hence, no plain optimal conditions for the optical data inversion are established. In fig. 28 we show the results obtained for the real part of the third-order harmonic generation susceptibility. The solid line in fig. 28 represents the experimental data. The dashed curve in fig. 28, which was calculated by using the conventional K-K relation by truncated integration of (328), consistently gives a poor match with the actual line. In contrast, we obtain a better agreement with a single anchor point located at ω1 = 0.9 eV, which is represented by the dotted line in fig. 28. SSKK and measured data for the real part of the susceptibility are almost indistinguishable up to 1.3 eV. In fig. 29 calculations similar to those presented above are shown, but for the imaginary part of the nonlinear susceptibility. In this case the anchor point is located at ω1 = 1 eV. From fig. 29 we observe that the precision of the data inversion is dramatically better when using SSKK rather than the conventional K-K relations. The


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Fig. 28. – Efficacy of the SSKK vs. the K-K relations in retrieving χ(3) (3ω) over a limited spectral range.

Fig. 29. – Efficacy of the SSKK vs. the K-K relations in retrieving χ(3) (3ω) over a limited spectral range.

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Fig. 30. – Efficacy of the SSKK relations in retrieving χ(3) (3ω) over the full spectral range.

Fig. 31. – Efficacy of the SSKK relations in retrieving χ(3) (3ω) over the full spectral range.


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presence of the anchor point greatly reduces the errors of the estimation performed with the conventional K-K relations in the energy range 0.9–1.4 eV. We then test the efficacy of the SSKK when the higher moments of the susceptibility are considered, and data on the whole available spectral range 0.4–2.5 eV are considered. The anchor point we select for the real part is at the lower boundary of the interval, thus simulating an experimental situation where we have information on quasi-static phenomena. In fig. 30 we present our results of the data inversion for the real part, which can be compared with those obtained with the conventional K-K relations shown in fig. 12, which nevertheless were obtained after an optimization procedure. These should be interpreted more in terms of self-consistency analysis than in terms of pure data inversion. We can see that the SSKK overperform the conventional K-K relations and provide virtually perfect data inversion for 0 ≤ α ≤ 2, while for α = 3 the finiteness of the range causes disagreement between the measured and retrieved data for photon energy ≤ 0.9 eV. In the α = 4 case, for which the theory does not prescribe convergence, we have reasonable agreement only in the higher-energy range. In the case of the imaginary part, the anchor point is on the low-energy side of the main spectral feature, just out of resonance. In fig. 31 we present the results of optical data inversion with the SSKK relations for the the imaginary part, which, with the same previously discussed caveats, should be compared with those obtained with conventional K-K relations and presented in fig. 13. In the case of the imaginary part we also observe that the SSKK procedure provides more precise data inversion than with conventional K-K one. The agreement between measured and retrieved data is excellent for 0 ≤ α ≤ 2, with no relevant decrease of performance for increasing α, while for α = 3 we have good agreement except for photon energy ≤ 0.7 eV. In the theoretically non-converging α = 4 case, somewhat surprisingly we still obtain a good performance of the SSKK relations for an extended data range. We can interpret this result heuristically as a manifestation of the faster asymptotic decrease of the integrands realized when SSKK relations rather than K-K relations are considered. Thus, we have observed how an independent measurement of the unknown part of the complex third-order nonlinear susceptibility for a given frequency relaxes the limitations imposed by the finiteness of the measured spectral range, the fundamental reason being that in the obtained SSKK relations faster asymptotic decreasing integrands are present. SSKK relations can provide a reliable data inversion procedure based on using measured data alone. We have demonstrated that the SSKK relations yield a more precise data inversion, using only a single anchor point, than the conventional K-K relations. Naturally, it is also possible to exploit the MSKK relations if higher precision is required. However, the measurement of multiple anchor points may be experimentally tedious. Finally, we may remark that the MSKK relations are valid for all holomorphic nonlinear susceptibilities of arbitrary order. As an example of such holomorphic third-order nonlinear susceptibilities we would mention those related to pump and probe . nonlinear processes [36], previously analyzed in subsect. (5 3). . 7 5. Application of sum rules in second harmonic generation processes. – Recently it has been shown how to use the constraints imposed by the sum rules relevant for the second harmonic generation susceptibility to derive efficient ways to estimate the (2) second-order static susceptibility χijk (0; 0, 0) for semiconductors without center of inversion [197]. Such quantity has been difficult to estimate with ab-initio as well as with semiempiric approaches because of the sensitivity to the constituent parameters (energy differences and matrix elements) and because of the unavoidable truncation error due to

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Fig. 32. – Second-order static susceptibility for some semiconductors. Abscissae: theoretical calculations in units of 10−8 esu. Ordinate: experimental data squares in units of 10−8 esu. Full squares: data from [197]; open squares: data from [198]. Adapted from [197].

the consideration of a finite set of eigenstates [158, 198, 199]. If we adopt a single-resonance simplified quantum model for solids, it is possible to write the second harmonic generation susceptibility as (361)

(2) χijk

(1)

∼

(2)

αijk ω0 − ω − iγ

+

(3)

αijk ω0 − 2ω − iγ (1)

(2)

+

αijk (ω0 − ω − iγ)

2

∗

+ (ω → −ω) ,

(3)

where ω0 is the resonance frequency, αijk , αijk , and αijk are tensorial parameters, and the last term indicates the terms obtained by changing ω into −ω and taking the conjugate of the first three terms. If we impose the sum rules (333) and (334) to expression (361), (1) (2) (3) we derive the values of the parameters αijk , αijk , and αijk , and obtain after lengthy calculations that the second harmonic susceptibility can be expressed as a product of linear susceptibilities in the form of Miller’s rule (346). The Miller ∆ can be expressed as (362)

∆ijk

1 = 3 2e

N V

−2 4

∂ 3 V (r) ∂ri ∂rj ∂rk

5 , 0

in agreement with expression (348) in the case n = 2 derived with the anharmonic oscillator model, when the quantum expectation value of the ground state substitutes the evaluation at the equilibrium position. If we take the static limit of expression (346), we obtain (363)

(2)

(1)

(1)

(1)

χijk (0; 0, 0) ∼ ∆ijk χii (0)χjj (0)χkk (0).

Since the static linear susceptibility is a well-known and easily measurable quantity for every material, it is then possible to obtain an estimate of the second-order static suscep-


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tibility once Miller’s ∆ can be computed. As thoroughly explained in [197], this can be accomplished by considering the widely available pseudopotential functions and subtracting the electrons’ Hartree screening. In fig. 32 we show that such an approach, although based on a very simplified scheme, greatly overperforms much more sophisticated and time-expensive calculations, since it is based and focused on the fundamental properties of the optical response functions. 8. – Conclusions In the present review we have made an attempt to present those relevant features of the optical response which derive from the causality principle and are common to all types of materials. It is shown that such general properties in the linear case only depend on the electron densities, with appropriate modifications to account for local field effects and for inhomogeneous media. In the nonlinear case the response functions are shown to obey a number of dispersion relations between their real and imaginary parts which depend on their asymptotic behaviour, when a specific theorem (Scandolo’s theorem) is obeyed, which ensures their holomorphic character in the half complex plane cut on the multiple complex space. The important cases of pump and probe systems and of harmonic generation processes are analyzed in detail and new Kramers-Kronig relations are given for the susceptibilities and for their powers. The number of Kramers-Kronig–type relations depends on the asymptotic behaviour for ω → ∞. Using the superconvergence theorem on such dispersion relations, and comparing the results with the correct asymptotic behaviour obtained from quantum response theory or from semiclassical models, new sum rules are obtained for the nonlinear susceptibilities to any order. Such sum rules have the common features that all even moments of the real part and all odd moments of the imaginary parts are null, except for the highest converging odd moment of the imaginary part of the susceptibility, which gives a result dependent on the expectation value on the ground state of appropriate derivatives of the potential. This gives a qualitative measure of the optical nonlinearity of the material. The Kramers-Kronig–type relations are shown to be useful for data inversions, required because in most solid materials only one optical function can be measured, and the range of frequency attainable is rather narrow. Subtractive Kramers-Kronig relations are derived to improve the retrieval process. Specific examples of the application of conventional, generalized, and subtractive Kramers-Kronig relations as well as of the verification of sum rules are given on experimental data of the third harmonic generation on polymers. The sum rules can be used to check any approximate theory of nonlinear phenomena, because they are necessary constraints which have to be obeyed. In a relevant example for the second harmonic generation, it is shown that sum rules allow a better understanding of Miller’s rule and provide tools for explaining critical experimental parameters. We believe that the role of dispersion theory and sum rules will be of increasing importance in the expanding field of nonlinear optics. ∗ ∗ ∗ The authors wish to thank Dr Erik M. Vartiainen for providing some of the data used in this paper. V. L., K.-E. P. and J. J. S. wish to express their gratitude for the

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Academy of Finland (grants Nos. 204109 and 204925). Appendix A. Derivation of the asymptotic behaviour of the pump and probe susceptibility In this appendix we wish to propose the derivation of the asymptotic behaviour of . . the third-order pump and probe susceptibility discussed in subsects. 4 3 e 5 3 using the . Kubo formalism presented in subsect. 4 4. We follow an approach slightly different from but equivalent to the original derivation presented in [19, 20], based on the study of the instantaneous nonlinear response of the system. We can express the pump and probe susceptibility by suitably specifying the general expression (222): (3)

(A.1) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ∞ (3) Gij1 j2 j3 (t1 , t2 , t3 ) × exp [iω1 t1 + iω2 t2 − iω2 t3 ] dt1 dt2 dt3 = = −∞

e4

∞

θ (t1 ) θ (t2 − t1 ) θ (t3 − t2 ) exp [iω1 t1 + iω2 t2 − iω2 t3 ] × 3 V (−i) −∞ N N N N ρ (0) dt1 dt2 dt3 . rjα3 (−t3 ), rjα2 (−t2 ), rjα1 (−t1 ), riα ×Tr

=−

α=1

α=1

α=1

α=1

We then perform the following variables change in the integration: (A.2)

tj =

j

τi ,

1 ≤ j ≤ 3,

i=1

so that expression (A.1) can be written as (3)

(A.3) χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ∞ e4 dτ1 dτ2 dτ3 θ (τ1 ) θ (τ2 ) θ (τ3 ) exp [iω1 τ1 − iω2 τ3 ] × =− 3 V (−i) −∞ N N N N rjα3 (−τ1 − τ2 − τ3 ) , rjα2 (−τ1 − τ2 ) , rjα1 (−τ1 ) , riα ρ (0) . ×Tr α=1

α=1

α=1

α=1

We follow the same computational strategy which has been used in the derivation of . the asymptotic behaviour of the linear susceptibility in subsubsect. 2 4.4 and which is fully presented in appendix B in the context of the n-th–order harmonic generation susceptibility. . Since, as thoroughly explained in subsubsect. 2 4.4, we have that ∞ (A.4)

k

θ (τ1 ) −∞

1 (−τ1 ) exp[iω1 τ1 ] ≈ −(−i)k+1 k+1 ; k! ω1


107

in order to deduce the asymptotic behaviour of expression (A.3) with respect to the variable ω1 , we have to expand in Taylor series around 0 the τ1 -dependent terms and consider the lowest-order non-vanishing contribution. We can obtain such expansion by repeatedly substituting in expression (A.3) the following equation: rjα (c − τ ) =

(A.5)

∞

α ak,j (c)

k=0

where α ak,j

(A.6)

(c) =

1 i

α H0 , ak−1,j (c) ,

(−τ )k , k!

α a0,j (c) = rjα (c) .

Along the line of the computation presented in appendix B, we obtain that for large values of ω1 the pump and probe susceptibility can be expressed as (A.7)

(3)

χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ∞ e4 1 N 1 θ (t2 ) θ (t3 − t2 ) exp [−iω2 t2 + iω2 t3 ] × =− 2 m 3!2 V ω14 −∞ " ! ∂ 2 V (r α ) α α ρ (0) dt2 dt3 + o ω1−4 . ×Tr rj3 (−t3 ) , rj2 (−t2 ) , α α ∂ri ∂rj1

This result corresponds to the n = 3 term presented in the appendix of ref. [19], with the difference of a factor 4 due to the different definition adopted for the nonlinear susceptibility, in the case of [19] based on the original formulation of Bloembergen [16]. We underline that if we consider the velocity gauge and adopt the general definition for the susceptibility presented in (319) and (321)-(323), we obtain the following expression for the asymptotic behaviour for the pump and probe susceptibility: (A.8)

(3)

χij1 j2 j3 (ω1 ; ω1 , ω2 , −ω2 ) = ∞ e4 1 N 1 1 θ (t2 ) θ (t3 − t2 ) exp [−iω2 t2 + iω2 t3 ] × =− 4 m 2!2 V ω22 ω14 −∞ " ! ∂ 2 V (r α ) α α ρ (0) dt2 dt3 + o ω1−4 , ×Tr pj3 (−t3 ) , pj2 (−t2 ) , α α ∂ri ∂rj1

which must be equivalent to expression (A.7) for gauge invariance. Appendix B. Derivation of the asymptotic behaviour of the n-th–order harmonic generation susceptibility In this appendix we propose a detailed derivation of the asymptotic behaviour of the n-th–order harmonic generation susceptibility. The asymptotic behaviour is substantially determined by the short-time behaviour of the physical system, as may be expected from

108


J. J. SAARINEN

and

the fact that the frequency-dependent susceptibility is its Fourier transform. The approach followed here [187] is similar to that used in previous works for the study of the integral properties of the second [193] and third [194] harmonic generation susceptibilities, but allows a natural extension to all orders n. We consider the previously derived expression (326): (n)

χij1 ,...,jn (nω; ω, . . . , ω) =

(B.1)

  ∞ n en+1 =− θ (τ1 ) . . . θ (τn ) exp iω (n + 1 − j) τj  × n V (m) ω n j=1 −∞ N N N α α α ×Tr pjn (−τn − . . . − τ1 ), . . . , pj1 (−τ1 ), ri . . . ρ (0) dτ1 . . . dτn . α=1

α=1

α=1

We then derive from the Heisenberg equations the following Taylor expansion for the time-dependent momentum operators p α (τ ) around the value zero of their argument: pjα (−τ ) =

(B.2)

∞

α bk,j

k=0

(−τ )k , k!

where α bk,j =−

(B.3)

1 i

α H0 , bk−1,j ,

α bj0,α = p0,j .

Inserting expression (B.2) in eq. (B.1) we obtain (B.4)

(n) χij1 ...jn

×

∞

en+1 (nω; ω, . . . , ω) = − n V (m) ω n k1 ,...,kn Bi,j 1 ,...,jn

k1 ,...,kn =0

(τ1 +. . .+τn ) kn !

∞ θ (τ1 ) . . . θ (τn ) × −∞

kn

...

τ1k1 k1 !

 exp iω

n

 (n+1−j) τj dτ1 . . . dτn ,

j=1

where we have introduced the following tensorial constants for every term corresponding to the set of upper indices {k1 , . . . , kn }: (B.5)

k1 ,...,kn Bi,j 1 ,...,jn

≡ Tr

N α=1

bα kn ,jn , . . . ,

N α=1

bα k1 ,ji ,

N

riα . . . ρ (0)

.

α=1

k1 ,...,kn We now concentrate on the term of the summation inside the integral (B.4) with Bi,j 1 ,...,jn as common factor, our purpose being to consider only those terms which give the lowest asymptotic decrease in ω and are then responsible for the asymptotic behaviour of the susceptibility. We expand the polynomials of the temporal variables τi as presented in (B.4) and obtain the sum of many contributions which, apart from their numerical factors k1 ,...,kn and the common Bi,j coefficient, can generally be written as 1 ,...,jn

(B.6)

θ (τ1 ) . . . θ (τn ) τip1 . . . τnpn eiω(n+1−j)τj ,

109


with exponents {p1 , . . . , pn } such that n

(B.7)

pj =

j=1

n

kj ,

j=1

thanks to the homogeneity in the sum of the exponents of the time variables τj . Using the well-known result of the distributions theory [41] ∞ (B.8)

θ (τi ) τip eilωτi dτi =

−∞

−i l

p

dp dω p

i℘

1 lω

+ πδ (lω) ,

we notice that the asymptotic behaviour in ω of the integral in (B.8) is ∝ ω −p−1 . This result can be used to integrate expression (B.6) variable by variable, so that we obtain ∞ (B.9)

 θ (τ1 ) . . . θ (τn ) τ1p1 . . . τnpn exp iω

n

 lj τj  dτ1 . . . dτn ∝

j=1

−∞

p1 +...+pn +n k1 +...+kn +n 1 1 = . ∝ ω ω

We notice that the power dependence of the Fourier transform of expression (B.9) is given by n plus the sum on the exponents of the temporal variables, which is fixed as k1 +. . .+kn k1 ,...,kn for every term with Bi,j as common factor. Therefore, each of these terms, carrying 1 ,...,jn k1 ,...,kn −2n−k1 −...kn out the integration in (B.4), results in being proportional to Bi,j ω . 1 ,...,jn Hence, the problem of finding the asymptotic behaviour of the n-th–order harmonic k1 ,...,kn generation susceptibility is equivalent to seeking the non-vanishing terms Bi,j which 1 ,...,jn have the minimum value of the sum of the upper indices. To exemplify the procedure we first carry out the calculations for the linear case n = 1. The linear susceptibility can be written as

(B.10)

(1) χij

e2 (ω) = − V mω

∞

iωτ1

θ (τ1 ) e −∞

Tr

N

α=1

pα j

(−t1 ),

N

riα

ρ (0) dτ1 .

α=1

m 0 The non-vanishing Bij with the minimum value of m is Bij : N N 0 (B.11) Bij = Tr pα riα ρ (0) = Tr {−iN δij ρ (0)} = −iN δij , j, α=1

α=1

so that we derive the well-known result of linear optics previously obtained in (82) using the length gauge: ∞ ! ! "" e2 (1) χij (ω) = θ (τ1 ) eiωτ1 (−iN δij ) + O τ10 dτ1 = (B.12) V mω −∞

=−

" ! −2 " ! ωp2 1 e N 1 = − δ + o ω δij + o ω −2 , ij 2 2 mV ω 4π ω 2

110


and

J. J. SAARINEN

! 0" where O ! −2 " τ1 represents all the powers of τ1 having exponents larger than zero, and o ω represents all the terms having a power decrease at infinity strictly faster than ω −2 . k1 ,...,kn having non-vanishing value and In the nonlinear case, n > 1, the factor Bi,j 1 ,...jn 2,0,...,0 minimum sum k1 +. . .+kn is unique for every n considered. We now prove that Bi,j 1 ,...,jn is such a term. The proof relies on the fact that the terms with k1 < 2 vanish and the others have a higher sum of the upper indices. In the case k1 = 0, the inmost commutator k1 =0,...,kn in the expression of Bi,j has a constant value and so the whole expression vanishes: 1 ,...jn (B.13)

k1 ,...,kn =0 Bi,j 1 ,...,jn

= Tr = Tr = Tr

N α=1 N α=1 N α=1

bα kn ,jn , . . . , bα kn ,jn , . . . ,

N α=1 N α=1

N

bα k1 =0,j1 , pα ji ,

N

α=1

riα

. . . ρ (0)

=

. . . ρ (0)

α=1

bα kn ,jn , . . . , [iN δij1 ] . . .

riα

=

ρ (0)

= 0.

k1 ,...,kn In the case k1 = 1, the inmost commutator of Bi,j is zero since it involves two 1 ,...,jn functions which depend on spatial variables only, and so the coefficient vanishes:

(B.14)

k1 =1,...,kn Bi,j 1 ,...,jn

= Tr

= Tr = Tr

N

α=1 N α=1 N α=1

bα kn ,jn , . . . ,

bα kn ,jn , . . . , bα kn ,jn , . . . ,

N α=1

bα k1 =1,j1 ,

N 1 α=1

i !

N

H0 , pα ji

∂V rjαi ∂rjαi α=1

" ,

N

N

riα

. . . ρ (0)

=

α=1

N α , ri . . . ρ (0) = α=1

riα

. . . ρ (0)

.

α=1

We notice that the relevance of the Coulombian e-e repulsive potential is always null since it is invariant for translations of the whole set of particles and so commutes with the sum of all the momenta of the particles. k1 ,...,kn with k1 = 2 and kj = 0, j > 1 is not We directly prove that the term Bi,j 1 ,j2 ,...,jn identically vanishing: (B.15)

k1 =2,...,kn =0 Bi,j 1 ,...,jn

= Tr

N

α=1

= Tr

N

α=1

bα kn =0,jn , . . . , pα jn , . . . ,

N α=1

N α=1

bα k1 =2,j1 ,

bα m1 =2,ji ,

N α=1

N α=1

riα

riα . . . ρ (0) =

. . . ρ (0)

N i ∂ 2 V (rα ) . . . ρ (0) = = Tr m α=1 ∂rjαi ∂riα α=1 n i ∂ n+1 V (rα ) n−1 (−1) N Tr ρ (0) , = m ∂rjαn . . . ∂rjαi ∂riα

N

pα jn , . . . ,

=


111

where, in the last equality, we have used the indistinguishability of the electrons. This k1 =2,...,kn =0 term Bi,j determines the asymptotic behaviour of the n-th–order harmonic 1 ,...,jn generation susceptibility because it does not vanish and has the minimum sum of the upper indices. Using this result in eq. (B.4) we obtain

(B.16)

(n) χij1 ...jn

×

en+1 (nω; ω, . . . , ω) = − n V (m) ω n

∞

k1 =0...kn

2,...,0 Bi,j 1 ,...,jn

τ12 2!

∞ θ (τ1 ) . . . θ (τn ) × −∞

  n & 2' +O τ exp iω (n + 1 − j) τj  dτ1 . . . dτn ,

j=1

where O({τ }2 ) denotes all the monomials with a total power degree in the temporal variables higher than 2. Eventually, by considering expression (B.9), we derive the dominant asymptotic behaviour for ω → ∞ of the general n-th–order harmonic generation susceptibility [187]: n

en+1 (n) n−1 (i) N (B.17) χij1 ...jn (nω; ω, . . . , ω) = × (−1) n m V (m) ω n N 31 " ! i i ∂ n+1 V (rα ) ρ (0) ×Tr + o ω −2n−2 = α α α ∂rjn . . . ∂rji ∂ri nω (n + 1 − j) ω j=2 n " ! (−1) en+1 N ∂ n+1 V (rα ) 1 = 2 Tr ρ (0) + o ω −2n−2 , α α α n+1 2n+2 n n! m V ∂rjn . . . ∂rji ∂ri ω ! " where o ω −2n−2 denotes the terms that have an asymptotic behaviour faster than ω −2n−2 . Appendix C. Convergence of Kramers-Kronig–type dispersion relations In this appendix we wish to present a detailed analysis of the convergence of the dispersion relations and the sum rules of the nonlinear susceptibility descriptive of harmonic generation processes [169]. We will show how a complex analysis allows to derive sum rules without exploiting the superconvergence theorem [6]. The convergence of K-K–type dispersion relations and sum rules are restricted by the integral along the arc A, presented in fig. 33. General K-K–type dispersion relations are derived by considering the function (C.1)

f (Ω) =

Ωm [χ(n) (nΩ)]k , Ω − ω

where m, n and k are integers and ω is located at the positive real axis. We consider both even and odd powers of the complex angular frequency Ω leading to different K-K–type relations. However, the sum rules are reduced to the same set for both even and odd powers. At high frequencies, the convergence of the arbitrary-order harmonic frequency generation susceptibility is proportional to ψω −(2n+2) [35, 187]. By expressing

112


and

J. J. SAARINEN

Fig. 33. – Integration path for the establishment of Kramers-Kronig relations.

the complex angular frequency in polar coordinates Ω = Reiϕ such that dΩ = Rieiϕ dϕ, the integral along the arc A is of the form (C.2) A

Ωm [χ(n) (nΩ)]k dΩ = Ω−ω

π

Rm eimϕ ψ k Rieiϕ dϕ = 2n+2 ei(2n+2)ϕ ]k (Reiϕ − ω) 0 [R π i[m+1−k(2n+2)]ϕ e dϕ. = iψ k Rm−k(2n+2) eiϕ − ω/R 0

The limit of the integral (C.2) at high frequencies defines the convergence of dispersion relations. For the convergent integral, the limit is finite when the radius R = |Ω| of the arc A approaches infinity. This restricts the highest power m. For values m−2k(n+1) < 0 the integral converges to zero (C.3)

lim

R→∞

k

iψ R

m−k(2n+2)

0

π

ei[m+1−k(2n+2)]ϕ dϕ = 0. eiϕ − ω/R

Powers higher than m = 2k(n + 1) are divergent. For the value m = 2k(n + 1), the integral along the arc converges to a certain purely imaginary value that depends on the model used to describe the medium as follows: (C.4)

lim

|Ω|→∞

A

π Ωm [χ(n) (nΩ)]k eiϕ k dϕ . dΩ = lim iψ iϕ R→∞ Ω − ω 0 e − ω/R

Let R be large enough such that R ≥ 2ω . Then, by Lebesgue’s convergence theorem, the integral along the arc A has a converging majorant (C.5)

eiϕ 1 1 eiϕ − ω/R ≤ 1 − |ω/R| ≤ 1 − 1/2 = 2,

which is a constant. In eq. (C.4), the order of the integration and the limiting process


113

can be changed as follows: (C.6)

lim

R→∞

iψ

k

π eiϕ eiϕ k dϕ = iψ lim dϕ = R→∞ eiϕ − ω/R eiϕ − ω/R 0 π dϕ = πiψ k . = iψ k

π

0

0

The integration around the closed semicircle presented in fig. 33 can be expressed as follows: * lim (C.7) f (Ω)dΩ = πiResΩ=ω f (Ω) = |Ω|→∞

C

= πiω m {[χ(n) (nω)]k } − πω m {[χ(n) (nω)]k }. On the other hand, the integration around the closed semicircle C can be split into parts

* (C.8)

lim

|Ω|→∞

∞

f (Ω)dΩ = ℘

f (ω )dω + lim

|Ω|→∞

−∞

C

f (Ω)dΩ. A

The Cauchy principal value integration can be reformulated with the aid of the symmetry relation equation (317). For the even integer m it holds (C.9)

∞

f (ω )dω = 2ω

℘

−∞

∞

ω m {[χ(n) (nω )]k } dω + ω 2 − ω 2

0

∞

+2i 0

ω m+1 {[χ(n) (nω )]k } dω ω 2 − ω 2

and for the odd integer m (C.10)

∞

∞

ω m+1 {[χ(n) (nω )]k } dω + ω 2 − ω 2 0 ∞ m ω {[χ(n) (nω )]k } dω . +2iω ω 2 − ω 2 0

f (ω )dω = 2

℘ −∞

In eqs. (C.7)-(C.10), one can separate the real and imaginary parts. For values m ≤ 2k(n + 1) − 1 with even integer m, the K-K–type relations are of the form (C.11)

(C.12)

ω m {[χ(n) (nω)]k } =

m

ω {[χ

(n)

2 ℘ π

∞

0

2ω (nω)] } = − ℘ π k

ω m+1 {[χ(n) (nω )]k } dω , ω 2 − ω 2 ∞

0

ω m {[χ(n) (nω )]k } dω , ω 2 − ω 2

and for odd integers m, the corresponding equations are (C.13)

ω m {[χ(n) (nω)]k } =

2ω ℘ π

0

∞

ω m {[χ(n) (nω )]k } dω , ω 2 − ω 2

114


2 ω m {[χ(n) (nω)]k } = − ℘ π

(C.14)

∞

and

J. J. SAARINEN

ω m+1 {[χ(n) (nω )]k } dω . ω 2 − ω 2

0

For the value m = 2k(n + 1), the imaginary part of the left-hand side of eq. (C.8) contains an extra term, which is caused by the integration along the arc A and the dispersion relation (C.11) is transformed as follows: (C.15)

ω m {[χ(n) (nω)]k } =

2 ℘ π

∞

0

ω m+1 {[χ(n) (nω )]k } dω + ψ k . ω 2 − ω 2

Sum rules for arbitrary-order harmonic frequency generation susceptibilities can be derived with the aid of obtained dispersion relations. For example, the two highest convergent even powers of eq. (C.12) are m = 2k(n + 1) and m = 2k(n + 1) − 2. By inserting these values into (C.12) one obtains ω 2k(n+1) {[χ(n) (nω)]k } = −

2ω ℘ π

ω 2k(n+1)−2 {[χ(n) (nω)]k } = −

2ω ℘ π

(C.16)

(C.17)

∞

ω 2k(n+1) {[χ(n) (nω )]k } dω , ω 2 − ω 2

∞

ω 2k(n+1)−2 {[χ(n) (nω )]k } dω . ω 2 − ω 2

0

0

Equation (C.17) is multiplied by ω 2 and subtracted from eq. (C.16), which can be expressed as follows: 0=−

(C.18)

2ω ℘ π

∞

0

(ω 2 − ω 2 )ω 2k(n+1)−2 {[χ(n) (nω )]k } dω , ω 2 − ω 2

leading to the sum rule

∞

(C.19) 0

ω 2k(n+1)−2 {[χ(n) (nω )]k }dω = 0.

In a similar manner, one obtains for the highest convergent odd powers m = 2k(n+1)−1 and m = 2k(n + 1) − 3 equations ∞ 2k(n+1) ω {[χ(n) (nω )]k } 2 dω , (C.20) ω 2k(n+1)−1 {[χ(n) (nω)]k } = − ℘ π ω 2 − ω 2 0 (C.21)

ω

2k(n+1)−3

{[χ

(n)

2 (nω)] } = − ℘ π k

0

∞

ω 2k(n+1)−2 {[χ(n) (nω )]k } dω . ω 2 − ω 2

Now, in turn, eq. (C.21) is multiplied by ω 2 and subtracted from eq. (C.20) leading to (C.22)

2 0=− ℘ π

which is of the form

0

∞

(ω 2 − ω 2 )ω 2k(n+1)−2 {[χ(n) (nω )]k } dω , ω 2 − ω 2


(C.23) 0

∞

115

ω 2k(n+1)−2 {[χ(n) (nω )]k }dω = 0.

A detailed derivation, which takes into account the parity of Ωm , shows that both even (C.19) and odd (C.23) powers reduce to the same set of sum rules but different K-K–type dispersion relations. The sum rule given by eqs. (C.19) and (C.23) can be reduced to the form ∞ (C.24) ω 2α {[χ(n) (nω )]k }dω = 0 with 0 ≤ α ≤ k(n + 1) − 1. 0

The other sum rules given by eqs. (333) and (334) can be obtained from similar mathematical procedures. REFERENCES [1] Kramers H. A., La diffusion de la lumi´ ere par les atomes, in Atti del Congresso Internazionale dei Fisici, Como, Vol. 2 (Zanichelli, Bologna) 1927, pp. 545-557. [2] De L. Kronig R., J. Opt. Soc. Am., 12 (1926) 547. [3] Thomas W., Naturwiss., 28 (1925) 627. [4] Reiche F. and Thomas W., Z. Phys., 34 (1925) 510. ¨hn W., Z. Phys., 33 (1925) 408. [5] Ku [6] Altarelli M., Dexter D. L., Nussenzveig H. M. and Smith D. Y., Phys. Rev. B, 6 (1972) 4502. [7] Landau L. D., Lifshitz E. M. and Pitaevskii L. P., Electrodynamics of Continuous Media, 2nd edition (Pergamon, Oxford) 1984. [8] Aspnes D. E., The accurate determination of optical properties by ellipsometry, in Handbook of Optical Constants of Solids, edited by Palik E. D. (Academic Press, New York) 1985. [9] Greenaway D. L. and Harbeke G., Optical Properties and Band Structure of Semiconductors (Pergamon Press, Oxford) 1968. [10] Nussenzveig H. M., Causality and Dispersion Relations (Academic Press, Inc., New York) 1972. [11] Bassani F. and Altarelli M., Interaction of radiation with condensed matter, in Handbook on Synchrotron Radiation, edited by Koch E. E. (North Holland, Amsterdam) 1983, pp. 463-605. [12] Cardona M. and Yu P. Y., Fundamentals of Semiconductors (Springer, Berlin) 1996. [13] Peiponen K.-E., Vartiainen E. M. and Asakura T., Dispersion, Complex Analysis and Optical Spectroscopy (Springer, Heidelberg) 1999. [14] Milonni P. W., J. Phys. B, 35 (2002) R31. [15] Feynman R. P., Phys. Rev., 76 (1949) 769. [16] Bloembergen N., Nonlinear Optics (Benjamin, New York) 1965. [17] Butcher P. N. and Cotter D., Elements of Nonlinear Optics (Cambridge University Press, Cambridge) 1990. [18] Boyd R. W., Nonlinear Optics (Academic Press, New York) 1992. [19] Bassani F. and Scandolo S., Phys. Rev. B, 44 (1991) 8446. [20] Bassani F. and Scandolo S., Phys. Status Solidi (b), 173 (1992) 263. [21] Peiponen K.-E., Phys. Rev. B, 35 (1987) 4116. [22] Peiponen K.-E., J. Phys. C, 20 (1987) 2785. [23] Peiponen K.-E., Phys. Rev. B, 37 (1988) 6463. [24] Scandolo S. and Bassani F., Phys. Rev. B, 45 (1992) 13257. [25] Hutchings D. C., Sheik-bahae M., Hagan D. J. and Van Stryland E. W., Opt. Quantum Electron., 24 (1992) 1.

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