coherent properties of wave fields in open waveguides

HEAT PHYSICS DEPARTMENT The Academy of Sciences of Uzbekistan KYRENSKY INSTITUTE of PHYSICS The Siberian Branch of the Academy of Sciences of the USSR

COHERENT PROPERTIES OF WAVE FIELDS IN OPEN WAVEGUIDES By Sadrilla Sayfullaevich Abdullaev (Field code: 01.04.03 − Radiophysics, including quantum radiophysics)

DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics and Mathematics

Advisor: Corresponding member of the Academy of Sciences of the Uzbek SSR, professor P.K. Khabibullaev Scientific Committee: Members of the scientific council of Kyrensky Institute of Physics

Krasnoyarsk 1980

About the dissertation My works on statistical optics of optical waveguides began in 1975 in the research group led by Dr. A.T. Mirzaev at the Radiophysics Department of Tashkent State University and later continued at the Heat Physics Department of Uzbekistan Academy of Sciences. This new field of research had been suggested to us by the world–leading experts in nonlinear and statistical optics, Professors A.S. Akhmanov and A.S. Chirkin, who, together with the Rector of the Moscow State University, Professor Rem V. Khokhlov, visited the Radiophysics Department in 1974 during the All-Union Conference on Coherent and Nonlinear Optics held in Tashkent. In 1975 I started my theoretical studies in this field. Since then there were no experts in this field in Uzbekistan. I had to learn a vast literature in this field and do research independently and alone. Having a background on quantum mechanics helped me to understand the problem of wave propagation in waveguides and the analogy between the classical and quantum motion of particles in potential fields and the ray and wave propagation in waveguides. This analogy allowed me to predict fundamentally new effects in the theory of wave propagation in waveguide media, the wave localization, and ray chaos in irregular waveguides, and also to develop the semiclassical and diagram methods to study the coherence and statistical properties of field in waveguides. There were also very useful discussions with colleagues working in this and other areas of physics, particularly, with Professor A.S. Chirkin from Moscow State University, Dr. F.K. Abdullaev of the Heat Physics Department, who was working on the physics of disordered solid states, and Professor G.M. Zaslavsky, a world–leading expert on nonlinear dynamics and chaos. Professor G.M. Zaslavsky, whom I met in May 1979, suggested that I complete my thesis in his group at the Kyrensky Institute of Physics of the Siberian Branch of the USSR Academy of Sciences in Krasnoyarsk and defend it there. The thesis was completed during my four months stay in the winter time in Krasnoyarsk. The dissertation was defended on February 13, 1981 at the Specialized Scientific Coincil of the Kyrensky Institute of Physics, Krasnoyarsk. The official referees were Professor Dr. Yu.A. Kravtsov, Lebedev Institute of Physics of the USSR Academy of Sciences, Moscow, Dr. V. Shapiro, Kyrensky Institute of Physics, Krasnoyarsk, and Professor Dr. A.S. Chirkin, the Moscow State University (as a referee organization).

The present manuscript is the translation of this dissertation. In this work the fundamentally new effects in the theory of wave propagation in waveguides have been predicted: (i) the transversal localization of optical beams in a system of coupled waveguides with random parameters (the optical analogy of Anderson localization) and (ii) classical nonlinear dynamics and chaos of rays in the wave propagation problems in inhomogeneous waveguide media. Besides these effects the asymptotical method for calculation of statistical characteristics of wavefields in waveguide media has been developed. These fundamental new trends in the wave propagation problems, started at the beginning 1980’s, were almost a decade earlier than the corresponding works in the West. At present, the wave beam localization or the Anderson localization of light is one of the forefront research areas in optics.

1

The studies on ray chaos in waveguide media is

another active research area in underwater acoustics and optics, known as ray and wave chaos.

2

The results obtained in this field until 1990s have been summarized in the re-

view by S.S. Abdullaev and G.M. Zaslavsky, Classical nonlinear dynamics and chaos of rays in problems of wave propagation in inhomogeneous media, Sov. Phys. Uspekhi, 34, 645 (1991) and in the monograph by S.S. Abdullaev, Chaos and Dynamics of Rays in Waveguide Media (Gordon & Breach, 1993). The translated version of the dissertation is identical to the original manuscript in Russian: the numbering of sections, equations, figures, and cited references are the same. Only books and some papers translated into Russian were replaced by the original editions in English. A few misprints have been also corrected. The scanned original text in Russian is given after the text in English.

Jülich, Germany, March 2014

Sadrilla S. Abdullaev

Translated by S.S. Abdullaev c S.S. Abdullaev, 2014

1 See a review by M. Segev, Y. Silberberg, and D.N. Christodoulides, Anderson localization of light, Nature Photonics, 7, 197 (2013). 2 See a review by A. L. Virovlyansky, D.V. Makarov, and S.V. Prants, Ray and wave chaos in underwater acoustics waveguides, Sov. Phys. Uspekhi, 55, 18 (2012).

2

Contents Introduction

5

1 Coherence properties of a electromagnetic field in dielectric waveguides 18 1

Correlation functions of a electromagnetic field in dielectric waveguides . . 18

2

Coherence properties of a in regular waveguides . . . . . . . . . . . . . . . 23

3

Spatial coherence of field and spatial oscillations of intensity in multimode waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Propagation of partially coherent waves in statistically irregular waveguides

41

1

Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2

Equations for the mode correlators . . . . . . . . . . . . . . . . . . . . . . 44

3

Propagation of partially coherent waves in parabolic waveguides . . . . . . 49

4

Fluctuations of wave intensity in waveguides with random bending of axis . 55

3 Wave propagation in a system of statistically irregular waveguides

60

1

Propagation of a wave in a system of regular waveguides . . . . . . . . . . 62

2

Localization of waves in a system of waveguides with random parameters . 64

3

Diffusional spreading of waves in a system of waveguides . . . . . . . . . . 67

4 Nonlinear dynamics of rays in inhomogeneous media

72

1

Equations of ray trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2

Ray trajectory for soliton–like n(x) profile . . . . . . . . . . . . . . . . . . 78

3

Formation of stochastic layer near the separatrix . . . . . . . . . . . . . . . 81

4

Stochastization of rays in a three-dimensional waveguide channel . . . . . . 83

Summary and Conclusions

86

3

1

Appendix 1

88

2

Appendix 2. Diagram technique for the mean field

90

3

Appendix 3. Diagram technique for the second moments of field

95

4

Appendix 4

99

Bibliography

101

4

Introduction The problems of transmission of information and images on long distances have received a considerable amount of attention duirng the last times. One of the most perspective way to solve this problem is a use of artificial and natural waveguide channels in which waves can propagate on very long distance without significant of energy. The dielectric waveguides of optical range are the most perspective for the ground devices of information and image transmissions [1, 2, 3, 4]. They have a practical value since the light propagation through them is protected from external disturbances. This opens a potential opportunity to use them for optical communication. Bundled fibers, consisting of large number coupled micro–waveguides, have a great opportunity for the image transmission [5, 118]. They also used in holography [6, 7, 8, 9]. With the help of fibers one can obtain holograms of objects located in inaccessible places. This gives an opportunity of application of holographic methods in technique, biology, and medicine. Optical waveguides are also of interest in their applications in the so–called integrated optics (for the fabrication of fast running miniature functional devices based on optical micro-waveguides) [3, 10, 11, 12, 13]. Natural waveguide channels for radio waves around the Earth appears due to its sphericity and layered structure of the ionosphere. These waveguide channels may be used for terrestial radio communication, communications with rockets and satellites, and between satellites (see [14] and references therein). The underwater sound communication in the ocean can be carried out through the underwater sound channels created due to non-monotonic dependence of the sound phase velocity on depth [15, 16, 17, 18]. The application of the above mentioned waveguide channels for optical, radio and acoustical communications requires the study of statistical properties, in particularly, correlation properties of wave fields propagating in waveguides. First of all this is related 5

with the fact that real sources of radiation (lasers, light–diodes etc.) used for the excitation of waveguides always have natural and technical noises and their radiations are partially coherent. The latter affects on the wave propagation in waveguides, for example, on the spatial distribution of intensity. Secondly, real waveguide channels are subject to various irregularities that limit propagation distance of wave and change its coherent properties and so on. Irregularities of dielectric waveguides arise mainly during the manufacturing process (for example, a distortion of the boundary surfaces between media with different refractive indices, bending of waveguide axis and etc.), and also are due to thermal fluctuations of media. Irregularities of ionospheric waveguide channels are caused by fluctuations of ion and electron concentrations, plasma turbulence, and also large–scale inhomogeneities that arise, for example, by difference of wave phase velocity on the sunlit side and in the shadow [14]. Irregularities in underwater sound channels are due to small–scale inhomogeneities (fluctuations of salinity and the water temperature) as well as large–scale inhomogeneities (for example, internal waves, the ocean surface waves) [17, 18]. On the other hand, the application of the holographic and correlation methods for the record and recovery of information and images [53, 119], transmitted through waveguide channels also require the study of correlation properties of wave fields, propagating in waveguide channels. The present thesis is devoted to the theoretical study of coherent properties of wave fields in regular and irregular waveguide channels. The problem of accumulating effects during the wave propagation in multimode waveguides with small periodic inhomogeneities is also considered in the frame of geometrical optics. Below we give a short survey of works devoted to the corresponding problems. It also includes the works by the author.

1

Coherent properties of wave fields in regular waveguide channels

Until now the general theory of propagation of partially coherent waves in waveguides was not developed although the corresponding problem in homogeneous space has been quite well advanced (see, e.g., [19, 20, 21, 22, 23, 24]. In Refs. [33, 34, 35, 39, 40, 41, 42])

6

devoted to the theoretical study of propagation of partially coherent optical fields in regular dielectric waveguides only specific type of waveguides or radiation sources were considered. Theoretical consideration of the coherent properties has been carried out based on the two approaches: geometrical optics [33, 39, 42] and mode approaches [34, 35]. In works [33, 39] the intuitive derivation of the equation for the degree of spatial coherence of field in a multimode optical fiber is given. The derivation has been based on the generalization of the well known the Van Cittert–Zernike theorem. According to the latter the normalized spatial coherence function γ (~r1 , ~r2 ) =

hE ∗ (~r1 ) E (~r2 )i

[h|E (~r1 ) |2 ih|E (~r2 ) |2 i]1/2

,

[where E (~r) is the field, ~r1 , ~r2 are points at the observation plane] of the wave field in a homogeneous medium excited by homogeneous and quasi-monochromatic source of radiation at the sufficiently large distance R from the source is determined by its angular size α:  γ (~r, ~r + ~x) = 2J1 (ξx/a) (ξx/a) ,

ξ = 2πaα/λ,

α = a/R,

(B.1.1)

where J1 (y) is the Bessel function of the first order, a is the source radius, λ is the wavelength in a medium. When the optical fiber with a circular cross section of radius a, and the refractive indices of core nc and cladding no is excited by homogeneous source of light the normalized coherence function of field at the large distance from the excitation plane is determined by the formula of type (B.1.1) with the parameter ξ given by  2πaα  ξ = , for α < αc , λ (B.1.2) 2 2 1/2  ξ = 2πaαc ≈ 2πa (nc − no ) ≡ V, for α > α , c λ λ

where αc is the critical angle of total internal reflection, α is the angular size of the source at the excitation plane. From (B.1.2) it follows that at the excitation of a fiber

by a source radiation with the angular spectrum exceeding αc , the coherence degree is determined only by the fiber parameter V . The generalization of Eq. (B.1.1) to the case of a multimode waveguide with the gradient profile of refractive index n(ρ), [~ ρ = (x, y)],has been given in Ref. [42]. Under 7

the excitement of waveguide by the spatial incoherent source of radiation the normalized coherence function is given by |γ (x, y; −x, y) | = where A(ρ) = [n2 (ρ) − n2 (a)]

1/2

2J1 (4πA(|y|)/λ) , 4πA(|y|)/λ

(B.1.3)

is the local numerical aperture of a waveguide. Eq. (B.1.3)

has been used to determine the profile of refractive index n(ρ) of waveguide. In spite of its simplicity, the geometric–optical approach is not able to take into account the influence of polychromatic radiation on the spatial coherence. The latter in the case of homogeneous media has been noted in Ref. [52]. This problem can be studied using more rigorous mode approach. Such a consideration has been carried out by Crosignani and Di Porto [34], Mehta with collaborators [35], and the author [43]. In Ref. [34] a cylindrical dielectric waveguide with a step–wise profile of refractive index is considered. It was established that the correlation function of field is a periodic function of the longitudinal coordinate z at z ≪ zc , and does not depend on z at z ≫ zc , where zc is the critical distance determined by the spectral width of radiation and temporal delays between modes. In Ref. [35] the coherent properties of a field in a parabolic waveguide are studied. In the paraxial approximation the closed expression for the correlation function of a field has been obtained. The latter is a periodic function of the coordinate z at all distances independent on the spectral content of radiation. The analog of the Van Cittert-Zernike theorem has been also established. The general theory of propagation of partially–coherent electromagnetic field in dielectric waveguides has been developed by the author [43]. In this work the general form of the spatial–temporal correlation function of stationary field in an arbitrary type of waveguides has been established, and based on this the coherent properties of a field at different distances has been studied. Particularly, it was shown that the coherence times τ1 and τ2 defined according to Wolf and Mandel, respectively, behave differently at the distances z ≫ zc ∼ |2π/[∆ω(Vn−1 − Vm−1 )]| (∆ω is the spectral width of radiation, Vn is

the group velocity of modes): τ12 grows quadratically with z while τ2 tends to the constant value. It was established the modulation of spectrum along z. The spatial coherence of fields is closely related to the problem spatial oscillations of

intensity in waveguides. The spatial of oscillations of field intensity has been first found 8

in numerical calculations carried out in Ref. [45]. It turns out that when the waveguide is excited by a spatially–incoherent radiation the intensity distribution in the transversal cross-section of multimode waveguide oscillates about the smooth distribution described by the profile of refractive index. The theory of oscillatory effects of field intensity in multimode waveguides has been developed by the author [44]. In this work the quasi-classical method of calculations of the spatial correlation function and field intensity in multimode waveguides has been proposed. The quasi-classical expressions for the correlation function and the field intensity in plane waveguides with the arbitrary profile of refractive index are obtained. It was shown that the intensity oscillations are irremovable and they are due to a finiteness of the transversal size of waveguides. The coherence radius has a finite value and its minimal value is determined by ρc ≈ a/V, where a is the effective width of waveguide, V is the dimensionless frequency of waveguide.

2

Propagation of partially-coherent waves in irregular waveguides

The coherent properties of fields propagating in inhomogeneous waveguides are studied less than the corresponding problems in randomly inhomogeneous boundless media (see, e.g., [25, 26, 27, 28, 76, 77]). The works devoted to this problems appeared only in a recent few years [54, 55, 56, 57, 58, 59, 60]. We analyze these works based on the approach given in Ref. [58]. The general form of the spatial–temporal correlation function of wave field in statistically irregular waveguides can be established by the expansion of the field E (~r, t) into the eigenfunctions of modes of the unperturbed waveguide un (~ ρ), E (~r, t) =

Z∞

dωe−iωt

X

anω (z)un (~ ρ),

(B.2.1)

n

0

where anω (z) are the mode amplitudes depending on the longitudinal coordinate z, ~r =

9

(x, y, z). Then the correlation function of the stationary wave field has the form [58], Γ (~ ρ1 , z; ρ~2 , z; τ ) = hE ∗ (~ ρ1 , z, t) E (~ ρ2 , z, t + τ )i Z∞ X ha∗nω (z)amω (z)iu∗n (~ ρ1 )um (~ ρ2 ), = dωe−iωτ 0

(B.2.2)

n,m

where h(· · · )i stands for the averaging over the ensemble of random irregularities of medium and realization of the initial field. From (B.2.2) it follows that the behavior of the correlation function along the z−axis depends on the mode correlators of type Pnm (z) = ha∗nω (z)anω (z)i.

(B.2.3)

The quantities Pnm (z) at n = m describe the power of an each mode, while at n 6= m the statistical correlation between different modes. In the absence of inhomogeneities the quantities Pnm (z) vary along z in the following way Pnm (z) = βnmω exp (−i [kn (ω) − km (ω)] z) .

(B.2.4)

In this case the quantities βnmω remain constant at all distances. In the presence of inhomogeneities in a medium the quantities βnmω are no longer constants and depend on the longitudinal coordinate z. Therefore, the problem of the behavior of the correlation function of field is reduced to the study of evolution of the quantities βnmω or Pnm (z) along the longitudinal coordinate z. During the propagation of fields in waveguides the wave encounters a multiple scattering on inhomogeneities of a medium. At large distances even small perturbations of medium may significantly disturb the initial pattern of the field. For the description of such effects it is necessary to go beyond the usual perturbation theory that takes into account only single scattering of waves. There are a large number of works devoted to the theory of statistically irregular waveguides (see, e.g., [62, 63, 64]). The one of the problems in the waveguide theory is the justification of one–dimensional transfer equations of the mode powers Pnn (z). The derivation of the power transfer equations without taking into account the inverse scattering has been carried out in Refs. [65, 66, 67] (see also a review [64]). As was shown in [66] that the power transfer equations can be rigorously justified in the so–called Khasminskii limit [71]: σ → 0, L → ∞, σ 2 L =const, where σ characterizes the intensity of medium fluctuations, L is the distance. 10

The backward scattering significantly restricts the applicability condition of the power transfer equations. The difficulties in the justification of power transfer equations consists in the following: in the statistical problem with two–point boundary conditions to which belong waveguide problems the principle of causality is not satisfied. The absence of the latter does not allow to obtain the closed equations for the second moments of field. These problems are discussed, for example, in [68]. On the other hand, in the derivation of the power transfer equations the non–diagonal in indices quantities Pmn (z) [n 6= m], that are responsible for the interference structure of field in a waveguide, are usually neglected. Generally speaking the neglect of them is not obvious and requires a justification (see [64]). The equations for the second moments with the non-diagonal moments Pmn (z) [n 6= m] are studied in Refs. [54, 55, 56, 57, 58] devoted to the investigation of the mutual coherence function in statistically irregular waveguides. In Refs. [54, 55, 56] the waveguides with a fluctuating impedance of one of walls are considered, while in [57] the waveguide filled with a random inhomogeneous medium is studied. In the last two works the equations for the mutual coherence function has been obtained by the method of local perturbation based on the parabolic equation of quasi-optics. The study of the equations for the mutual coherence function is actually reduced to the study of equations for the moments Pmn (z). In particularly, it was shown that the non–diagonal components Pmn (z) decay exponentially in the process of wave propagation, while the mode powers Pnn (z) satisfy to the linear transfer equations. In [56, 57] the spatial distribution of field intensity at the large distances has been also investigated. It has been shown that the mean field intensity is uniformly distributed over the waveguide cross section, except regions close to the waveguide boundary. Near the boundary with the zero boundary condition the intensity rapidly goes to zero, while near the impedance boundary it grows twice of its mean value. More detailed analysis of this phenomenon and it’s relation with the oscillatory behavior of the intensity has been considered in the above mentioned work by the author [44] (see, also Sec. 3 of Chapter 1). More general type of statistically irregular waveguides have been considered in Ref. [58]. In this work the equation for the moments Pmn (z) have been derived using the equation

11

of interacting modes −i

X dan Vnn′ (z)an′ (z), = kn an + dz ′ n

(B.2.5)

of the Katsenelenbaum’s cross-section method [69] (see also Ref. [70]), with random coefficients of mode couplings Vnn′ (z). By neglecting the backward waves in (B.2.5) and using a diagram technique the equations for Pmn (z) have been obtained. The applied diagram technique is the simplified version the Konstantinov–Perel diagram technique [72] (see, also [73]) employed to derive the kinetic equations in the solid state theory. In contrast to the diagram technique considered in Ref. [25, 62], this diagram technique uses the fact that the coordinate z is preferential. It leads to that the mass operators in the Dyson and the Bete–Salpiter equations are expanded into series of powers of the parameter δ = k0 σl|| , where l|| is the correlation length of medium fluctuations along the z− axis. The application of this technique to the wave propagation in boundless randomly inhomogeneous media is considered in the work by the author [75]. In Ref. [59] the spatial and temporal coherence function of sound field of a point– like source in the underwater sound channel with a statistically rough boundary have been studied by the numerical simulations. It was shown that the coherence radius of the random part of the field decreases with increasing the distance where the multiple scattering of a wave becomes predominant as well as with increasing the frequency of field. It was found that the envelope of the temporal coherence function of the narrow band noise signal in a waveguide has a sequence of maxima whose positions are determined by the distance between the sound source and the observation points. Such a behavior of the temporal coherence function has been early predicted in Refs. [34, 43]. The coherent properties of wave field in a parabolic waveguide with large scale random inhomogeneities have been considered in the work of the author [60]. The quasi-periodical decay law of the coherence radius of field was established. One should note that the formula for the mutual coherence function has been also independently found in Refs. [61, 78]. However, the coherent properties of field have been not studied in these works. The fluctuations of phase and amplitude of field in a waveguide with the random inhomogeneities have been studied in Ref. [85, 84, 86, 87]. In the works [86, 87] where the lens waveguides has been considered it was shown that at large distances the mean square of the fluctuations of amplitude and phase grow linearly with the propagation distance. 12

Similar results were obtained in Refs. [85, 84] where the slab waveguides with random boundaries were considered. The fluctuations of wave intensity in randomly irregular waveguides are studied in Refs. [88, 89, 90, 91, 92, 93]. In all these works the parabolic waveguides were considered. The case of weak wave fluctuations is considered in [88, 91]. The strong fluctuations of wave in waveguides with fluctuating focusing were studied in [89, 92], where it was shown the intensity fluctuations on the waveguide axis grow exponentially with distance, while the mean intensity remains constant. In Ref. [90] the fluctuations of thin laser beam in a waveguide channel with homogeneous fluctuations of media have been studied. The probability distribution functions of intensity fluctuations at the beam center and on the waveguide axis, the distribution of random width of the beam were found. It was shown that all moments of the beam width grow exponentially with a distance. The dispersion of intensity saturates at the beam center and exponentially decays on the waveguide axis. In the author’s work [93] the intensity fluctuations of field in a waveguide channel with random bending of its axis have been studied. It was shown that the beam width grows proportional to z at the large distance. The probability distribution function of intensity fluctuations were found. Particularly, it was shown that the dispersion of intensity on the waveguide axis decays inversely proportional to propagation distance.

3

Wave propagation in a system of irregular waveguides

In spite of that the wave propagation in a system consisting of identical and regularly– packed dielectric waveguides is studied sufficiently well (see, e.g., [5, 94, 96, 98]) to the corresponding problem in systems of waveguides with irregularities practically has not been paid attention. Perhaps, it is related with the complexity of this problem. Indeed, in a real system of waveguides beside inhomogeneities of each waveguides there are also irregularities caused by the fact that waveguides are non–identical and they can be irregularly located. For the first time the effect of these irregularities has been considered in the author’s works [103, 115]. The main result of these works have been the effect: the absence of tunnel spreading of wave beam across the transversal cross section of waveguide system when the level of random deviations of system’s parameters exceeds a certain critical value. 13

The absence of tunnel spreading of wave beam in a system of dielectric waveguides with random parameters is analog of the electron localization in random lattices. The latter has been first considered in the problem of disordered solid state physics [99, 100, 101, 102] (the localizations according to Anderson and Lifshitz).

4

Stochastic instability of rays in waveguides

Under smooth change of inhomogeneities of media over distances of order of the wavelength the wave propagation can be studied in the frame of geometrical optics. The ray statistics in statistically–irregular waveguide channels has been investigated in Refs. [79, 80, 81, 82, 116, 117]. The possibility of stochastic behavior of rays in a waveguide with regularly periodic (non–random !) inhomogeneities has been first shown in the work by the Chigarevs [104]. However, the δ−type periodic inhomogeneities of a medium considered in this work strongly limit the applicability domain of the geometrical optics. More detailed analysis of the nonlinear dynamics of rays has been carried out in the work by G.M. Zaslavsky and the author [113]. As has been shown in this work that the effect of large scale periodic inhomogeneities of media leads to the two principally new effects: modulation localization of rays and their stochastic instability. It has been also shown other mechanism of ray stochasticity in three–dimensional waveguides. The latter is possible even in the absence of inhomogeneities [varying] along the axis of a waveguide channel.

5

Experimental study of the coherence of fields in optical waveguides

To experimental measurements of the coherence of field in optical waveguides were devoted a relatively small number of works [36, 37, 38, 114]. In Refs. [36, 37, 38] the spatial coherence of laser radiation passed through an optical fiber with step–wise profile of refractive index has been measured. In [36, 37] where the fibers of very small length (up 10 cm) was used, it was observed a strong decrease of spatial coherence. In [38] the multimode fibers with the length up to 1 m were used. It was found insignificant decrease of the spatial coherence (up to 0.1). Similar results have been obtained in [114] where a laser radiation is passed through the gradient fibers with the length more than 200 m. In 14

this work the temporal coherence of radiation has been also measured. The dependence of the degree of temporal coherence γ(τ ) on the time delay τ has a series of maxima with amplitude less than one. The latter qualitatively confirms the results of the theoretical works [34, 43]. A small number of experimental works and obtained results is not sufficient to compare them with the results of theoretical works. However, one should note that the assumption made in those experimental works that the spatial coherence of field is independent on the temporal coherence is valid as was shown in the author’s work [43], only at the distances z < zc ∼ |2π/[∆ω(Vn−1 − Vm−1 )], where ∆ω is the spectral width of radiation, Vn is the group velocity of mode. At the distances z > zc the spatial coherence becomes dependent on the temporal coherence of radiation. This fact should be taken into account in the analysis of experimental results. [The content of the thesis] Let us turn to the content of the thesis. The main results of the thesis are given in publications [43, 44, 58, 60, 93, 103, 115, 113]. In the first chapter of the thesis the coherent properties of electromagnetic field in regular dielectric waveguides are studied. In particularly, the general form of the mutual coherent function of the stationary field is established and it’s properties at various limiting cases are studied. It was shown that in the waveguides with dispersion the correlation function of field becomes independent on the longitudinal coordinate z at the distances z ≫ zc , while in the waveguides without dispersion it is a periodic function of z at all distances from the place where a radiation is launched into a waveguide. The problem of spatial oscillations of intensity in multimode waveguides and their relation with spatial coherence of field is also studied. To study this problem the semiclassical method of calculations of the spatial correlation function of field in multimode waveguides is proposed. The semiclassical formulas for the spatial correlation function and the intensity distribution of field in slab waveguides with the arbitrary profile of refractive index are obtained. The wave propagation in randomly irregular waveguides is considered in the second chapter of the thesis. Using the diagram method the equations for the mean field and the second moments are obtained. The problem of correlation between different modes in waveguides with an equidistant location of propagation constants of modes and in the case 15

of waveguides with non-equidistant constants of modes are discussed. The evolution of coherent properties of field propagating in parabolic waveguides is studied. The fluctuation of wave intensity in a waveguide with the random bending of axis is considered. The last two chapters of the thesis are devoted to the description of the fundamentally new effects in the theory of wave propagation in waveguides. In the third chapter the propagation of an electromagnetic wave is a system of large number of coupled dielectric waveguides with random parameters is studied. The analogy between the wave propagation in such a system and a particle behavior in a disordered system of potential wells is described. Based on this analogy it is shown that when the width of random deviations of waveguide parameters exceeds a certain critical value it occurs a tunnel localization of a wave in the cross section transversal to the wave propagation direction. The estimation is given for the width of random deviations of waveguide diameters in the system of densely packed circular dielectric fibers that leads to the wave localization. The problem of diffusional spreading of wave beam in the system of waveguides in the presence of random inhomogeneities varying along its propagation axis is also studied. In the fourth chapter of the thesis the problem of stochastic instability of rays is discussed. There are two fundamental new effects are found in waveguides with the periodic inhomogeneities along its axis which are due to specific nonlinear interactions of rays with periodic inhomogeneities of medium. The first of them is the modulation localization of rays which is the analog of the nonlinear resonance in classical mechanics. The second effect consists of the formation of stochastic area of waveguide channel due to stochastic instability of rays. The phenomenon leads to the decrease of the effective size of the transversal width of waveguide. The areas of modulation localization of rays and the width of a stochastic layer are estimated for the waveguide with the soliton-like profile of refractive index. In three-dimensional waveguides another mechanism of ray stochasticity is considered which occurs even in the absence of inhomogeneities [varying] along the wave propagation axis. This mechanism of ray stochasticity is related with the destruction of one of the integrals of motion. The rules of a diagram technique to derive the equations of mean field and its second moments are described in Appendices. The long mathematical calculations are also given there. The numbering of formulas is carried out by three numbers separated by dots. The 16

first number indicates a chapter number, the second one corresponds to a section number, and the third one − a formula number.

17

Chapter 1

Coherence properties of a electromagnetic field in dielectric waveguides 1

Correlation functions of a electromagnetic field in dielectric waveguides

The electromagnetic field is described by real–value vector functions: electric vector ~ (r) (~r, t). To study the coherent properties of field it is E~ (r) (~r, t) and magnetic vector H convenient to work not with real–value functions but with the complex analytical signals through which it is simple to express observable quantities [19]. The analytical signal V (t) of the real-value function V (r) (t) is defined as the positive–frequency part of V (r) (t), V (t) =

Z∞ 0

where 1 Ve (ω) = 2π

dωe−iωt Ve (ω),

Z∞

dteiωt V (r) (t).

0

The real–valued field V (r) (t) is fully determined by the analytical signal, V (r) (t) = 2ReV (t). ~ r, t) and H(~ ~ r, t) be the complex analytical signals of electromagnetic field. We Let E(~ 18

introduce the correlation function matrix   EE EH Γ Γ ˆ= , Γ ΓHE ΓHH

(1.1.1)

where ΓAB , (A, B = E, H) is the 3×3 matrix whose elements are the correlation functions of type 



ΓEE ΓEH



ij

≡ ΓEE (~r1 , t1 ; ~r2 , t2 ) = hEi∗ (~r1 , t1 ) Ej (~r2 , t2 )i,

ij

≡ ΓEH (~r1 , t1 ; ~r2 , t2 ) = hEi∗ (~r1 , t1 ) Hj (~r2 , t2 )i.



(1.1.2)

From the elements of the correlation matrix one can make the quantities which describe the coherent properties of field as well as related to the physically observable quantities. For example, the scalar quantity obtained from the trace of the matrix (1.1.1) coincides with the mean density of field energy at the identical arguments ~r1 = ~r2 , t1 = t2 . In waveguides there is special interest to the vector correlation function ~Γ(I) (~r1 , t1 ; ~r2 , t2 ) defined by the expression ~Γ(I) (~r1 , t1 ; ~r2 , t2 ) = c hE~ ∗ (~r1 , t1 ) × H ~ (~r2 , t2 )i, 2π or ~Γ(I) (~r1 , t1 ; ~r2 , t2 ) = c εijk ΓEH r1 , t1 ; ~r2 , t2 ) , jk (~ 2π

(1.1.3)

At the identical arguments ~r1 = ~r2 , t1 = t2 , the quantity ~Γ(I) coincides with the mean value of the Poynting vector. Here εijk is the three-dimensional antisymmetric unit tensor. In (1.1.3) it is assumed summation over repeating indices. The characteristic feature of waveguides is the existence of the preferential direction parallel to the waveguide axis. The properties of medium in a regular waveguide do not change along its axis. We assume that the coordinate z coincide with the waveguide axis. The arbitrary random electromagnetic field in a waveguide can be presented as a superposition of fields of normal modes, E~ (~r, t) =

Z∞

dωe

0

~ (~r, t) = H

Z∞ 0

−iωt

Xh n

dωe−iωt

i ∗ ~ ~ anω Enω (~r) + bnω Enω (~r) ,

Xh n

i ∗ ~ nω (~r) + bnω H ~ nω anω H (~r) ,

19

(1.1.4)

~ nω (~r) and H ~ nω (~r) with random expansion coefficients anω and bnω . The vector functions E describe the field of the n−th normal mode at the frequency ω. In general the field components are related to each other, and they can be presented in the form ~ nω (~r) = ~unω (~ E ρ) exp (ikn (ω)z) , ~ nω (~r) = ~vnω (~ H ρ) exp (ikn (ω)z) ,

(1.1.5)

where ~unω (~ ρ) and ~vnω (~ ρ) describe the spatial distribution of field of the n−th mode in the transversal cross section of the waveguide, and ρ~ = (x, y) are the transversal coordinates. The propagation constants kn (ω) of waveguide modes take discrete value, while the ones of radiative modes take continuous values. The different modes satisfy the following orto–normality condition [2], c 2π

Z∞Z

∗ ~ nω ~ nω d2 ρ = δmn . E ×H

(1.1.6)

−∞

The condition (1.1.6) means the power passed along the z− axis through the transversal cross section of waveguide is normalized to unity. One should note that the first terms in the expansions (1.1.4) correspond to the modes propagating in the positive direction of the axis z, while the second terms correspond to backward waves. The expansions (1.1.4) allows one to express the correlation function through the second moments of random coefficients anω and bnω : ha∗nω anω i, ha∗nω bnω i, and so on. The latter remain unchanged during wave propagation in a regular waveguide and depend on the coherent properties of radiation source that excites a waveguide. We establish the relation between the correlation function of field in waveguide and the corresponding function of radiation source. Suppose that at the plane z = 0 the waveguide ~ (b) (~r, t) characterized is illuminated by the random electromagnetic field E~ (b) (~r, t) and H by the correlation function of type, (b)

(b)∗

Γij (~r1 , t1 ; ~r2 , t2 ) = hEi

(b)

(~r1 , t1 ) Ej (~r2 , t2 )i.

(1.1.7)

In waveguides the normal modes propagating in the positive direction of z are excited. The stochastic coefficients anω can be determined from the continuity condition of transversal

20

components of the electromagnetic field at the plane z = 0: X (b) ~ nωi (~ ρ, ω) = anω E Eei (~ ρ, 0), n

e (b) H i

(~ ρ, ω) =

X

~ nωi (~ anω H ρ, 0),

i = x, y,

n

(b) e (b) (~ ρ, ω) are the Fourier transforms the electromagnetic field of the ρ, ω) and H where Eei (~ i

external field at the plane z = 0. Note, that in the continuity conditions the reflected waves from the plane z = 0 were not taken into account since they practically does not affect on the coherence of field in a waveguide. Using the relation (1.1.6), we obtain anω

c = 4π

Z∞Z 

−∞

e (b) ~ ∗ E~ × H

nω



d2 ρ. z

The latter allows one to obtain the following expression for the correlation function of field, ΓEE r1 , t1 ; ~r2 , t2 ) = ij (~

Z∞

dω1

0

Z∞

dω2

X nm

0

ha∗nω1 amω2 i

∗ × Enω (~r1 )Emω2 j (~r2 ) exp [−i(ω2 t2 − ω1 t1 )] , 1i

(1.1.8)

where ha∗nω1 amω2 i

Z∞Z  c 2 Z∞Z 2 d2 ρ2 ε3ij ε3ls d ρ1 = 4π −∞

−∞

(b)∗ (b) ~ nω1 i (~ ~ mω2 s (~ ×H ρ1 , 0)H ρ2 , 0)hEei (~ ρ1 , ω1 )Eej (~ ρ2 , ω2 )i.

(1.1.9)

In (1.1.9) it is assumed a summation over repeating indices. The quantities with the angular brackets in the right hand side of (1.1.9) is the two-dimensional Fourier transform of the correlation function of the external field (1.1.7),  2 Z∞ Z∞ 1 (b) (b)∗ (b) dt1 dt2 ei(ω1 t1 −ω2 t2 ) Γij (~ ρ1 , 0, t1 ; ρ~2 , 0, t2 ) . hEei (~ ρ1 , ω1 )Eej (~ ρ2 , ω2 )i = 2π 0

0

For the correlation functions ΓHE , ΓEH , and ΓHH one can obtain the expressions similar

to (1.1.8). Furthermore, we suppose that the external field is stationary and spectrally pure, i.e., (b)

(s)

Γij (~r1 , t1 ; ~r2 , t2 ) = Γij (~r1 , ~r2 ) Γt (t2 − t1 ) . 21

(s)

where Γij (~r1 , ~r2 ) is the spatial correlation function and Γt (t2 − t1 ) is the temporal correlation function of the external field. The temporal correlation function Γt (t2 − t1 ) is determined by the spectral density G(ω) of the external field, Γt (τ ) =

Z∞

dω exp(−ωτ )G(ω).

0

The expression (1.1.9) takes the form ha∗nω1 amω2 i = δ(ω1 − ω2 )G(ω1 )βnmω1 , where the quantity βnmω is determined by the correlation function of the external field βnmω

Z∞Z  c 2 Z∞Z d 2 ρ1 d2 ρ2 ε3ij ε3ls = 4π −∞

−∞

(s) ~ nω1 i (~ ~ mω2 s (~ ×H ρ1 , 0)H ρ2 , 0)Γil (~ ρ1 , 0, t1 ; ρ~2 , 0, t2 ) .

(1.1.10)

Then the correlation function of electromagnetic field takes more simple form, ΓEE ij

(~r1 , t1 ; ~r2 , t2 ) =

Z∞

dωG(ω) exp [−iω(t2 − t1 )]

0

X

∗ βnmω Enωi (~r1 )Emωj (~r2 ).

(1.1.11)

nm

ˆ One can show Similar expressions can be obtained also for the other components of Γ. ˆ that the quantities βnmω will be the same for all components of the matrix Γ. One should note that the general form (1.1.11) of the correlation function of field has been first established in the author’s work [43]. In contrast to the corresponding formula obtained in Refs. [34, 35] Eq. (1.1.11) is more general and it describes the correlation properties of field in waveguides with the arbitrary profile of refractive index. In the conclusion of this section we establish the general form the correlation function (1.1.3) ΓI (~r1 , t1 ; ~r2 , t2 ). For simplicity we consider the case when only the TE– and the TM– modes propagate in the waveguide. Using the relations km Emωx (~r), k0 km Emωx (~r) = 2 Hmωy (~r), n k0

Hmωy (~r) =

km Emωy (~r2 ), k0 km Emωy (~r) = − 2 Emωx (~r2 ), n k0 Hmωx (~r) = −

22

TE–modes, TM–modes,

that follows from the Maxwell equation 1 , we obtain (I) Γ3

(~r1 , t1 ; ~r2 , t2 ) = (

Z∞

dωG(ω) exp [−iω(t2 − t1 )]

0

c 4πk0

X kn + km  ∗  ∗ βnmω Enωx (~r1 )Emωx (~r2 ) + Enωy (~r1 )Emωy (~r2 ) 2 nm,T E ) X kn + km  ∗  ∗ β Hnωx (~r1 )Hmωx (~r2 ) + Hnωy (~r1 )Hmωy (~r2 ) , (1.1.12) + 2 (ρ) nmω 2n nm,T M ×

where n(~ ρ) is the refractive index as a function of the transversal coordinate ρ~. In the first sum the summation is carried out over the TE–modes, while in the second sum −− over (I)

the TM–modes. At the coinciding arguments ~r1 = ~r2 , t1 = t2 , the function Γ3 coincides with the mean intensity of field at the point ~r, (I)

I (~r) = Γ3 (~r, t; ~r, t) . The quantity I (~r) as a function of ρ~ at the fixed z describes the spatial distribution of field intensity in the transversal cross section of waveguide.

2

Coherence properties of in regular waveguides

Now we turn to the analysis of the coherent properties of electromagnetic field. We will not consider the polarization properties of field, and therefore we keep in (1.1.11) only the diagonal component of the correlation functions. Using (1.1.5), we have Γ(~ ρ1 , z1 ; ρ~2 , z2 , τ ) =

Z∞

dωG(ω)e−iωτ

X

βnmω

n,m

0

× u∗nω (~ ρ1 )umω (~ ρ2 ) exp {i [km (ω)z2 − kn (ω)z1 ]} .

(1.2.1)

Consider the case when the waveguide is illuminated by spatially coherent light with the correlation function Γs (r1 , r2 ) = U ∗ (r1 )U (r2 ). In this case the quantities βnmω are factorized in indices n, m, and (1.2.1) takes the form Γ(~ ρ1 , z1 ; ρ~2 , z2 , τ ) =

Z∞

dωG(ω)e−iωτ Vω∗ (~ ρ1 , z1 )Vω (~ ρ2 , z2 ),

(1.2.2)

0

1 Furthermore,

we notate k0 for the wavenumber in a vacuum, k0 = ω/c, that should be distinguished from the propagation

constants of modes kn .

23

where Vω (~ ρ, z) =

X

bnω unω (~ ρ) exp [ikn (ω)z] ,

n

βnmω =

b∗nω bmω .

From (1.2.2) it follows that due to spectral dependence of mode parameters the correlation function cannot be presented as a product of the spatial correlation functions to the temporal one, i.e., in general the field in the waveguide is not a spectrally pure. The spatial coherence is conserved only in the case of a monochromatic light beam G(ω) = Aδ(ω − ω0 ). Thus, the spatial coherence of field in the waveguide as in the case of boundless space [52] depends on the spectral composition of radiation. We shall now analyze the case when the light beam is spatially non-coherent: Γs (ρ1 , r2 ) = J(r1 )δ(r1 − r2 ). In spite of this, the correlation function of the field inside the waveguide (1.2.1) will be nonzero for non–equal values of the transversal coordinates ρ~1 and ρ~2 . Consequently, the field in the waveguide produced by spatially incoherent light becomes partially coherent. This is related with the fact that the waveguide is excited only the part of angular spectrum of the incident radiation, γ e(~k1 , ~k2 ) =

Z∞Z

−∞

d 2 ρ1

Z∞Z

−∞

h i d2 ρ2 Γ(s) (~ ρ1 , ρ~2 ) exp −i(~k1 ρ~1 , +~k2 ρ~2 ) ,

limited by the inequalities, |~ki | ≤ k0 n(ρ) sin θc ,

(i = 1, 2),
θc = arccos knmin /k0 n(ρ) ,

(see [29, 30, 31]). The source of light with the effective angular spectrum   γ e(~k1 , ~k2 ) if |~ki | ≤ k0 n(ρ) sin θc , ~ ~ γ eef f (k1 , k2 ) =  0, if |~ki | > k0 n(ρ) sin θc ,

creates the field with a higher coherence than the field from the initial source of light. Therefore, the field created in a waveguide has the degree of coherence not less than the one of the source of light. We study the behavior of the correlation function at the different distances from the area where the radiation is coupled into the waveguide. Let the light beam has the 24

spectral density G(ω) with a narrow line of width ∆ω adjacent to the central frequency ω0 , (∆ω ≪ ω0 ). The values of unω (~ ρ) and βnmω vary slowly in the frequency interval (ω0 − ∆ω, ω0 + ∆ω) compared with the exponential factors in (1.2.1). Expanding the values of kn (ω) in powers of (ω − ω0 ) and retaining only two terms kn (ω) = k¯n + where k¯n = kn (ω0 ) and Vn−1 = (∂kn /∂ω)

ω=ω0

ω − ω0 , Vn

is the group propagation velocity of a wave

of mode n, we reduce Eq. (1.2.1) to the following form X 
 Γp (~ ρ1 , z, ; ρ~2 , z, τ ) = βnmω0 u∗nω0 (~ ρ1 )umω0 (~ ρ2 ) exp i k¯m − k¯n z − iω0 τ ×

n,m Z∞

dωG(ω) exp [i(ω − ω0 ) (τ − τnm (z))] ,

(1.2.3)

0

where



 z z τnm (z) = − Vn Vm is the time lag between the modes n and m at the distance z. The correlation function (1.2.3) depends on the integrals Cnm (z, τ ) =

Z∞

G(ω + µ)e−iµ(τ −τnm (z)) dµ,

−ω0

where µ = ω − ω0 . Consider separately the cases of waveguides with dispersion and without dispersion. In waveguides without dispersion the group velocities of modes are the same. This occurs in ideally focusing waveguides (for example, in the waveguide with the refractive index profile n(x) = 1/ cosh x). In parabolic waveguides only lower modes whose propagation constants linearly depend on the mode number have the same group velocities. In this case the quantities Cnm (z, τ ) do not depend on z since τnm (z) = 0. Thus in waveguides without dispersion the correlation function is a periodic function of the longitudinal coordinate z at all distances z. In waveguides with dispersion the group velocities of modes take different values. In this case the main contribution to the integral Cnm (z, τ ) is made by those values of τ and z for which the following inequality is satisfied |τ − τnm (z)| ≤ 25

2π = τ0 , ∆ω

where τ0 is the coherence time of the light beam from the external source. When τ = 0, the values of Cnm (z, 0) (n 6= m) will be nonzero at values of z < zcr ∼ 2π/∆ω(1/Vn − 1/Vm ). When z > zcr the values of Cnm (z, 0) will be close to zero and will tend to zero when z ≫ zcr . Thus the spatial correlation function Γ(~ ρ1 , z; ρ~2 , z, 0) at distances z < zcr is a [quasi-]periodic function of z with the [characteristic] period L ∼ 2π//(k¯n − km ), and at distances z ≫ zcr does not depend on z: Γ(~ ρ1 , z; ρ~2 , z; 0) = W

X

βnmω0 u∗nω0 (~ ρ1 )umω0 (~ ρ2 ),

(1.2.4)

n

where W =

Z

∞

dωG(ω). 0

The tendency of the spatial coherence to a constant value is associated with that at distances z ≫ zcr time lag between modes τnm exceeds the coherence time τ0 = 2π/∆ω, and thus different modes are not interfered. In this case the spatial coherence function and the intensity of the full field are formed by incoherent superposition of different modes. This phenomenon clearly shows the dependence of the spatial coherence on spectral composition of radiation. For the illustration the dependence of the degree of the spatial coherence function of field γ(x, 0; z) on the transversal coordinate x is plotted in Fig. 2.2 at the different distances z in the slab waveguide with the rectangular profile of refractive index (Fig. 2.1).

Figure 2.1:

We shall now investigate the temporal coherence function Γ(~ ρ, z, τ ) = Γp (~ ρ1 , z, ; ρ~2 , z, τ ). In waveguides without dispersion a coherence time as well as the temporal correlation function are [quasi-] periodic functions of z. Consider the waveguides with dispersion in detail. At values of z for which the time lags τnm (z) are considerably shorter than the 26

Figure 2.2: Dependence of the degree of spatial coherence γp (~ ρ, 0) on x for z ≪ zcr (curve 1) and at z ≫ zcr (curve 2). The waveguide parameters are: 2a = 10−3 µm, refractive indices n20 = 1.5, n21 = 1.41, radiation parameters: ω0 = 3.02 × 1015 Hz, ∆ω = 1010 Hz.

coherence time τ0 , the degree of temporal coherence of radiation is equal to the degree of temporal coherence of radiation that excites the waveguide. Indeed, from the condition τnm ≪ τ0 , it follows that Cnm (z, τ ) = eiω0 τ Γt (τ ), where Γt (τ ) is the temporal coherence of the incident light. It follows from Eq. (1.2.4) that γ(~ ρ, z, τ ) = Γ(~ ρ, z, τ )/Γ(~ ρ, z, 0) = Γt (τ )/Γt (0) = γt (τ ). Thus, the temporal coherence of the radiation penetrating the waveguide is conserved at distances z ≪ zcr . At large distances z ≫ zcr , the correlation function Γ(~ ρ, z, τ ) has a principal maximum at τ = 0 and a number of subsidiary maxima at values of τ = τnm . The amplitudes of subsidiary maxima are proportional to the coefficients of mode coupling βnmω0 , (n 6= m). They depend on the transversal coordinate ρ~ due the spatial dependence of the product u∗nω0 (~ ρ)umω0 (~ ρ).

Now we investigate the behavior of the effective coherence time at the differnt distance z. According to Born and Wolf [19], the coherence time of the field τ1 at the point ~r is determined by τ12 =

Z∞

τ 2 |Γ(~r; τ )|

2

, Z∞

|Γ(~r; τ )|2 ,

(1.2.5)

−∞

−∞

i.e., as the effective width of the dependence of Γ(~r; τ ) on τ . The appearance of subsidiary maxima of the temporal correlation function with increase z leads to the growth of τ1 27

with z. We determine a growth law of τ1 . Substituting Eq. (1.2.3) into Eq. (1.2.5) and performing simple calculations, we obtain τ12 (z) = (B + F z 2 )/D,

(1.2.6)

where B=

×

XX

n,m m,m′  Z∞

−ω0

An,m A∗n′ ,m′ exp [i (∆kn,m − ∆kn′ ,m′ ) z]

  o µ µ ∂2 n  dµ − 2 G ω0 + − ξ G ω0 + + ξ ∂ξ 2 2

F =

ξ=0

XX n,m

An,m A∗n′ ,m′

n′ ,m′

exp [i (∆kn,m − ∆km,m′ ) z] ×

Z∞

−ω0

D=

XX

n,m m,m′



1 1 1 1 − + − Vn Vm Vn′ V m′

Z∞

−ω0

 h µ  2 µi dµ G ω0 + dµ, exp i (τn,m − τn′ ,m′ ) 2 2

 h µ  2 µi ′ ′ dµ G ω0 + dµ, exp i (τ − τ ) n,m n ,m 2 2

At z ≪ zc , we have Z∞   i 2 h  η η ∂ τ12 (z) = − dη 2 G ω0 − − ξ G ω0 − + ξ ∂ξ 2 2 −ω0

×

−ω0

i.e.,

τ12

−1

 η  2  dη G ω0 + 2

=

(1.2.7)

∆kn,m = k¯n − k¯m .

An,m = βnmω0 u∗nω0 (~ ρ)umω0 (~ ρ),

Z∞



An,m A∗n′ ,m′ exp [i (∆kn,m − ∆kn′ ,m′ ) z] ×



h µi , exp i (τn,m − τn′ ,m′ ) 2

Z∞

−∞

ξ=0



τ 2 |Γt (τ )|2 dτ 

Z∞

−∞

−1

|Γt (τ )|2 dτ 

= τ12 (0),

coincides with the coherence time of the light from the external source. At large

distances z ≫ zcr , we have τ12 (z) = τ02 + M z 2 , where M =4

X n,m

|An,m |2



1 1 − Vn Vm

2 , 28

  2 X X  Ann + |An,m |2  . n n,m

(1.2.8)

Figure 2.3: Dependences of γ(z, τ ) on τ for various values of z.

Consequently, the effective coherence time of the light grows with increasing distance z quadratically. It is interesting to compare τ1 defined by the formula (1.2.5) with the coherence time defined according to Mandel [32]:

τ2 =

Z∞

|γ(~ ρ, z, τ )|2 dτ =

−∞

R∞

−∞

|Γ(~ ρ, z, τ )|2 dτ |Γ(~ ρ, z, 0)|2

,

(1.2.9)

which determines the area of under the curve |γ(~ ρ, z, τ )|2 versus τ . It is easily shown, that D , τ2 = P | n An,n |2

(1.2.10)

where D is given by (1.2.7). At z ≫ zc the coherence time τ2 goes to the constant value  2  Z∞ . X X |γt (τ )|2 dτ. (1.2.11) An,n  |An,m |2 τ2 =  1 + n n,m

−∞

It means that the area under the curve |γ(~ ρ, z, τ )|2 versus τ remains constant, although

positions of subsidiary maxima moves out from the center τ = 0 with increasing the distance z (see Fig. 2.3). The coherence times τ1 and τ2 behave differently only in the presence of non zero correlation between different modes, described by the quantities βnmω , (n 6= m). In the 29

absence of the correlation between different modes (βnmω = 0) the temporal correlation function and correspondingly the coherence time remain unchanged. In conclusion we briefly discuss the form of the spectral density of field in a waveguide. The enhancement of the effective coherence time τ1 is associated with a narrowing of the effective width of spectral line of the radiation at the distances z ≫ zc ,
(∆ωc )2 = 2π/τ12 = 2π/ τ12 (0) + M z 2 .

The latter means that the form of the effective spectral density narrows. According to Eq. (1.2.3) the spectral density G(ω; ρ~, z) of the light field in waveguide is given by G(ω, ρ~, z) = G(ω)

X n,m

ρ) cos [∆knm z − (ω − ω0 )τnm (z)] . ρ)umω0 (~ βnmω0 u∗nω0 (~

(1.2.12)

From (1.2.12) it follows that the decrease of spectral line is related with a modulation along z. The modulation of the spectrum of the radiation inside the waveguide when z ≫ zcr is explained by the Alford and Gold effect, [47, 46] (see also [34]), since the field at the point z is formed by the superposition of the fields of the discrete waveguide modes with the relative delays τn,m (z), which considerably exceed the coherence time τ0 of the radiation incident on the waveguide. One should note the modulation of spectrum occurs only in the presence of the correlation of modes.

3

Spatial coherence of field and spatial oscillations of intensity in multimode waveguides

When all modes of a multimode waveguide are excited equally, it is known that the spatial intensity distribution over the waveguide cross section I(ρ), (~ ρ = x, y), calculated in the geometric-optics approximation, describes the waveguide refractive index profile n(~ ρ) (see e.g., a review [51]). In circular waveguides this distribution has the form

and in plane waveguides

  I(ρ) ∼ n2 (p) − n2∞ ,  1/2 I(x) ∼ n2 (x) − n2∞ ,

where n∞ is the refraction index of the waveguide cladding n∞ = limρ→∞ n(ρ).

30

(1.3.1)

(1.3.2)

n(x) no

n+ nx

Figure 3.1:

Nevertheless, as numerical calculations have shown [45], the intensity distribution I(ρ) for incoherent excitation of a multimode waveguide has a more detailed structure than (1.3.1) or (1.3.2). Specifically, I(ρ) as a function of the perpendicular coordinate ρ oscillates around the smooth distribution (1.3.1) or (1.3.2). The latter describe only the average behavior of the intensity I(ρ). It is not necessary to carry out exact calculations in order to explain the spatial intensity oscillations. As will be shown below they may be explained within the bounds of the quasiclassical approximation. It turns out that the spatial intensity oscillations are closely connected with the spatial coherence of the waveguide field. To study this last effect a calculation was made of the spatial coherence function (I)

Γ3 (~ ρ1 , z; ρ~2 , z; 0) (1.1.12). For identical arguments ρ~1 = ρ~2 = ρ~, the correlation function (I)

(I)

Γ3 (~ ρ1 , z; ρ~2 , z; 0) describes the intensity distribution: I(~ ρ, z) = Γ3 (~ ρ, z; ρ~, z; 0). In the following we consider plane waveguides with a smooth variation of the index of refraction n(x) along the transverse coordinate x. The typical dependence of n(x) on x is shown in Fig. 3.1. Generalization of the results to the case of circular waveguides presents no difficulty. (I)

The general form of the correlation function Γ3 (~ ρ1 , z; ρ~2 , z; 0) was established in Section 1. For simplicity, taking into account only the TE-modes for which Ex = 0 (the medium is assumed to be homogeneous along the longitudinal coordinate y), we write

31

down (1.1.12) at t1 = t2 in the form Γ(x1 , x2 ; z) ≡

(I) Γ3 (x1 , z; x2 , z; 0)

X kn + km 2

n,m

c = 4πk0

Z∞

dωG(ω)

0

βnmω u∗nω (x1 )umω (x2 ) exp (i∆knm z) .

(1.3.3)

The orto-normality conditions of the wave functions unω (x) (1.1.6) can be written in the form: ckn 4πk0

Z∞

u∗nω (x)umω (x)dx = δnm ,

(1.3.4)

−∞

The quantities βnmω are determined by Eq. (1.1.10) which in this case has the following form βnmω ∼

Z∞

−∞

dx1

Z∞

dx2 u∗nω (x1 )umω (x2 )Γ(S) (x1 , x2 ).

(1.3.5)

−∞

In general, a direct calculation of (1.3.3) is a difficult problem. In the case of interest to us, however, it is possible to obtain closed analytical expressions for the spatial correlation function in the quasiclassical approximation. For excitation of the waveguide with a spatially incoherent field Γ(S) (x, x′ ) = J(x)δ(x− x′ ), the coefficients βnmω = 0, (n 6= m). Then for a quasi-monochromatic radiation source with frequency ω0 , (1.3.3) takes the form Γ(x1 , x2 ; z) =

c X kn βn u∗nω (x1 )unω (x2 ), 4πk0 n

(1.3.6)

where βn = βnnω . The multimode nature of the waveguide allows us to use the quasiclassical expressions for the wave functions un (x). The fundamental condition tor the applicability of the quasiclassical approximation in an inhomogeneous medium is the refractive index n(x) vary smoothly over distances comparable with the wavelength of the field λ = 2π/k0 . In a waveguide, which is an example of an inhomogeneous medium, the parameter that describes the degree to which the quasiclassical approximation is valid is the quantity ξ = (k0 apF )−1 , where a is the effective width of the waveguide, pF = [n2 (x) − n2∞ ]

1/2

, and the quantity n∞ is the larger

of the limiting values of n(x) as x → ±∞: n∞ = max {n+ , n− },

n± = lim n(x). x→±∞

32

The quasiclassical region is determined by the condition ξ ≪ 1. The condition ξ = 0 corresponds to geometric optics. Let us consider the dependence of the spatial correlation function (1.3.6) and the intensity I(x) = Γ(x, x) on the parameter ξ. Here it should be noted that the problem under consideration is analogous to the problem of oscillation effects in the density of matter in atoms and nuclei. Calculation of the intensity I(x) corresponds to the calculation of the spatial distribution of the electron density in atoms and the nucleon density in nuclei [50, 48, 49]. From the theory of oscillation effects (see, e.g., Ref. [48]), it follows that the intensity consists of regular Ireg (x) and oscillatory Iosc (x) parts. The regular part is an analytic function of ξ at ξ = 0, while Iosc (x) is not analytic there. The latter has the form of the combination of trigonometric functions of the type sin [α(x)ξ −1 + β(x)] with power series in ξ,  X  X Iosc (x) = sin α(x)ξ −1 + β(x) An ξ n + cos α(x)ξ −1 + β(x) Bn ξ n , n

(1.3.7)

n

where α(x) and β(x) are some functions of the coordinate x. That is, this part of the intensity describes the spatial oscillations in intensity with variation of the transverse coordinate x. In distinction from I(x), the spatial correlation function (1.3.6) does not have a part analytic in ξ at the point ξ = 0. Its series expansion in ξ has the form of (1.3.7). Since I(x) is obtained from Γ(x, x′ ) by matching of the coordinates x and x′ , we may say that the oscillation effects are directly connected with the spatial coherence of the field and are nonremovable. The fact that the correlation function is not analytic at ξ = 0 indicates the impossibility of describing the coherence properties of the field within the bounds of geometric optics. In this way, for calculating the spatial correlation function (1.3.6) in the quasiclassical approximation ξ = 0 it is sufficient to extract from the asymptotic form of Γ(x1 , x2 ) the principal terms in ξ. In the region of classical motion the quasiclassical wave function un (x) for the n−th mode in a plane waveguide with arbitrary profile of refractive index has the form  Z Rn  π −1/2 ′ ′ un (x) = cn [pn (x)] cos k0 pn (x )dx − , (1.3.8) 4 x 33

where pn (x) = [n2 (x) − kn2 /k02 ] Z Rn σn (x) = pn (x′ )dx′ ,

1/2

′ Rn

x

Ln (x) =

Z

Rn

′

[pn (x )]

−1

. Let us introduce the notations Z Rn 0 pn (x′ )dx′ , σn =

′

L0n

dx ,

x

=

Z

Rn

[pn (x′ )]

−1

dx′ ,

′ Rn

ǫn = kn2 /k02 ,

where Rn′ and Rn are respectively, the left and right turning points of the classical motion. The normalization constants cn determined by 1.3.4), are equal to cn =

4π k0 . c kn L0n

(1.3.9)

The propagation constants kn of the waveguide modes take on values in the interval [k0 n∞ , k0 n0 ] and are determined by the quantization rule   1 0 , n = 1, 2, . . . . k0 σn = π n − 2

(1.3.10)

Using (1.3.8) and (1.3.9), we can write (1.3.6) in the form Γ12 ≡Γ(x1 , x2 ) =

X n

βn L0m (pn1 pn2 )1/2

× {cos [k0 (σn1 − σn2 )] + sin [k0 (σn1 + σn2 )]} ,

(1.3.11)

where σni ≡ σn (xi ), pni ≡ pn (xi ), Lni ≡ Ln (xi ), (i = 1, 2). First let us consider the case where the distribution of power in the modes is described by a step function β(ǫ) ≡ βn =

  b,

if ǫF < ǫ < n20 ,

 0,

(1.3.12)

if n2∞ < ǫ < ǫF .

 1/2 In this case the quasiclassical parameter is ξ = 1/ k0 a(n( x) − ǫF ) . The calculation

of (1.3.11) is carried out by the method of Poisson summation (see Appendix 1). The first term in the expansion of the spatial correlation function in the parameter ξ is equal to (0)

(1)

(2)

Γ12 = Γ12 + Γ12 + Γ12 ,

(1.3.13)

where (0)

Γ12 =

b 1 sin [k0 (σF 1 − σF 2 )] , 1/2 4π (pF 1 pF 2 ) LF 1 − LF 2 34

(1.3.14)

(1) Γ12

=

(0) −Γ12

(2) Γ12

  ∗ sin k0 (σF 1 − σF 2 ) − (LF 1 − LF 2 ) [k0 σF0 ] /L0F + , (1.3.15) sin [(LF 1 − LF 2 )π/2L0F ] 8L0F (pF 1 pF 2 )1/2 b

  ∗ sin k0 (σF 1 + σF 2 ) − (LF 1 + LF 2 ) [k0 σF0 ] /L0F . =− sin [(LF 1 + LF 2 )π/2L0F ] 8L0F (pF 1 pF 2 )1/2 b

(1.3.16)

All quantities in (1.3.13) to (1.3.16) with subscript F are evaluated at ǫ = ǫF . The ∗

quantity [k0 σF0 ] is a periodic function of the argument k0 σF0 : 

k0 σF0

∗

= k0 σF0 − mπ, if (m − 1/2)π ≤ k0 σF0 ≤ (m + 1/2)π,  ∗ − π/2 ≤ k0 σF0 ≤ π/2.

For ξ ≪ 1 the expression (1.3.13) gives the main contribution to the spatial correlation function (1.3.6). From (1.3.14) to (1.3.16) it follows that Γ12 is a rapidly oscillating function of the transverse coordinates x1 and x2 . For x1 close to x2 , the principal contribution (0)

(1)

(2)

to (1.3.13) comes from the term Γ12 , which exceeds the other terms Γ12 and Γ12 by a factor of order ξ −1 . The field coherence radius ρc may be determined as the minimum (0)

distance between x1 and x2 for which Γ12 vanishes: k0 |σF (x + ρc ) − σF (x)| = π, or Z k0

x+ρc x



2

′

n (x ) − ǫF

1/2

dx = π.

(1.3.17)

Because of the slow variation of the index of refraction n(x) over an interval of order of the coherence radius ρc , we may obtain the following estimate ρc = πξa.

(1.3.18)

Thus the coherence radius of the field in the waveguide is considerably small in comparison of its effective width a. Because of the dependence of the parameter ξ on the transverse coordinate x, ρc also depends on x. We observe that (1.3.13) does not describe the true behavior of the correlation function at points where the quasiclassical approximation ξ ≪ 1 breaks down, in particular at the turning points where n2 (x) = ǫF . The coherence

radius ρc takes on its minimum value on the waveguide axis x = 0. For ǫF = n2∞ , that is, when an the waveguide modes are excited, the minimum value of ρc is equal to = πa/V, ρmin c 35

(1.3.19)

where V = k0 a(n20 − n2∞ )1/2 is the dimensionless waveguide frequency. Since the number of modes N ≈ V /π, then

ρmin = a/N . c

As an example, let us consider a waveguide with a quadratic variation of the refraction index n2 (x) =

  n2 (1 − ∆x2 /a2 ) , 0

 n2 = n2 (1 − ∆) , ∞ 0

for |x| < a,

(1.3.20)

for |x| > a.

In this case we may obtain the following expression for Γ(x, 0): −1/4  b x2 (0) (1) Γ (x, 0) + Γ (x, 0) = 1− 2 4π a ( "   1/2 #)   V x2 x 1 x x × sin 1− 2 , sin arcsin arcsin − 2 a a a 2 a −1/4  b x2 (2) Γ (x, 0) = − 1− 2 4π a ( " #) 1/2    V x x x2 x 1 × cos −π sec arcsin − 1− 2 arcsin . (1.3.21) 2 a a a 2 a

i∗ h √ For simplicity, we have set k0 σn0 2∞ = 0 in (1.3.21). This is valid if V = k0 n0 a ∆ is an

integer. The dependence of the correlation function Γ(x, 0) on x is presented in Fig. 3.2 for different values of V . Now we pass to consideration of the intensity distribution in a waveguide cross section. From (1.3.13) it follows that I(x) = I (0) (x) + I (1) (x) + I (2) (x),

(1.3.22)

where  1/2 b k0 n2 (x) − ǫF , 4π h i ∗ b 0 k σ I (1) (x) = − , 2 0 n ∞ 4πL0F h i∗ h i 0 0 cos 2k σ − 2L k σ /L 2 0 F F 0 n∞ F b . I (2) (x) = − 0 0 8pF LF cos(πLF /LF ) I (0) (x) = −

36

(1.3.23)

Figure 3.2:

The quantities I (0) (x) and I (1) (x) describe the regular part of the intensity, while I (2) (x) describes the oscillatory part. The quantity I (0) (x) coincides with intensity distribution (1.3.2) obtained in the geometric-optics approximation, and exceeds by a factor of ξ −1 the remaining terms in (1.3.22). The spatial intensity oscillations are due to the presence of the function cos[. . . ] in I (2) (x). The spatial period of the oscillations l, as well as the coherence radius, is determined by a condition of type (1.3.17) x+l Z ′ ′ 2k0 |σF (x + l) − σF (x)| = 2k0 pF (x )dx = 2π.

(1.3.24)

x

For l we obtain the following estimate

2

l/a = πξ ≪ 1.

(1.3.25)

From (1.3.24) and (1.3.25) it follows that the spatial period of the intensity oscillations coincides with the field coherence radius and possesses an the properties of the coherence radius. The oscillation period, as well as the coherence radius, is always nonzero, and its minimum value is determined solely by the waveguide parameters. The relative amplitude of the intensity oscillations is of the order of the quasiclassical parameter ξ ≪ 1. The 2 An

analogous estimate was obtained in Ref. [45]

37

Figure 3.3:

amplitude of the oscillations, as well as their period, depends on the transverse coordinate x. It should be noted that expression (1.3.22) does not describe the correct behavior of the intensity in regions where the quasiclassical condition ξ ≪ 1 breaks down. On the basis of the established properties, we may conclude that the spatial correlations of the field and the spatial intensity oscillations of the field in waveguides are interrelated. As an example we present the intensity distribution of the field in a waveguide with the profile (1.3.20). This has the form  1/2 bV x2 (0) I (x) = 1− 2 , 4πa a

" b I (2) (x) = − cos V 8πa(1 − x2 /a2 )

I (1 (x) = 0, π x x − arcsin − 2 a a



x2 1− 2 a

1/2 !#

,

i∗ h where we set k0 σn0 2∞ = 0. The dependences of I(x) on x for different values of V are displayed in Fig. 3.3.

The considerations described above allow us without much difficulty to treat the class of waveguides with constant index of refraction n(x) ≡ n0 =const and with boundary

38

conditions at the waveguide edges of the type u

x=0

∂u = 0. ∂x x=a

= 0,

(1.3.26)

Let us consider a waveguide of width a ≫ λ = 2π/k0 and with the conditions (1.3.26) at the edges x = 0 and x = a. The wave functions un (x), normalized by condition (1.3.4), have the form un (x) =



8π k0 c kn a

1/2

(1.3.27)

sin (k0 pn x) ,

where π pn = k0 a



1 n− 2



kn = k0 n20 − p2n

,


1/2

, n = 1, 2, . . . .

We will examine the case of uniform excitation of the N , (N ≫ 1) waveguide modes with intensity b. Using (1.3.6), (1.3.27), and the Poisson summation method, we may obtain the following expression for the intensity: I(x) =

b b sin(2k0 pN x − xπ/a) k0 bpN − − . 4π 8a 8a sin(xπ/a)

(1.3.28)

We note that expression (1.3.28), in distinction with (1.3.22), is exact. It follows from (1.3.28) that for all values of x, except for a narrow layer with width on the order of ∆x ∼ π/k0 pN ≈ a/N ≪ a near the waveguide edges, the intensity I(x) oscillates around the uniform distribution I (0) (x) = k0 bpN /4π with relative amplitude δI ∼ 1/(k0 apN ) ≈ 1/(πN ) ≪ 1 and oscillation period l ∼ 1/(k0 pN ) ≈ a/N . The minimum value of l is bounded from below by the wavelength in the medium λ = 2π/n0 k0 . Near x = 0 the intensity rapidly falls to zero in a narrow layer ∆x, while near x = a the intensity rises to the value 2I (0) (x). For an arbitrary distribution of the power among the modes, the spatial correlation function may be determined by the integral 2

Γ12

Zn∞ ∂Γ′12 (ǫF ) , dǫF β(ǫF ) = ∂ǫF n20

39

(1.3.29)

where Γ′12 (ǫF ) is the spatial correlation function for the case of a step distribution (1.3.12) in the power among modes, and β(ǫ) is the function describing the distribution of power among the modes. One may be convinced of the correctness of relation (1.3.29) by reference to formulas (IP.2)–(IP.4) in Appendix 1. It may be expected that nonuniform distribution of the power among the modes will result in an increase in the spatial coherence of the field in the waveguide.

40

Chapter 2

Propagation of partially coherent waves in statistically irregular waveguides 1

Basic equations

As was shown in Introduction, the correlation function of a field (B.2.3) in statistically irregular waveguides depend on the mode correlators Pnm (z) (B.2.3). The present and the next sections are devoted to the derivation of the equations for the mean field and the mode correlators Pmn (z). In contrast to Refs. [54, 56, 55, 57] where the parabolic equation of quasi-optics has been used we will use the equations of interacting modes (B.2.5) neglecting backward waves in them. Suppose that the waveguide is illuminated by the external partially–coherent field at the plane z = 0. We note that the equation (B.2.5) formally coincides with the Schrödinger equation in the energetic presentation for the quantum system affected by the external time–dependent perturbation. According to the invariant perturbation theory [74] the formal solution of Eq. (B.2.5) satisfying the boundary condition, an = a0n , z=0

(2.1.1)

has the form

an (z) = eikn z

X n′

41

[S(z)]nn′ a0n′ ,

(2.1.2)

where S(z is the transition matrix, determined by infinite series S(z) = I + i

ZZ

dz ′ V i (z ′ ) + i2

ZZ 0

0

dz2

ZZ2

dz1 V i (z2 )V i (z1 ) + · · · ,

(2.1.3)

0

where V i (z) is the matrix of wave mode interactions whose matrix elements are determined by i Vnm (z) = e−i(kn −km )z Vnm (z).

Here I is the unit matrix. For the conjugated amplitudes a∗n (z) we obtain X
∗ [S(z)]∗nn′ a0n′ . a∗n (z) = e−ikn z

(2.1.4)

(2.1.5)

n′

The quantities describing medium inhomogeneities are entered into the expression for

S(z). In the absence of back–scattered waves, the field at the plane z = 0 is statistically independent on random inhomogeneities of a medium located in the semi-space z < 0. Thus using (2.1.2) and (2.1.5) one can write down the following formal solutions for the mean amplitudes han (z)i and the correlators Pnm (z) (B.2.3), X h[S(z)]nn′ iha0n′ i, han (z)i = eikn z

(2.1.6)

n′

Pnm (z) = e−i∆knm z

X

n ′ m′

where ∆knm = kn − km .

h[S(z)]nn′ [S(z)]mm′ ihPn′ m′ (0)i,

(2.1.7)

The solutions (2.1.6) and (2.1.7) are actually given by the expansions in infinite series in powers of the perturbation Vnn′ (z). To study the solutions (2.1.6) and (2.1.7) we suppose that the coupling coefficients Vnn′ (z) are the Gaussian random processes with the zero mean value hVnn′ (z)i = 0 and the correlation functions Ψnn′ ,mm′ (τ ) = hVnn′ (z)Vmm′ (z + τ )i.

(2.1.8)

The analysis of the expansion series (2.1.6) and (2.1.7) is convenient to carry out using the diagram technique. The diagram technique for the mean field is described in Appendix 2 and for the second moments it is given in Appendix 3. As was shown in Appendix 2 the equation for the mean field has the form # " X 1 fnn′ (s)e e an (s) = W an′ (s) , ha0n i + s − ikn ′ n 42

(2.1.9)

where e an (s) is the Laplace transform of the average amplitude e an (s) =

Z∞

e−sz han (z)idz.

(2.1.10)

0

fnn′ (s) is the sum of all diagrams without free sections (see Appendix 2). The mass term W In contrast to the expansion (2.1.6), the mass term is expanded into powers of the parameter δ = k0 σl|| , where σ is the fluctuation magnitude of a medium, l|| is the longitudinal correlation length of fluctuations. At δ ≪ 1, one can retain in the mass term only the diagrams of the second order [see Eq. (2P.8)]. According to the rules of the diagram technique we have X

fnn′ (s) = W

where

e np,pn′ (u) = Ψ

(2.1.11)

p

e np,pn′ (s − kp ), Ψ

Z∞

dτ e−uτ Ψnp,pn′ (τ ).

(2.1.12)

0

At distances z ≫ (∆knm )−1 , z ≫ l|| , the solution of (2.1.9) has the form n o ′′ han (z)i = han (0)i exp i(kn + δkn′ )z − δkn z ,

(2.1.13)

where

δkn′ ′′

=−

δkn =

Z∞

0 Z∞

dτ

dτ

0

X

Ψnp,pn′ (τ ) sin(∆kpn τ ),

p

X

(2.1.14)

Ψnp,pn′ (τ ) cos(∆kpn τ ).

p

′′

We note that the expression for the attenuation constant δkn in the case of the waveguides with rough walls has been obtained in Ref. [62] taking into account back–scattered waves. In the waveguides with fluctuating impedance of one of walls it is derived in [55] in the parabolic approximation. As an example we consider a two–dimensional waveguide channel with the quadratic profile of refractive index in the presence of axis bending,   n2 (x, z) = n20 1 − α2 (x − f (z))2 , 43

where f (z) describes the random deviations of the waveguide axis from the straight axis z. Suppose, that hf (z)i = 0,

hf (z)f (z + τ )i = Φ(τ ).

Then according to (2.1.14) one can obtain the following expression for the decay constant of the n−th mode, ′′

D = k0 n20 α3

δkn = D(2n + 1),

Z∞

dτ cos(ατ )Φ(τ ).

0

Therefore, in the considered case the attenuation of modes grows linearly with the mode number n.

2

Equations for the mode correlators

We turn to study the equations for the mode correlators Pmn (z). For the Laplace transform, Penm (s) =

Z∞

e−sz Pnm (z),

0

we obtain the following equation (see Appendix 3) " # X 1 ′ ′ nm fnm Penm (s) = W Pnm (0) + (s)Pen′ m′ (s) . s + i∆knm ′ ′ nm

(2.2.1)

f n′ m′ (s) in Eq. (2.2.1) is the sum of all diagrams without free sections. Similar The kernel W nm n ′ m′ fnn (s) in the equations for the mean field, the quantities W fnm (s) are to the mass term W

n ′ m′ fnm (s) expanded into series of powers of δ = k0 σl|| . At k0 σl|| ≪ 1 one can retain in W

only diagrams of the second order,

W(2)

+

+

+

. (2.2.2)

According to the rules of the diagram technique the sum of the first and fourth diagrams is equal to (2)n′ m′

f W Inm

"

(s) = − δmm′

X p

e np,pn′ (s + i∆kpm ) + δnn′ Ψ 44

X p

#

e mp,pm′ (s + i∆knp ) , Ψ

(2.2.3)

and the sum of the second and third diagrams is f (2)n′ m′ (s) = Ψ e nn′ ,mm′ (s + i∆kn′ m ) + Ψ e nn′ ,mm′ (s + i∆knm′ ), W IInm

(2.2.4)

Equation (2.2.1) is equivalent to the following integro-differential equation XZ dPnm (z) (2)n′ m′ = −∆knm Pnm (z) + Wnn (z − z ′ )Pn′ m′ (z ′ ), dz n ′ m′ ∞

(2.2.5)

0

with the boundary condition

Pnm (z)

0 = Pnm . z=0

At the distances z ≫ l|| Eq. (2.2.5) can be reduced to X dPnm (2)n′ m′ fnm W = −∆knm Pnm (z) + (i∆knm )Pn′ m′ (z ′ ). dz n ′ m′

(2.2.6)

Equation (2.2.6) is the generalized transfer equation for the second moments Pmn (z). As was noted in Introduction the equations of type (2.2.6) have been also obtained in Refs. [54, 56] for the waveguides with fluctuating impedance of one of walls and in Ref. [57] in the waveguide filled with the random inhomogeneities. In contrast to the equations considered in these works, Eq. (2.2.6) can be used to study a wide class of random inhomogeneities of waveguide media. Beside, the diagram technique allows one to find the applicability conditions of these types of equations. This will be done at the end of this section. Below, we consider separately the equations for the mode powers Pnn (z) and the equations for the non–diagonal correlators Pnm (z). 2.1

Transfer equation

Neglecting in Eq. (2.2.6) the fast varying non–diagonal quantities Pmn (z), n 6= m and using (2)nn fmm Eqs. (2.2.3) and (2.2.4) for W (i∆knm ), we obtain the standard transfer equation X dPnn = −κn Pnn (z) + wnm (Pmm − Pnn ) , dz m Z∞ n o e nm,mn (∆knm ) = 2 dτ cos (∆knm ) Ψnm,mn (τ ), wnm = 2Re Ψ 0

κn = 2

∞ XZ α

dτ cos (∆knα ) Ψnα,αn (τ ).

0

45

(2.2.7)

The quantities wmn have the meanings of probabilities of energy transfer between modes, while κn describes the mode decay due to the scattering into radiative modes α. At κn = 0 Eq. (2.2.7) coincides with the equation obtained in [66]. 2.2

Equations for the non–diagonal correlators Pmn

We present the quantities Pmn (z) as a product of fast varying exponent to the slowly changing amplitude, Pnn (z) = P¯nn (z) exp (−i∆knm z) . Substituting the last expression into (2.2.6), multiplying the both sides of the equation by exp (i∆knm z), and neglecting by fast oscillating terms, we obtain X dP¯nm fnm n ′ m′ fnm W (−i∆knm )P¯n′ m′ ∆ (∆knm − ∆kn′ m′ ) , = Wnm (−i∆knm )P¯nm + dz ′ ′ nm

(2.2.8)

where ∆(x) is the function equal to unity at x = 0 and zero at x 6= 0. From (2.2.8) it follows that Pmn (z) depends of the [mutual] location of propagation constants kn of waveguides [in the space of mode number n]. Consider two possible cases of the location propagation constants. 1) Non-equidistant locations: ∆knm 6= ∆kn′ m′ .

In this case we obtain the equation for the only one function P¯nm (z), which has the

following solution 0 exp (−γnm z) , P¯nn (z) = P¯nn

(2.2.9)

where nm fnm γnm = W (−i∆knm ) =



X

(wnp + wmp )

p6=n,m

 e nn,nn (0) + Ψ e mm,mm (0) − 2Ψ e nn,mm (0) . + Ψ

(2.2.10)

At distances z ∼ (γnm )−1 different modes become de-correlated. For simplicity, furthermore suppose that Vnm (z) has the form Vnm (z) = Λnm f (z), where Lambdanm are constants, and f (z) is a dimensionless random function. Then Z∞ Z∞ −sτ e nn,mm (s) = Λnn Λmm dτ e dτ hf (z)f (z + τ )i. Ψ 0

0

46

In the case of large longitudinal correlation length l|| ≫ (∆knm )−1 , one can neglect by the transition probabilities wnm . Then γnm = (Λnn − Λmm )2 Lc ,

(2.2.11)

where Lc = hf (z)f (z + τ )i is the effective correlation length. Therefore, the de-correlation of modes occurs also in the absence of mode losses only due to the random phase shifts between modes. 2) Equidistant locations: ∆knm = ∆kn′ m′ . This case occurs, for example, in the ideally focusing waveguide with the refractive index n(x) = 1/ cosh(x), and in the parabolic waveguide. In this case the equation (2.2.8) can be written as X′ ′ ′
dP¯nm nm ′ unm = −γnm P¯nm + P¯n′ m′ − P¯nm , dz n ′ m′

(2.2.12)

where

′ γnm = γnm − n ′ m′

unm = 2Re

X′

′

′

nm unm ,

n ′ m′

n

o e nn′ ,mm′ (i∆knn′ ) . Ψ

The prime in the sum sign means that the summation is carried out over all modes that satisfy the equidistant condition ∆knm = ∆kn′ m′ . Unlike the previous case, in this case the non–diagonal correlators decay quasi-periodically. ′ The characteristic decay length of correlation is equal to zc ∼ γnm . For the example con-

′ sidered in Section 1, the quantity γnm equals to   p √ ′ γnm = 2n + 2m + 2 − nm − (n + 1)(m + 1) D.

We discuss the applicability conditions of transfer equations (2.2.6). In the derivation n ′ m′ fnm (−i∆knm ) only diagrams of the second order. This is of (2.2.5) we have taken in W

justified when the diagrams of the fourth and higher orders are small in comparison to

the diagrams of the second order. We estimate the diagrams of the fourth order. e nn′ ,mm′ (i∆k) and ∆k =min|∆knm |, (n 6= m). Let Φ(∆k) be a maximal value of ReΨ Then, one can obtain the following estimate for the diagram of the 4-th order: W (4) ∼ Φ2 (∆k)/∆k, 47

while for the diagram of the second order: W (2) ∼ Φ(∆k). The applicability condition of the power transfer equation has the form W (4) /W (2) ∼ Φ(∆k)/∆k ≪ 1.

(2.2.13)

Since Φ(∆k) ∼ σ 2 l|| , the condition (2.2.13) takes the form   1 ∆k ≪ 1. σ 2 l|| The last condition coincides with the applicability condition of the transfer equation given in [68]. Note that the condition (2.2.13) also means that ∆knm ≫ wnm ∼ Ψ(∆k), i.e., the probabilities of energy transfer between modes are small in comparison to the difference between the propagation constants of modes. 2.3

Asymptotic behavior of the mutual coherence function and the intensity distribution

In this section we discuss the behavior of the coherence of a field and the distribution of its intensity in randomly inhomogeneous waveguides at large distances. According to the results of Sec. 2.2 the behavior of the mutual coherence function Γ(~ ρ1 , ρ~2 , z) (B.2.2) at large distances is determined only by the mode powers Pnn (z). The latter satisfy the transfer equations (2.2.7). The solutions of these equations are different for the closed and open waveguides. In the closed waveguides with no dissipative losses the total power transmitted by all modes is conserved, X

Pnn (z) = const.

(2.2.14)

n

In this case, at the sufficiently large distances the power is uniformly distributed between all modes (see, e.g., [66]), Pnn (z) = const/N,

(2.2.15)

where N is the number of modes. As was shown in Sec. 3 of the first chapter, for the uniform power distribution over modes, one can obtain the general form of the mutual coherence function in plane waveguides with the arbitrary profile of refractive index (Eq. (1.3.13)). According to the results obtained in Chapter 1 the coherence radius tends 48

to its minimal value, ρc ∼ πaV −1 ,

(2.2.16)

determined only by the waveguide parameters, i.e., its width a and the dimensional frequency V (B.1.2). The intensity distribution of field in a plane transversal to the z− axis has an oscillatory structure about the smooth distribution determined by the profile of refractive index [Eqs. (1.3.22) and (1.3.28)] The relative amplitude of intensity oscillations is proportional to V −1 and its period is l = ρc . In open waveguides the total power transmitted by all modes is not conserved. However, at large distances a certain stationary distribution of power over modes is established, Pnn (z) = Bn e−σ1 z ,

(2.2.17)

where Bn is the power distribution of over modes that is independent on z, σ1 is the loss coefficient identical for all modes. The asymptotic distribution (2.2.17) does not depend on the initial power distribution over modes at the plane z = 0. The expression for the mutual coherence function take the form Γ(~ ρ1 , ρ~2 , z) = e−σ1 z

X

Bm u∗m (~ ρ1 )um (~ ρ2 ).

(2.2.18)

n

The analytical calculation of (2.2.18), in general, is a difficult problem: for an every specific case one should necessary to carry out numerical calculations. In the case of multimode waveguides one can use the quasiclassical formulas (1.3.29). However, for specific asymptotic distributions of type (2.2.17) obtained in Refs. [79, 80, 81] in the geometrical optics approximation we were not able to [analytically] calculate the integral (1.3.29). In spite of that one expects that the asymptotic value of coherence radius takes some finite value that exceeds its minimal value (2.2.16).

3

Propagation of partially coherent waves in parabolic waveguides

In many cases the real profiles of refractive index of waveguide channel can be approximated by quadratic law near the waveguide axis,   n2 (x, y) = n20 1 − α2 (x2 + y 2 ) . 49

(2.3.1)

The parabolic waveguides have a important practical value because of possibilities to transfer information and images through them with small phase distortions. In the following sections of this chapter we consider the propagation of partially coherent waves in a parabolic waveguide with random inhomogeneities. The wave propagation in a medium with large scale random inhomogeneities can be described by the parabolic equation, 2ik where

  ∂u + ∆⊥ u + k02 −n20 α2 ρ~ 2 + εe(~ ρ, z) u = 0, ∂z

k = k0 n0 ,

∆⊥ =

(2.3.2)

∂2 ∂2 + . ∂x2 ∂y 2

The random inhomogeneities of a medium are described by the fluctuating part of the dielectric permittivity εe(~ ρ, z). Furthermore, we assume that εe(~ ρ, z) is the Gaussian random

process.

According to the definition the mutual coherence function of monochromatic field is

given by Γ(~ ρ1 , ρ~2 ; z) = hu∗ (~ ρ1 .z)u(~ ρ2 , z)i.

(2.3.3)

We introduce the notations Γ(~ ρ1 , ρ~2 ; z) = hγ(~ ρ1 , ρ~2 ; z)i, γ(~ ρ1 , ρ~2 ; z) = u∗ (~ ρ1 , z)u(~ ρ2 , z),       ~ ρ~, z = εe R ~ + ρ~ , z − εe R ~ − ρ~ , z , εe+ R, 2 2 ~ = (~ R ρ1 + ρ~2 )/2,

ρ~ = (~ ρ1 − ρ~2 )/2. (2.3.4)

From Eq. (2.3.2) it easily obtains the equation for the function γ(~ ρ1 , ρ~2 ; z),   ∂γ 1 ~ · ρ~ γ = − k εe+ R, ~ ρ~, z γ. + (∇R · ∇ρ ) γ − kα2 R i ∂z k 2

(2.3.5)

In order to obtain the closed equation for Γ(~ ρ1 , ρ~2 ; z), it is necessary to separate the averaged value of the right hand side of (2.3.5). Using the Furutsu–Novikov formula (see, e.g., Ref. [27]) we obtain D

Z∞Z Z∞Z     E Zz ′ ′2 ~ ρ~, z γ R, ~ ρ~, z εe R, = dz dR dρ′2 +

0

−∞

−∞

D    ′ E ~ ρ~, z εe+ R ~ , ρ~ ′ , z × εe+ R, 50

*

  + ~ ρ~, z δγ R,  ′  . ~ , ρ~ ′ , z δe ε+ R

(2.3.6)

The equation (2.2.5) is equivalent to the following integral equation Zz Z∞Z Z∞Z     k 0 ′ ′2 ~ ρ~, z = γ0 R, ~ ρ~, z − i γ R, dz dR dρ′2 2 0 −∞ −∞    ′   ′  ′ ~ ρ~, z; R ~ , ρ~ ′ , z ′ εe+ R ~ , ρ~′ , z γ R ~ , ρ~ ′ , z , × g R,

(2.3.7)

  ~ ρ~, z; R ~ ′ , ρ~ ′ , z ′ is the Greene function of Eq. (2.3.5) without the right hand where g R,

side,

  ~ ρ~, z; R ~ ′ , ρ~ ′ , z ′ = g R, ×

h

k 2 α2 exp 4π sin2 [α(z − z ′ )]

~ · ρ~ + R ~ ′ · ρ~ ′ R



(

ikα sin [α(z − z ′ )]

)  i ~ · ρ~ ′ + R ~ ′ · ρ~ . (2.3.8) cos [α(z − z ′ )] − R

~ ρ~, z) ≡ Γ(~ Using Eqs. (2.3.5)–(2.3.8) one can obtain the following equation for Γ(R, ρ1 , ρ~2 , z): Zz Z∞Z Z∞Z ∂Γ 1 k02 2~ ′ ′2 i + (∇R · ∇ρ ) Γ − kα R · ρ~ Γ = −i dz dR dρ′2 ∂z k 4 0 −∞ −∞ D    E     ~ ρ~, z εe+ R ~ ′ , ρ~ ′ , z g R, ~ ρ~, z; R ~ ′ , ρ~ ′ , z ′ Γ R ~ ′ , ρ~ ′ , z ′ . × εe+ R,

(2.3.9)

We note that the last equation is valid under the condition kσl|| ≪ 1,

where σ is the magnitude of the fluctuations of media, l|| is the correlation length of fluctuations along the z− axis. Furthermore we consider the two types of random inhomogeneities of medium: homogeneous fluctuations of medium and random bending of the waveguide axis. In the both cases we assume that inhomogeneities of medium are δ−correlated along the z− axis. In the first case he ε (~ ρ1 , z) εe (~ ρ2 , z ′ )i = A (~ ρ1 − ρ~2 , z) δ (z − z ′ ) .

(2.3.10)

Then Eq. (2.3.9) is reduced to i

2 ∂Γ 1 ~ · ρ~ Γ = − k0 D (~ + (∇R · ∇ρ ) Γ + kα2 R ρ, z) Γ, ∂z k 4

where D (~ ρ, z) = A(0, z) − A (~ ρ, z) . 51

(2.3.11)

It is not difficult to show that the solution of Eq. (2.3.11) satisfying the boundary condition     ~ ρ~, z ~ ρ~ , Γ R, = Γ0 R, z=0

has the form

Z∞Z Z∞Z   k 2 α2 ~ ′ , ρ~ ′ dρ′2 Γ0 R dR′2 Γ (~ ρ1 , ρ~2 , z) = 2 2 4π sin (αz) −∞ −∞      k 2 Zz

 ~ ρ~, R ~ ′ , ρ~ ′ ; z − 0 × exp ikϕ R, dz ′ D ~v ρ~, ρ~ ′ ; z, z ′ , z ′ (2.3.12) ,   4 0

where

# "   ~ · ρ~ ′ + R ~ ′ · ρ~ ~ · ρ~ + R ~ ′ · ρ~ ′ R R ′ ~ ρ~, R ~ ′ , ρ~ ; z = α − , ϕ R, tan αz sin αz  
sin αz ′ tan αz ′ ′ ′ ′ ′ ~v ρ~, ρ~ ; z, z = ρ~ . − ρ~ cos αz 1 − sin αz tan αz

The obtained formula (2.3.12) in particular cases reconstructs known results. Particularly, in the absence of inhomogeneities εe (~ ρ, z) ≡ 0 we obtain the mutual coherence function in

a regular waveguide [35]. In the limit α → 0 one obtains the corresponding expression in boundless media (see, e.g., [28]).

The expressions of type (2.3.12) for the mutual coherence function in parabolic waveguides have been also independently obtained in Refs. [61, 78]. However, in these works the coherent properties of field were not studied. We investigate the behavior of the coherence radius of field. Assume that at the plane z = 0 the waveguide is excited by the quasi-monochromatic and spatially incoherent field,     ~ ρ~ = 4π I0 R ~ δ (~ Γ0 R, ρ) , k2   ~ is the spatial distribution of intensity at the plane z = 0. Then according where I0 R

to (2.3.12), we have

  kα~ ρ k 2 α2 W sin αz π 2 sin2 αz   z    ′ 2 Z ~ sin αz ′ k R · ρ~ , dz ′ D ρ~ − 0 ,z × exp ikα   tan αz 4 sin αz


~ ρ~,z = Γ R,

0

where

W (~a) =

Z∞Z

−∞

    ~ ′ exp −i~a · R ~ . dR′2 I R 52

(2.3.13)

From (2.3.13) it follows that in the propagation process the initial spatial incoherent field becomes partially coherent. The behavior of the spatial coherence radius at the distances z ∼ α−1 is determined by the function W (kα~ ρ/ sin αz). At these distances the coherence radius ρc is determined by the initial field as well as by waveguide parameters and it is a periodic function of the longitudinal coordinate z, ρc ∼ | sin αz|/(kαa), where a is the effective width of the initial field. At the large distances z ≫ α−1 the coherence radius is determined only by medium inhomogeneities. Formally, it means that the coherence radius ρc is determined by the decaying exponent in (2.3.13), i.e., it is given by

2 Zz   ′ k0 sin αz ′ , z = 1. dz ′ D ρ~ 4 sin αz

(2.3.14)

0

Assume that the structural function D(~ ρ, z) is determined by the Kolmogorov–Obukhov law: D(~ ρ, z) = Cε2 ρ5/3 . Then it is not difficult to obtain the estimation for ρc , ρc ≈

| sin αz| . (Cε2 k 2 z)3/5

(2.3.15)

The formula (2.3.15) for the coherence radius of field in waveguide differs from the corresponding one in a boundless medium by the presence of the term that is periodic in z (see [28]). Note, that it appears due to the effect of waveguide. One should expect that the quasiperiodic law of decay of the coherence radius is also valid for the fluctuations of medium described by other laws. Above we have considered the case of homogeneous fluctuations of medium. Below we consider the effect of random bending of the waveguide axis on the coherence of field. In this case a fluctuating part of the dielectric permittivity has the form εe (~ ρ, z) = 2n20 α2 ρ~ · ϕ ~ (z).

(2.3.16)

The random function ϕ ~ (z) = (ϕx (z), ϕy (z)) describes deviations of the waveguide axis from its unperturbed position at the plane z. Suppose, that h~ ϕ(z)i = 0,

hϕi (z)ϕj (z ′ )i = δij σ 2 l|| δ(z − z ′ ),

(i, j = 1, 2),

(2.3.17)

where δij is the Kronecker symbol. According to (2.3.16) and (2.3.17), we have D    E ~ ρ~, z εe+ R ~ ′ , ρ~ ′ , z ′ εe+ R, = 4α4 σ 2 l|| ρ~ · ρ~ ′ δ(z − z ′ ). (2.3.18) 53

Then the equation (2.3.9) takes the form i

∂Γ 1 ~ · ρ~ Γ = −ik 2 σ 2 α4 l|| ρ~ 2 Γ, + (∇R · ∇ρ ) Γ + kα2 R ∂z k

(2.3.19)

The last equation coincides with Eq. (2.3.11) with the structural function D (~ ρ, z) = 4σ 2 α4 l|| ρ~ 2 .

(2.3.20)

Using the solution (2.3.12), one can obtain the following expression for the mutual coherence function at large distances Z∞Z Z∞Z   k 2 α2 ′ ′2 ′ ′2 ~ dρ Γ0 R , ρ~ dR Γ, ρ~1 , ρ~2 , z = 2 2 4π sin (αz) −∞ −∞     k 2 σ 2 α4 l z
|| ′ 2 ′2 ′ ′ ~ ~ × exp ikϕ R, ρ~, R , ρ~ ; z − ρ~ + ρ~ − 2~ ρ · ρ~ cos αz .(2.3.21) 2 sin2 αz


According to (2.3.21), the behavior of the coherence radius is described by √ 2| sin αz| ρc ≈ . kσα2 (l|| z)1/2

(2.3.22)

From (2.3.15) and (2.3.22) it follows that the coherence radius tends to zero at large distances. However, as was established in Sec. 2 that the coherence radius in randomly inhomogeneous waveguide tends to a certain finite value. This discrepancy is related with the fact that in the derivation of the formula for the mutual coherence function (2.3.12) it was not taken into account a finiteness of the waveguide’s transversal width. Therefore, these obtained results has approximate nature and they are valid up to the distance zcr where wave beam reaches the waveguide boundary. The latter remark can be used for the approximate estimation of the critical distance zcr of the applicability of the above consideration. According to the results of Sec. 2 the coherence radius takes a minimal value ρc ∼ a/V, where a is the effective width of waveguide, V is the dimensionless frequency. The critical distance zcr can be found by equating the coherence radius determined by (2.3.14) to its minimal value, ρc (zcr ) ≈ a/V. 54

According to (2.3.22), for the waveguide with random bending of its axis, we have zcr ≈

V2 . k 2 a2 α4 σ 2 l||

(2.3.23)

For the waveguide with the profile of refractive index of type   n2 (1 − α2 ρ2 ) , ρ < a, 0 2 n (ρ) =  n2 (1 − α2 a2 ) , ρ > a, 0 the quantity V = k0 a (n20 − n2 (a))

1/2

= k0 n0 a2 α. Thus zcr ≈

4

a2 . α2 σ 2 l||

(2.3.24)

Fluctuations of wave intensity in waveguides with random bending of axis

The wave propagation in a waveguide with random bending of its axis can be described by the parabolic equation i

1 ∂u + ∆⊥ u − kα2 [~ ρ−ϕ ~ (z)]2 u = 0. ∂z 2k

(2.4.1)

Suppose that the random function ϕ ~ (z) = (ϕx (z), ϕy (z)) is the Gaussian random process. We assume that random bending along the transversal directions x, y are statistically independent and homogeneous along the z− axis, hϕi (z)ϕj (z ′ )i = δij σ 2 F (z − z ′ ),

(2.4.2)

where F (z) is a positive function (F (0) = 1). The equation (2.4.1) with the following substitute, u(~ ρ, z) = u¯(~ ρ, z) exp can be reduced to i

  

−i

kα 2

2

Zz 0

ϕ~ 2 (z ′ )dz ′

  

,

 1 kα2  2 ∂ u¯ ρ~ − 2~ ρ·ϕ ~ (z) u¯ = 0. + ∆⊥ u¯ + ∂z 2k 2

(2.4.3)

The last equation is analogous to the Schrödinger equation for the quantum harmonic ~ oscillator affected by the external time–dependent force F(z) = kα2 ϕ ~ (z). The coordinate z plays a role of time. Using the known expression for the Greene function of the 55

corresponding problem (see, e.g., [83]), we obtain u¯(~ ρ, z) =

Z∞Z

d2 ρ′ G(~ ρ, z; ρ~ ′ , 0)u0 (~ ρ ′ ),

(2.4.4)

−∞

where

( 
kα kα ′ G(~ ρ, z; ρ~ , 0) = ρ~ 2 + ρ~ ′2 cos αz − 2~ ρ · ρ~ ′ exp i 2πi sin αz 2π sin αz Zz Zz ~ (z ′ ) sin α(z − z ′ )dz ′ ~ (z ′ ) sin αz ′ dz ′ + 2αρ~ ′ · ϕ + 2αρ~ ′ · ϕ 0

0

− 2α2

Zz Zz 0

ϕ ~ (z ′ ) · ϕ ~ (τ ) sin α(z − z ′ ) sin ατ dz ′ dτ

0

)

.

Suppose that at the plane z = 0 the waveguide is excited by the coherent Gaussian beam u0 (~ ρ) =

p


I0 exp −ρ2 /2b2 .

Using (2.4.4) one can obtain the following simple formula for the intensity I(~ ρ, z) = |u(~ ρ, z)|2 :

where

 2 .  I(~ ρ, z) = I0 Λ(z) exp −Λ(z) ρ~ − V~ (z) b2 , 

 −1 Λ(z) = d2 sin2 αz + d2 cos αz , z Z ~ V (z) = α ϕ ~ (z ′ ) sin α(z − z ′ )dz ′ .

(2.4.5)

(2.4.6)

0

The quantity d = kαb2 is the square of the ratio of the external beam width b to the characteristic width of the waveguide w = (kα)−1/2 . From (2.4.5) it follows that the width of the beam propagating in the waveguide is a periodic function of z, b(z) =

b [Λ(z)]

1/2

 1/2 = bd−1 sin2 αz + d2 cos αz ,

and does not depend on the random bending of the waveguide axis. The beam center V~ (z), determined by (2.4.6), changes randomly. The mean value of the beam center coincides with the axis z, i.e., hV~ (z)i = 0. The mean square of displacement of the beam 56

center from the z− axis is equal to 1/2  z z Z Z h i1/2 F (z1 − z2 ) sin αz1 sin αz2 dz1 dz2  . ∆(z) = hV~ 2 (z)i = σα  0

(2.4.7)

0

The quantity ∆(z) at distances z ∼ α−1 , z ≪ l|| , where l|| is the longitudinal correlation length of random bending, grows linearly with increasing z: ∆(z) ∼ σαz, and at the

distances z ≫ α−1 , z ≫ l|| , ∆(z) grows diffusively,
1/2 ∆(z) ∼ σα l|| z .

(2.4.8)

We establish the probability density function of random intensity I(~ ρ, z) from the known probability distribution function of V~ (z). Let W (V~ , z) be the probability density function of the distribution of V~ (z). Using the relations (2.4.5) we transform the variables ¯ ψ), (Vx , Vy ) to the new ones (I,

q
Vx (z) = x + b ln Λ(z)/I¯ /Λ(z) cos Ψ, q
Vy (z) = y + b ln Λ(z)/I¯ /Λ(z) sin Ψ,

(2.4.9)

where I¯ = I/I0 is the normalized intensity. From (2.4.9) it follows, that b2 ¯ dIdΨ. 2Λ(z)I¯

dVx dVy =

From the condition of conservation of probabilities
¯ Ψ dIdΨ ¯ = W (V~ , z)dVx dVy , Pe I,

we obtain the density of probability distribution of intensity Z2π Z2π  2


 ¯ Ψ = dΨ b ~ I, ¯ Ψ ,z , P I¯ = dΨPe I, W V ¯ 2IΛ(z)

(2.4.10)

0

0

where Vx , Vy are determined by Eq. (2.4.9).

Since ϕ ~ (z) is the Gaussian random process, then V~ (z) is also the Gaussian random process with the dispersion ∆(z) and the probability distribution function    1 2 2 ~ exp −V /2∆ (z) . W V ,z = 2π∆2 (z)

Using (2.4.9)-(2.4.11) we obtain
¯ ρ~, z = P I;

¯1/κ−1

I

κ (Λ(z))


¯ ρ~, z = 0, P I;

1/κ

e−ρ

2 /2∆2



I0 

ρb

q 
ln Λ(z)/I¯ /Λ(z) , ∆2 (z)

57

(2.4.11)

I¯ < Λ(z), I¯ > Λ(z),

(2.4.12)

where I0 (y) is the modified Bessel function of the zero order, κ = 2∆2 (z)Λ(z)/b2 . At the unperturbed waveguide axis we obtain the simple expression ¯1/κ−1
¯ ρ~, z = I . P I; κ (Λ(z))1/κ

(2.4.13)

The moments of the intensity are 2 2 2 hI¯n i = e−ρ /2∆ Λn (z) κ

Z∞ 0

  . p 1 2 dξξe−(n+ 2 )ξ I0 ρσξ ∆2 (z) Λ(z) .

(2.4.14)

At the waveguide axis the moments take a quite simple form hI¯n i =

Λn (z) . nκ + 1

(2.4.15)

As seen from (2.4.15), the moments of the intensity at the unperturbed waveguide axis decay with increasing the distance. It is related with that the light beam in the propagation process deviates from the axis so often and the frequency its appearance at the axis decreases. At the waveguide axis the moments decay inversely proportional to z, hI¯n i ∼ z −1 .

(2.4.16)

We note that the distribution (2.4.13) coincides with the distribution of field intensity caused by the effect of random wandering of a thin laser beam in a medium with homogeneous random irregularities [90]. The relative fluctuation of the intensity at the waveguide axis ¯2 κ2 hI¯2 i − hIi = β 2 (z) = , ¯2 2κ + 1 hIi

(2.4.17)

at the distance z, where the condition

κ = 2σ 2 α2 zl|| Λ(z)/b2 ≪ 1,

(2.4.18)

is satisfied, grows proportionally to the square of the distance
β 2 (z) ≈ 4σ 4 α4 zl||2 Λ2 (z)/b4 z 2 .

(2.4.19)

The condition (2.4.18) indicates that the mean square displacement of the beam center from the waveguide axis is significantly smaller than the beam width. At the distances z where the beam center deviates from the waveguide axis to the distance exceeding the beam width, i.e., κ > 1, the relative fluctuations of intensity grow proportional to z, β 2 (z) ∼ z. 58

(2.4.20)

In conclusion, we discuss the applicability conditions of obtained results to the waveguides with finite transversal sizes. Naturally, that the described pattern of wave propagation is valid only up to distances zcr , where the dispersion of the beam center becomes of order of the waveguide width a: ∆(zcr ) ≈ a. From this it follows that zcr ≈

a2 . σ 2 α2 l||

(2.4.21)

This estimation coincides exactly with the estimate (2.3.24), obtained from the behavior of the coherence radius of field in a waveguide.

59

Chapter 3

Wave propagation in a system of statistically irregular waveguides The present chapter is devoted to study the wave propagation problem in a system consisting of large number of open waveguides. Such waveguide systems are of interest in integrated and fiber optics for fabrications of various integrated optical devices and in the transmissions of images [5, 94, 95, 97, 118]. One should note that the corresponding problem is also of interest in the study of short–radiowave propagation in the ionosphere where one could be formed a large number of waveguide channels [14]. To the theoretical study of systems consisting of large number of identical and regularly packed dielectric waveguides has been devoted Refs. [94, 98]. Particularly, in [94] the mode structure of electromagnetic field in a system of slab dielectric waveguides and in [98] the propagation of a light beam in a system of densely packed circular waveguides have been studied. The nature of wave propagation in such systems is mainly determined by the tunnel coupling between waveguides and consists in the following: a wave excited in one of waveguides of the system steadily transfer to neighboring waveguides and in the process of propagation spreads over the transversal cross section of the system. Naturally that real systems of waveguides are always exposed to the different kind of irregularities. These irregularities can be conditionally divided into two types: 1) Irregularities related with non identical parameters of waveguides and their irregular location in the system. (The waveguide parameters are transversal sizes, refractive indices and etc.).

60

2) Irregularities caused by fluctuations of dielectric permittivity and distortion of boundaries separating media with different refractive indices, etc. These irregularities can be formally considered as inhomogeneities varying along the propagation axis z of the waveguide system. It is interesting to study the effect of these irregularities on the nature of wave propagation in a waveguide system. We consider first the case of a system with the irregularities of the first type. Suppose that the waveguide parameters change randomly from one waveguide to another one. Then the propagation constants kn of different waveguides will be distributed randomly. Assume that the random quantities kn are characterized by the distribution function P (kn ) with the width W (Fig. 1).

Figure 1:

It is known [2] that the optical tunnel coupling between waveguides is most effective only when the propagation constants of modes of the corresponding waveguides are identical. Therefore, the spreading of a wave due to tunnel coupling in the system of waveguides under consideration is strongly suppressed. Moreover, as it will be shown in Sec. 2 when the width W of random distribution of the propagation constants exceeds a certain critical value Wc it takes place a localization of wave in an initially excited waveguide and the spreading of “energy” over the transversal cross section of the system with increasing the distance z does not occur. The expected effect of wave localization in a system of waveguides with random parameters is the main result of this chapter. If there are also inhomogeneities in a system varying along the z− axis then energy will be not localized in one waveguide at large distances z. However, the nature of this wave spreading over the transversal cross section will be different than in the case of tunnel 61

spreading. This problem will be considered in Sec. 3. We start our study by considering the wave propagation in a system of large number of identical and regularly packed waveguides.

1

Propagation of a wave in a system of regular waveguides

In this section we consider the wave propagation in a system of identical waveguides in some different point of view than it has been done in Ref. [98]. The amplitude i−th mode of the n−th waveguide ain (z) satisfies the system of equations [98] −i

X dain (z) ii′ i′ Vnn = kni ain (z) + ′ an′ (z), dz ′ ′ n 6=n,j

(3.1.1)

′

ii where Vnn ′ are the coefficients of tunnel coupling between waveguides that depends on

the degree of overlapping of the corresponding wavefunctions.

Figure 2:

Consider the system of single–mode slab waveguides (Fig. 2). In Eq. (3.1.1) taking into account only the coupling between neighboring waveguides, we obtain −i

dan = kn an + (Vn,n−1 an−1 + Vn,n+1 an+1 ) , dz 62

n = 0, ±1, ±2, . . . .

(3.1.2)

In Eq. (3.1.2) we replace the discrete variable n by the continuous variable x = nb, where b is the distance between neighboring waveguides. Then Eq. (3.1.2) takes the form −i

da(x, z) = k(x)a(x, z) + V (x) [a(x − b, z) + a(x − b, z)] , dz

(3.1.3)

where V (x) ≡ Vn.n−1 = Vn.n+1 ,

k(x) = kn ,

a(x, z) ≡ an (z).

Consider the process of wave spreading in the transversal plane at distances x considerably larger than the distance between waveguides: x ≫ b. Then Eq. (3.1.3) in finite differences can be reduced to the equation with partial derivatives, −i

∂a ∂ 2a = k(x)a + V (x)b2 2 . ∂z ∂x

(3.1.4)

The last equation is the continuous model of wave propagation in a system of dielectric waveguides. If waveguides are not identical then the propagation constants will be functions of the transversal coordinate x: kn = k(x) and for the irregularly located waveguides the coefficients of tunnel coupling V (x) will be a function of x. For the identical and regularly located waveguides we have k(x) ≡ k = const,

V (x) ≡ V = const.

Then the solution of Eq. (3.1.4) has the form Z∞ a(x, z) = G(x, z; x′ , 0)a(x′ , 0)dx′ ,

(3.1.5)

−∞

where

  (x − x′ )2 (1 + i)/2 exp ikz − i G(x, z; x , 0) = (2πV b2 z)1/2 4V b2 z ′

(3.1.6)

is the Greene function of the considered problem. From (3.1.5) and (3.1.6) it follows that a wave spreading in a system of waveguides takes place similar to the diffractional spreading of a beam in a free space. In this case the quantity
−1 2 zg ≈ 2V b2 b = (V )−1

(3.1.7)

plays the role of diffraction length. If the waveguide with the number n = 0 is excited at the plane z = 0 then the n−th waveguide is excited at the distance of order of z ∼

x2 /4V b2 = n2 /4V , from which it follows the law of wave spreading across waveguides, |n(z)| ≈ 2 (V z)1/2 . 63

(3.1.8)

One should note that (3.1.8) is valid only for the sufficiently large n. Similarly, one can also consider the system of circular dielectric waveguides (Fig. 3).

Figure 3:

2

Localization of waves in a system of waveguides with random parameters

Now we turn to the case when the parameters of different waveguides take random values. Suppose that the parameters do not change along the wave propagation axis z. Equation (3.1.1) formally coincides with the Schrödinger equation in the node presentation in the field consisting of potential wells with random parameters. Particularly, the waveguide system with random diameters of the waveguide core corresponds to the system of potential wells with random widths (Fig. 4), while the waveguide system with random refractive indices of the core corresponds to the system of potential wells with random depths (Fig. 5). In the both cases the discrete energy levels En in potential wells and the corresponding propagation constants kn take random values. Formally, the coordinate z of the waveguide system corresponds to time t for the system of potential wells. The coupling coefficients Vnn′ between different waveguides correspond to the matrix elements of transitions of a particle from one well to another one. The described analogy is not only a formal but it also has a common physical nature: 64

Figure 4:

Figure 5:

in the both cases the properties of system are determined by their wave character. The described problems are reduced to the known model of Anderson [99, 100, 101] studied in the physics of disordered solid states. The system of potential wells with random depths is described by the Anderson model (Fig. 5). The systems of slab and circular waveguides with random parameters are described by the one–dimensional and the two–dimensional models, respectively. The main feature of the model is a formation of localized states in near the well when the dispersion of random parameters exceeds a certain critical value. To the similar phenomenon leads also the disordered locations of potential wells in space (the Lifshitz model) that corresponds to the irregular locations of waveguides in the plane transversal to the wave propagation axis. One should note that the Anderson model does not allow to obtain the exact solution. Up to now there are no adequate mathematical tools that would well describe the localization phenomenon. However, the simple physical consideration explains quite well this phenomenon. It is concluded in the following. In the system with infinite number of regularly spaced potential wells due to tunnel transition of waves between potential wells it is formed the zone with the energy width ∆E ∼ V , where V is the characteristic value of the transition matrix Vnn′ between neighboring wells, and the wave function of states are de-localized across the entire system. The probability of a tunnel transition is large when the difference between energetic levels δE corresponding potential wells is considerably smaller than the width of the zone: δE ≪ ∆E (a resonance condition). In irregular system of potential wells the energy difference δE due to the random change [δa] of the potential well parameter a (a is a 65

certain parameter of a potential well), may exceed the width of the energy zone:   ∂E δa ≫ ∆E. δE = ∂a In this case the tunnel transition between neighboring potential wells does not occur due to the violation of the resonance condition. In order the state initially localized in one of potential wells to be spread across the entire system it is necessary the formation of a sequence of resonant wells. The formation of such a sequence is a less probable in the random system since the occurrence of the resonance condition between two neighboring potential wells is improbable. But the tunnel transitions between resonant wells located at large distances is strongly diminished due to the weak overlapping of their corresponding wave functions. Therefore, the localized states near potential wells are formed. As was shown by Anderson, the localized state is formed when the width of energy distribution W exceeds a certain critical value Wc . For the system of waveguides this result means that when the dispersion of a random change of the propagation constants W exceeds a certain critical value Wc , a wave beam launched into one of waveguides does not transfer to neighboring waveguides and do not spread across the transversal cross section of a system. In other words the wave [beam] is localized in the transversal section of the system of waveguides with random parameters. Interestingly to note that in one-dimensional systems the localization occurs at any small levels of fluctuations of energy (see, e.g., [101]). It means that in the system of slab waveguides the wave beam is always localized in one of waveguides. The exact values of Wc for the different two– and three–dimensional systems found by numerical simulations are given in [101]. The densely packed system of circular waveguides is equivalent to the plane triangular lattice (Fig. 6). For this system Wc = 9.4V . To estimate the lower boundary of the dispersion of random changes of diameter of the waveguide core that leads to the wave localization we calculated V given in [2] for the coupling of HE11 -modes in single–mode waveguides. The refractive indices of the core and the cladding are taken equal to nc = 1.01 and n0 = 1.0, respectively, the diameter 2d = 2.7µm, and the wavelength λ = 0.63 µm. For these values of waveguide parameters we have Wc d = 9.4V d = 10−1 exp (−R/d) /

p R/d,

where R is the distance between the centers of waveguides. On the other hand the 66

Figure 6:

magnitude of dispersion of kn has an order W d ∼ 10∆d/d, where ∆d/d is the relative dispersion of the core diameter. Then it follows the estimation for the critical dispersion of d, 

∆d d



. 10−2 exp (−R/d) / c

p R/d.

For R > 2d one has (∆d/d)c . 10−3 . Such deviations of diameters may occur during fabrications of dielectric waveguides.

3

Diffusional spreading of waves in a system of waveguides

In the previous section we have considered the case when waveguide parameters do not change along the wave propagation axis z. Obviously, that in real waveguide systems there are inhomogeneities varying along the z− axis. In the presence of the latter a wave will not be localized in one of waveguides at all distances z. The spreading of waves across the transversal cross section occurs due to the scattering of waves on inhomogeneities of medium changing along the z− axis. In the absence of the tunnel transition of a wave between waveguides the energy exchange between waveguides takes place due to two possible mechanisms: by direct scattering of mode of one waveguide into another one of other waveguide, and re-scattering of wave through radiative modes. In the first case, the energy exchange occurs between closely located waveguides since the probability of such an exchange is proportional to the degree of overlapping the corresponding wavefunctions. In the second case it is also possible the energy exchange between distantly located waveguides. 67

For the quantitative analysis of the above analysis we consider a system consisting of single–mode waveguides. Suppose that due to random distribution of waveguide parameters the tunnel transition of a wave between waveguides are negligibly small. Then the wave propagation in a system of waveguides with varying along the z− axis inhomogeneities can be described by the following equations, X daµ = kµ aµ + Veνµ (z)aν , dz ν Z∞Z k0 u∗µ (~ ρ)uν (~ ρ)e ε(~ ρ, z)d2 ρ, Veνµ (z) = 2 −i

(3.3.1)

−∞

where indices µ, ν at values µ, ν = n, m notate the waveguide numbers, and at values µ, ν = α, β notate the radiative modes of the waveguide system. The quantities Veνµ (z) describe transitions between modes due to the scattering of wave on the fluctuations of

dielectric permittivity εe(~ ρ, z). The matrix elements of the transition between waveguides Venm (z) strongly decay with increasing the distance between corresponding waveguides due to exponentially decay of the degree of overlapping of their wavefunctions un (~ ρ) and

um (~ ρ). Suppose that εe(~ ρ, z) is the Gaussian random process with the zero mean value and the

correlation function,

Bε (~ ρ1 , ρ~2 , z2 − z1 ) = he ε(~ ρ1 , z1 )e ε(~ ρ2 , z2 )i.

(3.3.2)

Naturally to assume that the fluctuations of the medium εe(~ ρ, z) near the distantly located waveguides are statistically independent. This implies that the characteristic scale l⊥ of

variation of the correlation function (3.3.2) in the variable ρ~1 −~ ρ2 is of order of the distance between waveguides b, l⊥ ∼ b.

Equation (3.3.1) coincides with Eq. (B.2.5) which has been analyzed in the chapter 2. Consider first the behavior of the mean field. In Eq. (2.1.9) we make the following change of the mean amplitude of the n−th waveguide han (z)i, han (z)i = han (z)i exp(−ikn z). Then Eq. (2.1.9) at the distances z ≫ l|| where a wave has undergone a multiply scattering,

68

takes the form −i

dhan (z)i X i(km −kn )z = e ham (z)iWnm , dz m

Wnm =

Z∞ X 0

(3.3.3)

Ψnν,ν,m (τ ) exp (i∆kν,n τ ) dτ,

ν

k2 Ψnν,ν,m (τ ) = hVenν (z)Venν (z + τ )i = 0 4

Z∞Z

d 2 ρ1

−∞

Z∞Z

d 2 ρ2

−∞

× Bε (~ ρ1 , ρ~2 , τ )u∗n (~ ρ1 )um (~ ρ2 )uν (~ ρ1 )u∗ν (~ ρ2 ).

The quantities Wnm decay fast with increasing the distance between waveguides. Therefore, Wnm describes only transitions between neighboring waveguides. On the other hand, the coupling between the mean amplitudes of different waveguides occurs only for the equal propagation constants of the corresponding waveguides. Due to the randomness of waveguide parameters the probability of that the propagation constants of a wave in neighboring waveguides take equal values is small. Therefore, at the distances z ≫ W −1 = |kn − km |−1 the coherent parts of the amplitudes of a wave in different waveguides propagate independently and decay exponentially, han (z)i = exp(−Wnn z)han (0)i.

(3.3.4)

In spite of the absence of the coherent spreading of waves it occurs the diffusional spreading of wave across the transversal section of the waveguide system. To convince in that, we consider the equation for the mean powers Pn (z) (2.2.7), X dPn = −2κn Pn + wnm (Pm − Pn ) , Pn = h|an (z)|2 i, dz m Z∞ Wnm = 2 dτ cos (∆knm τ ) Ψnm,mn (τ ), 0

κn =

∞ XZ α

dτ cos (∆knα τ ) Ψnα,αm (τ ).

(3.3.5)

0

The quantities wnm describe the probabilities of power transition between different waveguides. According to (3.3.3) and (3.3.5) wnm and Ψnm,mn (τ ) are non–zero only for the transition between neighboring waveguides. The quantities κn describe a decay rate of the wave in the n−th waveguide due to a scattering into the radiative modes. 69

From (3.3.5) it follows that the power transition from one waveguide to its neighbors −1 occurs at the distances z ∼ wnm . We estimate wnm . Suppose that

Bε (~ ρ1 , ρ~2 , τ ) = σ 2 e−τ

2 /l2 ||

(3.3.6)

A(~ ρ1 − ρ~2 ),

where σ is the fluctuation intensity of medium, A(0) = 1. Then, according to (3.3.5), we have

√

π 2 2 −(∆knm )2 l||2 /4 k σ l|| e I, 4 0 Z∞Z Z∞Z 2 I= d ρ1 d2 ρ2 un (~ ρ1 )u∗m (~ ρ2 )A(~ ρ1 − ρ~2 )u∗n (~ ρ2 )um (~ ρ1 ).

wnm =

−∞

(3.3.7)

−∞

The matrix of the tunnel transition V equals to V = k0

Z∞Z

d2 ρun (~ ρ)u∗m (~ ρ).

−∞

The quantity I satisfies the following condition I≤

Z∞Z

2

d ρ1

−∞

Z∞Z

ρ2 )um (~ ρ1 ) = ρ2 )u∗n (~ d2 ρ2 un (~ ρ1 )u∗m (~

−∞

1 |V |2 . 2 k0

(3.3.8)

Therefore 2 2 1 wnm ≤ σ 2 l|| e−(∆knm ) l|| /4 |V |2 . 4

(3.3.9)

In the absence of the tunnel coupling between waveguides the random differences ∆knm of the propagation constants of field in neighboring waveguides exceed the quantity V : |∆knm | > V . In this case we have 2 2 1 wnm . σ 2 l|| |V |2 e−V l|| /4 ≪ |V |. 4

(3.3.10)

In the case of weak inhomogeneities (σ ≪ 1) the condition (3.3.10) is satisfied in the both cases: for the large scale inhomogeneities, V l|| ≫ 1; and the small longitudinal correlation lengths, V l|| ≪ 1. Therefore, the diffusional spreading of wave is much weaker that the one due to tunneling one. Consider the diffusional spreading of a wave on the example of the system of slab waveguides (Fig. 2). We replace the equation in finite differences (3.3.5) by the continuous one in partial derivatives, ∂P (x, z) ∂ 2 P (x, z) = −κP (x, z) + wb2 , ∂z ∂x2 70

(3.3.11)

where P (x, z) = Pn (z),

κ = κn ,

w = wn,n−1 = wn,n+1 ,

x = nb,

n = 0, ±1, ±2, . . . .

The diffusion equation (3.3.11) has the solution 1 e−κz P (x, z) = (2πwb2 z)1/2

Z∞



(x − x′ )2 dx P0 (x ) exp − 2wb2 z ′

′

−∞



.

The diffusional spreading law of waves across waveguides has the following form |n(z)| ∼ 2(wz)1/2 . Up to now we have considered the case when the energy transition between waveguides occurs through a direct scattering of a waveguide mode to other mode of the neighboring waveguide. This effect is taken into account by the diagrams of the second order in nm fnm W (s) (Appendix 3). In this case the transitions are possible only between neighboring

waveguides. As was noted above there is a possibility of transitions between distant

waveguides due to re-scattering of wave through radiative modes. These transitions can nm fnm be described by taking into account in W (s) the diagrams of the fourth order. Here we

will not consider this effect because it is small in comparison to the one considered above. Only we note that the coupling through radiative modes is described by the following diagrams of the 4-th order, α

n

m

n

α

m

, n

β

m

n

β

m

In conclusion we note that under the diffusional spreading of a wave the fields in different waveguides become de-correlated. Indeed, as was shown in Sec. 2 of Chapter 2, the non–diagonal correlators Pnm (z) describing in the considered case the statistical correlations between fields in different waveguides decay. Therefore, in the absence of tunnel spreading of a wave the initial spatially coherent field becomes incoherent during propagation in a system of waveguides with random parameters. The presence of the tunnel 71

spreading can prevent the decrease of the coherence of field. Therefore, a measurement of the spatial coherence of field can used as an indicator of the possible localization of waves in a system of waveguides with random parameters.

72

Chapter 4

Nonlinear dynamics of rays in waveguide channels Up to now we considered the effect of random inhomogeneities of medium on wave propagation process in waveguides. It is also important to reveal the influence of regular inhomogeneities of waveguide channel on wave propagation. We present an example of a regular and moreover periodic large-scale inhomogeneity in the ionosphere. It is known that electromagnetic waves propagate with different velocities in the region illuminated by the sun and in the shadow region. Thus, a wave propagating around the earth encounters a periodic inhomogeneity with a period of the order of the earth’s radius. If the wave propagation time is long, even small inhomogeneities can lead to cumulative effects, which manifest themselves in changes of the wave intensity and of the wavefront structure. It is clear that calculation of effects of this kind is beyond the scope of ordinary (in a certain sense) perturbation theory. The purpose of the present paper is to analyze cumulative effects of wave propagation in a periodically inhomogeneous medium. The real parameters of the medium are here such that geometric-optics approximation can be used [14, 15]. The problem of wave propagation in natural media can thus be reduced to a corresponding ray dynamics problem, or to the equivalent problem of the motion of a nonlinear oscillator under the influence of a periodic perturbation. Below we consider two fundamentally new effects. The first can be called modulation localization of the ray as a result of a unique nonlinear interaction of the ray with the periodic inhomogeneity of the medium. This is the analog of nonlinear resonance in 73

classical mechanics. The second effect can be called stochastic instability of the ray. Its gist is that under certain conditions the action of a periodic (not random!) inhomogeneity stochastizes the motion of the representative point of the ray in a plane perpendicular to the propagation direction. This phenomenon is analogous to stochastic instability in nonlinear mechanics [105]. It disturbs a certain region of the waveguide channel. The rays are radiated out of this region, and the effective width of the waveguide channel is decreased.

1

Equations of ray trajectory

We describe the ray trajectory using the Hamiltonian formalism [106]. Let the z axis coincide with that of the waveguide channel and let the coordinate ~r = (x, y) be in a plane transverse to the z axis. Then the ray coordinates (x, y, z) satisfy the Hamilton equations d~p ∂H =− , dz ∂~r

d~r ∂H = , dz ∂~p

(4.1.1)

where H is the Hamilton function H = −[n2 (~r, z) − p~ 2 ]1/2 ,

(4.1.2)

p~ is the momentum and equals 1/2  ˙ ˙2 p~ = n~r/ 1 +~r ,

˙ ≡ d~r/dz. ~r

(4.1.3)

Here n = n(~r, z) is the refractive index. We represent n in the form n2 (~r, z) = n2 (~r) + ǫv(~r, z),

(4.1.4)

where n(r) corresponds to the regular case (homogeneous in z), and the perturbation ǫv takes into account the influence of the inhomogeneity. The quantity ǫ ≪ 1 is a dimensionless parameter of the perturbation. Taking its smallness into account, we can write (4.1.2) in the form H = H0 (~r, p~) + ǫV (~r, p~, z), H0 (~r, p~) = −[n2 (~r) − p~ 2 ]1/2 , V (~r, p~, z) = v(~r, z)/2H0 . 74

(4.1.5)

The unperturbed motion of the ray is determined by the Hamiltonian H0 . The problem of the ray trajectory is thus reduced in the inhomogeneous case to the equivalent problem of the action of a nonstationary perturbation on a particle that executes finite motion described by the Hamiltonian H0 . The role of the time is assumed in this case by the variable z, and it is the inhomogeneity along this variable which produces the perturbation. We begin the investigation of the problem with the simplest planar case, when n is independent of y. Equations (4.1.1) and (4.1.5) then become p˙ = −∂H/∂x,

x˙ = ∂H/∂p,

p = px ,

H0 = −[n2 (x) − p2 ]1/2 .

H = H0 + ǫV (x, p, z),

(4.1.6)

We describe first the unperturbed motion of the ray. A typical plot of n(x) vs the transverse coordinate x is shown in Fig. 1. The values of n± determine the corresponding asymptotic forms of n(x) as x → ±∞. We assume next for simplicity n+ = n− = n∞ .

(4.1.7)

It is easy to show that the unperturbed phase trajectories, defined by Eqs. (4.1.6) and (4.1.7) at V ≡ 0, take the form shown in Fig. 2. Trajectories of type 1 correspond to finite motions of the ray along the x axis. These are in fact the rays propagating in the natural waveguide chanel. Trajectories of type 3 correspond to infinite motion of the ray along x. The two trajectory types are demarcated by the separatrix 2. Let E be the energy of the equivalent particle corresponding to the value of the integral of motion H0 (x, p) = E. It follows then from (4.1.5) and (4.1.7) that on the separatrix we have E = −n∞ ≡ Es .

(4.1.8)

−n0 < E < −n∞ ,

(4.1.9)

−n∞ < E < 0.

(4.1.10)

In the finite-motion region

and in the region of infinite motion

75

Figure 1: Typical dependence of the refractive index n(x) on x.

Figure 2: Trajectory of rays in the phase plane (p, x) in the unperturbed case: l − waveguide rays, 2 − separatrix, 3 − radiated rays.

We introduce in the region (4.1.9), where the motion is periodic, the action and angle variables (I, ϑ): 1 I= 2π

I

S(x, I) =

pdx, Zx

ϑ=

pdx,

∂S(x, I) , ∂I

p=

In terms of the new variables H0 = H0 (I), and

p n2 (x) − E.

ω(I) = dH0 /dI

(4.1.11)

(4.1.12)

is the nonlinear frequency of oscillations of the ray along x relative to the z axis of the

76

waveguide channel. In addition, we can write the Fourier expansion x=

X

xm eimϑ ,

p=

m

X

pm eimϑ .

(4.1.13)

m

Using the variables I and ϑ, we rewrite according to (4.1.11) the equations of motion (4.1.6) in the form H = H0 (I) + ǫV (I, ϑ, z), ϑ˙ = −∂V /∂I.

I˙ = −∂V /∂ϑ,

(4.1.14)

As already noted, we are interested in the case of a perturbation potential V that is periodic in z. Then V (I, ϑ, z) =

∞ 1 X Vms (I)eimϑ+isκz + c.c., 2 m,s=−∞

(4.1.15)

where κ is the "frequency" of the perturbation (2π/κ is the spatial period of the perturbation), and c.c. stands for terms that are the complex conjugates of the preceding ones. It is seen from (4.1.14) and (4.1.15) that the strongest influence of the perturbation takes place in the resonant case, i.e., upon satisfaction of the condition mω(I) + sκ = 0.

(4.1.16)

This case is called nonlinear resonance and was described in detail earlier [105]. We present directly the result for the motion of a ray in the vicinity of one resonance. Let I0 be that value of I at which the condition (4.1.16) is satisfied for definite numbers m and s. It follows then from (4.1.14) (Ref. [105]) that ϑ satisfies the equation ϑ¨ + Ω2 sin ϑ = 0, 2 2 dω(I0 ) Vms . Ω = ǫm dI

(4.1.17)

Equation (4.1.17) is the pendulum equation and describes nonlinear periodic modulation of the phase, ϑ of the ray with frequency Ω. The principal condition for the validity of (4.1.17) is of the form ǫ ≪ 1, α

α=

i.e., a sufficiently strong nonlinearity. 77

dω I , dI ω

(4.1.18)

From (4.1.17) and (4.1.14) we get the region of localization of the nonlinear resonances [105]: . dω(I ) 1/2 0 ∆I = max |I − I0 | = 4 ǫVms , dI

1/2 dω(I0 ) dω(I0 ) . ∆I = 4 ǫVms ∆ω = max |ω(I) − ω(I0 )| = dI dI

(4.1.19)

The physical meaning of the foregoing results was the following. As already noted, in the absence of perturbation the ray trajectory oscillates along the x axis at a frequency ω(I). In the vicinity of the resonant frequency ω(I0 ) there is superimposed on this motion an additional ray modulation along z. The modulation amplitude is determined by expressions in (4.1.19). The amplitude in turn determines also the region of ray localization in the plane perpendicular to z. An additional waveguide channel is thus produced along the trajectory of the unperturbed ray that corresponds to the action of I0 , and has an effective dimension ∆I. The rays trapped in this channel oscillate in it, with frequency Ω, about the unperturbed trajectory. This leads in turn to a periodic modulation of the group velocity of the wave field. To verify this, we consider the connection between the frequency ω(I) and the group velocity of the corresponding wave of the unperturbed problem. In the employed notation kz = k0 |H0 | ,

k0 = ν/c,

(4.1.20)

where ν is the cyclic frequency of the wave field in vacuum, and kz is the wave number of the field along the z-axis. From (4.1.20) we have 1 |H0 | dkz d|H0 | dI = + k0 . = vg dν c dν dν The quantization conditions in the waveguide yield I c 1 (m = 1, 2, . . . ). pdx ≈ m, I= 2π ν We get therefore, taking (4.1.12) into account, vg =

c . |H0 | + ω(I)I

(4.1.21)

In the unperturbed case [ǫ = 0] I˙ = 0 and the action I, hence also vg does not depend on z. In the presence of a perturbation due to inhomogeneities, we have from (4.1.21) 78

and (4.1.14), dvg dvg ˙ ǫ 2 dω(I) ∂V = I = vg I dz dI c dI ∂ϑ m = ǫαω(I0 )vg2 Vms (I0 ) sin(mϑ + sκz). c

(4.1.22)

It follows from (4.1.22) that in the inhomogeneous case vg is also periodically modulated in z with a modulation period Ω and with a modulation amplitude 1/2 vg2 I0 ∆ω 4 2 dvg dω(I0 ) ∆I = vg I0 ǫVms (I0 ) . ∆vg = = dI c dI c

(4.1.23)

A similar appearance of modulation oscillations can be postulated also for the phase

velocity vp = c/|H0 | of the wave. In the general case, the number of regions in which modulation localization of the beam takes place is connected with the number of possible resonances of type (4.1.16). We shall discuss this in greater detail in Sec. 3.

2

Ray trajectory for soliton–like n(x) profile

We consider by way of example a waveguide channel with the following refractive-index profile: µ2 , cosh2 (x/a)

n2 (x) = n2∞ +

(4.2.1)

where µ = (n20 − n2∞ )1/2 characterizes the depth of the corresponding potential well, and a is its effective width. Substituting (4.2.1) in (4.1.11) and (4.1.12) we have 1/2 1 2 H0 (I) − n2∞ , I = Is (1 − β), β≡ µ H0 (I) = −[n2∞ + µ2 (1 − I/Is )2 )]1/2 ,

ω(I) =

Is = a/µ

Is − I , 2 a |H0 (I)|

 (1 − β 2 )1/2 cos ϑ , x = a sinh β sin ϑ p = µβ(1 − β 2 )1/2 , 2 (cos ϑ + β 2 sin2 ϑ)1/2 −1



(4.2.2)

ϑ = ω(I)z + ϑ0 . It follows from (4.2.2) that the separatrix corresponds to the action Is , with H0 (Is ) = −n∞ , H0 (0) = −n0 ,

ω(Is ) = 0, ω(0) ≡ ω0 = 79

µ . an0

(4.2.3)

At |Is − I| ≪ 1 we obtain from (4.2.2) the behavior of the frequency near the separatrix  1/2 Is − I 2 ω(I) ≈ 2 = [H0 (Is ) − H0 (I)]1/2 . (4.2.4) a n∞ a2 n∞ The expansion of the momentum p in a Fourier series (see Appendix 4) has the form 2 1/2

p = µβ(1 − β )

∞ X

Am sin [(2m + 1)ωz] ,

m=0

(2m − 1)!! Am = (−1)m 3m (1 − β 2 )m F 2 m!



 1 3 2 m + , m + , 2m + 2, 1 − β . 2 2

(4.2.5)

From (4.2.1)-(4.2.4) we obtain asymptotic expressions for the spectrum: Am ∼ (1 − β)m , β → 1, (I → 0),  1/2 (1 − β 2 )m 2 β → 0, (I → Is ). Am ∼ π mβ

(4.2.6)

According to the second equation of (4.2.6) the ray-oscillation spectrum is cut off exponentially near the separatrix at numbers m ≥ N , where N ≈ 1/β =

µ2 ω2 = 2 0 ≫ 1. [H0 (Is ) − H0 (I)] ω (I)

(4.2.7)

We take the perturbation to be periodic deviations of the waveguide axis from the straight z axis. In this case n(x, z) = n(x − f (z)), where f (z) describes the deviation of the axis from the coordinate x = 0 in the plane z[=[const]. At small deviations (ǫ = |f |/a ≪ 1) we have n2 (x − f (z)) = n2 (x) + f (z)

∂n2 (x) + ··· . ∂x

Retaining the first two terms, we obtain for the expression for the perturbation ǫV (x, z) =

dp 1 ∂H02 ǫ = −f (z) . v(x, z) = f (z) 2H0 2H0 ∂x dz

Using the expansion (4.2.5), we represent the perturbation in the form  ∞  X 1 2 1/2 ǫV (x, z) = −µωβ(1 − β ) f (z) × Am ei(2m+1)ωz + c.c. . m+ 2 m=0 Let f (z) = f0 cos κz 80

, where 2π/κ is the spatial period of the perturbation. For the matrix elements in (4.1.15) we then obtain 2 1/2

Vms = −Is ωβ(1 − β )



1 m+ 2



Am ,

ǫ = f0 /a.

(4.2.8)

The resonance condition takes in this case the form (2m + 1)ω = κ.

(4.2.9)

The distance between the nearest resonances is δω = |ωm+1 − ωm | =

2κ . (2m + 1)(2m + 3)

(4.2.10)

In particular, near the separatrix, where large values of m are possible, we have δω ∼

ω2 . κ

(4.2.11)

We estimate now the width of the modulation localization of the ray in two limiting cases: near the region of small ray oscillations and near the separatrix. At small oscillations, the greatest influence on the behavior of the ray is exerted according to (4.2.9) by the resonance with m = 0, i.e., ω(I) = κ. Taking into account the asymptotic form (4.2.6) for the spectrum Am as β → 1, as well as (4.1.18), we obtain  1/4 2 κ ∆I 1/2 κan0 1− , = 4ǫ Is µn∞ ω0 1/4  ∆I dω(I) κ ∆ω 1/2 κan∞ . = 4ǫ = 1− ω0 ω0 dI µ ω0

(4.2.12)

We note that nonlinear resonance sets in under condition (4.1.18) when the nonlinearity is large enough. This means [according to (4.2.2) and (4.1.18)] that α≈1−

ω(I) κ =1− > ǫ. ω0 ω0

(4.2.13)

If the values κ are so close to ω0 that condition (4.2.13) does not hold, then the main resonance must be considered in a different manner. It follows thus from (4.2.12) and (4.2.13) that the regions of modulation localization of the ray have a certain lower bound on the order of ǫ3/4 . Near the separatrix it is possible also for high harmonics of the ray oscillations to enter into resonance. For a resonance m ≫ 1 we can obtain the following estimates of ∆I and

81

Figure 3: Stochastic layer in phase plane of rays (shaded region).

∆ω: 1/2  1/4  ω n∞ 2 , ǫ π ω0 n0 1/2  1/4  ω n0 2 ∆ω . =4 ǫ ω0 π ω0 n∞

∆I =4 Is

(4.2.14)

We shall use these expressions in the next section.

3

Formation of stochastic layer near the separatrix

It was shown earlier [105, 107, 108] that the perturbation produces in the vicinity of the separatrix a so-called stochastic layer, in which the particle trajectories are random. The main feature of this phenomenon is that the stochastic layer is produced under periodic perturbations of arbitrary form and magnitude, and only the width of the layer is determined by the character of the perturbation. No analogous property was observed up to now for ray trajectories. At the same time, formation of a stochastic layer in which random motion of the rays takes place can have, as will be made clear below, important physical consequences. It was already noted in Sec. 3 that the analysis of modulation localization of a ray is valid when other resonances are far enough from the considered one. Near the separatrix, the distance δω between the resonances is determined by (4.2.11). As the separatrix is approached ω → 0, the quantity δω decreases rapidly. In this case the regions of the nonlinear resonances can overlap. It is known (the Chirikov criterion [105, 109, 110]) that overlap of resonances causes the trajectories to become stochastic. This condition is

82

written in the form K = (∆I/δI)2 > 1.

(4.3.1)

Let us consider inequality (4.3.1), using formulas (4.2.14) and (4.2.11). We have  1/2  2   2 ω0 3 n0 κ K=4 ǫ . (4.3.2) π ω0 ω n∞

It follows therefore that at all values of ǫ and κ the value of K increases as the separatrix is approached, ω → 0. Therefore, starting with a certain value ω ¯ , K reaches unity and the criterion (4.3.1) begins to be satisfied. Thus, a stochastic layer (shaded region in Fig. 3) is produced in the vicinity of the separatrix and its boundary is determined from the condition K = 1. Hence #1/3 "   2 1/2  n0 2 κ ω ¯ . = 4 ǫ ω0 π ω0 n∞

(4.3.3)

With the aid of (4.3.3) and (4.2.4) we obtain the width of the stochastic layer in terms of variables I and H "   2  2 #1/3 1/2  ¯ δI n∞ κ 2 ǫ , = 4 Is π ω0 n0 2 #2/3  ¯ 2 "  1/2  2  ¯ n∞ 2 δI δH κ = 4 . = ǫ H0 (Is ) − H0 (I) Is π ω0 n0

(4.3.4)

If the initial state of the ray is such that its action I lies in the region (4.3.4), this means that its motion in space along z is of the diffusion type. The diffusion causes the ray to reach the region near the unperturbed separatrix and to be "radiated" out of the waveguide region. The described phenomenon is similar to the existence of a "loss cone" of particles in magnetic traps. Thus, the action of an inhomogeneity as a perturbation leads to a decrease of the effective width of the waveguide channel. We note also that field modes with higher numbers end up in the region of the stochastic layer. Radiation of the field from the stochastic layer means therefore also the filtering of the high waveguidechannel modes. It was shown earlier [107, 108] that the particle traverses the stochastic-layer width within a short characteristic time of the order of the period of the small oscillations. In our case it means that the emission of the ray from the stochastic layer of the waveguide

83

region takes place over a length l=

2πan0 2πan0 2π = 2 . = ω0 µ (n0 − n2∞ )1/2

(4.3.5)

In real situation, µ reaches values 10−1 n0 , and the emission length exceeds by an order of magnitude the width a of the waveguide channel. It must also be noted that over the length l defined by (4.3.5), information is lost concerning that part of the initial wave front which is produced by rays that are far enough from the axis.

4

Stochastization of rays in a three-dimensional waveguide channel

We consider now a waveguide channel in which the refractive index depends on both transverse coordinates x and y. The ray motion is then analogous to the motion of a particle with two degrees of freedom in a potential well. In the presence of periodic perturbations along the z axis, just as in the case of a flat waveguide channel, modulation localization of the beam will be observed, as well as stochastization of the rays near the separatrix. In contrast to the planar case, the presence of two degrees of freedom in the considered waveguide channel can give rise to another ray-stochastization mechanism, which is possible even in the absence of perturbations along the z axis. The last phenomenon is connected with the stochastic destruction of one of the integrals of motion of the ray because of interaction of different degrees of freedom [111] (see also the review [112]). Interaction of two degrees of freedom is possible at certain dependences of the refractive index n(~r) on the transverse coordinates x and y. We consider by way of example a waveguide channel with a refractive index of the form  2   x + y2 1 3 2 2 2 2 n (x, y) = n0 − , (4.4.1) + 3 x y− y a2 a 3 where a has the dimension of length and is of the order of the width of the waveguide channel. The profile (4.4.1) can be used to approximate the refractive index near the waveguide axis, where n(~r) takes a maximal value. For rays propagating near the waveguide axis and at small angles to this axis, the following conditions are satisfied: 2   x + y2 1 3 2 2 2 a2 + a3 x y − 3 y ≪ n0 , |~p| ≪ n0 ,

˙ p~ = n0~r. 84

(4.4.2)

When the conditions (4.4.2) are satisfied, the Hamiltonian (4.1.2) can be written in the paraxial approximation H = −n0 + H ′ ,

  
1 x2 + y 2 1 3 2 1 2 2 2 . + 3 x y− y x˙ + x˙ + H = 2n0 2 a2 a 3 ′

(4.4.3)

At n0 = 1 and a = 1, H ′ coincides with the Hamiltonian of the Henon-Heiles model [111]. A numerical analysis of the model motion in that paper shows that at values E ′ = H’ smaller than a certain critical Ec′ the particle trajectories correspond to a periodic stable motion. Starting with energy values E ′ > Ec′ the trajectories become stochastic, owing to the absence of a second integral of the motion. The latter means that the trajectories of the rays corresponding to high modes with wave numbers k, kz = k0 |H| < kzc = k0 |n0 − Ec′ |,

(4.4.4)

are stochastic, where trajectories of rays for which k0 |n0 − Ec′ | < kz < k0 n0 , are periodic functions of the longitudinal coordinate z. The region of stochastization of the waveguide rays with respect to H are determined by the condition −|n0 − Ec′ | < H < −n∞ ,

n∞ = lim n(x, y). |~ r |→∞

(4.4.5)

Just as in the case of a planar waveguide channel with periodic inhomogeneities, the stochastization causes the rays to be diffusively "radiated" out of the waveguide- channel stochastization region, and decreases the effective width of the layer. Conclusion. The presented results were obtained within the framework of geometric optics, so that we must discuss the question of the restrictions imposed by wave effects. The main condition for the applicability of geometric optics is a smooth variation of the refractive index n(~r) over distances of the order of the wavelength. An additional restriction is imposed by the nonlinearity of the ray oscillations about the waveguidechannel axis. We note that the system considered by is equivalent (from the wave viewpoint) to a quantum nonlinear system. The coordinate z of the waveguide channel corresponds to the time of the quantum system. The simplest effect of violation of the quasiclassical 85

description of a nonlinear system is connected with the spreading of the wave packet as a result of the nonlinearity. The time of this spreading is tc ∼ 2π/[~dω(I)/dI]. For wave optics this means that the geometric-optics approximation ceases to be valid starting with the distance z > zc ∼ 2πk0 /[dω(I)/dI]. For the example considered in Sec. 3 we have   zc ∼ 2πk0 a2 n2∞ + µ2 (1 − I/Is )2 /n2∞ .

The minimum value of zc is of the order of the diffraction length for a beam with a radius of the order of the effective width of the waveguide channel. The wavelengths considered by us are much shorter than the waveguide-channel width: a ≫ λ = 2π/k0 , therefore zc ≫ λ. Thus, the stochastization of the rays sets in over distances shorter than zc . However, the question of the range of applicability of the quasiclassical approximation in quantum mechanics, or of geometric optics in wave optics, in those cases when a stochastic instability develops, is at present quite complicated [112] and will not be discussed here.

86

Summary and conclusions The main results of the thesis are the following: 1. The coherent properties of electromagnetic field in dielectric waveguides are studied. It was established the general form of the spatial–temporal correlation function of field in a waveguide illuminated by the external stationary source of radiation with arbitrary coherence. The behavior of the spatial and temporal coherence of field at different distances from the area where the radiation is coupled to waveguide is investigated. 2. The quasiclassical method of calculations of the spatial correlation function and spatial distribution of field intensity in multimode waveguides is proposed. In the quasiclassical approximation the general formulas for the spatial correlation function and intensity distribution in the case plane waveguides excited by a spatially incoherent radiation were obtained. Based on these formulas the problem of spatial oscillation of field intensity in the section transversal to the waveguide axis is studied. 3. Using the diagram technique the propagation of partially coherent field in statistically irregular waveguides is studied. The kinetic type equations are derived for the expansion coefficients of the mutual coherence function of field in the eigenfunctions of normal modes of unperturbed waveguides. The general behavior of the spatial coherence function at large distance is investigated. 4. The propagation of partially coherent field in parabolic waveguides with large scale inhomogeneities is considered. In the approximation of the δ−correlated Markovian process the formulas for the mutual correlation function of field are obtained. The quasiperiodical decay law of the coherence radius is found. The fluctuations of field intensity in a waveguide with random bending of its axis are studied.

87

5. For the first time a wave propagation problem in a system consisting of large number of dielectric waveguides with random parameters is studied. In such a system the effect of tunnel localization of wave in the plane transversal to wave propagation direction is found. It occurs when the magnitude of random deviations of waveguide parameters exceeds a certain critical value. The considered effect is the optical analog of the electron localization in a disordered lattice. The diffusional spreading of a wave in such a system in the presence of random inhomogeneities varying along the wave propagation direction is studied. 6. The nonlinear dynamics of rays in waveguide channels with periodic inhomogeneities along the wave propagation direction is investigated. The new fundamental effects are found. The first of them is the modulation localization of rays that is an analog of the nonlinear resonance in classical mechanics. The second effect is the formation of stochastic area in a waveguide channel due to the stochastic instability of rays. This phenomenon leads to decrease of the effective transversal width of waveguide channel. It is shown the possibility of ray stochasticity in three–dimensional waveguide channels that are homogeneous along the waveguide axis.

Acknowledgments Using the opportunity, it is the author’s pleasant duty to express his sincere gratitude to his supervisor, the corresponding member of the Academy of Sciences of Uzbekistan, P.K. Khabibullaev for supervising the thesis, the academician of the Academy of Sciences of Uzbekistan S.Kh. Sirazhdinov for the support and advises, doctor of physical and mathematical sciences G.M. Zaslavsky and candidate of physical and mathematical sciences F.Kh. Abdullaev for their help, valuable advises, and fruitful discussions.

88

Appendix 1

We will make use of the Poisson summation rule b X

f (n) =

n=a

Zb+δ ∞ X

(IP.1)

dnf (n) cos(2πns),

s=−∞ a−δ ′

0 where 0 < δ, δ ′ < 1, δ is selected from the condition k0 σN = π(N + δ − 1/2), δ ′ = 1/2,

and N is the number of the highest excited mode. Then ǫ1 = n20 , k0 σǫ01 = 0 and pǫ01 = 0 (see Ref. [49]). Using (IP.1), we write (1.3.11) in the following form: (0)

(1)

(2)

Γ12 = Γ12 + Γ12 + Γ12 , where (0) Γ12

=

N Z +δ

dn

δ′

(1) Γ12

L0n (pn1 pn2 )1/2

N Z +δ ∞ X = dn s=−∞

(1) Γ12

βn

=

∞ X

δ′ N Z +δ

dn

s=−∞

δ′

cos [k0 (σn1 − σn2 )] ,

βn L0n

(pn1 pn2 )

(0)

1/2

βn L0n

(pn1 pn2 )1/2

cos [k0 (σn1 − σn2 ) + 2πsn] − Γ12 , sin [k0 (σn1 + σn2 ) + 2πsn] .

Extending the quantization rule (1.3.10) to continuous values of n and differentiating it with respect to n, we obtain dǫ 2π =− 0 . dn Ln k0 Using the last relation and (1.3.10), we obtain 2

(0)

Γ12

Zn∞ β(ǫ) k0 dǫ cos [k0 (σǫ1 − σǫ2 )] , =− 2π (pǫ1 pǫ2 )1/2 n20

89

(IP.2)

2

(1)

Γ12 = −

k0 2π

∞ X

s=−∞

Zn∞ (−1)s dǫ n20

β(ǫ) (pǫ1 pǫ2 )1/2


 (0) cos k0 σǫ1 − σǫ2 + 2sσǫ0 − Γ12 ,

(IP.3)

2

(2)

Γ12 = −

k0 2π

∞ X

Zn∞ s (−1) dǫ

s=−∞

n20

β(ǫ) (pǫ1 pǫ2 )1/2


 sin k0 σǫ1 + σǫ2 + 2sσǫ0 .

(IP.4)

We carry out a further calculation for the step function β(ǫ) (1.3.12). Using the relation
d [k0 (σǫ1 ± σǫ2 + 2sσǫ0 )] k0 =− Lǫ1 ± Lǫ2 + 2sL0ǫ , dǫ 2

we perform in (IP.2) to (IP.4) a single integration by part. Then the main contribution to Γ12 is given by the term outside the integral at the upper limit, the leading term in ξ , (0)

b 1 sin [k0 (σF 1 − σF 2 )] , 1/2 4π (pF 1 pF 2 ) LF 1 − LF 2 ∞ 0 X 1 b (0) s sin [k0 (σF 1 − σF 2 + 2sσǫ )] − Γ12 , (−1) = 0 1/2 4π (pF 1 pF 2 ) s=−∞ LF 1 − LF 2 + 2sLF

Γ12 = (1)

Γ12

(2) Γ12

∞ 0 X 1 b s cos [k0 (σF 1 + σF 2 + 2sσǫ )] . (−1) =− 4π (pF 1 pF 2 )1/2 s=−∞ LF 1 + LF 2 + 2sL0F

Quantities with the subscript F are evaluated at ǫ = ǫF . Making use of the sums i h x ∞ (2mπ − b) sin a + X y sin(a + sb) π (−1)s = , x + sy y sin(πx/y) s=−∞ h i x ∞ X π cos a + y (2mπ − b) cos(a + sb) = , (−1)s x + sy y sin(πx/y) s=−∞ where (2m − 1)π ≤ b ≤ (2m + 1)π, m = 0, 1, 2, . . . , and x/y is not an integer, we obtain Eqs. (1.3.13)-(1.3.16).

90

Appendix 2

Diagram technique for the mean field We rewrite Eq. (2.1.6) in the form han (z)i =

X n′

Gnn′ (z)ha0n i,

(2P.1)

where Gnn′ (z) is the Greene function of the corresponding problem in the presentation of eigenfunctions of normal modes of the waveguide, Gnn′ (z) = eikn z h[S(z)]inn′ .

(2P.2)

To derive the equation for Gnn′ (z) its convenient to work with its Laplace transform, Z∞ e (2P.3) Gnn′ (s) = e−sz Gnn′ (z)dz. 0

According to (2.1.3) in Eq. (2P.2) there are the mean values of the products of different number of the random functions Vnn′ (zi ) with arguments zi in the interval [0, z]. Using that the random functions Vnn′ (z) are the Gaussian random functions with hVnn′ (z)i = 0, these products can be presented as the sum of all possible combinations of the products of the mean values hVnn′ (z1 )Vmm′ (z2 )i. For the even number of functions, we have hV (z1 )V (z2 ) · · · V (z2N )i = hV (z1 )V (z2 )i · · · hV (z2N −1 )V (z2N )i + · · · + hV (z1 )V (z2N )i · · · hV (z2 )V (z2N −1 )i,

(2P.4)

where there are (2N − 1)!! terms. The mean value of the product of odd number of functions V (z) is equal to zero. The each term in the expansion of the Greene function in a series obtained by Eq. (2.1.3) and (2P.4) it is convenient to present by a separate diagram. Below we describe the rules of the diagram technique. 91

A. The rules of diagram technique For the graphical presentation of each terms in the expansion of the function Gnn′ (z) we introduce a horizontal segment [0, z] (see Fig. 1) n′

n1

ni

n

r -

r -

r -

r

0

z1

z2

zi

-

r r -

r

zN

z

Figure 1 The integration variables zi are given by points which take the values in the following interval 0 6 z1 6 z2 · · · 6 zN 6 z. Since the mean value of the product of the odd number of functions V (zi ) vanishes, then the number of integration points N should be even. The integration points zi (i = 1, 2, . . . , zN ) will be called internal, in contrast to the external points 0 and z. The number N is an order of the diagram. The segment between two sequential points correspond to the modes. The external segments (0, z1 ) and (zN , z) correspond to initial n′ and final n modes, respectively, while the segments (zi , zi+1 ) connecting internal points present the intermediate modes ni . The mean values of type hVni ni−1 (zi )Vnj nj−1 (zj )i are presented by dashed lines connecting internal points zi and zj . The number of diagram of the 2N −th order is equal to (2N − 1)!!. Figure 2 shows a several diagrams of the 2−nd (a), 4-th (b,c), and 8-th (d) orders.

(a)

(b)

(c)

(d) Figure 2:

92

We formulate the rules of writing down of analytical expressions corresponding the arbitrary diagram. These rules will be called the diagram technique rules on the z−presentation. To write an arbitrary term it is necessary: 1. To each solid segment located between points zi and zi+1 corresponds the function Gn (zi+1 − zi ) = exp [ikn (zi+1 − zi )] . 2. To each dashed line connecting internal points zi and zj corresponds the term hVni ni−1 (zi )Vnj nj−1 (zj )i. 3. One should perform summation over intermediate modes n and integrate with internal points zi as a coupled integral Zz2 Zz ZzN dzN dzN −1 · · · dz1 . 0

0

0

4. The obtained expressions is multiplied by iN and taken a Laplace transform.

n1

n´ z1

0

n2 z2

n3 z3

n z4

z

Figure 3:

We give an example of analytical presentation of the diagram shown in Fig. 3: ∞ Zz2 Zz3 Zz4 Zz XXXZ (4) −sz e ′ (s) = G dze dz4 dz3 dz2 dz1 nn n1

n2

n3 0

0

0

0

0

× eikn (z−z4 ) eikn3 (z4 −z3 ) eikn2 (z3 −z2 ) eikn1 (z2 −z1 ) eikn′ (z−0) × hVnn3 (z4 )Vn2 n1 (z2 )ihVn3 n2 (z3 )Vn1 n′ (z1 )i.

In the zero approximation, we have e(0)′ (s) = G nn

1 δnn′ . s − ikn

enn′ (s) is given only In the second order of the perturbation theory the contribution to G

by one diagram shown in Fig. 2a. According to the diagram rules we have # " Z X e ′ Ψ (u) 1 1 du np,pn e(2)′ (s) = , G i2 nn s − ikn p C 2πi s − u − ikp s − ikn′ 93

(2P.5)

where e np,p′ n′ (u) = Ψ

Z∞

dτ e−uτ hVnp (z)Vp′ n′ (z + τ )i.

(2P.6)

0

During the calculation of (2P.6) the integral have been used Z∞ 0

dze−sz

Zz 0

dzn · · ·

Zz2

dz1 e−[an zn +···+a1 z1 ] =

0

1 1 1 . ··· s s − an s − (an + · · · + a1 )

(2P.7)

e(2)′ (s) gives a hint to another, a more convenient formuThe expression (2P.5) for G nn

lation of the rules of diagram technique which can be called the diagram technique in “s−”presentation. These rules are the following. 1. Each segment located between two neighboring internal points is characterized by “energy” s and mode number n. To this segment corresponds the quantity e(0) G n (s) =

1 s − ikn

2. The dashed line is characterized by “energy” u, which is “radiated” from or “absorbed” by the vertex. To the dashed line connecting the vertices (n, p) and (p′ , n′ ) e np,p′ ,n′ (see Fig.4). corresponds the term Ψ u

n´

p´

p

n

Figure 4:

3. To each vertex corresponds the term (−i) and the energy conservation law s = s′ + u is satisfied u -

s



-

s−u

4. One should carry out a summation over intermediate modes ni and the integration over the “energy” u of dashed lines by multiplying to 1/(2πi).

94

u

(s,n´)

(s-u,n1)

v

(s-u-v,n2)

(s-v,n3)

(s,n)

Figure 5:

As an example we write down the diagram shown in Fig. 5:  X X X Z du Z dv 1 1 (4) 4 e ′ (s) = i G nn C 2πi C 2πi s − ikn′ s − u − ikn1 n1 n2 n3 e nn3 ,n2 n1 (v)  1 e n3 n2 ,n1 n′ (u) Ψ Ψ . × s − u − v − ikn2 s − v − ikn3 s − ikn B. The Dyson equation Let us turn to the classification of diagrams. All diagrams can be divided into classes according to the presence in them so–called “free sections”, which divide the diagram into two parts by vertical line without crossing any dashed lines. In Fig. 2 a several diagrams with the different number of free sections were shown: with no free sections (a,b), with e(0) one (c), and two (d). To the solid line corresponds the Greene function G n (s). e f (s) be We formulate directly the equation for the full Greene function G(s). Let W

the sum of all diagrams without free sections and solid segments at the initial and final states,

W

(2P.8)

Then the Dyson equation for the full Greene function has the form # " X fnn′′ (s)G en′′ n′ (s) . enn′ (s) = G e(0) W δnn′ + G n

(2P.9)

n′′

f (s) is performed in a series of powers of the paOne should note that the expansion W

rameter δ = k0 σlk where σ− is the intensity of medium fluctuations, lk is the longitudinal correlation length of medium fluctuations. 95

Using (2P.1) and (2P.9), we obtain Eq. (2.1.9).

96

Appendix 3

Diagram technique for the second moments of field 1

Diagram technique rules

We rewrite the Laplace transform of the quantities Pnm (z) (2.1.7) in the form

where n ′ m′

fnm (s) = M

Z∞

X

n ′ m′ fnm M (s)Pn′ m′ (0),

(3P.1)

e−sz−i(kn −km )z h[S(z)]∗nn′ [S(z)]mm′ i.

(3P.2)

Penm (s) =

n ′ m′

0

The quantity (3P.2) contains two expansions of type (2.1.3). Therefore for the diagram f one introduces not only one segment as in the case of the presentation of the quantity M

mean field, but two horizontal segments of the interval [0, z] (Fig. 5) n′

s z1′

0

s n z2′

m′

z

m s z1

0

s z2 Figure 5

z

The lower segment is used for the expansion of S(z) and the upper segment for S ∗ (z). f into a series, obtained using (2.1.3), (2P.4) and The each terms of the expansion of M

(3P.2), are presented by separate diagrams. In the case, beside the diagrams with dashed lines connecting points on the same horizontal segments, it appears also diagrams with 97

dashed lines connecting points on the different horizontal segments. During the investigation of such diagrams we encounter integrals containing two independent groups of coupled integrals of type Zz

dzp · · ·

dz1

zj′

Zz

′

dzk′ · · ·

Zz2

(3P.3)

dz1 ,

0

0

0

0

where zi (i = 1, 2, · · · , p) and

Zz2

(j = 1, 2, · · · , k) are the integration variables represented

by dots on the lower and the upper segments, respectively. Such integrals should be rewritten in the form of orderly coupled integrals. The integrals of type (3P.3) can be always rewritten as the sum of certain number of ordered integrals. The number of the orderly coupled integrals obtained from (3P.3) equals to (p + k)!/(p!k!). For example, the double integral Zz

dz1

0

Zz

dz1′

0

can be written as sum of two orderly coupled integrals Zz 0

dz1

Zz1

dz1′ +

0

Zz

′

dz1′

0

Zz1

dz1 .

0

Figure 6 shows the diagrams corresponding to these integrals. z1′ @ @

z1′ @

@

@ @ z1 (a)

z1 (b)

Figure 6

We introduce the notation of neighbor points. It is determined by the distance between vertical lines, plotted through the corresponding points. In the diagram shown in Fig. 7 the neighbors of the point z1 are 0 and z2 . z1

0

z2 Figure 7

We formulate the diagram technique rules in the “s−”presentation. 98

1) To each pair of solid horizontal segments located between neighboring points and characterized by “energy” s corresponds the quantity 1 . s − i∆knm

f(0) (s) = M nm

2) The dashed line, similar to the case of the mean field, characterized by the “energy” u, which absorbed by or radiated from the pair of the solid segments. To the dashed line corresponds the quantity n′

n @

-

e mm′ ,nn′ (u) Ψ

@ u R @ @ @

m′

@ @

m

3) To each internal point on the upper segment corresponds the multiplier (−i), while to the lower segment corresponds the multiplier (+i). 4) The summation over all intermediate modes pi and the integration over “energy” u of dashed lines with the weight 1/2πi are carried out. As an example we write the diagram shown in Fig. 8. The expression corresponding this

n1 n2

n´ u

n

v

m´

m Figure 8

diagram is given by n ′ m′ fnm M (s)

2

=

XX n1

n2

×

i(−i)

3

Z

C

du 2πi

Z

C

e m′ m,n2 n1 (u) Ψ s − u − v − i∆kn1 m

f Equation for the function M



1 s − u − i∆kn′ m  e nn2 ,n1 n′ (v) Ψ 1 . s − v − i∆kn2 m s − i∆knm

dv 1 2πi s − i∆kn′ m′

The classification of diagrams is carried out as in the case of the mean field (see Apf is derived similar to the Dyson equation for the Greene pendix 2). The equation for M 99

function (see Ref. [73]). It is written in the form " # X 1 ′ ′ ′′ ′′ ′ ′ fn m (s) = f n m (s)M fn′′m ′′ (s) , M W δnn′ δmm′ + nm nm n m s − i∆knm n′′ m′′

(3P.4)

n ′ m′ fnm (s) is the sum of all diagrams without free sections, where W

W

+

+

+

Using (3P.1) and (3P.4) we obtain Eq. (2.2.1).

100

+

+ . . .

. (3P.5)

Appendix 4

Using the expansion (1 − x)

−1/2

=

∞ X Γ(n + 1/2) n=0

we write

n!Γ(1/2)

xn ,

∞ X Γ(n + 1/2) sin x (1 − β 2 )n sin2n+1 x. U (x) = x 2 2 cos x + β sin n=0 n!Γ(1/2)

Next, using the expansion sin

2n+1

we obtain U (x) =

∞ X

∞ (2n + 1)! 1 X (−1)m sin [(2m + 1)x] , x = 2n 2 m=0 (n − m)!(n + m + 1)!

Am sin [(2m + 1)x] ,

m=0

Am = (−1)

m 2(1

− β 2 )m Γ(m + 1/2)Γ(m + 3/2) F [Γ(1/2)]2 Γ(2m + 2)



 3 1 2 m + , m + , 2m + 2, 1 − β . 2 2

We investigate the asymptotic form of Am as β → 0. We use for this purpose the integral representation of the hypergeometric function Z1 Γ(γ) uα−1 (1 − u)γ−α−1 (1 − ux)−β du. F (α, β, γ, x) = Γ(β)Γ(γ − β) 0

Then 2 Am = (1 − β 2 )m π

Z1  0

u(1 − u) 1 − u + β 2u

m+1/2

du . 1−u

At β = 0 the integral diverges. We calculate the integral in Am by the saddle-point method in the asymptotic limit β → 0. We write it in the form "  m+1/2 # Z1 1 u(1 − u) eϕ(u) du, ϕ(u) = ln . 1 − u 1 − u − βu 0

101

At m ≫ 1 and mβ ≪ 1 we get Z1 0

eϕ(u) du ≈

 π 1/2 1 . 2 mβ

Thus  1/2 1 2 (1 − β 2 )m . Am ≈ π mβ Starting with certain m ≥ N , the spectrum of Am decreases exponentially. For the characteristic number N we have 1 ≈ 1. N ≈ 2 ln(1 − β ) β 2

102

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