Phenomenological three-layer model for surface second ... - CiteSeerX

Phenomenological three-layer model for surface second-harmonic generation at the interface between two centrosymmetric media Pierre F. Brevet Laboratoire d' Electrochimie, Ecole Polytechnique Fidhale de Lausanne, CH-1015 Lausanne, Switzerland

A three-layer model is presented to account within a single framework for the different contributions to the surface secondharmonic response at the interface between two centrosymmetric media. It is shown that the optical relative permittivity for the intermediate layer can be incorporated in the geometrical factors of the second harmonic response. The problem of the bulk non-local contribution is then discussed within the same formalism and the associated response incorporated into the effective susceptibility tensor. Three particular cases are given as application of the model to experimental procedure.

1. Introduction Surface second-harmonic generation (SSHG) has been a field of growing interest in recent years, principally due to its inherent surface ~pecificity.'-~Surface science has found there a powerful tool to study interfacial changes, not only at free surfaces, like vacuum/solid or air/liquid interfaces, but also at buried interfaces like metal/electrolyte or liquid/liquid interfaces. Experimental work carried out on these systems has brought new insights into fundamental surface science and SSHG now starts to be applied as an analytical probe, as shown by the work of Tohda et al. at liquid/liquid interfaces6 From the theoretical point of view, two approaches have been undertaken. Bloembergen et al. derived their model for a thick non-linear slab at the interface between two linear media as early as the end of the 1 9 6 0 Here, ~ ~ we understand 'thick' as large compared to the wavelength of light. Since the thickness of the slab was subsequently allowed to be of the order of, or even smaller than, the wavelength of light, simplifications in the field amplitude were deduced. Results gave a true account for the SH generation within a thick optically non-linear slab as well as for a thin slab, although the SH response in the slab was reduced to only one component of the multiple reflections among the four components of the non-linear polarisation. However, it was then considered that for SSHG, a sheet of non-linear polarisation could be used instead of a non-linear slab of vanishing thickness.' Introducing standard linear Fresnel coefficients of reflection and refraction, Sipe and then Mizrahi and Sipe, proposed a phenomenological model, more dedicated to real experimental geometrie~.~,'It was indeed argued in this work that the extensive use of Fresnel coefficients facilitated the understanding of the SH generation. This latter model also accounted for the non-local contributions as well as multiple reflections in the vicinity of the non-linear polarisation sheet. However, results were given for air/medium systems thus only introducing the non-local contribution from one single phase. It was later pointed out by Guyot-Sionnest and Shen" that all previous derivations lacked a term in the non-local description of the non-linear surface polarisation source, namely the gradient of the electric quadrupole susceptibility tensor. This deficiency was thus remedied and the missing part of the non-linear source reinstated, but the formalism presented was different than the one introduced by Mizrahi and Sipe. Nevertheless, and although it is possible to clearly account for all the local and non-local contributions of the polarisation source, a complete treatment within the phenomenological model never appeared in the literature.

Recent reviews on SSHG, although discussing the different contributions, have not done The phenomenological model yielded the theoretical background for the electric dipole response, but non-local contributions from the bulk were discussed with other more adequate models. Therefore to date, the work of Mizrahi and Sipe is the only attempt to derive the complete formalism with the phenomenological model. However, it was not derived for buried interfaces since one non-local contribution was included. Also, it did not include the contribution due to the susceptibility tensor gradient. It is thus the aim of this work to develop a unique framework for the phenomenological model within which a complete account for all the contributions is given. This lack of uniqueness has been easily overcome in previous studies since the two contributions were usually of different magnitudes. Experiments were, most of the time, clearly separating the local electric dipole contribution from the non-local electric quadrupole contribution. At liquid surfaces and interfaces, in the case when large non-linear active adsorbates are used, the non-local electric quadrupole contribution is negligible. However, when a fraction of a monolayer is present, it is possible to get interfering contributions from both origins. In the past, only the magnitude of these contributions was of interest and thus it could be discussed with a mixed approach. SSHG has been shown to be an extremely surface sensitive technique, therefore it is reasonable to believe that in the near future we may require a unified approach. This question will surely arise in analytical applications when quantitative considerations on sensitivity will be required. Finally we would like to emphasize that the model described in this work has never been presented before in detail, although it is closely related to the phenomenological model derived by Mizrahi and Sipe. Also, it includes the optical constants of the inner slab within corresponding Fresnel coeficients in a rather straightforward way as compared to previous formulation^.'^ The importance of the inclusion of the optical relative permittivity has already been emphasised for the case of Langmuir-Blodgett films, for e~ample.'~.'The geometric description of the system at hand is given in the next section. A three-layer geometry is used where the inner slab is a linear slab of vanishing thickness embedding the sheet of non-linear polarisation. This system offers the possibility to give definite optical relative permittivity values to the region of non-linear activity, their occurrence in the geometric factors then arising naturally. Such a model is desirable in resonant SSHG studies of monolayers. This is the case at air/liquid or liquid/liquid interfaces, for example, where adsorbate monolayers are used to study J . Chem. SOC., Faraday Trans., 1996,92(22), 4547-4554

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interfacial properties, e.g. adsorption isotherms or surface pH.16.17In Section 3, we give the surface polarisation source expression and give the corresponding SH field amplitude. We take the opportunity here to present the theory within the convention proposed by the International System of units. In Section 5, we derive the SH response in the electric dipole approximation. The derivation is rather close to the one given by Mizrahi and Sipe, although the three-layer model leads to different expressions for the geometrical parameters. We emphasise here that we exclusively discuss SSHG and not SHG from a thick slab, since the inner slab has a vanishing thickness in our model. The problem of the SHG generation from a thick slab has already been accounted for in previous work through phase shifts upon reflection." In Section 4, we discuss the problem of the non-local origin of the non-linear polarisation source, both for the field gradients and the susceptibility tensor gradients within this model.

2. Geometric description The geometry we describe now is general, and can be extended to any system where the source of non-linear polarisation is taken in a medium with a thickness much smaller than the harmonic wavelength. This case can therefore be applied to adsorbed monolayers on solid substrates or liquids, and is the topic we will focus on. In this case, the effect of the optical spectrum of the monolayer can easily be introduced. Such a surface spectrum is available through Kramers-Kronig analysis, as shown for a monolayer of p-nitrophenol by Higgins et a1." The three-layer model is described in Fig. 1 and consists of two half spaces constituting the optically linear media 1 and 2 of optical relative permittivities E:, 8 7 and c:, t$ at the fundamental 0 and harmonic D = 2 0 frequencies, respectively. In the interfacial region, between z = t , and z = t,, t, < 0, a sheet of non-linear polarisation lies at z = 0. The thickness of the slab is small compared to the wavelength of light, thus we have t, + t, < A. The slab is characterised by its own optical relative permittivities E" and E" at the fundamental and harmonic frequencies respectively. The laboratory frame is oriented with the surface normal along the .? axis, pointing upwards, and the -2 axis along the interface. The j axis is then chosen to define a direct (-2, 9, 2) frame. The three-layer system is now illuminated by the fundamental wave from the upper side, which impinges at the interface

with an angle of incidence 8.: Due to the laws of reflection and refraction, the fundamental beam propagates in the inner slab with the angle 8" and in medium 2 with the angle 0;. In the case where Re(&:) > Re(&"),total internal reflection (TIR) is obtained on the fundamental beam at the interface between medium 1 and the inner slab. If Re(&")> Re(&:), TIR is obtained at the interface between the inner slab and medium 2. The different harmonic beams propagate with angle 87, 87 or 8" in medium 1, 2 and the inner slab respectively. TIRSSHG at the interface between the inner slab and medium 2 is thus obtained if the condition sin 0; > 1 is fulfilled.,' The TIR condition can be achieved on the fundamental beam but not on the SSH beam, depending on the dispersion in the different media. In this particular case, the SSH beam will still exhibit a large enhancement due to the linear fundamental TIR geometry and will still be observed in transmission if medium 2 is transparent. Multiple reflections at both the medium l/inner slab and inner slab/medium 2 interfaces can easily be taken into account, but following Mizrahi and Sipe, we restrict the derivations to the first reflection only. We also stress here that we assume isotropic optical relative permittivities. If this is an obvious condition to set in media 1 and 2, on the other hand, this could be a drastic assumption for the inner slab constant if it is constituted by a monolayer of oriented organic molecules for example. Finally, we underline here that all the derivations are taken in the International System of units.

3. The non-linear polarisation source We start from the general form of the non-linear polarisation source, as found in the l i f e r a t ~ r e . ~ In ~ ~homogeneous, ~'~'~~ non-magnetic, centrosymmetric media, and for incident monochromatic plane waves, the complete form of this polarisation P(:d(r, t ) is the superposition of three contributions: one local contribution, also known as the electric dipole contribution, and two non-local contributions of electric quadrupole origin. In this model, we truncate the polarisation P:;(r, t ) to the first term involving the electric quadrupole contributions. It has long been recognized that many different conventions were used throughout in the literature. This multiplicity has also been an impediment to comparison of the quantities either calculated or measured. In order to clarify the situation, we make clear here that we follow the IUPAC convention established for these electromagnetic quantities.,, Hence the polarisation is :

0 2

t

medium 1

z = t,

-D ox 0 oy

z = t,

medium 2

Fig. 1 General description for a phenomenological three-layer model of surface second harmonic generation at the boundary between two centrosymmetric media. Linear media 1 and 2 are separated by a thin layer of linear material embedding the sheet of non-linear polarisation. Optical constants as well as the different angles are given.

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J . Chem. SOC.,Faraday Trans., 1996, Vol. 92

this form being valid for both the time- and the frequencydependent functions. c0 is the vacuum permittivity. Unfortunately convention (1) introduces l/n ! as numerical factors for processes of order higher than the linear polarisation. This factor is cumbersome to carry over. Hence, we introduce new tensors, x("), related to the tensors lo("), the true IUPAC compliant tensors for the nth non-linear optical orders, used in eqn. (1). The new tensors f " ) , which we are going to use throughout this work, are:

The different fields used in this work, either the electric field or the polarisation, follow the convention with a 1/2 factor. Hence, we write : 1 q r , t) = 2 [E(v, t )

+ E*(r, t ) ]

(3)

see Willets et ~ 1 for. example. ~ ~ Finally, to avoid the complications of discontinuities for the second-order susceptibility

where the double points underline the product between a tensor and two vectors. We have emphasised the zdependence of the electric field vector for clarity but at this stage of the derivation, it only indicates that the fields are taken within the inner slab. In eqn. (7), we have also used the component IC along the plane of the interface of the fundamental wavevector k". This component is taken as IC 11 Ox and the two-dimensional space vector R lies in the plane of the interface, that is in the (2, j ) plane. We have:

t , + t,