Reflection and Transmission of a Plane TE-Wave at a Lossy, Saturating, Nonlinear Dielectric Film

Reflection and transmission of transverse-electric (TE) electromagnetic waves at a single nonlinear homogeneous, isotropic, nonmagnetic layer situated between two homogeneous, semi-infinite media has been the subject of intense theoretical and experimental investigations in recent years. In particular, the Kerr-like nonlinear dielectric film has been the focus of a number of studies in nonlinear optics (Chen & Mills, 1987; 1988; Leung, 1985; 1988; Peschel, 1988; Schurmann & Schmoldt, 1993).


Introduction
Reflection and transmission of transverse-electric (TE) electromagnetic waves at a single nonlinear homogeneous, isotropic, nonmagnetic layer situated between two homogeneous, semi-infinite media has been the subject of intense theoretical and experimental investigations in recent years.In particular, the Kerr-like nonlinear dielectric film has been the focus of a number of studies in nonlinear optics (Chen & Mills, 1987;1988;Leung, 1985;1988;Peschel, 1988;Schürmann & Schmoldt, 1993).
As Chen and Mills have pointed out it is a nontrivial extension of the usual scattering theory to include absorption (Chen & Mills, 1988) and it seems (to the best of our knowledge) that the problem was not solved till now.In the following we consider a nonlinear lossy dielectric film with spatially varying saturating permittivity.In Section 2 we reduce Maxwell's equations to a Volterra integral equation ( 14) for the intensity of the electric field E(y) and give a solution in form of a uniform convergent sequence of iterate functions.Using these solutions we determine the phase function ϑ(y) of the electric field, and, evaluating the boundary conditions in Section 3, we derive analytical expressions for reflectance, transmittance, absorptance, and phase shifts on reflection and transmission and present some numerical results in Section 4.
It should be emphasized, that the contraction principle (that is used in this work) (Zeidler, 1995) includes the proof of the existence of the exact bounded solution of the problem and additionally yields approximate analytical solutions by iterations.Furthermore, the rate of convergence of the iterative procedure and the error estimate can be evaluated (Zeidler, 1995).Thus this approach is useful for physical applications.
The present approach can be applied to a linear homogeneous, isotropic, nonmagnetic layer with absorption.In this case the problem is reduced to a linear Volterra integral equation that can be solved by iterations without any restrictions.The lossless linear permittivity as well as the Kerr-like permittivity can be treated as particular cases of the approach.
Referring to figure 1 we consider a dielectric film between two linear semi-infinite media (substrate and cladding).All media are assumed to be homogeneous in x− and z− direction, isotropic, and non-magnetic.The film is assumed to be absorbing and characterized by a complex valued permittivity function ε f (y).A plane wave of frequency ω 0 and intensity E 2 0 , with electric vector E 0 parallel to the z-axis (TE) is incident on the film of thickness d.Since the geometry is independent of the z-coordinate and because of the supposed TE-polarization fields are parallel to the z-axis (E =(0, 0, E z )).We look for solution E of Maxwell's equations rotH = −iω 0 εE rotE = iω 0 µ 0 H that satisfy the boundary conditions (continuity of E z and ∂E z /∂y at interfaces y ≡ 0 and y ≡ d) and where (due to TE-polarization) H =( H x , H y ,0).Due to the requirement of the translational invariance in x-direction and partly satisfying the boundary conditions the fields tentatively are written as ( ẑ denotes the unit vector in z-direction) where E(y), p = √ ε c k 0 sin ϕ, k 0 = ω 0 √ ε 0 µ 0 , q c , q s , and ϑ(y) are real and E r = |E r | exp(iδ r ) and We assume a permittivity ε(y) of the three layer system modeled by A particular nonlinearity in (2) of cubic type (r = 0) can be met in the context of a Kerr-like nonlinear dielectric film, while the case when r > 0 corresponds to the saturation model in optics (see (Bang et al., 2002;Berge et al., 2003;Dreischuh et al., 1999;Kartashov et al., 2003)).
The problem to be solved is to find a solution of Maxwell's equations subject to (1) and ( 2).With respect to the physical significance of ( 1) and ( 2) some remarks may be appropriate.Though ansatz (1) widely has been used previously (Chen & Mills, 1987;1988;Leung, 1985;1988;Peschel, 1988;Schürmann & Schmoldt, 1993) it should be noted that it is based on the assumption that the time-dependence of the optical response of the nonlinear film is described by one frequency ω 0 .Phase matching is, e.g., assumed to be absent so that small amplitudes of higher harmonics can be neglected.The permittivity function (2) also represents an approximation.The dipole moment per unite volume and hence the permittivity is not simply controlled by the instant value of the electric (macroscopic) field at the point (x, y, z), due to the time lag of the medium's response.Further more the response is nonlocal in space.-The model permittivity (2) does not incorporate these features.Nevertheless, experimental observations (cf., e.g.(Peschel, 1988)) indicate that (2) has physical significance (with ε R = ε I = r = 0).Finally, Maxwell's equations, even for an isotropic material, imply that all field components are coupled if the permittivity is nonlinear.The decomposition into TE-and TM-polarization is an assumption motivated by mathematical simplicity.To apply the results below to experiments it is necessary to make sure that TE-polarization is maintained.

Nonlinear Volterra integral equation
By inserting (1) and (2) into Maxwell's equations we obtain the nonlinear Helmholtz equations, valid in each of the three media (j = s, f , c), where Ẽj (x, y) denotes the time-independent part of E(x, y, t).
Scaling x, y, z, p, q c , q s by k 0 and using the definition of ε ε 0 in equation (2), equation (3) reads where the same symbols have been used for unscaled and scaled quantities.Using ansatz (1) in equation ( 4) we get for the semi-infinite media

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Reflection and Transmission of a Plane TE-Wave at a Lossy, Saturating, Nonlinear Dielectric Film For the film (j = f ), we obtain, omitting tildes, and Equation ( 7) can be integrated leading to where c 1 is a constant that is determined by means of the boundary conditions: Insertion of dϑ(y)/dy according to equation ( 3) leads to As for real permittivity, real q s (transmission) implies c 1 = 0.
Setting I(y)=aE 2 (y), a = 0, multiplying equation ( 10) by 4E 3 (y), and differentiating the result with respect to y we obtain Equation ( 12) can be integrated with respect to I(y) to yield where κ 2 = ε 0 f − p 2 and c 2 is a constant of integration.In the case a = 0 (it corresponds to the linear case with absorption in Eq. ( 2)) we obtain the following analog of ( 13) where I(y) denotes E 2 (y).Later on for the case a = 0 under I(y) we always understand E 2 (y).
The homogeneous equation d 2 I(y)/dy 2 + 4κ 2 I(y)=0 which corresponds to Eq. ( 13) has the solution that satisfies the boundary conditions at y = 0 so that the general solution of equation ( 13) reads (Stakgold, 1967) where the constant c 2 must be determined by the boundary conditions.
In the case a = 0 the general solution of equation ( 14) reads The Volterra equations ( 16), ( 17) are equivalent to equation (3) for 0 < y < d for a = 0 and a = 0, respectively.According to equations ( 16) and ( 17) I(y) and Ĩ0 (y) satisfy the boundary conditions at y = 0. Evaluating some of the integrals on the right-hand side, equations ( 16) Reflection and Transmission of a Plane TE-Wave at a Lossy, Saturating, Nonlinear Dielectric Film www.intechopen.comand ( 17) can be written as and respectively, with [on the evaluation of c 2 see Appendix B] (in the case a = 0) where Ĩ0 (y) is given by equation ( 15), and with (in the case a = 0) where Ĩ0 (y)=|E 3 | 2 cos(2κy).
Iteration of the nonlinear integral equations ( 16) and ( 17) leads to a sequence of functions I j (y),0 < y < d.Subject to certain conditions it can be shown that the limit exists uniformly in 0 < y < d and represents the unique solution of ( 16) and ( 17).The error of approximations can be expressed in terms of the parameters of the problem (see ( 68) and ( 69) in Appendix A).
Iterating ( 16) and ( 17) once by inserting I 0 (y) according to ( 20) and ( 23), the first iteration I 1 (y) reads (a = 0) and (a = 0) I 1 (y) is used for numerical evaluation of the physical quantities defined in the following section.

Reflectance, transmittance, absorptance, and phase shifts
Conservation of energy requires that absorptance A, transmittance T, and reflectance R are related by with for a = 0 and a = 0, respectively.

Numerical evaluations
A numerical evaluation of the foregoing quantities is straightforward.It is useful to apply a parametric-plot routine using the first approximation I 1 (y).If the parameters of the problem (a, r, ε R , ε I , ε s , ε 0 f , ε c , p, d, ω 0 ) satisfy the convergence conditions ( 48) and (49) (see Appendix A) the results obtained for I 1 (y) are in good agreement with the purely numerical solution of equation ( 13) and ( 14) (cf.figure 3).26) and dashed curve to the numerical solution of the system of differential equations ( 6), ( 7).
(i) Prescribe the parameters of the problem such that ( 48) and ( 49) are satisfied.
(ii) Prescribe a certain upper bound (accuracy) of the right-hand side R j (see (66) in Appendix A) and perform a parametric plot of R j (with I(0) as parameter) with j=1.If R 1 is smaller (or equal) than (to) the prescribed accuracy for all aE 2 0 (or E 2 0 ) of a certain interval, accept I 1 (y) as a suitable approximation.
(iii) If R 1 exceeds the prescribed accuracy calculate I 2 (y) according to (50) and check again according to step (ii) or enlarge the accuracy so that R 1 is smaller (or equal) than (to) the prescribed accuracy.
The reason for the satisfactory agreement between the exact numerical solution and the first approximation I 1 (y) (cf.figure 3) is due to the foregoing explanation.
If a|E 3 | 2 (or |E 3 | 2 ) is fixed (as in the numerical example below), and thus aE 2 0 (or E 2 0 ) according to (33) (or (35)), the inequality (68) (or ( 69)) can be used to optimize the iteration approach with respect to another free parameter, e.g., d or r or p, as indicated.
Using the first approximation the phase function can be evaluated according to equation ( 8) as (for all values of a) where for a = 0 and a = 0, respectively.Thus, the approximate solution of the problem is represented by equations ( 26) or ( 27) and ( 41).The appropriate parameter is I(0)=aE 2 (0) or I(0)=E 2 (0), since E 0 in equation ( 42) can be expressed in terms of I(0) as shown in ( 33) or ( 35).
For illustration we assume a permittivity according to (a = 0) where ε 0 f , γ, b, d, r are real constants.For simplicity, ε I is also assumed to be constant.Results for the first iterate solution I 1 (y, aE 2 0 ) are depicted in figures 2, 3. Using I 1 (y, aE 2 0 ), the phase function ϑ 1 (y, aE 2 0 ), absorptance A 1 (d, aE 2 0 ) and phase shift on reflection δ r1 (y, aE 2 0 ) are shown in figures 4, 5 and 6, respectively.The left hand side of condition ( 48) is 0.572 for the parameters selected in this example.Results for R, T and the phase shift on transmission can be obtained similarly.

Summary
Based on known mathematics we have proposed an iterative approach to the scattering of a plane TE-polarized optical wave at a dielectric film with permittivities modeled by a complex continuously differentiable function of the transverse coordinate.
The result is an approximate analytical expression for the field intensity inside the film that can be used to express the physical relevant quantities (reflectivity, transmissivity, absorptance, and phase shifts).Comparison with exact numerical solutions shows satisfactory agreement.
(ii) The quality of the approximate solutions can be estimated in dependence on the parameters of the problem.
On the other hand the conditions of convergence explicitly depend on the permittivity functions in question and thus have to be derived for every permittivity anew (cf.(Serov et al., 2004;2010) and ( 65)).We introduce in the Banach space C[0, d] bounded integral operators N 1 , N 2 , N 3 , N 4 , N 5 , N 6 by

Acknowledgment
where with the values N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , I 0 which are defined as We are in the position now to show that if and This inequality holds if and thus if It means that for this value of ρ continuous map F transfers ball S ρ (0) in itself.Hence, equation ( 16) has at least one solution inside S ρ (0).For uniqueness of this solution it remains to prove that F is contractive (see (Zeidler, 1995)).To prove the contraction of F we consider Hence The following estimations hold Using Thus, from equation ( 58), one obtains so that F is contractive if Thus, the uniform convergence follows.
If we denote by m the left-hand side of the inequality (48) the solution I(y) of ( 16) can be approximated by the iterations I j (y) as follows (see (Zeidler, 1995)): where j = 0, 1, 2, ... and I 0 is defined in (20).
Let us remark that for the sufficient condition (48) to hold parameters must be chosen such that (48) holds even if r is small (Equation ( 18) represents the exact solution I(y) if ( 48) and (49) are satisfied).I(y) can be approximated by the first iteration I 1 (y) with the error 1−m I 0 , where m denotes the left-hand side of (48).Condition (48) must hold for a particular r > 0. In the limit r → 0 equation ( 18) transforms to equation where I 0 (y) is the same as (23) with the constant c 2 which is equal to Equation ( 66) is equivalent to (in the case of lossless medium) (41) in (Serov et al., 2004).Equation ( 66) is uniquely solvable in the ball of radius ρ if the following conditions are satisfied (they are consistent with the corresponding conditions from (Serov et al., 2004;2010)): In order to obtain a condition of the type (48) for all 0 < r < 1 (uniformly) combination of N 3 , N 4 , N 5 and part of I 0 within the estimations is necessary.It seems impossible to obtain a condition of the type (48) uniformly with respect to all nonnegative r.It is possible only to obtain such kind of condition uniformly for 0 < r < 1 or for 1 < r < ∞ independently.In this respect, some mathematical complications arise that are not the main point of this paper.
Estimation of I 0 (cf.Appendix C) gives where constants c 1 and c 2 are defined by ( 21) and ( 22) for a = 0, and by ( 24) and ( 25) for a = 0, respectively.Combining ( 65) and ( 67) we obtain the error of approximation (for a = 0) where j = 0, 1, 2, .... Since for linear case (a = 0) equation ( 17) is the linear Volterra integral equation this equation has always a unique solution and the following error of approximation holds: where j = 0, 1, 2, ..., I 0 is estimated in (67), m = N 1 + N 2 + N 6 and I j are defined by for a = 0 and a = 0, respectively.

Appendix C
With ε R (x) ∈ C 1 [0, d] and ε I (x) ∈ C[0, d] one obtains where a = 0 and ε ′ R denotes the first derivative of ε R .For a = 0 the estimate of I 0 (82) transforms to the following one with c 1 and c 2 from ( 24) and (25).
For ε R , given by (44), we obtain

Fig. 1 .
Fig.1.Configuration considered in this paper.A plane wave is incident to a nonlinear slab (situated between two linear media) to be reflected and transmitted.

Fig. 3 .
Fig. 3. Dependence of the field intensity I 1 (y, aE 2 0 ) inside the slab on the transverse coordinate y for a|E 3 | 2 = 0.1.The other parameters are as in figure 2. Solid curve corresponds to the first iteration of equation (26) and dashed curve to the numerical solution of the system of differential equations (6), (7).

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Fig. 5. Absorptance A 1 depending on the layer thickness d and on the incident field intensity aE 2 0 for the same parameters as in figure 3.

Fig. 6 .
Fig. 6.Phase shift on reflection δ r1 depending on the layer thickness d and on the incident field intensity aE 2 0 for the same parameters as in figure 3.

Financial
support by the Deutsche Forschungsgemeinschaft (Graduate College 695 "Nonlinearities of optical materials") is gratefully acknowledged.One of the authors (V.S.S.) gratefully acknowledges the support by the Academy of Finland (Application No. 213476, Finnish Programme for Centres of Excellence in Research 2006-2011).