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264 Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997 Enhanced Nonlinear Optical Response from a Stee and Gently Sloping Low Relief Surface: What is the Dierence? R.Z. Vitlina, G.I. Surdutovich, V. Baranauskas DSIF/FEEC, Unicamp, Cx.P.6101, Campinas - SP Received February 2, 1997 The second-order eective nonlinear susceptibility of a low relief interface between two media with nonlinear susceptibilities is calculated analytically for the cases of steep and gently sloping elongated periodical irregularities. An enhancement of the nonlinearity occurs for steep relief irregularities only under a suciently large ratio of the linear dielectric permittivities of these media. I. Introduction Heterogeneous composite materials such as optical bers, amorphous and polycrystalline semiconductors, two-dimensional structures on a surface and etc. are widely used nowadays in optoelectronics and dierent devices for propagation of light. Interest in composites arises since their optical properties can dier dramatically from those of the constituent components. One of the rst eective - medium theories to deal with optical properties of a composite material comprised of spherical mesoscopic impurities embedded in a host medium was given by Maxwell Garnett [1], who considered the linear response of metallic inclusion particles and was able to explain the colors of metallic colloids. In several articles [2-5] the analysis was extended to the case of nonlinear optical media. It was shown that due to the local eld eects a composite can possess a greater (enhanced) eective nonlinear susceptibility than any of its components. In the case of a third-order susceptibility the predicted enhancement was recently observed experimentally for a composite structure of alternating sub-wavelength-thick layers[6]. When two components are intermixed over a distance much smaller than the wavelength of light but still large compared with the interatomic distance then the linear and nonlinear susceptibilities of each constituent material are essentially the same as those of a bulk sample of the material. At the same time, the propagation of light through the composite can be described by spatially averaged values of these susceptibilities, i.e. a concept of the eective equivalent homogeneous medium may be introduced. Here we calculate the eective second - order nonlinear susceptibility, , of a new structure: a low relief interface between two materials possessing second - order nonlinearities. We obtained analytical solutions for the cases of steep and gently sloping periodic two-dimensional (2D) low reliefs. Both lowness and periodicity are essential for obtaining such solutions. Since the electric eld in composites is nonuniformly distributed between the two constituents, then the eective nonlinearity may be enhanced due to the concentration of the electric eld in a component of greater nonlinearity. We demonstrate that such an enhancement takes place if the more nonlinear component has the smaller linear refractive index. Low - relief - surface Let us examine the reection of light from a periodic elongated low-relief interface between two nonlinear media with linear permittivities 1 and 2 and second-order susceptibilities ^1 and ^1 . The term low here means that the period ` and height h of the irregularities are much smaller than the wavelength of the R. Z. Vitlina et al. incident radiation, whereas the relation between ` and h may be arbitrary. A particular case of such a relief (steep triangular irregularities with h `) is given in Fig. 1. For linear media this problem was considered 265 earlier in [7], where it was shown, that a low - relief surface, in terms of electromagnetic wave reection, to be equivalent to a homogeneous but anisotropic thin lm. Figure 1. A low -relief layer with periodic irregularities of a triangular shape on the interface between ambient medium with (2) permittivity 1 and susceptibility (2) 1 , and a medium with permittivity 2 and susceptibility 2 . The sublayer of a thickness `2 d has f2 (z ) = ` . The developed approach is analogous to going over from the microscopic to the macroscopic Maxwell equation in a medium. The ne details of the relief do not aect the reected radiation. In fact, the reection is only inuenced by a certain limited amount of information about the relief. Really, spatially nonuniform elds, which depend on the shape of the relief, vary over distances comparable to the parameters `; h << . Therefore, the quasistatic approximation is sucient for calculating the \microelds". These elds satisfy the Laplace equation, and their spatially varing part is known to fall o exponentially with distance from the interface. Since the minimum rate of this decay is of the order of the reciprocal dimensions of the relief, only spatially uniform elds survive outside a narrow surface layer [7,8]. As far as wave zone begins at distances which are comparable to the wavelength, where the microelds have decayed, the relief and its shape aect the scattering only through the \relief-averaged" elds. Thus, the problem is reduced to calculation of the eective permittivity tensor ^ef of a lm of thickness h, including all irregularities. This tecnique allows the numerical calculation of all the components of the effective tensor for arbitrary ratio of the parameters `; h. For a relief with plane faces in the limiting cases of \gen- 266 Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997 tly sloping;; (` h) and steep (` h) irregularities this may be done analytically. The particular case of reection of a normally incident wave from a low relief interface between two linear media has been considered in [9] in a somewhat dierent context. Here, we interested in a process of surface second harmonic generation (SSHG) from a low relief surface in these two limiting cases and so should calculate the eective second-order susceptibility ^(2)ef of the interface layer. Similar to the approach used in Ref.[6], we introduce the eective susceptibility ^(2)ef in the following way P~ (2!) = ^(1)ef E (2!) + ^(2)ef E~ (!)E~ (!); (1) where P~ (2!); E~ (!), E~ (2!) are Fourier transforms of the polarization and the electric eld at the frequencies 2! and !; ^(1)ef = ^ef ; 1=4 and and (2)ef are the eective susceptibilities of the equivalent homogeneous surface layer. a) Gently sloping irregularities: h ` According to the theoretical analysis of [7] for linear media in this case we may apply the perturbation theory in the parameter h=`. Let the 2D surface relief be described by a periodic symmetric function '(x) = '(x + `), '(x) = '(;x). To rst order in the parameter h=`, only diagonal components of a tensor ^ef are nonzero [7]: 1 + 2 1 ; A h ; ef = xx 2 ` 2 h 1 2 ef zz = + 1 ; A ` ; 1 2 ef yy = 1(1 ; c) + 2c ; (2) where constant A depends on the form of the relief, R` 1 c = h` 0 '(x)dx is the concentration of material 2 in the interface layer. The zeroth order terms, under h=` ! 0; correspond to a smooth boundary, i.e. to the substitution of the inhomogeneous two - layered (with permittivities jep1; 2 and thickness h=2 each) lm by the homogeneous lm with the permittivity tensor ^ef . The presence of this eective lm results only in an arbitrary change in the common phase of the amplitude reection coecients, which is irrelevant. Similarly, for the nonlinear media the presence of the eective lm in the limiting case h=` ! 0 leads to an eective nonlinear susceptibility tensor with components: (2) (2) 1xij + 2xij ; 2 ! ef (2) (2) (2)ef 1zij 2zij zz zij = 2 + ; i; j = x; y; z: (3) 1 2 Such renormalization also results in a change only in the common phase of the amplitude of SSHG. Therefore, gently sloping irregularities on the interface cannot induce any enhancement eect and, on the whole, their inuence on SSHG is negligibly small. ef (2) = xij b) Steep symmetric irragularities:h ` We divide the irregularity layer of height h (Fig.1) into sublayers of thickness d in such a way that the condition ` d h holds. The inequality d h allows us to ignore the slope of the faces with respect to the z axis in evaluating of the component xx and zz of one of the sublayers, while the inequality ` h allows us to treat the set of irregularities in a sublayer of thickness d as a ne - layered medium with a normal directed along x-axis. As a result, the components of the sublayers eective permittivity tensor, which is diagonal for symmetric irregularities, may be written as xx (z ) = f (z )1+2 f (z ) ; 2 1 1 2 yy (z ) = xx(z ) = 1 f1 (z ) + 2f2 (z ); (4) where 1 and 2 are the permittivities of the ambient and substrate media, and f1 (z ) and f2 (z ) are volume fractions of these constituents in the sublayer. Now, when the eective permittivity tensor of the sublayer is known, we may write out the all components of the tensor ^(2) (z ) of this sublayer: c 2 (2) xxx (2!; z ) = xx (2!; z )[xx (!; z )] Pxxx (!; 2!; z ) R. Z. Vitlina et al. 267 xxi(2!z ) ixx (2!; z ) ijx(2!; z ) xij (2!; z ) ijk (2!; z ) = = = = = xx(2!; z )xx (!; z )Pxxi(!; 2!; z ) [xx(!; z )]2Bixx (!; 2!; z ) xx(!; z )Cijx(!; 2!; z ) xx(!; z )Bxij (!; 2!; z ) Aijk (!; 2!; z ) (5) where fn (z ) nijk n (2! ) n=1;2 n=1;2 X fn (z ) X fn (z ) Bijk = D nijk ijk = 2 ( ! ) ( ! )n (2!) nijk n=1;2 n n=1;2 n X fn X fn(z ) Tijk = P (6) nijk ijk = 2 n (! ) n (2! )n (! ) n=1;2 n=1;2 As a result, we have a homogeneous sublayer with an eective linear permittivity ^(z ) and nonlinear susceptibility ^2(z ) at the height z . In this way, we reduce the problem of calculation of the properties of the interface layer of a thickness h to the calculation of the inhomogeneous ne-layered structure consisting of sublayers with known parameters. The calculation of the dielectric permittivity tensor ^ef of the h - layer may be performed in a straightforward manner [7]: Aijk = X 1Z ef = xx h fn (z )nijk h=2 ;h=2 Cijk = X ;1 1 xx (z )dz ; (ef zz ) h Z h=2 ;h=2 ;zz1 (z )dz; (7) whereas the component ef yy is given by Eq. (2) since for 2D relief it depends only on concentrations of the materials in a layer and does not depend on the shape of the irregularities. The calculation of the eective tensor (2)ef may be performed as a generalization of a known procedure for two component ne-layered alternating structure in [6] for the case of multiple (actually innite) number of components. The nal result is expressed through the sublayer quantities, Eqs. (5). The most important tensor components are presented below in an explicit form: zz (2!)[zz (!)]2 Z h=2 Azzz (!; 2!; z )dz ef (2 ! ) = zzz h ;h=2 zz (2!; z )[ ; zz (!; z )]2 Z h=2 1 2 ef zzz (2! ) = h ;h=2 xx(2!; z )[xx (!; z )] Pxxx(!; 2!; z )dz Z h=2 ef 2 xx (2!; z ) C (!; 2!; z )dz [ zz (! )] ef xzz (2!) = xzz h ;h=2 2zz (!; z ) Z h=2 ef xx(2!; z )xx (!; z ) D (!; 2!; z )dz zz (! ) ef (2 ! ) = xxz xxz h ;h=2 zz (!; z ) Z h=2 ef xx(!; z ) T (!; 2!; z )dz ef zz (! )zz (2! ) (8) ef (2 ! ) = zxz zxz h ;h=2 zz (!; z )zz (2!; z ) To obtain concrete results it is necessary to assume a certain model of the shape of the irregularities. As an example, we present results of the calculations for a triangular relief. In this case the function '(x) has the form: 2h ; l x ;l=2 x 0 '(x) = 2lh ; 4l + ; x 0 x l=2 l 4 268 Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997 From Eqs. (7) we obtain [7] 1 2 ln 2 ; ef = 1 2 ; ef = 22 (9) ef xx = zz 2 ; 1 1 yy 2 ln ;21 If one makes the simplifying assumptions that the linear dielectric constants are not frequency dependent, i.e. 1(2!) = 1 (!) and 2 (2!) = 2(!), the nal formulas become more visible. We present explicit expressions for the most enhanced components of the eective tensor ^(2)ef : xxx (2)ef 1 (2) ; = 21 (2) 1xxx + 22 2xxx 1 zzz (2)ef = 21 1ln;12 1 2 2 !3 ( (2) (2) 1zzz 2zzz + 1 2 ) (10) For = 21 = 10 the estimation for enhanced components gives (2) (2) (2) (2) xxx 52xxx ; zzz 32zzz : ef The reason for the greatest enhancement of the (2) xxx component is evident, since a steep relief presents maximal screening for the electric eld in the x-direction. As far as there is no screening at all of the eld along the y-axis, values of the components with indices iyy; yiy; yyi(i = x; z ) are the smallest. Conclusion We have demonstrated that gently sloping low relief irregularities do not inuence the nonlinear properties of the interface between two media. In contrast, quite a sizeable enhancement of the second-order nonlinear properties of the interface layer with steep low relief periodic irregularities may be achieved. The most enef (2)ef hanced are the (2) components of the efxxx and zzz fective tensor. The degree of the enhancement increases with an increase in the ratio = 1 =2 of the linear permittivities of the ambient and substrate media, while the electric eld amplitude becomes concentrated in the more nonlinear component. The nal characteristics of SSHG strongly depend on the specic form of the substrate medium tensor ^(2) 2 . Of future interest is the case of the asymmetric steep irregularities. If due to symmetric properties of the tensor ^(2) SHG for a at sur2 face is absent, then all SSHG eects will be connected with a presence of low-relief asymmetric irregularities. d Acnowledgments We are grateful to Prof. R.W. Boyd for useful discussions and Dr. Steven Durrant for reading of the text. This work is supported by the Brazilian Research Council (CNPq). References 1. Maxwell Garnett, Phil.Trans.R.Soc.,London, 203, 385 (1904). 2. J.W. Haus, R. Ingura and C.W. Bowden, Phys. Rev. A. 40, 5729 (1989). 3. D.S. Chelma and D.A.B. Miller, Opt. Lett. 11, 522 (1986). 4. J.P. Sipe and R.W. Boyd, Phys.Rev.A 46,1614 (1992). 5. R.W. Boyd and J.E. Sipe, JOSA B11, 297 (1994). 6. G.L. Fisher, R.W. Boyd, R.J. Gehr, S.A. Jenekle, J.A. Osaheni, J.E. Sipe and L.A. Weller- Brophy, Phys. Rev. Lett. 74, 1871 (1995). 7. R.Z. Vitlina and A.M. Dykhne, Sov.Phys. JETP, 72(6),983 (1991). 8. R.Z. Vitlina, Sov. Phys. Poverhnost 3, 50 (1991). 9. D.E. Aspnes, Phys. Rev. B41, 10334 (1990).