WKB Analytic Modeling for Non-Uniform Thin-Films, Holograms, and Gratings

Transcription

1 WKB Analytic Modeling for Non-Uniform Thin-Films, Holograms, and Gratings Tomasz Jannsona, Steve Kupieca, Kal Spariosua, and Lev Sadovnikb a)physjcal Optics Corporation, 2545W. 237th Street, Torrance, CA b)waveband Corporation, 375 Van Ness Avenue, Torrance, CA ABSTRACT The WKB (Wentzel-Kramers-Brillouin) method, well-known in quantum mechanics, is applied in the second-order approximation into non-uniform Bragg structures, such as rugged dielectric thin films, sinusoidal gratings, and holograms. In this paper, the analytic WKB problem solution will be presented including numerical results. 1.0 INTRODUCTION The WKB (Wentzel-Kramers-Brillouin) method is a very useful method for the analysis of non-uniform media in quantum mechanics, in particular, and in wave propagation theory, in general. In this presentation, we apply the second-order WKB method into non-uniform Bragg structures such as rugged thin films, sinusoidal gratings, and holograms. We assume in this structures sinusoidal refractive index distribution with z-dependent slowly-varying envelopes of: index modulation, spatial period, average index. In such a case, the optical wavelength in wave propagation is formally equivalent to Planck's constant in quantum mechanics. As a first step, we solve coupled-wave equations with slowly-varying coupling coefficients, in the form of so-called "eikonal" solution. In the 2nd-order WKB approximation, we obtain generalized coupled-wave formulas for diffraction efficiency, with characteristic turning points for non-uniform (reflection) rugged Bragg thin films. 2.0 SLOWLY-VARYING KOGELNIK CONSTANTS In our modeling, two basic Kogelnik's parameters [U, the coupling constant, v, and off-bragg parameter,,arc slowly-varying z-dependent variables: v=v (z) = (z) (2-1) This is a natural generalization of Kogelnik's coupled-wave theory [U where these quantities are constants. Here, z- coordinate is perpendicular to the surface. We assume lossless case and sinusoidal modulation of dielectric constant, where: C=Eo+C1 (2-2) i.e., Co is the bias dielectric constant, and Cl is its modulation. Using optical notation, we have ii = = and (ii + n)2 = CO + El = 2iizn E (2-3) 88 SPIE Vol X/97/$1O.OO

2 By analogy to quantum mechanics, where modified Planck's constant /l is introduced, we also introduce the notation: therefore, x= (2-4) 2ic v(z) = irtb(z) (2-5) X(CSCR) - which is the generated Kogelnik's formula (see; Ref. [1], Eq. (4-2)), for non-uniform index modulation. In this relation, CS and CR are constants, as in Johnson's modeling [21, where they are direction cosines of the true wavevectors: k0 and k, where, k=k0=ii (2-6) Therefore, the v(z)-dependence is only due to z-dependence of index modulation which, for dichromated gelatin (DCG) holograms, for example, can be a consequence of non-uniform hardening and/or sensitization. The Kogelnik off-bragg parameter, can be described, as (z)= - )(Z)X T (2-7) = K(z) K(O) (2-8) and Eij = (O), while the grating vector, K, satisfies the following Bragg relation: k ko =K; K=1 (2-9) where A is grating constant. Therefore, the non-uniform off-bragg parameter represents non-uniform shrinkage/swelling in z-direction which is typical for DCG hologram processing. We see that the off-bragg consists of two terms, the 1st one represents Bragg selectivity for uniform hologram with thickness T and the second one represents some non-uniform correction to this selectivity. In Figure 2-1, exemplary Bragg gratings are presented, and in Figure 2-2, Bragg relation is illustrated. It should be noted, that, in this generalized Kogelnik model, Fresnel reflections are ignored (as in Ref. [1]); i.e., bias index, ii is the same inside and outside the structure. 89

3 (a)_ 4 N S (b) (d) )- f Shrinkage I I j L(z) z Figure 2-1 Illustration of exemplary Bragg gratings: uniform and non-uniform, with: R-incident beam amplitude, S-diffracted beam amplitude, A-grating constant, A;1-grating constant horizontal component, including: (a) uniform slanted transmission grating; (b) non-uniform (due to shrinkage) slanted transmission grating as modification of (a); A11 is presented; (c) uniform reflection grating; (d) highly non-uniform reflection-lippman grating (or, dielectric multi-layer film). (a) K (b) K -y (c) K K k CR k 2it.... K k =k=n sinê= K =(K,K) =(KcosKn);K= = e, CR = cos(3 e) C = cos(e + Figure 2-2 Illustration of the Bragg relation, including (a) uniform hologram with three basic vectors k0, k, and K; (b) Bragg triangle; (C) grating vector; (d) auxiliary formulas. 90

4 3.0 NON-UNIFORM COUPLING EQUATIONS Let us consider Kogelnik-Johnson coupling equations for a =0 (i.e., for purely-phase holograms): CR R' = i' S (3-1(a)) where x= ;v= CS S' = ii S= ix R xxi (CS CR) 2 (3-1(b)) ; = -- (3-2) dr, ds dz where R =, S =, (3-3) dz For uniform case, Eq(s). (3-1(a)) coincide with Kogelnik's coupled-wave equations (see; Ref. [1], Eqs. (21) and (22). For non-uniform case, however, X' i3, are slowly-varying (in respect to X) functions of z: x = x(z), = (3-4) After respective differentiating and substituting Eq. (3-1(a)) into (3-2(b)), or vice versa, we obtain the following second-order ordinary differential equations. and R" + R' + x2 R = 0 (3-5(a)) C5 X) CRCS i' ( 2 x i-1- I=0 (3-5(b)) C5 x) CRCS C5 xcs) We see that quite unlike the uniform case [1], Eqs. (3-5(a)) and (3-5(b)) have no identical forms. We can modify Eq.(3-5(b)) in the form: where = ( 2 S" + S' ie + SI x + ie' -- Is= 0 (3-6) x) LCRCS x) T - For our WKB purposes, we p introduce two auxiliary parameters: (3-7) T T then, Eq. (3-6) takes the form: where in the non-uniform case: X p) u2 p) (3-8) =(z),p=p(z) (3-9) 91

5 4.0 "EIKONAL" FORM OF THE WKB SOLUTION Let us assume the following "eikonal" form of the WKB solution: S = exp () (4-1) where X is a small value in the common sense of the WKB method. In other words, the WKB method is some kind of quasi-geometrical approximation (small X ) for the non-uniform case. (In an analogous sense, modified Planck's constant, h, is small). We have;. S'=çS (4-2) where the "eikonal" ç is expanded in a power series: I Ii\2 2 S =c S+) (c) S, etc. (4-3) 'x\2 c=co --ci I--) c2. (4-4) 1 \1J Substituting Eq(s). (4-1) through (4-3) and using Eq. (4-4) we require zero-value of all terms with the same power of X, independently (this step is strictly analogous to that in quantum mechanics). Then, we obtain the following set of differential equations; + = 0 (4-6) ö =0 (4-7) Comparing these equations to those in quantum mechanics, we should remember that the stationary Schrodinger equation is equivalent to the Helmholtz equation. Hence, the situation in quantum mechanics is much easier, and it is equivalent to = 0 and p =0, in our case. We can solve the set of Eq (s). (4-6) to (4-7) as follows. First we find co from Eq. (4-7) and substitute it into Eq. (4-8). Second, we find ci from Eq. (4-8) and substitute it into the next equation (which is not written here). Solving Eq. (4-7) gives us the first order approximation; solving Eq. (4-8), the second-order approximation; etc. In the fully uniform case, we usually cut the solution off at the second-order approximation. Eq. (4-6) which is the square-root equation with respect to go, yields: = p2 (4-8) thus, simple integration of Eq. (4-8) yields: In the fully-uniform case, Eq. (4-9) takes the form: g0(z) = Jc (z)dz + const. (4-9) 92

6 For our purposes, we should also calculate the derivative of Eq. (4-8): ç0(z) = co Z + const. (4-10) 2 o 2 j(2 2 1J%;i J+P (4-11) After simple but long calculations, we obtain the following solution of Eq. (4-7): 2gi =+ + + (4-12) J2 + U2 2 + U2 1),J2 + ' X' +UXU' U' Now, we prove that the differential dg, is the total differential (i.e., it depends only on boundary values of ç, for z = 0 and z = T). To prove this, we present Eq. (4-12) in the following form: U, where 2ç1'=±A+B+C (4-13) U' A U (4-14).j2+U2 J2+U2 B=- (4-15) U E x ' + i +1) Introducing W = so, w' = X U X U) Eq. (4-13) takes the form: Substituting v = 2 + v2 (so, r' = 2' + 2u.u') Eq. (4-16) has the form: w A= (4-17) ji + w2 therefore: 1 V (4-18) 2v w u' 1i' 2i = + (4-19) Ji+w2 u 2i where w, V are described above, hence: 93

7 2i1n 0/v0)2 + in [/v i+2/v2 (4-20) 1 + (/v)2 + Ji + where = (0) and V0 = V. Introducing new variables: X=, X0=- (4-21) V V we have - In the second order approximation (usually used in quantum mechanics), 81!1[1 xo2+ x+i+x2 (4-22) 2 \1+X2 xo+fi+xo2 S(z) = exp[ co(z)] exp[ci(z)] (4-23) and substituting Eq(s). (4-9) and (4-20) into Eq. (4-23), we obtain the general solution of the WKB problem in the second-order approximation: where S(z) = G(z)S1 expc(z) exp 'Yi (z)dz) + s2 exp[ C(z)] exp (i Y2 (z)dz) (4-24) G(z) = 11 + ø2/v2 (4-25) i+ Iv ) C(z) = [ /V 2,2 ln (4-26) Ivo + + = + IJ + V and (4-27) of course, Y2(Z) = T (4-28) = Si si2 (4-29) CR 5.0 GENERAL SOLUTION FOR TRANSMISSION HOLOGRAMS In this case, Eq. (4-29) has the form: sin2 </2 + 2 > (i +XT2)(l + x02) (5-1) j(i + XT2)(l + x02) 94

8 where XO=-, XT- (5-2) VO VT VT = v(t) + V2> = 5T2 + V2 dz (5-3) We can see that solution of Eq. (5-1) only depends on boundary values for z =0, z =T, except for Eq. (5-3), where the average J2 + V2 value is to be calculated after integrating over the functions =(z) and V = V(z). We can show that solution of Eq. (5-1) satisfies the conservation energy principle: as well as the symmetry principle: RI2 +1=1 (5-4) Tl(0,T) = ii(t,0) (5-5) which guarantees that diffraction efficiency measurement is independent on the side of the hologram. In the fullyuniform Kogelnik's case, E and V are constants. Hence, x and x = x, < + V2> =/2 + V2, and: I + XOXT - J(i +XT2 )(i +x02) = 0 (5-6) so Eq. (5-1) reduces to the form: which is identical to Kogelnik's classical solution. sin2(2 + V2)2 TI 2 (5-7) l+x 6.0 NUMERICAL RESULTS FOR TRANSMISSION HOLOGRAMS In order to obtain some numerical results, we need to apply the (z) function defined by Eq. (2-8), and then Eq. (2-7) for the off-bragg parameter. In the case of non-uniform shrinkage-swelling, we have: Let us consider two elementary cases (see Figure 6-1) A) LINEAR Then, Eq. (6-1) becomes: A = A(z) (6-1) B) QUADRATIC Then, Eq. (6-1) becomes: A(z)=A0+bz;b=tana= AT AO (6-2) 95

9 az2 2(AT _ Ao) 2 T2 We also introduce the relative shrinkage/swelling coefficient, p, in the form: A(z) = A0 + ; a = (6-3) A AT-AO =21 AT AO (AT+AQ)/2 (AT+AO)) (6-4) A A (a) Figure 6-1 Illustration of linear and quadratic non-uniform axes (. = 0.55 j.tm; ii = 1.55; n/ii =0.05; T = 20.tm; = 300). For our numerical purposes, we have selected the following (uniform) input data: X = 0.55 pm, ii = 1.55, n/ii = 0.05, T = 20 pin, and incident angle: f3 = 300. There will be four (4) non-uniform cases: Al) LINEAR, T = constant = 20 Jim, and = 0; 0.01; 0.02; 0.05; 0.07; 0.1 (b) (Bi) QUADRATIC, same as (Al) (A2) LINEAR, (Al) = constant = 0.05; T = 10 tim, 20 m, 30 rim, 40 tim, 100 jim (B2) QUADRATIC, same as (A2). The numerical results are illustrated in Figures 6-2 and 6-3, including the uniform case (Figure 6-2) and the nonuniform case (Figure 6-3). For the uniform (transmission) case, the following numerical parameters have been assumed (see Figure 2-2, for notation): thus, E= 30', T= 20 Jim, 0.55 jim, A= ldjim, , Cs = 0.76, Cr = 0.94, n = 0.012; (6-5) = 8.49icz[jim] (6-6) v=v= (6-7) 96

10 For the non-uniform case, we assume the "uniform' parameters: (6-5),(6-6), (6-7), as well as: = ic C V = V0 = - (6-8) (6-9) The non-uniformity parameter,, is defined through Eq. (2-7), as For &z = 100 nm,, we obtain T= - 1CLA1C T A 2 Jto2 (6-10) (6-11) which is equivalent to E=1. In general, we obtain Eq. (6-8). In Figure 6-3, fl(ax) relation has been illustrated for E= 0.3; 0.7, and 1.0. We can observe the characteristic shift of the maximum position. -a49ir, 7o = (sin{,/c2+ v02])2 1+.Q 2 Figure 6-2 Illustration of ii(&) relation for uniform case, where & is in tm. 97

11 e= 0.3 (a) 0.7 (b) (c) Figure 6-3 Same as in Figure 6-2, but for the non-uniform case. 7.0 GENERAL SOLUTION FOR REFLECTION HOLOGRAMS In this case, the situation is much more complicated because the parameter, v, is now purely imaginary (C <0), and boundary conditions are not so simple. Therefore, we cannot give the solution in one compact form. Thus, we have: 98

12 For X0 1 sh2r + sin2 I 2 Ii 2 \2 sh R1 Xo cosl X0 sin!) (7-1) and for X0 > 1 sin2 I + sh2r (7-2) sin2 i (J x 1 cohr + X0 dhr) where R, and I, are the real and imaginary parts of the expression. where - T2 2 Z=<vrr>CT (73) and with CT = in - XTr - ixtr (74) Xr ixor XTr=, Xor= VrT Vor it M T VrT = (7-6) 4ICS ICR Also, in this case, we can prove that solution of Eqs. (7-1) through (7-6) satisfies the conservation energy principle as well as the symmetry principle. For constant and Vr, this expression reduces to Kogelnik's solution. 8.0 NUMERICAL RESULTS FOR REFLECTION HOLOGRAMS Numerical results for reflection holograms can be obtained in an analogous way to those for transmission, or by using non-analytic methods as in Ref. [3]. 9.0 VALIDITY OF THE WKB APPROXIMATION The situation is similar to that in quantum mechanics. The WKB approximation holds, if: ()2 << (9-1) However, ö' is of the order of JE2 + v2, which for transmission holograms has large values, even near the Bragg condition (because the v-parameter has to be large for high diffraction efficiency holograms). For reflection holograms, however, is negative, and for v = E2,'= 0; the 0 = oo. Fortunately, the quantity 0 does not exist in the final formulas of Eq.(7-l) and (7-2). Therefore, the presence of turning points does not lead to singularities in the final formulas for reflection holograms. 99

13 10.0 SUMMARY It has been demonstrated that the well-established WKB method can be applied to non-uniform holograms, gratings, and thin films, with vertical (z-dependent) non-uniformity. The advantage of this method is its analytic character (i.e., an analytic, direct solution exists as a generalization of Kogelnik's coupled-wave theory REFERENCES 1. H. Kogelnik, "Coupled-Wave Theory for Thick Hologram Gratings," BSTY 4, pp (1969). 2. K.C. Johnson, "Coupled Sector Wave Diffraction Theory," AppI. Phys.,pp (1980). 3. T. Jannson, I. Tengara, Y. Qieo, G. Savant, "Lippman-Bragg Broadband Holographic Mirrors," JOSA A pp , January