Explicit analysis of anisotropic planar waveguides by the analytical transfer-matrix method

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1 96 J. Opt. Soc. Am. A/ Vol., No. / November 004 Liao et al. Explicit analysis of anisotropic planar waveguides by the analytical transfer-matrix method Weijun Liao, Xianfeng Chen, Yuping Chen, Yuxing Xia, and Yingli Chen Department of Physics, The State Key Laboratory on Fiber-Optic Local Area Communication Networks and Advanced Optical Communication Systems, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 0040, China Received April 3, 004; revised manuscript received May 0, 004; accepted May 4, 004 The propagation properties of light in anisotropic optical planar waveguides with different index distributions are investigated with the analytical transfer-matrix method. Dispersion equations are analytically deduced by the method in terms of different index profiles. It is shown by examples that this method exhibits good accuracy compared with numerical methods while still holding physical insight. 004 Optical Society of America OCIS codes: , , 60.80, , INTRODUCTION As integrated optical components are fabricated with increasingly complicated processes, it becomes more and more essential to analyze the propagation characteristics of optical waveguides with accurate and simple tools. Besides isotropic optical waveguides, anisotropic waveguides have also been widely applied in optical devices. In past decades, some analytical and numerical methods have been presented to solve the propagation characteristics of anisotropic waveguides. Gia Russo and Harris used a ray-approach method to characterize an anisotropic waveguide. A WKB method was proposed by Ctyroky and Cada to analyze inhomogeneous anisotropic planar waveguides. A 4 4 matrix algebra was introduced 3 by Yeh and developed 4 by Visnovsky to investigate light propagation in an arbitrarily anisotropic medium. Kolosovsky et al. also proposed a method on the electromagnetic field distribution 5 to characterize anisotropic graded-index slab waveguides with arbitrary parameters. By the method of selecting zero elements in a characteristic matrix, 6 Walpita obtained the solutions for step- and graded-index planar waveguides. Coupledmode theory was improved by Tsang and Chuang to analyze reciprocal anisotropic waveguides. 7 Additionally, the hybrid modes in planar uniaxial waveguides were calculated based on a rigorous electromagnetic model. 8 Besides analytical methods, some numerical approaches have been suggested to find the propagation constants of anisotropic waveguides. The finite-element method (FEM) has been well developed for analysis of anisotropic optical waveguides. 9,0 Employing a unique vector potential, the FEM can avoid spurious and nonphysical modes for anisotropic waveguides. The FEM was also adopted to investigate anisotropic waveguides with off-diagonal elements in the permeability tensor and with arbitrary cross sections. 3 In addition to the FEM, the finite-difference time-domain method 4 and the Lanczos Fourier expansion (LFE) approach 5 were introduced to study asymmetric anisotropic waveguides. A finite-difference (FD) scheme 6 was also used to analyze complicated anisotropic structures of arbitrary index profiles with consideration of the refractive indices in the vicinity of index discontinuities. Furthermore, the surface integral equation method reported by Gaal 7 was used to determine the propagation character of an anisotropic embedded waveguide by minimizing the quadratic difference of the longitudinal field components in the cladding and the core along the dielectric interface. However, the analytical methods can usually obtain only approximate results, while the main deficiency of the numerical methods is a lack of physical insight. Researchers keep seeking other methods with simplicity, accuracy, and physical insight. Recently, an equivalent attenuation vector (EAV) method based on transfermatrix analysis was proposed 8 and improved 9 to characterize an isotropic waveguide with good accuracy. In this paper, we begin by analyzing three- and four-layer anisotropic waveguides with an analytical transfermatrix method, and then we apply an improved EAV method to analyze an anisotropic waveguide with gradedindex distribution and discontinuous profile. It is proved by examples that the method can be applied in anisotropic waveguides reliably and accurately.. THEORY AND APPLICATIONS For anisotropic planar waveguides the principal axes of anisotropy are parallel to the rectangular axes in the coordinate system of the waveguide, as shown in Fig., the dielectric tensor can be depicted in diagonal form as x 0 0 0n 0 n y 0 () 0 0 n, z 0 is the permittivity in free space and n x, n y, and n z are the refractive indices of an electric field polarized parallel to the x, y, and z axes of the coordinate system, respectively. In the following analysis, we take the z axis /004/96-09$ Optical Society of America

2 Liao et al. Vol., No. /November 004/J. Opt. Soc. Am. A 97 n Hzh z h sin h n z n sin h cos h z H z 0 Fig.. Principal axes of anisotropy in the coordinate system of a waveguide structure. H zh cos n H (3) z0. 0z Fig.. Coordinate system of a three-layer anisotropic waveguide and index distribution in each layer. as the propagation direction and assume that the refractive index varies with the x axis and that the waveguide materials are lossless. A. Three-Layer Anisotropic Waveguide As shown in Fig., for a three-layer anisotropic waveguide, n ix, n iy, and n iz (i 0,,) are the refractive indices of each layer. The thickness of the guiding layer is h, and the superstrate and the substrate are semiinfinite. The electromagnetic field of a guided wave oscillates in the guiding layer and exponentially attenuates in the superstrate and the substrate. Guided modes in a planar optical waveguide are divided into two types: transverse magnetic () mode and transverse electric (TE) mode, according to the polarization direction of the electromagnetic waves. For the mode, the magnetic field can be expressed as H z x A exp p x, x 0 B cos x C sin x, 0 x h, D expq h x, x h () n z k 0 n x /, n x p n 0z k 0 n 0x /, n 0x q n z k 0 n x /, n x k 0 /, is the wavelength in vacuum, and is the propagation constant for the mode. Based on thin-film theory and the boundary constraints of electromagnetic fields, we can obtain a matrix equation by employing an analytical transfer matrix: From Eq. (), we have n cos z h sin h A q n z n sin h cos h z p D. (4) n 0z Solving the above matrix equation, we can finally get the eigenequation of the three-layer anisotropic waveguide for the mode: that is, tan h n z n z h m tan n z tan n z q n z n 0z q n z p p n 0z q p n z n 0z ; (5) m 0,,,... (6) For the TE mode, the electric field can be expressed as A exp pte x, x 0 E z x B cos TE x C sin TE x, 0 x h, D expq TE h x, x h (7) TE (k 0 n y ) /, p TE ( k 0 n 0y ) /, q TE ( k 0 n y ) /, and is the propagation constant for the TE mode. Similarly, we can get the analytical transfer-matrix equation A qte cos TE h TE sin TE h TE p D TE sin TE h cos TE h (8) and the TE-mode eigenequation

3 98 J. Opt. Soc. Am. A/ Vol., No. / November 004 Liao et al. TE pte qte f h m tan TE tan TE i i h i sin i h i i m 0,,,... (9) Equations (6) and (9) are the well-known result for the guided-mode dispersion equation of the three-layer step anisotropic waveguide for and TE modes. By introducing, p,q, p,q for mode TE, p TE,q TE for TE mode, (0) f i n iz for mode for TE mode i 0,,, () we can finally write the dispersion equation for the threelayer anisotropic waveguide in unified form as h m tan f f 0 p tan f f q m 0,,,... () B. Four-Layer Anisotropic Waveguide The mode analysis for a four-layer anisotropic waveguide is given as follows. As shown in Fig. 3, there are two guiding layers. The guided wave may propagate in both guiding layers when the propagation constant is between k 0 n and k 0 n 3 or merely in the first guiding layer when is between k 0 n and k 0 n while the electromagnetic field in the second layer is evanescent, n i (i,,3) is written as n ix for the mode and n iy for the TE mode. The two situations will be treated separately in the following discussion.. k 0 n 3 k 0 n Just as in the three-layer case, we can obtain the matrix equation of the four-layer configuration by multiplying analytical transfer matrices in two guiding layers: p 0 f 0 M M 0, (3) p 3 f 3 M i i p 0 p 3 cos i f i sin i h i cos i h i i,, i n iz k 0 n ix / n ix for mode TE i k 0 n iy / for TE mode i,, p 0 n 0z k 0 n 0x / for mode n 0x, p TE 0 k 0 n 0y / for TE mode p 3 n 3z k 0 n 3x / for mode n 3x, p TE 3 k 0 n 3y / for TE mode f i n iz for mode i 0,,,3. for TE mode The form of M i is the so-called oscillatory analytical transfer matrix. Similarly, we get the eigenvalue equation of the four-layer anisotropic waveguide: h m tan f p 0 f 0 tan f p f m 0,,,..., (4) p 3 /f 3 ). Af- p is obtained by multiplying M and ( ter some algebraic transformation and employing p tan tan f p 3 f 3 h, (5) we can rewrite Eq. (5) as h m tan f p 3 m 0,,,..., f 3 (6) tan ( p / ). Adding the left- and righthand sides part of Eqs. (4) and (6) gives the eigenvalue equation of the four-layer anisotropic optical waveguide: h h s m tan f f 0 p 0 tan f f 3 p 3 m 0,,,..., (7) (s) tan ( f /f )( / )tan. Fig. 3. Index distribution of a four-layer anisotropic waveguide.. k 0 n k 0 n The electromagnetic wave is evanescent in the second layer when is between k 0 n and k 0 n ; then i, is equal to (n z /n x )( k 0 n x ) / for the mode or ( k 0 n y ) / for the TE mode. Based on the mathematical relations sin(ix) i sinh(x) and cos(ix) cosh(x), the analytical transfer matrix in the second layer in the evanescent case is

4 Liao et al. Vol., No. /November 004/J. Opt. Soc. Am. A 99 f cosh h sinh h M. sinh h cosh h f Here the form of M is called an evanescent analytical transfer matrix. Similarly, p is expressed as p tanhtanh ( f /f 3 )( p 3 / ) h. Finally, the eigenvalue equation in such a situation is rectified as h m tan f p 0 f 0 tanh f p f m 0,,,... (8) C. Graded-Index Anisotropic Waveguide In Subsections.A and.b, the guided-mode equation in step three- and four-layer anisotropic waveguides has been investigated by an analytical transfer-matrix method. The graded-index waveguide can be regarded as a step multilayer waveguide if we divide the graded-index profile into numerous step layers. In the following, we employ the analytical transfer-matrix method in the case of an anisotropic waveguide by introducing the improved EAV concept. The improved analysis adopts an EAV to characterize the evanescent field beyond the turning point. Unlike WKB analysis, this method establishes that the phase loss at a turning point is exactly equal to, independent of the wavelength, the propagation constant, and the refractive-index profile, which is also proved in the quantization condition. 0 For a graded-index anisotropic waveguide, we take the refractive-index profile as the following arbitrary form, shown in Fig. 4: n x x n sx n x f x/d, x 0 n cx, x 0, n y x n sy n y f x/d, x 0 n cy, x 0, n z x n sz n z f 3 x/d 3, x 0 n cz, x 0, n sx, n sy, n sz and n cx, n cy, n cz are the refractive indices of the substrate and the cover, respectively, n x, n y, and n z are index modulations, and f (x/d ), f (x/d ), and f 3 (x/d 3 ) are index profile functions. As a practical example, this kind of index profile can be found in an annealed proton-exchanged (APE) waveguide or a Ti-diffused waveguide in lithium niobate. x t is assumed as the turning point, and then the propagation constant can be calculated as k 0 n x (x t ) (for the mode) or k 0 n y (x t ) (for the TE mode). Furthermore, we assume that x s is close enough to the substrate that n x (x s ) and n y (x s ) are equal to n sx and n sy, respectively. There are l divided layers from x 0 to x t and m divided layers from x t to x s with uniform thickness h and constant refractive indices n x (x i ), n y (x i ), and n z (x i ) for each layer, as shown in Fig. 4. Fig. 4. Graded-index anisotropic waveguide with arbitrary index profile. It is known that the electromagnetic wave has an exponential convergence in the superstrate (x 0) and an oscillatory behavior in the guiding area from x 0 to x t, every divided piece can be characterized by an oscillatory analytical transfer matrix. The electromagnetic field is evanescent from x t to x s, each divided slice can be characterized by an evanescent analytical transfer matrix. According to transfer-matrix theory, we obtain the matrix equation p l lm c f c M i i j jl M 0, (9) p s f s f i cos i h sin i h i M i i 0,,,...,l, i sin i h cos i h f i f i cosh j h sinh j h j M j j sinh j h cosh j h f j with i j j l, l,...,l m, i n iz k 0 n ix / n ix for mode TE i k 0 n iy / for TE mode i 0,,,...,l, i n jz k 0 n jx / n jx for mode TE i k 0 n jy / for TE mode j l, l,...,l m,

5 00 J. Opt. Soc. Am. A/ Vol., No. / November 004 Liao et al. p c p s p c n cz k 0 n cx / for mode n cx, p TE c k 0 n cy / for TE mode p s n sz k 0 n sx / for mode n sx, p TE s k 0 n sy / for TE mode f k,f c,f s n kz,n cz,n sz for mode for TE mode k 0,,,...,l m. Now the EAV p t is used to describe the evanescent field from the turning point to the substrate, and Eq. (9) becomes p l c f c i M i p t 0. (0) fx t p t is determined by the following iterative equations: p t p l, p j j p j j f j f j p j j tanh j h f j f j tanh j h p lm p s, j l, l,...,l m, () Similar to the conclusion for the four-layer anisotropic waveguide in Subsection.B, we can obtain the following dispersion equation for a graded-index profile: l i0 i h s N tan f 0 p c f c 0 tan fx t p t f l t N 0,,,... () Because t 0 and tan f(x t )/f l ( p t / t ) /, Eq. () can be further written as l i0 s i0 i h s N tan f 0 f c p c l i tan p i 0, (3) i tan f i i tan i f i i, (4) i, (5) p i i tan tan f i p i f i i i h i,,...,l. (6) Furthermore, when the second-order minor term is neglected, Eqs. (4) and (6) can be rewritten as l s i0 p i p i f i i i /f i i /f i, (7) p i /f i p i /f i p i i. (8) h When the divided layers are infinitely increased, the thickness h of each layer will approach zero. According to the theory of integration and differential calculus, l i0 i h in Eq. (3) can be expressed as xx lim h 0 t x x0 (x)h t 0 (x), and s in Eq. (7) can be written as lim h 0 xx t h x0 px hfx h p x h x h x/fx x h/fx h h h 0 x t pxfx Similarly, Eq. (8) is expressed as px h/fx h px/fx lim h 0 h f i p x x /fx. p x x, f x and, furthermore, d p(x)/f(x)/ (x) p (x)/ f (x). Then Eq. (3) can be transformed into 0 x tx 0x t pxfx x p x x /fx N tan f 0 f c p c 0 x n zx n x x k 0 n x x / TE x k 0 n y x / N 0,,,..., (9) for mode. for TE mode p(x) is determined by the following differential equation; d px/fx x p x, px t p t, f x (30)

6 Liao et al. Vol., No. /November 004/J. Opt. Soc. Am. A 0 fx n z x for mode for TE mode. As the subscripts i and i indicate the neighboring segments of layers in the guiding region, shown in Eqs. (4) and (7), s is interpreted as phase contributions from the scattered subwaves. 0 If we set i /f i i /f i, which exists in the step-index profile, s 0 will be obtained, consistent with the above analysis of the step-index case. Thus Eq. (9) shows a clear physical meaning: The two terms or the left-hand side present half of the phase contribution of the guiding waves and the subwaves, and the second and third terms on the right-hand side offer half of the phase loss at the interface and the turning point, respectively. When the sum of phase contributions and losses is equal to N, the optical wave will be guided and propagate in the waveguide. For the situation when two turning points occur, as shown in Fig. 5, the dispersion equation can be straightforwardly presented, according to the above analysis, as x t x tx xt x t pxfx p x x /fx N N 0,,,... (3) D. Discontinuation in a Graded Anisotropic Waveguide A discontinuous index profile is also considered, as shown in Fig. 6. Figure 6(a) demonstrates a condition a discontinuous point is situated in the oscillatory region, which provides an extra phase contribution, referred to as d. From Eqs. (4) (6), we can find that, at a given point x, px xtan fx px h tan fx h x xh, (3) and the phase contribution of scattered subwaves of the neighboring layers on the left and the right of point x is px x tan tan x h fx px h. fx h x (33) When h 0, we obtain px 0 fx 0px 0, (34) fx 0 px x tan x 0 fx 0 tan fx 0 px 0 x 0. (35) Then the extra inner phase contribution at the discontinuous point x d can be calculated as Fig. 5. points. Graded-index anisotropic waveguide with two turning d tan p x d x d tan f x d p x d f x d x d, (36) Fig. 6. Discontinuous index profile in a graded-index anisotropic waveguide.

7 0 J. Opt. Soc. Am. A/ Vol., No. / November 004 Liao et al. n z x d x d n x x k 0 n x x d / d x d for mode, x d TE k 0 n y x d / for TE mode f x d n z x d for mode for TE mode. Finally, just as for the continuous graded-index profile, the dispersion equation for the discontinuous distribution can be obtained: x t x tx xt x d pxfx p x x /fx x t pxfx x d p x x /fx d N N 0,,,... (37) p(x) is determined by the following differential equation: d px/fx px t p t, x p x, f x p x d f x d p x d. (38) f x d Another situation is depicted in Fig. 6(b), the effective index is between n (x d ) and n (x d ). This situation can be solved very similarly to the continuous graded-index profile. However, in this case, half of the phase loss at the discontinuous point is tan f (x d )/f (x d ) p t / (x d ) instead of /, for (x d ) is not equal to zero, which can be easily found from Eq. (). Thus the dispersion equation for such a situation can be written as x x d pxfx xt p x x /fx N tan f x d p t f x d x d x t N 0,,,... (39) p(x) is determined by the following differential equation: d px/fx x p x f x, p x d f x d p t f x d (40) If the discontinuous point x d is beyond the turning point, it can also be found that p (x d ) f (x d )/ f (x d )p (x d ), which can be directly obtained evolved from Eqs. () by setting h 0. It needs to be noted that Eqs. (9), (3), (37), and (39) are approximate, for we have neglected the second-order Table. Mode Comparison of Calculated Values of b for a Four-Layer Structure Present Method M FD-BPM (x 3.9 nm) FD (W5 m, x 3.9 nm) minor terms, that is, we consider just the phase contribution of the first-order scattered subwaves and the higherorder subwaves are ignored. This approximation is proved to be reasonable and reliable by various numerical approaches in Section NUMERICAL RESULTS To verify the reliability of the current method, we consider diversified waveguide structures, as adopted in Refs. 5 and 6. To realize a practical numerical approach, the thickness of the divided layer presented in our analysis cannot be infinitely small, so a uniform thickness of 5 m is introduced for all simulations. For a fourlayer anisotropic waveguide, at a wavelength of nm, the refractive index of each layer is given by n 0,x,y,z.456, n x.67, n y n z.6738, n x.5630, n y n z.56, and n 3x,y,z.0, and the thicknesses of the first guiding layer, h, and the second guiding layer, h, are 59 and 600 nm, respectively. The calculated values of b, which is defined as (N eff n 0x )/(n x n 0x ) for each mode, are shown in Table together with the results calculated with the M, FD- BPM, and FD methods of Ref. 6. From the table, we can see that the results of the present method are almost the same as those of the other methods. In the numerical approach, we found that the and 0 guided modes just cover the two situations in the analysis of the fourlayer structure. An annealed proton-exchanged (APE) waveguide in lithium niobate that has a graded-index profile is used as another example to testify to our theory. The protonexchange process increases the extraordinary index in the guiding layer while decreasing the ordinary index; thus one can stimulate only modes in Z-cut samples and TE modes in X- and Y-cut substrates. After annealing, both the extraordinary and ordinary indices redistribute into a graded profile. Here we assume that the index distribution in an APE waveguide sample has the following form: x n se n e n e exp x D, x 0, n ce, x 0 x n so n o n o exp x D, x 0. n co, x 0

8 Liao et al. Vol., No. /November 004/J. Opt. Soc. Am. A 03 For a 63.8-nm He Ne laser, n se., n so.86, n e 0.0, n o 0.004, n ce n co.0, and D D 5 m. Comparisons of the effective index N eff calculated with the present method and with the other methods of Ref. 6 are shown in Table. It is easily found that the values calculated with the present method have a very good accuracy. To further verify the universal applicability of the method, we introduce a complicated discontinuous refractive-index profile in a Ti-diffused proton-exchanged (TIPE) LiNbO 3 waveguide with SiO buffer and air cover. At the wavelength of 63.8 nm, the index profile of the TIPE SiO waveguide sample is given by Table 3. Calculated Values of Effective Index N eff for an Anisotropic TIPE LiNbO 3 Waveguide with Discontinuous Profile TE Mode Present Method M (3000 layers) FD (W 5 m, x 0 nm) TE TE TE TE n e x nc, x n b, x 0 n es 0.04 expx /3, 0., 0 x n es 0.04 expx /3, x n o x x n b, x 0 n os expx /3, 0.04, 0 x n os expx /3, x nc, Table. Calculated Results of Effective Index N eff for an Anisotropic Graded-Index Profile Waveguide Anisotropy Mode Present Method M (000 layers) FD (W 5 m, 500 grids) Z-cut, Y-propagation, APE-LiNbO X-cut, TE Y-propagation, TE APE-LiNbO 3 TE TE Fig. 7. Normalized mode index b ( k 0 n c )/(k 0 n x,max k 0 n c ) with normalized frequency V k 0 d(n x,max n c ) / of modes in a symmetric anisotropic waveguide. n es., n os.86, n b.45, and n c.0. To reliably cope with the discontinuous profile in the above theory, we need to make some prior assumptions on the effective index: above the discontinuous segment, in the discontinuous segment, and under the discontinuous segment. Each assumption should be implemented with the corresponding dispersion equation to cover all potential stimulated modes. For an X-cut, Y-propagating sample, the results of TE modes calculated by the current method and the other methods of Ref. 6 are demonstrated in Table 3. As shown in the table, the current analytical method agrees quite well with the accurate numerical schemes. We finally analyze a symmetric discontinuous anisotropic waveguide with n x (x) (x/d) /, n y (x) n z (x) (x/d) /, and n c n s.46, d x d. modes of the waveguide are calculated at 860 nm. Just as in the last calculation, the propagation constants in the discontinuous distribution are presupposed for diversified situations, and every condition is considered. The simulated results of a normalized mode index b ( k 0 n c )/(k 0 n x,max k 0 n c ) with a normalized frequency V k 0 d(n x,max n c ) /, in comparison with the Lanczos Fourier expansion (LFE) technique, 5 are demonstrated in Fig. 7. It can be found that the current analytical method is also in good agreement with the accurate numerical approach. 4. CONCLUSIONS In summary, a compact method based on an analytical transfer-matrix method and an improved EAV assumption is suggested to analyze an anisotropic planar waveguide. The eigenvalue equations are proposed by the method for both and TE modes in explicit and analytical forms. A multilayer structure, graded-index distributions with one and two turning points, and discontinuous profiles are separately investigated. It is shown by examples that this method can obtain accurate results compared with numerical methods, while it holds considerable physical insight. The proposed phase contribution of scattered subwaves and phase contribution at a discontinuous point may serve to reliably model and design complicated anisotropic waveguide configurations.

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