Derivation of Eigen value Equation by Using Equivalent Transmission Line method for the Case of Symmetric/ Asymmetric Planar Slab Waveguide Structure

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1 ISSN Vol. 15 No.1 (011) Journal of International Academy of Physical Sciences pp Derivation of Eigen value Equation by Using Equivalent Transmission Line method for the Case of Symmetric/ Asymmetric Planar Slab Waveguide Structure S. K. Raghuwanshi and V. Kumar Department of Electronics Engineering Indian School of Mines Dhanbad India R. R. Pandey Department of Electronics Communication Engineering Inst. of Engineering &Industrial Technology Durgapur (Received December 010) Abstract: The symmetric/asymmetric planar slab waveguide is simplest waveguide structure to be analyzed. These waveguides are used in optical communication systems. In this paper we have derived the Eigen value equations by using the transmission line (TL) method for the case of symmetric/asymmetric planar slab waveguide structure. Earlier also the equations have been derived but no where the intermediate steps of solution found by author knowledge. The derived results have been exactly matched with the existing results found into the literatures. 1. Introduction The symmetric/asymmetric planar waveguide structures have impact on WDM optical communication systems. The transmission line (TL) method has great application to analyze waveguide structure having arbitrary refractive index profile 1-. There is large number of current research papers on application of TL method of waveguide analysis The asymmetric waveguide is somewhat tough to analyze due to their asymmetric mode field profile. The asymmetric waveguide have found certain advantage over the symmetric waveguide structure due to their easiness 4. In section- we have shown the calculation of Eigen value equation for symmetric planar slab waveguide followed with asymmetric waveguide structure in section-3. Maxwell Equations Let consider the planar slab optical waveguide as shown in Fig.1 having refractive index variation in -direction and direction of wave propagation in

2 114 S. K. Raghuwanshi V. Kumar and R. R. Pandey direction. Consider z j t (time harmonic e ) the (1.1) E 0 (1.) H following Maxwell equations H t E t Fig 1: A thin wave guiding element with coordinate For thete mode case propagating in z direction as shown in Fig.1 have Ex 0 H y 0 and Ez 0 hence from equations (1.1) and (1.) (1.3) E y x (1.4) H z j Hx j 0 {n( x) Ey x (1.5) E y 0 Hx. Derivation of Eigen Value equation for Symmetric Planar Slab Dielectric Waveguide In this section the derivation of Eigen value equations have been done for odd/even TE modes by Equivalent T L method. Let us define new variables as follows (.1) v Hz (.) I E y 0 Hx Equation (1.4) can be written as

3 Derivation of Eigen value Equation by using Equivalent Transmission (.3) 115 V I x j 0 and equation (1.3) can be written as I j 0V x (.4) with (.5) 0 0 {n( x)} The characteristics impedance in equations (.3) and (.4) is given by (.6) Z j 0 The planar layer of thickness can be represented by an equivalent T-circuit as shown in Fig. with series and parallel elements given by (.7) d Z s Z tanh (.8) Zp Z 1 sinh( d ) Fig : Equivalent T-circuit of TE modes of planar waveguide layer of thickness. If the layer is homogeneous and infinite of thickness and by using the following relations (.9) (.10) lim sinh ( d ) d e d e d lim tanh ( d ) d e d / e d / e d / e d / 1

4 116 S. K. Raghuwanshi V. Kumar and R. R. Pandey Then layer may be represented by its characteristics impedance (.6). At the center of the film region where thickness eq. d d e e (.11) lim sinh ( d) 0 d 0 d / d / (.1) ( / ) e e lim tanh d d / d 0 d / e e This condition will lead Zs Z p 0. It means that at the center of waveguide the layer may be represented by an open circuit. Thin waveguide shown in Fig. 1 can be represented in equivalent form for a region as shown in Fig. 3. Fig 3: Equivalent T network representation of symmetric slab waveguide ( x 0) as shown in Fig. 1 In the Fig. 3 the impedance of the various branches are shown as follows F F d (.13) ZS Z t a n h 4 F F 1 (.14) Z P Z si n ( d / )

5 Derivation of Eigen value Equation by using Equivalent Transmission (.15) Zs Zs Here subscripts and superscripts film substrate respectively. Where (.16) 117 ZF represent the series parallel k0 n f j 0 and k0 ns Z j 0 s (.17).1 Derivation of Eigen value equation for Odd Modes The Eigen value equation for odd modes can be obtained when the impedance seen from either side of terminal A B in Fig.3 is same as Z SF Z (.1.1) F Zs P or (.1.) 1 d Z F t a n h Z F Zs s i n h ( d / ) 4 In this equation (.1.3) is given by k0 nf jk where k0 n f Simple algebraic manipulation gives (.1.4) d tan (odd m o d e) s Here we have used the following relations

6 118 S. K. Raghuwanshi V. Kumar and R. R. Pandey d d d e e s i n h d /4 d /4 e e c o s ( d / 4) (.1.5) and also s 0 ns. This equation is exactly same for the Eigen value equation of odd TE mode of planar slab symmetric waveguide having thickness d 1.. Derivation of Eigen value equation for Even Modes To derive the Eigen value equation for even mode we define the new parameters (..1.) d YSF Y 5 t a n g 4 (..) Y Fp Y F (..3.) Y s Y s 1 s i n h ( d / ) Here YF 1 ZF Ys 1. Zs Fig 4: Equivalent T- network representation of even mode symmetric slab waveguide.

7 Derivation of Eigen value Equation by using Equivalent Transmission 119 The Eigen value equation can be obtained when the impedance seen from either side of terminal A-B in Fig. 4 is same. (..4.) YSF Y F Y s P Substituting the equations (..1)-(..3) into eq. (..4) and after some trivial algebra it can shown 1 (..5.) d s t a n ( Even mode) 3. Derivation of Eigen Value equation of Asymmetric Planar Slab Dielectric Waveguide In this section we derive the rigorous and exact Eigen value equation for the case of asymmetric planar slab waveguide structure having the refractive index variation1 (3.1) nc n( x) n f ns x 0 d x 0 x d Fig 5: Equivalent T network representation of asymmetric slab waveguide. where

8 10 S. K. Raghuwanshi V. Kumar and R. R. Pandey (3.) F F 1 ZS Z si n ( d ) (3.3) F F 1 ZP Z s i n h ( d ) (3.4) s Z s Z c c (3.5) Z Z and k0 n F f Z j0 s k0 ns s (3.6) Z j0 j0 c k0 ns c Z. j0 j0 After some trivial algebraic manipulation and by using eq. (.1.3) it can also be shown F j d (3.7) ZS t a n F 1 (3.8) Z p. j si n( d) 0 0 Minus in eq. (3.4) is due to limit. Finally the Eigen value equation can be derived from the following expression when the impedance seen from either side of terminal A-B in Fig. 5 is same c F F ( Z ZS ) Z p F s (3.9) ZS Z. c F F ( Z Z ) Z S p After some trivial mathematical manipulation leads the following results

9 Derivation of Eigen value Equation by using Equivalent Transmission (3.10) d c tan d tan ( ) s [ c sin( d ) cos( d )] This implies (3.11) 11 tan( d ) c s. s c 1 This is exactly same Eigen value equation for the asymmetric planar slab waveguide structure Conclusion First time we have derived the exact Eigen value equation for the case of symmetric/asymmetric planar slab waveguide. We have shown the intermediate step of calculation with substantial assumption. In most of the papers being published on TL method have not clarifies the trivial calculation to achieve some specific expression. The derivation presented into this paper is useful to the beginners who want to gain inside into TL method. One can easily extended these results to simulate the mode field profile mode cutoff condition dispersion relation of multilayer dielectric waveguide structure. References X. Qian and A. C. Boucouvalas Synthesis of symmetric and asymmetric planar optical waveguides IET Optoelectron. 1(4) (007) C. D. Papageogiou and J. D. Kanellopoulos Equivalent circuits in Fourier space for the study of electromagnetic fields J. Phys. A: Math. Gen. 15 (198) P. Baccarelli P. Burghignoli F. Frezza A Galli P. Lampariello G. Lovat and S Simone Paulotto Fundamental modal properties of surface waves on meta-material grounded slabs IEEE Trans. On Microwave Theory and Tech. 33(4) (005) A. C. Boucouvalas and C. D. Papageogiou Cut-off frequency in optical fibers of arbitrary refractive index profile using the resonance techniques IEEE J. Quantum Electron. 18(1) (198) D. Papageogiou and A. C. Boucouvalas Propagation constants of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance techniques Electron. Lett. 18(18) (198) Xin Qian and A. C. Boucouvalas Propagation characteristics of single mode optical fibers with arbitrary complex index profiles IEEE J Quantum Electron. 40(6) (004)

10 1 7. S. K. Raghuwanshi V. Kumar and R. R. Pandey A. C. Boucouvalas and Xin Qian Mode dispersion and delay characteristics of optical waveguides using equivalent TL circuits IEEE J Quantum Electron. 41(7) (005) Xin Qian and A. C. Boucouvalas Analysis of leaky modes and Bragg fibers using transmission line equivalent T-circuits IEEE Photonics Tech. Lett. 17(5) (005) A. C. Boucouvalas and C. A. Thraskias Accurate optical fiber refractive index reconstruction from near field IEEE CSNDSP 3 (008) A. C. Boucouvalas and Xin Qian Optical fiber refractive index profile synthesis from near field IEEE GLOBECOM Optical Networking and Systems. (003) Jin-Hong Lin and Cha o-kuang Chen An inverse algorithm to calculate the refractive index profiles of periodically segmented waveguide from the measured near field intensities IEEE J Lightwave Technol. 0(1) (00) S. K. Raghuwanshi Comparative study of asymmetric versus symmetric planar slab dielectric optical waveguides Indian J. of Phys 84(7) (010)

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