Microwave Science an Technology Volume 2007, Article ID 85181, 5 pages oi:10.1155/2007/85181 Research Article Numerical Analysis of Inhomogeneous Dielectric Waveguie Using Perioic Fourier Transform M. Moraian an M. Khalaj-Amirhosseini Receive 22 September 2007; Accepte 24 December 2007 Recommene by Yue Ping Zhang A general metho is introuce to obtain the propagation constants of the inhomogeneous ielectric waveguie. The perioic Fourier transform is applie to the normalize Maxwell s equations an makes the fiel components perioic. Then they are expane in Fourier series. Finally, the trapezoial rule is applie to approximate the convolution integral which leas to a set of couple secon-orer ifferential equations that can be solve as an eigenvalue-eigenvector problem. The normalize propagation constant can be obtaine as the square roots of the eigenvalues of the coefficient matrices. The propose metho is applie to the ielectric waveguie with a two-layere ielectric profile in the transverse irection, an the first four-confine TE moes are obtaine. The propagation constants for the mentione ielectric waveguie are also erive analytically an are then compare with those erive by the propose metho. Comparison of results shows the efficacy of the propose metho. Copyright 2007 M. Moraian an M. Khalaj-Amirhosseini. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. 1. INTRODUCTION Dielectric waveguie with various or inhomogeneous ielectric layers profile has been aresse in many research literatures. The accurate etermination of propagation characteristic for such a transmission meia is essential in esign an optimization of various types of optical an millimeter wave evices. Several computation methos have been propose to obtain the propagation characteristic of the planar grae-inex ielectric waveguie. Some approximate analytical methos such as WKB [1], perturbation metho [2], IBMOM [3], an GIBMOM [4] are effective for computing the propagation characteristic of these types of waveguies. Other methos that use computationally intensive methos such as sine/cosine expansion methos [5, 6], finite ifference methos [7], an finite element methos [8] are also effective for this purpose. A proper metho for computing the propagation characteristic of inhomogeneous ielectric waveguie is presente, which is base on the perioic Fourier transform. The perioic Fourier transform has been establishe in several literatures. In [9] the perioic Fourier transform was propose to investigate the scattering problems in a perioic structure with finite extent, while in [10]it was use to analyze a series of transition in ielectric waveguie. In this paper, a metho in [10] will be evelope to etermine the propagation constants of inhomogeneous ielectric waveguies. To show the valiity of the propose metho, the metho is applie to a two-layere ielectric waveguie. But, the application of the propose metho is not essentially restricte to the multilayer ielectric waveguie profile, an can be easily applie to any inhomogeneous ielectric waveguie profiles. 2. WAVE PROPAGATION IN GRADE-INDEX WAVEGUIDE In this section, the propagation constant for the TE waves in inhomogeneous slab ielectric waveguie is erive. Cartesian coorinates Oxyz is introuce an oriente such that Oz is parallel to the irection of propagation an Ox is in the irection of the ielectric profile variation as shown in Figure 1. The geometry an the fiels are also uniform an have no variation in the y-irection. The value of ielectric constant changes in the transverse irection as a function of x. The ielectric is assume to be linear, isotropic an its permeability set is equal to the free space permeability. The fiels are time harmonic an complex variables. They have only x an z epenency. The Electric fiel intensity E, Magnetic fiel intensity H, an Electric flux ensity D are normalize by 4 μ0 4 /ε 0, ε0 4 /μ 0,anε 0 μ0 /ε 0, respectively.in these relations, ε 0 an μ 0 are the permittivity an permeability of free space, respectively.
2 Microwave Science an Technology x + y ε s ε r x) h z All of the transforme relations are perioic in terms of x, because of the perioic nature of perioic Fourier transform. So they can be approximately expane in the truncate Fourier series. By introucing the 2N +1) 1column matrices which have the Fourier series coefficients of all the electromagnetic fiels components, the equations yiel the following relations: Figure 1: Cross section of the investigate inhomogeneous ielectric profile. The ielectric constant can be ecompose into two parts, namely, constant an nonconstant parts an expresse as εx) = ε s + ε nc x). 1) The reciprocal of relative permittivity istribution is enote by ηx) = 1/εx)anecomposesas ηx) = η s + η nc x). 2) The first part is equal to 1/ε s, while the last part is not equal to 1/ε nc x) since ε nc x) iszerointheregionwhichisfille with ielectric waveguie slab. Although many relations which are mentione here are the same as those in [10], they will be mentione here for simplicity. From normalize Maxwell s equations, one has the following relations: z H xx, z) x H xx, z) = ik 0 D y x, z), z E yx, z) = ik 0 H x x, z), 3) x E yx, z) = ik 0 H z x, z). In these equations, k 0 is the wave number in free space. The ielectric constant has inhomogeneous profile in the transverse x irection an the require constitutive relation is in the form D y x, z) = εx)e y x, z). 4) Applying the perioic Fourier transform to 3), the following equations will be achieve: ) z H x x, α, z) x + iα H x x, α, z) = ik 0 D y x, α, z), z Ẽyx, α, z) = ik 0 H x x, α, z), ) x + iα Ẽ y x, α, z) = ik 0 H z x, α, z), D y x, α, z) = ε s Ẽ y x, α, z)+ 2π/ ε nc x, α ξ)ẽ y x, ξ, z)ξ. 2π 0 5) z h x α, z) ik 0 Xα) h x α, z) = ik 0 y α, z), 6) z ẽyα, z) = ik 0 h x α, z), 7) Xα)ẽ y α, z) = ik 0 h z α, z), 8) y α, z) = ε s ẽ y α, z)+ 2π/ ]] [[ εnc α ξ)ẽy ξ, z)ξ. 2π 0 9) In the above equations, the matrices Xα) an[[ ε nc ]]α) are erive by α Xα) )n,m = δ n,m + n λ ) 0, 10) k 0 ]] ) [[ εnc α) n,m = ε nc,n m α). 11) Here, δ n,m is the Kronecker elta an ε c,n α) is the nth-orer Fourier coefficients of ε c x, α) an it is given by ε nc,n α) = 1 = 1 0 ε nc x, α)e in2π/)x x ε nc x)e iα+n2π/))x x. 12) Taking L + 1 equispace sample points between [0, 2π/]an approximate the convolution integral by applying the trapezoial rule to it, the following will eventually arrive: y αl, z ) = ε s ẽ y αl, z ) + 1 ]] [[ εnc αl )ẽy α0, z ) 2L L 1 ]] +2 [[ εnc αl m )ẽy αm, z ) + [[ ε ]] nc αl L )ẽy αl, z )). m=1 13) Introuce L +1) 2N +1) 1columnmatricestoexpress the iscretize Fourier coefficients at L + 1) sample points by ẽ y α 0, z) ẽ y z) =.. 14) ẽ y α L, z)
M. Moraian an M. Khalaj-Amirhosseini 3 So 6) 8)canberewrittenintheform with z h xz) ik 0 X h z z) = ik 0 y z), z e yz) = ik 0 h x z), X e y y) = h z z), y z) = [[[ ε ]]] e y z), 15) X ) α 0 0 0. 0........ Xα) =, 16)....... 0 0 0 X ) α L [[[ εnc ]]] = 1 2L [[[ ε ]]] = εs I + [[[ ε nc ]]], 17) a a a a......., 18) q q q q where a enotes [[ ε nc ]]α 0 ), a enotes 2[[ ε nc ]]α 1 ), a enotes 2[[ ε nc ]]α L+1 ), a enotes [[ ε nc ]]α L ), q enotes [[ ε nc ]]α L ), q enotes 2[[ ε nc ]]α L 1 ), q enotes 2[[ ε nc ]]α 1 ), an q enotes [[ ε nc ]]α 0 ). Finally, from 15), one can obtain the following seconorer ifferential equation as with 2 z 2 ẽyz) = k 2 0C e ẽ y z) 19) C e = [[[ ε ]]] X 2. 20) Similar to the TM moes, the normalize Maxwell s equations an the constitutive relations are as follows: z E xx, z) x E zx, z) = ik 0 H y x, z), 21) x H yx, z) = ik 0 D x x, z), 22) x H yx, z) = ik 0 D y x, z), 23) D x x, z) = εx)e x x, z), 24) D z x, z) = εx)e z x, z). 25) Following the same proceure with the TE case, we obtain the following relations z e xz) ik 0 Xe z z) = ik 0 h y z), z h yz) = ik 0 x z), Xh y z) = z z), x z) = [[[ η ]]] 1 ex z), z z) = [[[ ε ]]] e z z). 26) The expression of [[[η]]] is riven just by replacing the notation ε by η in 11), 12), 17), an 18). From 26), one can obtain another couple ifferential equation as with 2 z 2 h yz) = k 2 0C h h y z) 27) C h = [[[ η ]]] 1 I X [[[ ε ]]] 1X ). 28) The propagation constant for TE an TM moes can be obtaine by solving couple secon-orer ifferential equations 19) an27) as an eigenvalue-eigenvector problem. The following example emonstrates how to use this metho to fin the propagation constant for a typical ielectric waveguie with two-layere ielectric profile. 3. DESIGN EXAMPLE In this section, the propose metho is applie to a ielectric waveguie with a two-layere profile [11]. The profile of ielectric waveguie has the following form: 1 x<0 6.25 0 x λ 0 ε r = 2.25 0 <x 2λ 0 1 x>0. 29) The height of the mentione ielectric waveguie is equal to 2λ 0, λ 0 is the wavelength in free space. The first layer has the ielectric constant equal to 6.25 an the secon one has the ielectric constant equal to 2.25. Table 1 shows the exact normalize propagation constants for four-confine TE moe in this waveguie. These normalize propagation constants are erive analytically [11]. Normalize propagation constants for the mentione waveguie are also erive by the propose metho an by selecting the number of integration segments an truncation orer equal to 30 an 20, respectively. Figure 2 shows the relative error rate of the obtaine propagation constants versus the truncate orer N an with the fictitious perios = 3λ 0. Comparison of the numerical results shows the convergence of the propagation constants versus the truncation
4 Microwave Science an Technology Table 1: The exact normalize propagation constant for four-confine moe in two-layere ielectric waveguie. β 0 β 1 β 2 β 3 2.46190937255032 2.34488648040691 2.13982061988205 1.82913990191955 10 2 10 1 10 3 10 2 Error rate 10 4 10 5 Error rate 10 3 10 4 10 5 10 6 10 6 10 7 5 10 15 20 25 30 Truncation orer N 10 7 4 6 8 10 12 14 16 18 20 22 24 Truncation orer N First Secon Thir Fourth Figure 2: Error rate of the four-confine TE moe versus truncation orer N. = 0.2λ = 0.5λ = 1λ = 2λ = 5λ Figure 3: Error rate of the funamental TE moe versus truncation orer N. orer N. The values of error rates are also increase as the orer of TE moes is increase. Figure 3 shows the error rate forfirsttemoeversustruncationorern an the fictitious perios = 0.2λ 0,0.5λ 0, λ 0,2λ 0,an5λ 0.Thenumber of integration segments is also equal to 30 in Figure 3. The convergence of error rate for the fictitious perios = 0.2λ 0, 0.5λ 0,anλ 0 is fast, while for other fictitious perios, there is a slow convergence an they have worse accuracy limit. Figure 4 shows the error rate for the first TE moe versus number of integration segments an with the fictitious perios = 0.2λ 0,0.5λ 0, λ 0,2λ 0,an5λ 0. The number of truncation orer is also selecte equal to 15 uring the numerical proceure. As Figure 4 shows, the convergence of error rate for the fictitious perios = λ 0,2λ 0 an 5λ 0 is much faster than the fictitious perios = 0.2λ 0 an 0.5λ 0. It shows also that increasing the number of integration segments afteracertainnumberwillnolongeraltertheerrorrate.while for small value of this number an for the fictitious perios = 0.2λ 0 an 0.5λ 0, there is a really worse accuracy limit. So by selecting a large fictitious perio, the convergence of error rate with respect to L becomes better, while the convergence of the relative error rate with respect to N becomes worse. Error rate 10 1 10 2 10 3 10 4 10 5 10 6 10 7 5 10 15 20 25 30 35 40 45 50 55 60 Number of integration segment L = 0.2λ 0 = 0.5λ 0 = λ 0 = 2λ 0 = 5λ 0 Figure 4: Error rate of the funamental TE moe versus number of integration segment L. 4. CONCLUSION In this paper, the perioic Fourier transform has been applie for fining the propagation constant of inhomogeneous ielectric waveguies. The propose metho is straightforwar an oes not nee any bounary conition. The normalize fiel quantities became perioic by apply- ing the perioic Fourier transform to them. First, the truncation orer Fourier transform of fiel quantity was erive. Next, by applying the trapezoial rule to approximate the convolution integral, a set of couple secon-orer ifferential equations was foun. Then they were solve as an
M. Moraian an M. Khalaj-Amirhosseini 5 eigenvalue-eigenvector problem for fining the propagation constant of an inhomogeneous ielectric waveguie. To valiate, a two-layere ielectric waveguie was selecte as an objective an propagation constant for four-confine TE was compute by the propose metho. There is a goo agreement between the results of the propose metho an those from the literature. REFERENCES [1] R. Srivastava, C. Kao, an R. Ramaswamy, WKB analysis of planar surface waveguies with truncate inex profiles, Journal of Lightwave Technology, vol. 5, no. 11, pp. 1605 1609, 1987. [2] A. Kumar, K. Thyagarajan, an A. Ghatak, Moes in inhomogeneous slab waveguies, IEEE Quantum Electronics, vol. 10, no. 12, pp. 902 904, 1974. [3] A. Weisshaar, Impeance bounary metho of moments for accurate an efficient analysis of planar grae-inex optical waveguies, Lightwave Technology, vol. 12, no. 11, pp. 1943 1951, 1994. [4] J. Li an A. Weisshaar, Generalise impeance bounary metho of moments for multilayer grae-inex ielectric waveguie structures, IEE Proceeings: Optoelectronics, vol. 143, no. 3, pp. 167 172, 1996. [5] Y. Tu, I. C. Goyal, an R. L. Gallawa, Analyzing integrate optical waveguies: a comparison of two new methos, Applie Optics, vol. 29, no. 36, pp. 5313 5315, 1990. [6] A. Weisshaar an V. K. Tripathi, Moal analysis of step iscontinuities in grae-inex ielectric slab waveguies, Journal of Lightwave Technology, vol. 10, no. 5, pp. 593 602, 1992. [7] K. Bierwirth, N. Schulz, an F. Arnt, Finite ifference analysis of rectangular ielectric waveguie structures, IEEE Transactions on Microwave Theory an Techniques, vol. 34, no. 11, pp. 1104 1114, 1986. [8] B. Rahman an J. B. Davies, Finite-element solution of integrate optical waveguies, Lightwave Technology, vol. 2, no. 5, pp. 682 688, 1984. [9] J. Nakayama, Perioic Fourier transform an its application to wave scattering from a finite perioic surface, IEICE Transactions on Electronics, vol. E83-C, no. 3, pp. 481 487, 2000. [10] K. Watanabe an K. Kuto, Numerical analysis of optical waveguies base on Perioic Fourier transform, Progress in Electromagnetics Research, vol. 64, pp. 1 21, 2006. [11] P. Yeh, Optical Waves in Layere Meia, JohnWiley& Sons, New York, NY, USA, 2005. AUTHOR CONTACT INFORMATION M. Moraian: Department of Electrical Engineering, Iran University of Science an Technology Narmak, Tehran 16844, Iran; m moraian@ee.iust.ac.ir M. Khalaj-Amirhosseini: Department of Electrical Engineering, Iran University of Science an Technology Narmak, Tehran 16844, Iran; khalaja@iust.ac.ir
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