This paper provides a numerical simulation for determining the complex propagation 
constants of integrated optics waveguide. The waveguide under consideration may 
consist of any number of layers with complex indices due to gain and loss. A general-
purpose mode solver MAPLE program has been developed to implement the transfer 
matrix method, and then it applied to solving the multilayer waveguide dispersion 
equation in complex plane. Additionally, our program can be used to determine the 
electromagnetic mode structure including modal power, spatial distribution, mode size 
parameters, and the position of the modal peak power. Therefore, all necessary 
parameters for a wide range of laser devices can be calculated

Ali  A. Amara

Sabah M.M. Ameen

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 111  J.Thi-Qar Sci           Vol.1 (4)                          May/2009            ISSN  1991- 8690                                                               1661  - 0968 الترقيم الدولي                                                  Key word: Optical waveguide, Transfer matrix, Mode solver.        Computation of propagation characteristics in complex plane for multilayer optical waveguide Sabah M.M. Ameen                  Ali A. Amara Physics Dept.- College of Science - University of Basrah - Basrah, Iraq.  ABSTRACT     This paper provides a numerical simulation for determining the complex propagation constants of integrated optics waveguide. The waveguide under consideration may consist of any number of layers with complex indices due to gain and loss. A general-purpose mode solver MAPLE program has been developed to implement the transfer matrix method, and then it applied to solving the multilayer waveguide dispersion equation in complex plane. Additionally, our program can be used to determine the electromagnetic mode structure including modal power, spatial distribution, mode size parameters, and the position of the modal peak power. Therefore, all necessary parameters for a wide range of laser devices can be calculated.   111  J.Thi-Qar Sci           Vol.1 (4)                          May/2009            INTRODUCTION     Multilayer planar waveguide structures have been widely used in the implementation of variety of optical devices including distributed feedback lasers [1], TE-TM converters [2], waveguide polarizers [3], Bragg reflectors [4], directional couplers [5], antiresonance reflecting optical waveguide (ARROW) structures [6], broad-area semiconductor lasers [7], [8].     In order to optimize the performance of integrated optical devices using such waveguides, it is important to know its propagation characteristics and the transverse modal power distribution. Since, analytical solutions does not exist for such structures, one may uses either approximate methods such as the perturbation method [9], the WKB method [10], variational method [11],  graphical methods [12], mode-matching method [13], and Cauchy integration method (CIM) [14]-[17] or numerical methods [18], [19] to solve the wave equation.     The perturbation method was used for lossless 5-layer structure, lossy 4-layer structure, and for a metal-clad waveguide to determine the propagation constants and the resulting propagating mode profiles. The previous technique can not be applied successfully and then can not easily be extended to multilayer structures, since their approach is analytic and the formulas involved become cumbersome.      Numerical methods, that can efficiently and accurately solve the wave equation for the propagating modes, are thus of obvious importance since they are used as a basic tool in the design technique.     Traditional numerical zero-search algorithms such as the downhill method [20], [21], Newton's method [22] and the one-dimensional scan method in the complex plane [23] needs an initial guess value close to the actual root. Therefore, these methods are not efficient and reliable, especially for a general-purpose mode solver.     Among these methods, transfer matrix method (TMM) [24], [25] is one of the primary tools for multilayer waveguide analysis. The theory of TMM can easily generate the dispersion equation for TE and TM modes supported by such structures in a straightforward manner. Multilayer waveguide can consist of any combination of lossless and lossy (dielectric, semiconductor, metallic) and active (including uniaxially  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009           anisotropic quantum well) layers. The guided mode propagation constants of the structure correspond to zeros of the dispersion equation.     A program was needed for solving the wave equation for guided modes that consist of many layers with complex refractive indices. The imaginary part of refractive index describes gain and loss in the layers. The program developed should be run on a personal computer within an acceptable time and accuracy.     In this paper, we develop a MAPLE program for the solution of the characteristic equation in the complex plane numerically corresponding to TE and TM modes of a multilayer planar optical waveguide. The TMM based program is extended to compute the mode field profile and its evolution, mode size, and its peak position. In section 2, we will discuss the necessary theory of the method, and in section 3, we will present some numerical results to show the validity of our calculations. Conclusions will drown in section 4. Transfer Matrix method (TMM)     Maxwell's curl equations for source-free, time-harmonic fields in anisotropic media are: HjE o       (1a)  EjH ro       (1b) Where o is the free space permittivity,  is the angular frequency, and r is a relative permittivity.     Consider a planar multilayer waveguide as shown in Fig. 1. For mode propagating along the ẑ  direction in the homogeneous i-th layer, the field components Ex, Ez, Hy will vanishes for TE mode, whereas Hx, Hz, Ey will vanishes for TM mode. With these assumptions the wave equation for the i-th layer reduces to    1,222,22,0)()( iiiyioiy xxxxFnkxFx  (2) Where TMHTEEFyyy ,,  ,  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009           re im is the complex propagation constant of the mode, ko  is the free space wavenumber. The layers have complex refractive indices n=nre+jnim, where the imaginary part is due to gain or loss. The effective index neff and the absorption WG are given, respectively, by [24]  neff re o        (3a)  WG im        (3b) The general solution of the wave equation in each homogeneous layer (i) is well known ))(exp())(exp()(, iiiiiiiy xxBxxAxF      (4) Where 222ioi nk  , Ai and Bi are the complex field coefficients that vary from layer to layer, and xi is the position of the interface between layer i and i+1. By imposing the continuity conditions of the field and its derivatives for each interface, it is easy to find [24], [26] iiiiiBATBA11        (5) Where )exp(1)exp(1)exp(1)exp(1211111iiiiiiiiiiiiiiiiiiiiiddddT Where di is the i-th layer thickness and  TMnnTEiii,/,1221 , One can relate field coefficients in the cladding (Ac and Bc) with the coefficients in the substrate (As and Bs) as follows: ccssBATBA        (6) Where 5512211112ttttTTTTT cN   and N is the total number of layers.  122  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                For the guiding modes, the fields should be evanescent in the cladding and the substrate layers, so Ac=0 and Bs=0 that results the characteristic equation [24] 0)(11 t          (7)     For multimode waveguide, there is a z-dependent phase difference between modes due to modal dispersion. As a result the combined mode field for multimode waveguide is given by superposition of all the guided mode fields as follows [25]  Mmmyy zjxFzxF mtotal1)exp()(),(       (8) Where M is the total number of modes. Numerical Results and Discussions     A program has been developed using MAPLE symbolic computational language to handle the transfer matrix method. The built-in floating point arithmetic root finding method through fsolve function to determine the complex roots of the characteristic equation (Eq. 7) that corresponds to the propagation and loss constants directly without needing to find the analytical derivatives for the characteristic equation as required in CIM [12]-[15].     In this section, we present some numerical results to test the validity of our program. We have computed the propagation and loss characteristics, the modal field profile, the position of power maxima, and the mode size of TE and TM modes that propagates in multilayer planar optical waveguides.      Our program was applied to 6-layer lossy waveguide. The typical values of the various parameters are assumed as follows [17]: ns=3.172951, n1=3.16455, n2=3.22534, n3=3.39583, n4=3.5321-j0.08817, n5=3.39614, n6=3.38327, nc=1.0, d1=0.6 , d2=1.6 , d3=0.518 , d4=0.6 , d5=0.2 , d6=0.1 , o=1.523 , which corresponds to the multi-mode region. The results are perfectly agree with those reported in [17] as indicated in Table 1.    The dispersion equation of the multilayer planar waveguide (Eq. 7) has been numerically solved to calculate the effective index and loss of the guided modes and plotted as a function of ko for TE and TM modes as shown in Figs. 2 and 3, respectively.    Upon normalizing the time-averaged power per unit length in the lateral direction, the  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009           field profile (Eq. 4) of each guided mode and its evolution (Eq. 8) can be obtained by successively applications of the transfer matrix to the field at the reference interface plane. Having found the roots, the coefficients Ai and Bi of each layer can be calculated from Eq. 6.  A1 is equal to zero and B1 has to be determined. Since all coefficients are proportional to B1 , B1 reflects the normalization, which has been accomplished according to Ref.[25].     In Figs. 4 and 5, we have plotted the modal power profiles 2yE  and 2yH  as a function of x for TE and TM modes for ko=2.7 and 4.0 -1, respectively. Such plots are extremely useful in computing the power peak position as a function of ko (see Fig. 6) and mode size which is defined as the value of x where the power value reduces to 1/e of its peak value (see Fig. 7). The modal spatial power distribution are also useful in overlap integral calculations for estimating the efficiencies in electrooptic or acoustooptic interactions using such waveguides, coupling efficiency calculations, etc.     Mode field evolution of this structure can be determined along its propagation direction using Eq. 8 and plotted for TE- and TM-modes for different values of ko as shown in Fig. 8. The propagation constants are tabulated in Table 2. The calculations are repeated in the case of absence the imaginary part of the refractive index (lossless case) for comparison purposes and also shown in the same figure. The waveguide support three modes with effective refractive index (neff) as shown in Table 2 for the complex case and in Table 3 for the real case.     Also, Figs.8 shows that the lightwave are not coupled between the layers as it propagate along the structure because of the attenuation that it suffers due to the waveguide loss which uses lossy materials. Whereas, for lossless case, the coupling between the different modes are very effected on the power evolution. Moreover, the coupling lengths between the different propagating modes can be estimated directly from this figure. Conclusions     In this paper, we have presented a general-purpose mode solver MAPLE program to analyze multilayer planar optical waveguide. We have used TMM and some numerical methods in the analysis. The method not only converges rapidly but is also capable of  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009           giving results of specified accuracy. The program can be used in effective index and loss, field or power profiles, and mode size calculations                                                                                                         Table 1. Complex propagation constants of guided modes in a six layer planar optical waveguide. Table 2. Complex propagation constants (/ko) of guided modes for different wavenumbers.  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                        Table 3. Propagation constants (/ko=neff) of guided modes for different wave-numbers for the real case.  Fig.1. index profile for 6-layer planar optical waveguide.    121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                         Fig. 2. Propagation and loss characteristics of 6-layer planar waveguide for TE modes.  Fig. 3. Propagation and loss characteristics of 6-layer planar waveguide for  TM modes  121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                         Fig. 4. Modal power distribution in 6-layer planar waveguide for TE and TM modes at ko=2.7 m-1. Fig. 5. Modal power distribution in 6-layer planar waveguide for TE and TM modes at ko=4.0 m-1.   121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                         Fig. 6. Peak position of the power profile as a function of ko for TE and TM modes.   121  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                         Fig. 7. Mode size as a function of ko for TE and TM modes.   111  J.Thi-Qar Sci           Vol.1 (4)                          May/2009                                                         Fig. 8. The evolution of the TE and TM modes in 6-layer planar waveguide for three different values of ko=2.7, 3.4 and 4.0 m-1. The figure is plotted for the case of complex and real propagation constant.   111  J.Thi-Qar Sci           Vol.1 (4)                          May/2009           References  1.   S. M. Sadeghi, W. Li, X. Li, and W.P. Huang, 2006. " Coherently Tunable Mid-Infrared          Distributed Feedback Lasers," IEEE J. Quantum Electron., 42 (8), 752-759. 2.    S. Bhandare and R. No´e, 2001. "Optimization of TE–TM mode converters on X-cut, Y-    propagation LiNbO3 used for PMD compensation," Appl. Phys. B , 73, 481–483. 3.    J. Bornemann, and V.A. Labay, 1995. "Ridge Waveguide Polarizer with Finite and Stepped-          Thickness Septum," IEEE J. Microwave Theory and Techniques, 43 (8), 1782-1787. 4.   Z. Zhaowei, T. Chun, M. R. Mokhtar, P. Petropoulos, D. J. Richardson, and M. 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Computation of propagation characteristics in complex plane for  multilayer optical waveguide

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Computation of propagation characteristics in complex plane for multilayer optical waveguide

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