IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 11, NOVEMBER 2005 2331 Hermite–Gauss Series Expansions Applied to Arrayed Waveguide Gratings Víctor García-Muñoz and Miguel A. Muriel, Senior Member, IEEE Abstract—An accurate model is presented to calculate the trans- mission characteristics of an arrayed waveguide grating under the Fourier optics model using the properties of the Gauss–Hermite beams. Index Terms—All-optical devices, expansion methods, Fourier optics. I. INTRODUCTION T HE arrayed waveguide gratings (AWGs) are key devices in wavelength-division-multiplexing optical networks. The AWGs are planar-type components that are superior to other types of wavelength multi/demultiplexers in terms of spatial separation of frequencies and in compactness. The model based on Fourier optics presented by Takenouchi [1] and extended by Muñoz [2] produces a simple expression of the field profile in all the AWGs regions of interest. The calculi on the simulations become closed expressions when the field profile in the waveguides is taken to be Gaussian. The Gaussian approximation (GA) is not accurate enough for the description of the main device characteristics, so mode solvers must be used, with a great computationally effort, in order to obtain good ac- curacy. The objective of this letter is to improve the calculations of the transmission characteristics of the AWG in the Fourier op- tics model via a series expansion of the field in the waveguides, taking advantage of the properties of the Hermite–Gaussian polynomials (GH) in order to keep the closed expressions of the fields in the regions of interest. The GH set constitute an orthonormal basis well suited for the description of compact support signals like the field of a slab waveguide; this feature permits a very accurate descrip- tion of the field in the waveguides using a few terms of an Her- mite–Gauss series expansion (GH-SE). The field profile at the end of propagation along the free propagation regions (FPRs) of the device (Fig. 1) can be obtained by the Fourier transform (FT) of the input distribution. The GHs are eigenfunctions of the FT; hence, if the input distribution is expressed as a GH-SE then the output distribution will be expressed as a GH-SE too. These two characteristics allow a simple yet accurate description of both the field of the waveguides and the field on the propaga- tion regions using GHs functions. This letter in organized as follows. First, GH-SEs are used to describe the AWG’s waveguides fields. Second, the series ex- Manuscript received March 22, 2005; revised July 18, 2005. This work was supported by the Spanish Ministerio de Educacion y Ciencia under the Project Plan Nacional de I+D+ITEC2004-04754-C03-02. The authors are with ETSI Telecommunicatión, Universidad Politécnica de Madrid, Madrid 28040, Spain. Digital Object Identifier 10.1109/LPT.2005.857976 Fig. 1. Diagram of the AWG showing the location of the fields. The number of waveguides shown is smaller than the real one. pansions are included in the Fourier model obtaining analytical expressions for the field in the different parts of the AWG. II. GH-SE OF THE FIELD OF A RECTANGULAR OPTICAL WAVEGUIDE A. Waveguide Field In the model presented in [1], the AWG is taken as a two-di- mensional device (in the plane) where the transversal field is considered equal to the field of a slab waveguide. In the case of a slab waveguide, the field equation can be solved exactly [3]. The spatial distribution of the transverse-electric (TE) fundamental mode is called real mode and follows the expression: (1) (2) where is the power normalization coefficient, and and are the eigenvalues of the TE mode transcendental equation. B. Calculus of the Optimal Waist Parameter (WP) The field profile of the waveguide is expanded in terms of GH-SE. The field profile is an even function, so only the even terms of the expansion are needed (3) where being even and are the coeffi- cients of the expansion (CE). The expression for the th order GH function is (4) 1041-1135/$20.00 © 2005 IEEE