JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 11, NOVEMBER 1997 2179 Planar Optical Waveguides with Arbitrary Index Profile: An Accurate Method of Analysis I. C. Goyal, Rajeev Jindal, and Ajoy K. Ghatak Abstract—We present here an improvement of the existing modified airy function (MAF) method. The results of our study show that the improvement gives extremely accurate propagation constants and also the modal fields for planar waveguides with arbitrary index profile. Index Terms—Airy functions, arbitrary index profile, eigen- values/eigenfunctions, improved MAF method, planar optical waveguide. I. INTRODUCTION I N order to determine the characteristics of the devices and components formed by optical waveguides, one needs to know accurately the propagation constants and the fields of the modes supported by the waveguide. The exact analytical so- lution is possible only for some specific profiles. So, to obtain the propagation constant and the field of practical waveguides one resorts to some approximate method, viz. perturbation, variational, JWKB, modified airy function (MAF), etc., or numerical methods. With the exponential growth of the fast computers, numerical methods are no doubt most convenient to use but one cannot get much physical insight as with the closed-form expressions obtained by using approximate methods. The perturbation method is based on a closely related problem which yields an exact solution and we usually resort to first-order perturbation; even then it is extremely difficult to calculate the perturbed eigenfunction as it would involve the summation over an infinite series. On the other hand, the variational method gives good estimate for the lowest order mode by choosing an appropriate trial function and carrying out an optimization; the method becomes quite cumbersome when one has to apply it to higher order modes. The JWKB method is widely used but the solution obtained by the JWKB method diverges around the turning points. The MAF method originally given by Langer [1] has been successfully applied to planar optical waveguides with arbitrary index distribution in the core [2]-[4]. The method gives more accurate results than the JWKB method for most of the profiles and also do not diverge around the turning point [5]. As such, the MAF method represents a powerful tool to analyze optical waveguides for which the exact solution is not known. We may also mention here that now computer codes are available (like MAPLE, MATHEMATICA, etc.) which make direct calculations of Ai(a;) and Bi(a;) and, therefore, while carrying out numerical Manuscript received November 19, 1996; revised July 15, 1997. The authors are with the Department of Physics, Indian Institute of Technology, New Delhi 110 016 India. Publisher Item Identifier S 0733-8724(97)08144-9. calculations, one can play around with Airy Functions (and their derivatives) with as much ease as the trigonometric functions. The MAF method though more accurate than the JWKB method still shows small errors in the calculations of eigen- value and the eigenfunction. In this paper, we present a significant improvement over the existing MAF method. II. THEORY In the analysis of optical waveguides one needs to solve the equation dx 2 =0 where T 2 {x) = k 2 o n 2 {x) - (i 2 (1) (2) and depends on the refractive index profile. This is supposed to be a known function of x apart from an eigenvalue parameter /?, the propagation constant. In the MAF method, we assume the solution to (1) to be of the form ( tf (a;) = { or ( (3) where Ai(x) and Bi(a;) are the Airy's functions [5] and are the solutions of the following equation: y"(x)-xy(x)=0. Substituting (3) in (1), one obtains } 2 +T 2 (x)] = 0. Now, we choose £(x) so that the solution of which gives (4) (5) (6) where x 0 is the turning point, i.e., F 2 (:ro) = 0. Using (5) in (4) and neglecting the F" term, which is the only approximation we make in the MAF method, (4) becomes 2F'{x)Ai'{x)Z'{x) + F(x)Ai'(x)£"(x) = 0. (7)