Volume 34, number 1 OPTICS COMMUNICATIONS July 1980 VARIATIONAL ANALYSIS OF DIRECTIONAL COUPLERS WITH GRADED INDEX PROFILE * Anurag SHARMA, Enakshi SHARMA, I.C. GOYAL and A.K. GHATAK Department of Physics, Indian Institute of Technology, Delhi, New Delhi- 110029. bTdia Received 10 March 1980 A variational analysis of graded optical directional couplers is presented. Fabrication of such directional couplers by an electron-beam writing method has recently been reported. We have shown that our analysis gives better results than the WKB method which has been used previously to analyse such couplers. Further, our analysis involves much less alge- braic and nunlerical work than the WKB method. 1. Introduction 2. Theory Recently Suhara et al. [1] have presented an anal- ysis of a graded-index directional coupler fabricated by electron beam writing on chalcogenide amorphous semiconductor films [2,3]. The corresponding re- fractive index profile is given in sect. 2. In their anal- ysis, the two dimensional wave equation has been separated by applying the method of the effective- index-of refraction into two one-dimensional equa- tions. One of the resulting equations corresponds to a step-index planar waveguide, solutions for which are well-known, while the other equation corresponds to a graded index planar waveguide, solutions for which have been obtained by using a WKB analysis. How- ever, a directional coupler is usually a few-mode wave- guiding structure (ideally only the first two modes should propagate) and therefore a WKB analysis is not expected to give accurate results. With this view, we have examined the accuracy of these results by solving the corresponding wave equation numerically and have found the WKB results to be highly inaccu- rate. We have also shown that a variational analysis gives much more accurate results and involves much less algebra and numerical effort than the WKB analysis. * Work partially supported by Department of Science and Technology, India and Counsil of Scientific and Industrial Research, India. The refractive-index distribution of a directional coupler fabricated by the electron-beam writing method can be written as [1] n 1 y > T n(x, y) = n20 + An 2 f(xlo) 0 < v < T n 3 y<0 where f(x/o) = exp [-(x + d) 2/02 ] + exp [-(x - d) 2/02 ] ; Tis the thickness of the waveguiding layer; o is a characteristic length; nl, n20 and n 3 are the refractive indices of the air, the guiding layer and the substrate, respectively, and 2d is the separation of the centres of the two waveguides constituting the directional coupler (see fig. 1). The refractive-index increment An 2 is assumed to be small so that An 2 ~ n20 and the following relations hold: 17_ _ 2 where k is the free space wave number. Since there is a gradation of refractive index in the (1) 39