MoM solutions. In Section II, a few results at a moderately high frequency were presented, which showed excellent agreement between the standard MoM code and the hybrid method. These hybrid results, while interesting, are not new given the wide use and maturity of this technique. It was necessary, however, to briefly document the hybrid method in order to show how we could extend the current technique by using asymptotics. It was shown that, given a simple Hankel function substitution, asymptotic analysis could be used to evaluate many of the highly oscillatory integrals that resulted from the hybrid formulation. Results showed that the asymp- totics reduced the computation time by roughly an order of Ž . magnitude over the normal hybrid solution see Fig. 5 , with no appreciable loss in accuracy. Finally, it is important to note that the solution run time of the hybrid method with the asymptotics is generally independent of the incident wave frequency. REFERENCES 1. W.D. 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Carin, Scattering by 2-D strips using an asymptotic hybrid formulation combining the method of mo- ments and physical optics techniques, Microwave Opt Technol Ž . Lett 7 1994 , 542546. 13. C.A. Balanis, Advanced engineering electromagnetics, Wiley, New York, 1989. 14. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical recipes in FORTRAN, Cambridge University Press, Cambridge, England, 1986. 15. L.B. Felsen and N. Marcuvitz, Radiation and scattering of waves, Prentice-Hall, Englewood Cliffs, NJ, 1973. 16. I.A. Stegun and R. Zucker, Automatic computing methods for special functions. Part IV. Complex error function, Fresnel inte- Ž . grals, and other related functions, J Res Nat Bur Stand 86 1981 , 661683.  1999 John Wiley & Sons, Inc. CCC 0895-247799 MODAL BEHAVIOR, CUTOFF CONDITION, AND DISPERSION CHARACTERISTICS OF AN OPTICAL WAVEGUIDE WITH A CORE CROSS SECTION BOUNDED BY TWO SPIRALS Vivek Singh, 1 S. P. Ojha, 1 and L. K. Singh 2 1 Department of Applied Physics Institute of Technology Banaras Hindu University Varanasi 221005, India 2 Dr. R. M. L. Awadh University Faizabad, India Recei  ed 18 September 1998 ABSTRACT: This paper presents a theoretical treatment of the modal characteristics, cutoff condition, and dispersion characteristics of a new type of optical wa eguide whose guiding region is bounded by two spirals. A segment of the core cross section can be considered as a distorted planar wa eguide in which a cur  ature and a flare ha e been introduced. Using the boundary conditions for the proposed wa eguide under the weak guidance condition, the modal characteristic equation and modal cutoff equation ha e been deri  ed. Dispersion cur  es ha e also been obtained for some low-order modes.  1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 21: 121124, 1999. Key words: optical wa eguide; spiral shape; cutoff conditions; normal- ized dispersion cur  es INTRODUCTION The last few decades have seen a tremendous advance in optical communication systems and integrated optical tech-   nology 1  4 . The phrases ‘‘smart structures’’ and ‘‘smart materials’’ have come into fashion. Currently, the study of waveguides with smart structures, like those having elliptical, rectangular, triangular, pentagonal, Piet  Hein, cardiodic,   hypocycloidal, and other cross sections 5  17 have been extensively studied by many researchers. Optical waveguides with smart materials like chiral materials and liquid crystals   have also been investigated elaborately 18  20 . To investi- gate these unconventional waveguides, one may use various analytical and numerical methods such as the perturbation     method 21 , the variational method 12 , the finite-element     method 22 , the boundary-element method 23 , the point-  matching method 9 , etc. Among these, we have chosen here the analytical method with only a few approximations which are physically and mathematically legitimate.   In two recent communications 24  25 , the present investi- gators studied the modal characteristics, cutoff conditions, and the normalized dispersion curves of two new types of waveguides, and some interesting results about the modal behavior and dispersion curves were obtained. Here, in the present communication, an analytical analysis under weak guidance has been made to study the modal behavior, cutoff condition, and dispersion characteristics of a new type of waveguide having a core cross section bounded by two spirals. Choosing an appropriate new coordinate system, we find the MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999 121