IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-19, NO. 1, JANUARY 1983 15 Analytical Approximations for the Propagation Characteristics of Dual-Mode Fibers SHAIKH IQBAL HOSAIN, ENAKSHI KHULAR SHARMA, ANURAG SHARMA, AND AJOY K. GHATAK Abstract—In this paper we present accurate analytical approximations for the modal fields of dual-mode optical fibers with power law profiles based on a scalar variational analysis. We propose single parameter and two parameter trial functions and use these to study the dispersion characteristics and estimate the value of the normalized frequency cor- responding to zero intermodal dispersion, which defies the operating point for such fibers. Our results show that the relatively simple singte parameter field for the LPll mode gives a good fit to the field inside the core and estimates the propagation constant fairly well, but is inade- quate to calculate the dispersion characteristics. On the other hand, the two parameter field estimates all these characteristics with a high degree of accuracy and enables one to accurately compute the normal- ized frequency for zero intermodal dispersion as well as dispersion tolerance around this value. INTRODUCTION O VER the last few years, dual-mode optical fibers have emerged as a promising alternative to single-mode fibers for high bit rate long haul systems [11-[6]. This is because they can have a relatively larger core radius which facilitates jointing and splicing and still provides transmission bandwidths comparable to single-mode fibers. The essential feature of these high bandwidth dual-mode fibers is that by a suitable choice of fiber parameters, the mode propagation velocities of the LP ol and LP l l modes can be equalized giving modal dispersion free propagation. Hence, design considerations of such fibers require an accurate knowledge of the propagation characteristics such as the field distributions, propagation con- stants, and group velocities of the LP ol and LPll modes, as a function of power law profile parameters. Most of the earlier work [l]-[3] on design consideration of dual-mode fibers made use of the analytical approximation of the modal fields and propagation constants (based on the scalar variational analy- sis) reported by Okamoto and Okoshi [7]. However, as also pointed out by Kitayama et al. [4], the use of such an analysis for computing the modal dispersion characteristics of dual- mode fibers gives incorrect results. Hence, most authors have now resorted to a direct numerical solution of the scalar or vector wave equation [4], [SI. In this paper, we present extremely accurate analytical ap- proximations for the propagation characteristics of the dual- mode fibers which are also based on the scalar variational analy- Manuscript received March 8, 1982; revised August 15, 1982. This work was supported in part by the Council of Scientific and Industrial Research (India) and the University Grants Commission (India). S. I. Hosain is with the Department of Physics, Indian Institute of Technology, New Delhi, India, on leave from the Department of Phys- ics, Ravenshaw College, Cuttack, India. E. K. Sharma, A. Shama, and A. K. Ghatak are with the Department of Physics, Indian Institute of Technology, New Delhi, India. sis. For the LP o l mode accurate approximations for the field and propagation constant (using the variational method) have already been reported [8], [9]. In this paper we propose single parameter and two parameter approximations for the LP l l mode. We also show that the single parameter field for the LPl l mode gives a good discription of the modal field in the core, but is inadequate to calculate the fields in the cladding, fractional power in the core, and the dispersion characteristics. In particular, it is necessary and sufficient to use the two param- eter trial field to calculate the V value for the zero intermodal dispersion and also for the calculation of the dispersion toler- ance in the operating V value. These studies should be of great interest in the analysis of dual-mode fibers which are of great interest in high bandwidth optical communication systems. 11. THEORY A. Variational Expression for Propagation Constant First, we briefly summarize the essential features of the vari- ational procedure. For a graded index fiber with any power law profile, the refractive index distribution is given by ni R>l (1) where R = r/a, a being the fiber core radius; n l and n 2 are the refractive indexes on the fiber axis and in the cladding, respec- tively, q is the profile exponent, and 6 is the grading parameter. The variational expression for the dimensionless propagation constant U 2 for a LPl, mode is given by [lo] U 2 = + V R 4 )$ t (R)1 2 RdR \\p t (R)\ 2 RdR + J djh du RdR R dR p t (R)\ 2 RdR (2) where V 2 =a 2 k 2 ,(n ? -n;) A 2 =a 2 k i n?6. fi is the propagation constant and $,(R)is the variational trial field. For refractive index profiles which are continuous at R = 1, L e., when there is no index jump at the core-cladding