JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 2, FEBRUARY 1999 369 Analysis of “Quasi-Modes” in Periodic Segmented Waveguides Daniel Ortega, Associate Member, IEEE, J. M. Aldariz, John M. Arnold, and J. Stewart Aitchison, Member, IEEE, Member, OSA Abstract— A three-dimensional (3-D) explicit finite difference beam propagation method (3-D EFD-BPM) has been used to study the modal characteristics and losses of periodic segmented waveguides (PSW’s). Results for the variation of the mode width and depth, as a function of the duty-cycle and period, are presented and compared with those obtained using the equivalent continuous waveguide model at a wavelength of 1.55 m. The radiation losses associated with the modulation of the refractive index are studied and we show that a 3-D representation of a PSW is necessary to evaluate the radiation loss. Index Terms—Fabrication, finite difference method, modeling, optical losses, optical propagation, optical strip waveguide. I. INTRODUCTION R ECENTLY there has been an increasing amount of interest in the fabrication and application of periodic segmented waveguides (PSW’s). These waveguides consist of a periodic modulation in the refractive index, along the direction of propagation. Each period of the PSW is composed of two regions with different refractive indices: a doped region, with an increase in the refractive index of , and an undoped region with the same refractive index as the substrate (see Fig. 1). Typically these waveguides are buried structures and have been demonstrated in LiNbO 3 by both Ti indiffusion [1] and proton exchange [2]. Interest in PSW’s has arisen due to the increased flexibility for the design of integrated optical components. By proper control of the period and duty-cycle it is possible to realize a range of novel components. One of the most important properties of PSW’s is that they allow independent control over the vertical and lateral mode confinement [3]. This effect has been used to demonstrate two-dimensional (2- D) mode tapers in proton-exchanged LiNbO 3 [4] and InP waveguides [5]. In the case of LiNbO 3 the lateral change in size was produced by an adiabatic taper in the waveguide width and the vertical change in size as a result of the change in duty-cycle of the PSW. The advantage of this type of taper is that it can be produced in a single fabrication step. Other applications of PSW’s include wavelength filters [6], wavelength demultiplexers [7], polarization filters and in second harmonic generation (SHG) [8]. Manuscript received May 1, 1998; revised August 27, 1998. The work of D. Ortega was supported by the Faculty of Engineering, University of Glasgow, U.K. The authors are with the Department of Electronics and Electrical Engi- neering, University of Glasgow, Glasgow G12 8QQ U.K. Publisher Item Identifier S 0733-8724(99)01523-6. Fig. 1. A schematic representation of a PSW. The arrow indicates the propagation direction of the light along the PSW of period and duty-cycle (the ratio of the length of a segment and the period of the guide). n is the increase in the refractive index introduced as a consequence of the doping. The modal properties of a PSW are determined by its dimensions, period, duty-cycle and the transverse distribution of the refractive index. Previous reports have proposed that the behavior of a PSW can be approximated by a continuous waveguide whose refractive index step is given by [9] (1) where is the duty-cycle. This relationship has proved useful for the design of PSW; however, it does not provide informa- tion about the radiation losses associated with the modulation of the refractive index. In this paper, a three-dimensional (3-D) EFD-BPM [10] was used to study the optical field of step index periodic segmented waveguides (PSW) with a index distribution chosen to resemble that of Ti : LiNbO 3 channel waveguides. We will show that the EFD-BPM can be used in the study of PSW’s and to describe the behavior of the optical field along the direction of propagation. One key difference between PSW’s and continuous waveguides is in the definition of a mode. A waveguide mode is normally associated with a field distribution which remains unchanged along the direction of propagation. However, in the case of a PSW the mode shape changes periodically as a function of distance. It is important to be able to understand the behavior of these “quasi-modes” and to be able to account for radiation losses associated with the periodic modulation in refractive index. 0733–8724/99$10.00  1999 IEEE