Conclusions: zyxwvutsrqponmlkjih We have found that the calculated losses in the Fresnel equation formalism for longitudinally varying semi- conductor single-mode rib waveguides depend significantly on the value chosen for the reference refractive index no. While for strongly guiding structures this dependence is modest as long as no is chosen in a fairly wide region zyxwvutsrq (2 +0.3) in the vicinity of the refractive index of the guided modes po/k,, for weakly confining structures the propagation losses vary rapidly with no. In the latter case, however, excellent agree- ment with the two-dimensional effective index method is attained for n, zyxwvutsrqponmlkjihg = po/ko. Acknowledgments: We thank G. Chik for his continued inter- est and encouragement. zyxwvutsrqpon D. Yevick also thanks Bell Northern Research, Corning Glass and STUF, Sweden for financial support. D. YEVICK* Department of Electrical Engineering I21 Electrical Engineering East Pennsylvania State University University Park, PA 16802, USA 16th June 1989 C. ROLLAND Bell Northern Research, Advanced Technology Laboratory PO Box 3511, Station C , Ottawa, Ontario, Canada KIY 4H7 B. HERMANSSON Swedish Telecommunication Administration S-123 86 Stockholm, Sweden * Teoretisk Fysik, Lunds Universitet, Solvegatan 14A, S-223 62 Lund, Sweden References YEVICK, D., and HERMANSSON, B.: ‘New formulations of the matrix beam propagation method: application to rib waveguides’, IEEE Trans., QE-25, 1989, pp. 221-229 YEVICK, D., and HEXMANSSON, B.: ‘Split-stepfinitedifference analysis of rib waveguides’, Electron. Lett., 25, 1989, pp. 461462 YEVICK, D., and HERMANSSON, zyxwvutsrqponml E.: ‘An efficient fast-Fourier trans- form based beam propagation technique’, submitted to lEEE J. Quantum Electron. YEVICK, D., and GLASNER, M.: ‘An analysis of forward wide-angle light propagation in semiconductor rib waveguides’, submitted to Electron. Lett. WICK, D., and HERMANSSON, B.: ‘A numerical investigation of the mode coupling among the local normal modes of sinusoidally modulated parabolic gradex-index planar waveguides’, Optical and Quantum Electron., zyxwvutsrqponm 16, 1984,pp. 331-337 DANIELSEN, P., and YEVICK, D.: ‘Propagatingbeam analysis of bent optical waveguides’, J. Opf. Comm., 3, pp. 9498 HERMANSSON, E., YEVICK, D., and SAJONMAA, I.: ‘A propagating beam analysis of two-dimensional microlenses and tbree- dimensional taper structures’, J. Opt. Soc. Am. A, 1, 1984, pp. 663-671 ANALYSIS OF STRONGLY GUIDING RIB EXPERIMENT WAVEGUIDE S-BENDS: THEORY AND Indexing terms: Waveguides, Optical waveguides, Optoelec- tronics, Inteqrated optics We have measured the losses associated with strongly con- fined GaAs/AIGaAs rib waveguide S-bent structures and obtained excellent agreement with a full three-dimensional numerical simulation based on the beam propagation method. Introduction: In a series of recent papers, we have analysed several mathematical methods for modelling both paraxial and wide-angle light propagation through strongly and weakly guiding rib waveguide str~ctures.’-~ The results of the paraxial propagation studies for strongly guiding longitudi- nally varying structures for which the effective index approx- imation is invalid have, however, not yet been verified experimentally. In this letter we compare measured and calcu- lated optical losses in GaAs/AlGaAs rib waveguide S-bends, and demonstrate the validity of our computational techniques. The results of this study should be of considerable importance in the design of future integrated devices containing strongly guiding structures. Numerical method: While a number of methods have recently been applied to electric field propagation through semicon- ductor rib waveguides, we will consider here only the split- step fast Fourier transform procedure. This direct method is based on the following simple expression relating a single pol- arisation component of a monochromatic elecric field at a given transverse plane to the electric field at an adjacent trans- verse plane displaced by a longitudinal distance Az: W , Y, + A4 = exp [ -i 21 Z+AZ x exp [ -i 21 x exp [-ikono Az]E(x, y, z) + O(AZ)~ (1) The symbol V: represents the transverse Laplacian operator, while k, and no are the vacuum wavevector and the reference value of the refractive index in the Fresnel equation. To apply eqn. 1, we successively fast-Fourier-transform the electric field, multiply by the first exponential operator, which is diagonal in Fourier space, and finally inverse fast-Fourier-transform and multiply by the second operator. If desired, a filter may be imposed on the propagation operator to limit the phase change and therefore ensure the single-valuedness of this ~perator.~ In our implementation of eqn. 1, we both replace nZ(xi, yj, z), where xj and y, are grid point co-ordinates, by an average of the squared refractive index over the transverse region closest to the given grid point, and absorb the electric field which propagates to the computational window bound- aries.1.2.6 The remaining power is determined and graphed as a function of propagation distance by computing the sum of the squared electric fields at each grid point.’ Experimental technique: We have fabricated a series of S-bends and straight waveguides in MBE-grown epilayers on (100) n+ GaAs substrates. The epilayer structure is a l’lpm- thick GaAs guiding layer, above a 3.0pm AI0.,,Ga,.,,As cladding layer. The 2.3 pm-wide rib waveguides are reactive- ion-etched to a depth of 0.85pm into the guiding layer. The centre of the rib waveguide S-bends follows a shape given by x(z) = h{z/L - sin (2nz/L)/2n}, where h is the displacement between the two parallel waveguide sections and L is the tran- sition length along the longitudinal direction.’ In both the mask design and the subsequent numerical modelling, we were careful to ensure that the waveguide width remained constant at all transverse planes within the S-bend. The transition lengths of the S-bends were 25, 50, 100, 200, 300, and 400pm while h was kept fixed at IOpm. The optical loss was calculated by measuring the Fabry- Perot fringe contrast of the cleaved waveguide sample.’ In contrast to Reference 8, however, the transmission resonances were generated by temperature-tuning the frequency of a semiconductor diode laser instead of varying the optical cavity length at a fixed frequency. Accordingly, we first focused light from a i = 1.3pm distributed feedback laser into a single-mode fibre which was suitably twisted to generate TE-polarised light at the output. The rib waveguide was excited by positioning the fibre end 50pm from the waveguide face. Subsequently, the guided light was collected with an NA = 0.85 microscope objective and measured with a Ge detector. The near field was simultaneously viewed on an infra-red vidicon camera. Conventional lock-in detection tech- 1256 zyxwvutsrqpon ELECTRONICS LE77ERS 31st August 1989 Vol. 25 No. 18