put, and connecting guides are designed to be single mode. Because the guides are deeply etched and the spacing between single-mode guides is one guide width, crosstalk between the guides is neglected. Reflections in the longitu- dinal direction and resonance within the MMI section are not considered. In the optimization of these devices the two MMI section lengths and the phase shift values are considered to be adjustable parameters. The final output intensity depends on several variables, and derivatives are not available to assist convergence. A modified Powell method is therefore employed, in which the direction of largest decrease during the previous iteration is discarded. The one dimensional directions selected with the Powell algorithm are optimized with Brent’s method. Tolerances on all parameters can be adjusted to produced optimal parameters within specified error. A. 2X2x1 CASCADED DEVICE OPTIMIZATION The 2x2x1 cascaded device shown in Fig. 2 is modeled, where L1,2 are the lengths of the first and second MMI regions, respectively, w is the width of both MMI regions, and g1,2 are the phase shifts in the interconnecting guides. The simple 2x2x1 case is a three variable optimization problem in terms of the two lengths L1,2 and the difference between the phase shifts 1 - 42 with w held constant. Because local maxima are sought in the optimization, approxi- mations for the individual parameters from either approximation discussed in Theory can be used as seed values. The small number of variables allows the generation of a surface of output intensity as a function of adjustable parameters; the smoothness of the surface and location of maxima can therefore be observed. Only a single input guide (guide 1) is excited, and by virtue of the symmetry, the response with an input to the other guide is identical. 1st MMI 2nd MMI foligo t 2 { 8.0 pm Rg | ra Ate UN a nea ety od Fic. 2. 2X2xX1 CASCADED SWITCHING DEVICE WITH TWO INPUTS AT +2.0 UM FROM THE CENTER OF THE DEVICE. To determine whether numerical techniques could be used to optimize the device design, the output intensity of the device as a function of L1,2 is calculated with g1 - g2 held constant. One example surface is plotted in Fig. 3, where the difference between the phase shifts is fixed at 1/2. The optimization surface is continuous with a single local maximum. For a variety of fixed phase differences, surfaces with similar characteristics but lower peak intensity were generated (not shown), so when all three parameters were varied, a unique local maximum remained. Therefore, numerical optimiza- tion should be effective for this and similar models. AM bf hy a) MER SEES FARK Normalized Intensity : \y LEROY trees ZEIT ZESOIRN\\ fs RKB LESAN ERS SERN RO Sees 110 Le (um) Fic. 3. OPTIMIZATION SURFACE OF L1,2 FOR INPUT INTO GUIDE 1; PHASE SHIFTS ARE FIXED AT @1 - @2 = 1/2 FOR INPUT INTO GUIDE 2. L1,2 and gl - g2 have been optimized with the Powell algorithm for maximum output power in the single output guide. The width W of the multimode region was fxed so that the desired number of single-mode input guides, with spacings equal to the guide widths, could be accommodated. A more general optimization which takes into account variations in the widths and separations of the different guides and sections within the constraints of single mode input, intermediate, and output guides, overall device length, etc. is an area of future work. B. 2X2x1 DESIGN APPROXIMATIONS Because simple approximations of the self-imaging lengths are often used in device design, a comparison of the per- formance of devices with approximated lengths to that of devices with optimum lengths is shown in Table I. As men- tioned above, these approximated lengths were used as seed values in the optimization. A comparison between two vo