MMI Devices with Weak Guiding Designed in Three Dimensions Using a Genetic Algorithm

We discuss the design of weakly guided multimode interference (MMI) devices using a genetic algorithm. For devices exhibiting a nonnegligible vertical waveguide offset, such as those produced using ion exchange in glass, three-dimensional modeling is required to properly evaluate the device performance. A combination of semivectorial finite difference modeling in two transverse dimensions and mode propagation analysis (MPA) in the propagation direction is used to evaluate the merit of each trial design. An example is provided of a 1 x 4 power splitter designed for ion exchange, which shows considerable improvement over that obtained by self-imaging theory. ©2004 Optical Society of America OCIS Codes: (130.3120) Integrated optics devices; (230.7370) Waveguides References and links 1. L.H. Spiekman, Y.S.Oei, E.G. Metaal, F.H. Green, I. Moerman, and M.K. Smit, “Extremely small multimode interference couplers and ultrashort bends on InP by deep etching,” IEEE Photon. Technol. Lett. 6, 1008-1010 (1994). 2. T. Rasmussen, J.K. Rasmussen, and J.H. Povlsen, “Design and performance evaluation of 1-by-64 multimode interference power splitter for optical communications,” J. Lightwave Technol. 13, 2069-2074 (1995). 3. A. Bakhtazad, J.N. McMullin, C.J. Haugen, and R.G. DeCorby, “MMI multiplexer for dual-channel erbium-doped waveguide amplifiers,” Opt. Express 9, 178-183 (2001), http://www.opticsexpress.org/abstract.cfm?uri=OPEX-9-4-178. 4. D. Hah, E. Yoon, and S. Hong, “An optomechanical pressure sensor using multimode interference couplers with polymer waveguides on a thin p-Si membrane,” Sens. Act. A 79, 204-210 (2000). 5. A. Irace and G. Breglio, “All-silicon optical temperature sensor based on multi-mode interference,” Opt. Express 11, 2807-2812 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2807. 6. P.A. Besse, M. Bachmann, H. Melchior, L.B. Soldano, and M.K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004-1009 (1994). 7. L.B. Soldano and E.C.M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615-627 (1995). 8. S.E. Yliniemi, B.R. West, T.T. Aalto, P. Madasamy, N. Peyghambarian, and S. Honkanen, “Buried ionexchanged glass waveguides featuring low birefringence with a broad range of waveguide widths,” in Integrated Optics and Photonic Integrated Circuits, G. C. Righini and S. Honkanen, eds., Proc. SPIE 5451, no. 87 (2004). 9. D.F. Geraghty, D. Provenzano, M.M. Morrell, J. Ingenhoff, B. Drapp, S. Honkanen, A. Yariv, and N. Peyghambarian, “Polarisation-independent Bragg gratings in ion-exchanged glass channel waveguides,” Elect. Lett. 36, 531-532 (2000). 10. P. Madasamy, B.R. West, M.M. Morrell, D.F. Geraghty, S. Honkanen, and N. Peyghambarian, “Buried ion-exchanged glass waveguides: Burial-depth dependence on the waveguide width,” Opt. Lett. 28, 11321134 (2003). 11. Q. Wang, J. Lu, and S. He, “Optimal design of a multimode interference coupler using a genetic algorithm,” Opt. Comm. 209, 131-136 (2002). 12. R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Amer. 68, 583-592 (1978). 13. A.R. Gupta, “Optimization of access waveguide width of multimode interference (MMI) couplers,” Opt. Commun. 221, 99-103 (2003). (C) 2004 OSA 14 June 2004 / Vol. 12, No. 12 / OPTICS EXPRESS 2716 #4048 $15.00 US Received 18 March 2004; revised 2 June 2004; accepted 3 June 2004 14. B.R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, Optical Sciences Center, The University of Arizona, 1630 E. University Blvd., Tucson, AZ, 85721, are preparing a manuscript to be called “Accurate modeling of ion-exchanged glass waveguide structures.” 15. C. M. Kim and R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581-1589 (1989). 16. J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, and H. Melchior, “Multimode interference couplers for the conversion and combining of zeroand first-order modes,” J. Lightwave Technol. 16, 1228-1239 (1998).


5.
A. Irace and G. Breglio, "All-silicon optical temperature sensor based on multi-mode interference," Opt. Express 11, 2807-2812 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2807. 6. P.A. Besse, M. Bachmann, H. Melchior, L.B. Soldano, and M.K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994 , conventional thought is that waveguides exhibiting weak guiding cannot produce efficient MMI devices. As several material systems used for integrated optics do result in weak guiding, it is important to investigate the resultant changes in design procedures. One particular technology that has recently found commercial applications is ion exchange in glass. Ion-exchanged waveguides exhibit low propagation loss and compatibility with single-mode fiber, and cost relatively little to produce. In addition, very low birefringence is possible across a wide range of waveguide widths [8], resulting in devices that show very low polarization dependent loss [9]. It has been shown [10] that when the ionic species being exchanged have substantially different mobilities, the depth of the waveguide after burial is dependent upon the waveguide width. This will have an adverse effect upon MMI devices, which contain transitions between narrow access guides and the multimode section. Proper analysis of these transitions requires that both vertical and transverse field variations be modeled. Such an approach is unique to MMI design, as most authors prefer to eliminate vertical field variation through such techniques as the effective index method. Section 2 of this paper reviews the design of an MMI 1 x N power splitter using selfimaging theory. In section 3, we discuss the challenges posed by using a weakly guided multimode section, with particular emphasis on those fabricated by ion exchange, and their effects on the results of self-imaging theory. We then propose, in section 4, an alternate design procedure using a genetic algorithm (GA). This algorithm is used in section 5 to design a 1 x 4 power splitter in ion exchange, and it is shown that polarization dependent loss and power imbalance in the output waveguides can be essentially eliminated in the design. A previous effort to use a GA for MMI optimization has been reported [11]. This procedure differs considerably from that reported in the present text in that the variation of mode fields in the vertical direction is not accounted for, and that the position of each access waveguide is related through an effective MMI width. Both of these approximations lose validity in the case of ion exchanged waveguides.

Self-imaging theory of multimode interference
For a strongly guided, step-index multimode waveguide, it can be assumed that the penetration depth of each transverse mode into the cladding is equal and negligible. Thus, within the paraxial approximation, the distribution of propagation constants β ν is quadratic, , 3 ν is the mode number and L π = π / (β 0β 1 ) is the "beat length" between the two lowestorder modes [7]. At the input to the multimode section, the field distribution of the input access waveguide can be expanded in the eigenmodes of the multimode guide, where the a ν are the expansion coefficients, calculated using the overlap integrals The summation is over all guided and leaky modes, although the latter may be neglected in all practical cases, as their a ν are relatively small for practical access waveguide widths. As each mode propagates in z, it accumulates a phase term equal to exp(-i β ν z). Using Eq.
(1), we find that the field distribution at the output plane of the multimode guide is (4) where a global phase factor has been omitted. The application of Eqs.
(2) -(4) is commonly referred to as Mode Propagation Analysis (MPA). If the input access waveguide is symmetric with the multimode section, analysis of Eq. (4) shows that the shortest N-fold self-imaging distance is (5) from the input [7]. A 1 x N power splitter is shown in Fig. 1.
The output access waveguides are equally spaced and arranged symmetrically about the axis of the multimode waveguide. The x-coordinate of the center of the i th waveguide is then (6) where W e is the effective width of the multimode waveguide, equal to the physical width in the strongly guided case. The optimal width of the input access guide is somewhat difficult to determine. A waveguide that is too wide will not excite enough transverse modes of the multimode section (as a general rule, for a 1 x N splitter, at least N + 1 guided modes must be excited in the multimode waveguide in order to resolve the desired output field [12]). On the other hand, too narrow a guide will launch more power into the substrate modes of the multimode section, increasing the excess loss of the device. The optimal width of the output waveguide has been the subject of previous study [13], and can be estimated as the widest guide that exhibits negligible coupling to adjacent guides.

Weakly guided MMI devices
When the index difference between core and cladding in the multimode guide is small or contains a gradient as with diffused waveguides, higher-order transverse modes have greater effective widths. As a result, the distribution of effective indices is sub-parabolic, as can be inferred from [7]. The gradual de-phasing of the modes during propagation will blur the selfimages and produce both a longitudinal and transverse shift in the optimal locations of the output waveguides. In addition, the width of the multiple self-images shows a slight increase as distance from the axis of symmetry increases.
Determining the optimum access waveguide width for buried ion-exchanged MMI devices is quite difficult, due to the vertical offset between waveguides of dissimilar width [10]. Generally speaking, the widths need to be increased in order to balance transverse resolution against vertical insertion loss.

Description of the genetic algorithm
The GA described in this paper seeks the best MMI design by simultaneously optimizing several geometrical parameters. It is assumed that the fabrication process and width of the MMI section are pre-determined. Referring to Fig. 1, the parameters to be determined are the length of the MMI section (L MMI ), width of the input guide (W in ), and positions (x i ) and widths (W i ) of the output guides. By symmetry, many of these parameters can be eliminated; for N even, only N / 2 positions and N / 2 widths need be determined, while for N odd, (N -1) / 2 positions and (N + 1) / 2 widths are required, for a total of N + 2 parameters in total.
Before the GA begins, a library is created containing the quasi-TE and -TM mode fields for all possible access waveguide widths (in increments of 0.1 µ m) as well as the fields and propagation constants β ν of the multimode waveguide. Modeling of the ion exchange process is described in [14], and a semivectorial finite difference method described in [15] is used to solve for all eigenmodes and propagation constants. This is done to eliminate the need to perform process modeling or optical mode solution of every waveguide during execution of the GA. The absence of such modeling within each generation of the GA drastically reduces the computational effort required to optimize the device design. Note that the process modeling is performed only in transverse slices. Strictly speaking, this does not account for longitudinal ion diffusion at the transition regions, but this effect exists over a distance of only a few microns. As such, there will be no appreciable adiabatic tapering, and perhaps a very slight improvement in power coupling. The 0.1 µ m resolution in guide width reflects the finite lithographic precision. Next, the power in the multimode section for an arbitrary symmetric input is calculated over a coarse grid using Eqs. (2)-(4), and plotted as in Fig. 2, in order to estimate the optimum device length to within a few tens of microns.
Each of the N + 2 parameter ranges is binary-encoded to arbitrary precision in a "gene" (as above, the practical limit of precision is on the order of 0.1 µ m). The genes are concatenated to form a "chromosome." In this way, all geometrical parameters to be optimized are encoded in a single binary sequence. For the waveguide widths, the value encoded in the gene is rounded to the nearest 0.  Table 1. The splitter is designed first by using the calculated β 0 and β 1 (averaged over polarization), and Eqs. (1)-(6). Modeling shows a very poor excess loss of 2.088 dB, an imbalance of 1.225 dB, and a polarization dependent loss of 0.016 dB (note that this particular ion exchange process inherently exhibits very low birefringence). The results are summarized in Table 2.
Next, we use the GA to design the same device. As mentioned in Section 4, we are free to choose the relative importance of each device property. For this example, we assume that the device application requires as small a power imbalance and polarization dependent loss as possible, with less strict requirements on excess loss. In this case, we set C XL = 0.5, C IB = 1, and C PDL = 10. The search ranges and resolution of each parameter are shown in Table 3. After 250 generations with 15 trial devices per generation, the GA converged to the parameters shown in Table 2. The imbalance is 0.007 dB, and polarization dependent loss is 0.001 dB. Excess loss is 1.901 dB, dominated by the vertical mismatch at the output plane. Figure 3 contains simulated TE power cross-sections of the end of the multimode guide using self-imaging theory (top) and the GA (bottom), showing a significant improvement in the latter. Comparing Fig. 3 and Table 2, it can be seen that, although the outermost of the four self-images are higher in power, imbalance is minimized by an intentional transverse misalignment of the output waveguides. The excess loss due to this misalignment is ≈ 0.5 dB, producing a total excess loss for the GA-designed device that is still lower than that of the self-imaging theory design.