Finite difference beam propagation method is an accurate numerical procedure, used here to explore the switching dynamics of a nonlinear coherent directional coupler. The coupling lengths derived from this simulation are compared with coupled mode theories. BPM results for the critical power follow the trend of the coupled mode theories, but it lies in between two coupled mode theories. Coupled mode theory is sensitive to numerical approximations whereas BPM results practically do not depend on grid size and longitudinal step size. Effect of coupling-region-width and core-width variations on critical power and coupling length is studied using BPM to look at the aspects of optical power-switch design. 1. Introduction Beam propagation method [1] is a versatile tool to investigate or model various optical phenomena in photonic devices. Earlier, the method was popular to find the guided modes in dielectric waveguide structures [2]; even in microstructure fibers like holey fibers it can be used to find the modes [3]. With suitable boundary conditions, it gave proper estimates of the propagation constant in leaky structures like arrow waveguides [4]. The method was used in linear directional couplers to simulate switching of a CW signal and filtering TE/TM mode signals [5]. With growing interest in nonlinear fibers, it was the only tool to model the pulse propagation [6] in conjugation with split-step Fourier transform. It has been also used to model various other nonlinear phenomena like second harmonic generation [7], leakage loss in buried silicon substrate [8], and so forth. Beam propagation method can be incorporated either in the finite element or finite difference framework. In cases where the interfaces are parallel to the axes, the two methods are almost the same in terms of accuracy, but the latter is sometimes preferred due to its simplicity in implementation. Nonlinear directional coupler (NLDC) is a very useful device in photonic circuits; it can be used as all optical modulator, switches, logic gates, and so forth. Jenson [9] first showed that total power exchange in a linear coherent coupler is lost in a nonlinear coherent coupler above a critical input power in one guide. This fact was used to design power-controlled all optical switches [10, 11]. The device can also be used as phase-controlled switches [12] and optical modulator [13]. So, the knowledge of the dependence of the critical power and phase variations on the NLDC parameters is important. There are numerous studies on this device; analytic methods include coupled mode theory [9,
References
[1]
D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE Journal of Quantum Electronics, vol. 25, no. 2, pp. 221–229, 1989.
[2]
W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE Journal of Lightwave Technology, vol. 10, no. 3, pp. 295–305, 1992.
[3]
K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE Journal of Quantum Electronics, vol. 38, no. 7, pp. 927–933, 2002.
[4]
W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technology Letters, vol. 8, no. 5, pp. 652–654, 1996.
[5]
W. P. Huang, C. L. Xu, and S. K. Chaudhuri, “Application of the finite-difference vector beam propagation method to directional coupler devices,” IEEE Journal of Quantum Electronics, vol. 28, no. 6, pp. 1527–1532, 1992.
[6]
G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, Calif, USA, 2001.
[7]
P. S. Weitzman and U. Osterberg, “Modified beam propagation method to model second harmonic generation in optical fibers,” IEEE Journal of Quantum Electronics, vol. 29, no. 5, pp. 1437–1443, 1993.
[8]
J. Yamauchi, K. Ose, J. Shibayama, and H. Nakano, “Leakage loss and phase variation of a buried directional coupler on a silicon substrate,” IEEE Photonics Technology Letters, vol. 18, no. 17, pp. 1873–1875, 2006.
[9]
S. M. Jensen, “The nonlinear coherent coupler,” IEEE Journal of Quantum Electronics, vol. 18, no. 10, pp. 1580–1583, 1982.
[10]
M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Optical and Quantum Electronics, vol. 24, no. 11, pp. S1237–S1267, 1992.
[11]
S. Trillo and S. Wabnitz, “Anisotropic and Nonlinear Optical Waveguides,” C. S. Someda and G. Stegeman, Eds., Elsevier, Amsterdam, The Netherlands, 1992.
[12]
S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coherent coupler,” Applied Physics Letters, vol. 49, no. 14, pp. 838–840, 1986.
[13]
A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” Journal of the Optical Society of America B, vol. 8, pp. 1914–1931, 1991.
[14]
X. J. Meng and N. Okamoto, “Improved coupled-mode theory for nonlinear directional couplers,” IEEE Journal of Quantum Electronics, vol. 27, no. 5, pp. 1175–1181, 1991.
[15]
I. M. Uzunov, R. Muschall, M. G？lles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, “Pulse switching in nonlinear fiber directional couplers,” Physical Review E, vol. 51, no. 3, pp. 2527–2537, 1995.
[16]
A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” Journal of the Optical Society of America B, vol. 8, pp. 986–992, 1991.
[17]
A. D. McAulay and X. Xu, “Finite element analysis of optically controlled non-linear directional coupler switches,” Optics and Laser Technology, vol. 26, no. 4, pp. 251–258, 1994.
[18]
B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, “Finite element analysis of nonsynchronous directional couplers,” Fiber and Integrated Optics, vol. 13, no. 3, pp. 331–336, 1994.
[19]
K. Yasumoto, H. Maeda, and K. Nakamura, “Beam-propagation method analysis of a three-waveguide nonlinear directional coupler,” Microwave and Optical Technology Letters, vol. 9, no. 6, pp. 319–323, 1995.
[20]
Y. Chung and N. Dagli, “Analysis of Z-invariant and Z-variant semiconductor rib waveguides by explicit finite difference beam propagation method with nonuniform mesh configuration,” IEEE Journal of Quantum Electronics, vol. 27, no. 10, pp. 2296–2305, 1991.
[21]
H. F. Chou, C. F. Lin, and G. C. Wang, “An iterative finite difference beam propagation method for modeling second-order nonlinear effects in optical waveguides,” IEEE Journal of Lightwave Technology, vol. 16, no. 9, pp. 1686–1693, 1998.
[22]
P. R. Berger, P. K. Bhattacharya, and S. Gupta, “A waveguide directional coupler with a nonlinear coupling medium,” IEEE Journal of Quantum Electronics, vol. 27, no. 3, pp. 788–795, 1991.
[23]
A. K. Ghatak and K. Thyagrajan, Introduction to Fiber Optics, Cambridge University Press, 1999.
[24]
S. Juengling and J. C. Chen, “Study and optimization of the eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE Journal of Quantum Electronics, vol. 30, no. 9, pp. 2098–2105, 1994.
