Source: https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-20-21427
Timestamp: 2019-04-19 06:21:21+00:00

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A deep insight into the inherent anisotropic optical properties of silicon is required to improve the performance of silicon-waveguide-based photonic devices. It may also lead to novel device concepts and substantially extend the capabilities of silicon photonics in the future. In this paper, for the first time to the best of our knowledge, we present a three-dimensional finite-difference time-domain (FDTD) method for modeling optical phenomena in silicon waveguides, which takes into account fully the anisotropy of the third-order electronic and Raman susceptibilities. We show that, under certain realistic conditions that prevent generation of the longitudinal optical field inside the waveguide, this model is considerably simplified and can be represented by a computationally efficient algorithm, suitable for numerical analysis of complex polarization effects. To demonstrate the versatility of our model, we study polarization dependence for several nonlinear effects, including self-phase modulation, cross-phase modulation, and stimulated Raman scattering. Our FDTD model provides a basis for a full-blown numerical simulator that is restricted neither by the single-mode assumption nor by the slowly varying envelope approximation.
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Fig. 1 Staggered space arrangement (left) and leapfrog time ordering (right) of discretized electromagnetic-field components using a cubic Yee cell.
Fig. 2 Positions of electric field components in the FDTD grid. The values of components Eβ and Eγ at the node (i, j, k) are calculated by averaging their four values over the nodes (i ± 1/2, j ± 1/2, k) and (i ± 1/2, j, k ± 1/2), respectively.
Fig. 3 Relative orientation of FDTD (α, β, γ) and crystallographic (x, y, z) axes adopted in the paper. For ϑ = π 4 , the FDTD axes α, β, and γ coincide, respectively, with the , [1̄10], and  crystallographic directions. The inset shows the TM and TE polarizations and an arbitrary linear polarization determined by the angle φ.
Fig. 4 Three one-dimensional Yee cells corresponding to the simplified FDTD model. The electric and magnetic fields along the propagation direction α are ignored.
Fig. 5 Input–output characteristics for 100-, 200-, and 500-μm-long silicon waveguides pumped by TM- and TE-polarized, 1.4-ps Gaussian pulses. Other parameter values are given in the text.
Fig. 6 Efficiency of SPM-induced polarization rotation for different input SOPs of a 1.4-ps Gaussian pulse. The output polarizer is perpendicular to the input polarization state of the pulse. We choose τc = 0.8 ns; other parameters are given in the text.
Fig. 7 Poincaré-sphere representation of SOP variations along a 1.4-ps Gaussian pulse at the output of a 0.5-mm-long silicon waveguide. Numbers near the traces staring from the equator show φ values for the input SOPs (see Fig. 3); φ 0 ≈ 35°. Traces starting from the poles correspond to circularly polarized input pulses. Here, I 0 = 400 GW/cm2; other parameters are the same as in Fig. 6(a).
Fig. 8 Switching efficiency of a CW beam for a 350-fs Gaussian pump pulse as a function of (a) input polarization angle φ of the CW beam, (b) waveguide length L, and (c) pump's peak intensity Ip 0. Panel (d) shows switching windows for three different waveguide lengths; dashed curve shows the Gaussian pulse that is used in the definition of transmittance. In panels (b)–(d), φ = π/4. In all panels input CW intensity is 1 GW/cm2 and τ c = 0.8 ns; other parameters are given in the text.
Fig. 9 Peak intensity of TM- and TE-polarized signals amplified via SRS by TM- and TE-polarized pumps. Both input signal and input pump are assumed to be 1.4-ps Gaussian pulses with peak intensities of 1 and 100 GW/cm2, respectively; carrier frequencies are 200 and 215.6 THz; gR = 76 cm/GW; other parameters are given in the text.

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