We propose the nonlinear Fourier Modal Method (FMM) [J. Opt. Soc. Am. B 31, 2371 (2014)] as a convenient and versatile numerical tool for the design and analysis of grating based next generation all-optical devices. Here, we include several numerical examples where the FMM is used to simulate all-optically tunable functionalities in sub-wavelength periodic structures. At first, we numerically investigate a 1-D periodic nonlinear binary grating with amorphous TiO<sub>2</sub>. We plot the diffraction efficiency in the transmitted orders against the structure depth for normally incident plane wave. Change in diffraction efficiencies for different incident field amplitudes are evident from the plots. We verify the accuracy of our implementation by comparing our results with the results obtained with the nonlinear Split Field-Finite Difference Time Domain (SF-FDTD) method. Next we repeat the same experiment with vertically standing amorphous Titanium dioxide (TiO<sub>2</sub>) nanowire arrays grown on top of quartz which are periodic in two mutually perpendicular directions and examine the efficiencies in the direct transmitted light for different incident field amplitudes. Our third example includes analysis of a form birefringent linear grating with Kerr medium. With FMM we demonstrate that the birefringence of such a structure can be tuned by all-optical means. As a final example, we design a narrow band Guided Mode Resonance Filter (GMRF). Numerical experiments based on the nonlinear FMM reveal that the spectral tunability of such a filter can be obtained by all-optical means.
Using the Fourier Modal Method for gratings with Kerr media [J. Opt. Soc. Am. B 31, 2371 (2014)] we demonstrate that low energy Optical Bistability for normally incident light field can be observed by strong nonlinear light-matter interactions in a Silicon Nitride waveguide-grating with 2-D periodicity. Finite divergence of the incident light beam has been taken into account in our numerical study and the gratings are designed to observe bistable behavior in direct transmitted light inside the optical telecommunication C-band (1520 nm-1570 nm). The waveguide grating structures are fabricated from PECVD synthesized Silicon Nitride thin film on top of quartz with standard electron beam lithography and reactive ion etching techniques. We aim to demonstrate this phenomenon experimentally using a tunable narrow line-width pulsed laser. Our resonant nanostructures may find applications in free space all-optical information processing and all-optical switching.
In this work the split-field finite-difference time-domain method (SF-FDTD) has been extended for the analysis of two-dimensionally periodic structures with third-order nonlinear media. The accuracy of the method is verified by comparisons with the nonlinear Fourier Modal Method (FMM). Once the formalism has been validated, examples of one- and two-dimensional nonlinear gratings are analysed. Regarding the 2D case, the shifting in resonant waveguides is corroborated. Here, not only the scalar Kerr effect is considered, the tensorial nature of the third-order nonlinear susceptibility is also included. The consideration of nonlinear materials in this kind of devices permits to design tunable devices such as variable band filters. However, the third-order nonlinear susceptibility is usually small and high intensities are needed in order to trigger the nonlinear effect. Here, a one-dimensional CBG is analysed in both linear and nonlinear regime and the shifting of the resonance peaks in both TE and TM are achieved numerically. The application of a numerical method based on the finite- difference time-domain method permits to analyse this issue from the time domain, thus bistability curves are also computed by means of the numerical method. These curves show how the nonlinear effect modifies the properties of the structure as a function of variable input pump field. When taking the nonlinear behaviour into account, the estimation of the electric field components becomes more challenging. In this paper, we present a set of acceleration strategies based on parallel software and hardware solutions.
We introduce a numerically feasible method for rigorous modeling of crossed diffraction gratings with isotropic
third order nonlinear materials. The approach is based on an iterative solution of the crossed grating problem
with anisotropic linear materials. Several numerical experiments are performed to demonstrate the versatility
and numerical stability of our computation scheme. Resonance waveguide gratings made of isotropic cubic
nonlinear materials are investigated numerically using this newly developed technique. A polarization-sensitive
shift of resonance peak with variation of light intensity is numerically demonstrated.