28 May 2004 Surface nonlinear wave convolution on thin film photonic crystal waveguides
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In signal and system analysis, the concept of convolution is very important. It can be implemented optically using non-linear-optic effects by interacting, or mixing, two contra-directed surface waves in the non-linear thin-film photonic crystal waveguide. Complex electric field amplitudes are derived using Maxwell's equations that take the polarization into account and general solutions are then found by using the Variation of Parameters method. In reality, it is the non-linear part of the polarization that allows the mixing of two surface waves to generate a third wave. In order for this optical mixing process to take place effectively, both frequency- and phase-matching conditions must be satisfied. Both collinear and non-collinear interactions are being considered. All derived electric field amplitudes are intensities that may be regarded as signals. Convolution may then be implemented using the above non-linear-optic effects. Because predicted signals are often discussed in terms of their Fourier transforms, the convolution concept is being studied in both frequency- and time-domains. Convolution kernels are being looked into graphically and the convoluted wave is being analyzed according to the Convolution theorem and the Input-Output Description of a system. Finally, the optical implementation of the Fourier transform using a lens is briefly looked into.
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Wing-Ying Kwong, "Surface nonlinear wave convolution on thin film photonic crystal waveguides", Proc. SPIE 5355, Integrated Optics: Devices, Materials, and Technologies VIII, (28 May 2004); doi: 10.1117/12.526368; https://doi.org/10.1117/12.526368

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