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HW7_Solution - This problem is the continuation of the...

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This problem is the continuation of the Gaussian beam diffraction problem you studied earlier. Once you calculate the plane wave amplitudes by FFT you need to advance the phase of each component by the appropriate factor. Then the reflection coefficient for each plane wave component needs to be calculated. Remember that for a TE polarized plane wave incident on a planar interface 1 2 z ε 1 μ 0 x ε 2 μ 0 θ i k x k z k 1 1 1 2 2 0 0 0 1 2 0 1 2 1 cos cos 1 z z t r z z n n η θ η θ μ η η η η ε ε = = = = = 2 1 2 1 2 1 2 1 1 1 cos cos 1 1 cos cos r y z z t i i y z z t i E z z n n R E z z n n θ θ θ θ = = = + + 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 2 2 2 1 cos cos cos 1 cos cos cos 1 x i t z x i i t x z x t k k n n k k R n n k k k k k k k k k k θ θ θ θ θ θ = = = = + = = = Remember k x is the same in both media due to phase matching. Furthermore [ ] 2 ( 1) for 1 2 2 ( 1) for 1 2 xn xn N k n n L N k n N n N L π π = = + + where N is the number of grid points and L is the width of the computational window. Therefore for 1 2 N n
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