ECSE 527 Optical Engineering: Assignments 4 Solutions Question 1 i) For propagation over a distance zthe electric field evolves according to E(z)=E0exp(-j kz) Since for x-directed polarization kx=2π n0/λand for y-directed polarization ky=2π ne/λ, and the propagation distance is dwe can write: ( )()( )()( )()()λπλπλπ/2exp/2exp/2exp0000dnnjdEdnjEdEdnjEdEexeyx−−=−=−=So we can write in matrix-vector form: ( )( )()()( )( )⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−=⎥⎦⎤⎢⎣⎡00/2exp0010yxeyxEEdnnjdEdEλπand where we have omitted the phase term that is common to both. So the Jones matrix is given as above. ii) For an mthhalf-wave plate designed for wavelength λ0we have thickness: d|(n0-ne)|=(2m+1)λ0/2 ()⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+−λλπ012exp001mjSo we just have to substitute this value of dinto the matrix to obtain: iii) A linear polarizer at -45° has a Jones matrix: ⎥⎦⎤⎢⎣⎡−−111121So the combination of the HWP and linear polarizer is ()()()⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+−−⎟⎠⎞⎜⎝⎛+−−=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+−⎥⎦⎤⎢⎣⎡−−=λλπλλπλλπ00012exp112exp12112exp001111121mjmjmjMiv) We have +45° polarized incident light, so the output electric field is given by
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