Birefringent Media.nb

$E E E E E E x y x 2 x 2 x y 2 RowBox[{Cell[TextData[{Electromagnetic Wave Propagation in Birefringent Media, For an electromagnetic wave propagating along the z-axis, the electric field can be broken up into two components , Cell[BoxData[E ], Background -> RGBColor[1, 0, 1]], along the x direction and , Cell[BoxData[E ], Background -> RGBColor[1, 0, 1]], along the y direction. These orthogonal components of the electric field can be thought of as a two level system. Thus, many of the interesting quantum phenomena that occur in two level systems can be observed using polarization. Following the treatment in Born and Wolf, the two components of the electric field can be expressed in the form: , Cell[BoxData[E = a cos[ωt - kr + δ ]], Background -> RGBColor[1, 0, 1]], , Cell[BoxData[E = a cos[ωt - kr + δ ]], Background -> RGBColor[1, 0, 1]], A parameter τ can be defined as: τ=ωt-kr and the electric field components can be written as follows: , Cell[BoxData[{-- = cos(τ) cos(δ ) - sin(τ) sin(δ ), , -- = cos(τ) cos(δ ) - sin(τ) sin(δ )}], Background -> RGBColor[1, 0, 1]], Eliminating time from the equations and defining the parameter , Cell[BoxData[δ = δ - δ ], Background -> RGBColor[1, 0, 1]], yields the following equation for a conic: , Cell[BoxData[(--) + (--) - 2 -- -- cos(δ) = sin δ], Background -> RGBColor[1, 0, 1]], In the following notebook, the phases can be set for each component of the electric field or different indices of refraction can be defined for each of the polarizations as would occur in a birefringent medium. The outputs of the notebook are a parametric plot of the electric field displayed in 2D and a parametric plot of the electric field in 3D. The 3D plot illustrates the evolution of the electric field versus position along the z-axis. }], Background -> RGBColor[1., 0., 1.]], Cell[]}] x y x 1 1 y 2 2 a 1 1 a 2 2 2 1 a a a a 1 2 1 1 1 2$

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$"Propagation Model Parameters" ; << Graphics`Animation` λ = 1.55 * 10^-6 ; "Wavelength" ; n _ 1 = 1.5 ; "Index along the x-axis" ; n _ 2 = 1.45 ; "Index along the y-axis" ; a _ 1 = 1 ; "Amplitude of the E-field in the x-direction" ; a _ 2 = 1 ; "Amplitude of the E-field in the y-direction" ; δ _ 1 = π/4 ; "Phase delay of E-field in the x-direction" ; δ _ 2 = π/4 ; "Phase delay of E-field in the y-direction" ; k _ 1 = n _ 1 (2 π)/λ ; "Wavenumber in x" ; k _ 2 = n _ 2 (2 π)/λ ; "Wavenumber in y" ; E _ 1[z_] := a _ 1 Cos[k _ 1 z + δ _ 1] ; "Component of the electric field vector in x" ; E _ 2[z_] := a _ 2 Cos[k _ 2 z + δ _ 2] ; "Component of the electric field in y" ; ParametricPlot[{E _ 1[z], E _ 2[z]}, {z, 1.05 * 10^-6, 10.5 * 10^-6}, Frame -> True, FrameLabel -> {Ex, Ey, Electric Field Vector, o}] ParametricPlot3D[{E _ 1[z], E _ 2[z], z * 5 * 10^5}, {z, 0, 1.5 * 10^-5}]$

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Converted by Mathematica (September 4, 2003)