By A. Schuster, J. Nicholson

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**Example text**

N2 + 1 DOP P= n = 1…20 Refractive Index n The Mueller matrices for reflection and transmission assume simplified forms: Normal Incidence (i = 0°): ⎛1 ⎜ ⎛ n −1 ⎞ ⎜0 MR = ⎜ ⎟ ⎝ n +1 ⎠ ⎜0 ⎜ ⎝0 2 0 0 0⎞ ⎟ 1 0 0⎟ 4n , MT = 2 0 −1 0 ⎟ n ( + 1) ⎟ 0 0 −1 ⎠ ⎛1 ⎜ ⎜0 ⎜0 ⎜ ⎝0 0 0 0⎞ ⎟ 1 0 0⎟ , 0 1 0⎟ ⎟ 0 0 1⎠ where n is the refractive index of the medium. For incident unpolarized light of intensity I0 the reflected and transmitted intensities are 2 4n ⎛ n −1 ⎞ IR = ⎜ I 0 , IT = I , ⎟ 2 0 ⎝ n +1 ⎠ ( n + 1) and IR + IT = I0.

The Mueller matrix M = Mpol(θ) ⋅ MWP(φ) and the output intensity are then − sin 2θ sin φ ⎞ cos2θ sin 2θ cos φ ⎛ 1 ⎜ cos 2θ ⎟ 2 cos 2θ sin 2θ cos 2θ cos φ − sin 2θ cos 2θ sin φ 1 ⎜ ⎟, M = 2 2 − sin 2θ sin φ ⎟ sin 2θ cos φ 2 ⎜ sin 2θ sin 2θ cos2θ ⎜ ⎟ 0 0 0 ⎝ 0 ⎠ I ( θ, φ) = (1 / 2)( S0 + S1 cos 2θ + S2 sin 2θ cos φ − S3 sin 2θ sin φ ). org/terms Polarization 28 The Observable Polarization Sphere When an incident beam propagates through several polarizing elements, the mathematical manipulations to determine the polarization state of the output beam are tedious.

The first is the Mueller matrix for a rotated ideal linear polarizer which is cos 2θ sin 2θ ⎛ 1 ⎜ 2 cos 2θ sin 2θ cos 2θ 1 ⎜ cos 2θ M POL ( θ) = ⎜ 2 sin 2θ sin 2θ cos 2θ sin 2 2θ ⎜ ⎜ 0 0 0 ⎝ 0⎞ ⎟ 0⎟ ⎟. 0⎟ 0 ⎟⎠ In particular, the Mueller matrix reduces to the following special forms for θ = 0°, 45°, 90°, and 135°: M LHP M LVP ⎛1 ⎜ 1 ⎜1 = 2 ⎜0 ⎜ ⎝0 1 1 0 0 ⎛ 1 −1 ⎜ 1 −1 1 = ⎜ 2⎜ 0 0 ⎜ ⎝0 0 0 0 0 0 0⎞ ⎟ 0⎟ , 0⎟ ⎟ 0⎠ 0 0 0 0 M L+45P ⎛1 ⎜ 1 ⎜0 = 2 ⎜1 ⎜ ⎝0 0⎞ ⎛1 ⎟ ⎜ 0⎟ 1 0 , M L-45P = ⎜ 0⎟ 2 ⎜ −1 ⎟ ⎜ 0⎠ ⎝0 0 0 0 0 1 0 1 0 0 −1 0 0 0 1 0 0 0⎞ ⎟ 0⎟ , 0⎟ ⎟ 0⎠ 0⎞ ⎟ 0⎟ .