All matrices below are multiplied right to left in the order light meets the elements…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Polarisation and anisotropic media

Optics and Photonics, Ch Polarisation states, polarisers and the Jones and Stokes calculus

Problem

All matrices below are multiplied right to left in the order light meets the elements (the rule of Chapter 1). ((a)) Two thin converging lenses, in air (n=1n=1), of focal length f1=20 cmf_1=20\text{ cm} and f2=10 cmf_2=10\text{ cm}, lie in contact (negligible separation). A paraxial ray leaves an object with height y0=2.0 mmy_0=2.0\text{ mm} and angle θ0=0.010 rad\theta_0=0.010\text{ rad} and meets lens 1 first, then lens 2. Find the ray's height and angle immediately after lens 2, using the thin-lens matrix (10−1/f1)\begin{pmatrix}1&0\\-1/f&1\end{pmatrix} and the system matrix T=T2T1\mathbf{T}=\mathbf{T}_2\mathbf{T}_1. ((b)) Light linearly polarised along xx, Jones vector (1,0)(1,0), meets an ideal linear polariser with axis at θ1=30∘\theta_1=30^\circ first, then one at θ2=60∘\theta_2=60^\circ. Using J=JP(θ2)JP(θ1)\mathbf{J}=\mathbf{J}_P(\theta_2)\mathbf{J}_P(\theta_1) in the same order rule, find the output irradiance relative to the input. ((c)) A sheet of crown glass (n=1.50n=1.50) reflects unpolarised light at Brewster's angle; the reflected beam of total irradiance Itot=1.00I_{\text{tot}}=1.00 (arbitrary units) is purely ss-polarised (Chapter 6). A polariser set at 0∘0^\circ, 45∘45^\circ, 90∘90^\circ, 135∘135^\circ to the plane of incidence transmits I0∘=0I_{0^\circ}=0, I45∘=0.50I_{45^\circ}=0.50, I90∘=1.00I_{90^\circ}=1.00, I135∘=0.50I_{135^\circ}=0.50. Write the Stokes vector (S0,S1,S2,S3)(S_0,S_1,S_2,S_3) of the reflected beam and its degree of polarisation. ((d)) Two coherent beams of the same polarisation, amplitudes E1=3.0E_1=3.0 and E2=4.0E_2=4.0 (arbitrary units) and phase difference δ=60∘\delta=60^\circ, arrive together (as in Chapter 4). Find the resultant irradiance by phasor addition, ER2=E12+E22+2E1E2cos⁡δE_R^2=E_1^2+E_2^2+2E_1E_2\cos\delta, and compare it with the simple sum of the two separate irradiances.

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