Two collimated beams of the same frequency travel collinearly and overlap at a point

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Polarisation and anisotropic media · Interference and diffraction

Optics and Photonics, Ch Two-beam interference and interferometry

Problem

Two collimated beams of the same frequency travel collinearly and overlap at a point. Beam 1 is right-circularly polarised with real amplitude E0E_0; beam 2 is left-circularly polarised with the same real amplitude E0E_0 but carries an extra phase δ\delta relative to beam 1, coming from a longer path in one arm of an interferometer. Under CV3 the Jones vectors are E1=E02(1−i)\mathbf{E}_1 = \dfrac{E_0}{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix} and E2=E02eiδ(1i)\mathbf{E}_2 = \dfrac{E_0}{\sqrt2}e^{i\delta}\begin{pmatrix}1\\i\end{pmatrix}. The total field is E=E1+E2\mathbf{E} = \mathbf{E}_1 + \mathbf{E}_2, and the irradiance is proportional to E†E\mathbf{E}^\dagger\mathbf{E} (the constant 12cϵ0n\tfrac12 c\epsilon_0 n is dropped since only the ratio matters here). ((a)) Show that the cross (interference) term of E†E\mathbf{E}^\dagger\mathbf{E} is 2 Re(E1†E2)2\,\mathrm{Re}(\mathbf{E}_1^\dagger\mathbf{E}_2), the same combination as the dot product of the two Jones vectors with the first one conjugated. ((b)) Evaluate this term for the right- and left-circular pair above and show it is zero for every value of δ\delta, so no fringes form as δ\delta is scanned.

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