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Past exam of the materials science course of the University of Cambridge / 2019 / Paper 1 / 2 / b / Solution 2026-10-20

An optically isotropic film changes the phase of both transverse polarization components equally, so crossed polarizers still extinguish the transmitted light. The etched rod-and-gap composite has form birefringence: its effective permittivity depends on the direction of the electric field, even when the constituent material is intrinsically isotropic. In a simple subwavelength lamellar model,

ϵ_{∥} = f ϵ_{r} + (1 - f) ϵ_{g}, ϵ_{⊥} = (\frac{f}{ϵ _{r}} + \frac{1 - f}{ϵ _{g}})^{- 1} .

(1)

The precise effective-medium formula changes with rod geometry, but the anisotropy remains. Resolving incident light along the two principal directions produces a phase retardation. An ideal lossless specimen between crossed polarizers transmits

I = I_{P} sin^{2} (2 α) sin^{2} (π Γ/ λ), Γ = (n_{slow} - n_{fast}) t,

(2)

where

I_{P}

is the intensity after the first polarizer. Extinction occurs when a principal direction follows either polarizer; transmission is strongest near

4 5^{\circ}

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