Consider a one-dimensional imaging system whose exit pupil is a uniform slit

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Fourier optics and image formation · Fourier optics and image formation

What an Instrument Can See, Ch The transfer function of an imaging system

Problem

Consider a one-dimensional imaging system whose exit pupil is a uniform slit, transmission 11 for −a≤s≤a-a\le s\le a and 00 outside, used incoherently at wavelength λ\lambda. Its coherent cutoff is νc,coh=a/(λzi)\nu_{c,\mathrm{coh}}=a/(\lambda z_i) and its incoherent cutoff is νc=2νc,coh\nu_c = 2\nu_{c,\mathrm{coh}}, exactly as for the chapter's circular pupil. Now let this slit carry a pure defocus, W(s)=W020(s/a)2W(s) = W_{020}(s/a)^2. ((a)) Show that for two pupil points shifted by ±Δ/2\pm\Delta/2 about a coordinate ss, the phase difference k[W(s+Δ/2)−W(s−Δ/2)]k[W(s+\Delta/2)-W(s-\Delta/2)] is exactly linear in ss, and give its slope. ((b)) Hence show that, writing ν^=ν/νc\hat\nu=\nu/\nu_c and Δ=2aν^\Delta=2a\hat\nu, the optical transfer function of this pupil reduces to H(ν^)=sin⁡ ⁣[β ν^(1−ν^)]β ν^,β=8πW020λ,0≤ν^≤1.\mathcal{H}(\hat\nu) = \frac{\sin\!\left[\beta\,\hat\nu(1-\hat\nu)\right]}{\beta\,\hat\nu},\qquad \beta = \frac{8\pi W_{020}}{\lambda},\qquad 0\le\hat\nu\le1. ((c)) Evaluate H(0.5)\mathcal{H}(0.5) for a quarter wave of defocus, W020=λ/4W_{020}=\lambda/4, the same amount used in the chapter's own worked example, and state whether it is positive.

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