A grating of lines per millimetre sits in the front focal plane of a lens of focal length

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 lens as a Fourier transformer

Problem

A grating of ν0=100\nu_0 = 100 lines per millimetre sits in the front focal plane of a lens of focal length f=180 mmf = 180\ \mathrm{mm}, illuminated at λ=600 nm\lambda = 600\ \mathrm{nm}. The screen recording the back-focal-plane pattern sits L=360 mmL = 360\ \mathrm{mm} from the grating (the front focal plane to the lens, plus the lens to the back focal plane), and a calculation for this setup reads: "the transforming property gives xn=nλLν0x_n = n\lambda L\nu_0, with LL the object-to-screen distance, so x1=(600×10−9)(0.360)(100×103)=2.16×10−2 m=21.6 mmx_1 = (600\times10^{-9})(0.360)(100\times10^3) = 2.16\times10^{-2}\ \mathrm{m} = 21.6\ \mathrm{mm}." ((a)) Identify the error in this reasoning. ((b)) State the correct formula and compute the correct value of x1x_1.

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