A thin lens of focal length has an amplitude transmittance object placed immediately…

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

What an Instrument Can See, Ch The lens as a Fourier transformer

Problem

A thin lens of focal length ff has an amplitude transmittance object tA(x1,y1)t_A(x_1,y_1) placed immediately against it (the object-to-lens separation is negligible compared with any other length in the problem), illuminated by a unit-amplitude monochromatic plane wave at normal incidence, wavelength λ\lambda. ((a)) Starting from the paraxial (Fresnel) free-space propagation integral and the quadratic lens phase tL(x1,y1)=exp⁡[−ik(x12+y12)/2f]t_L(x_1,y_1)=\exp[-ik(x_1^2+y_1^2)/2f], derive the field U(x2,y2)U(x_2,y_2) a distance ff behind the lens (its back focal plane). Show that it is, up to an overall constant and one quadratic phase factor which you should identify, the exact two-dimensional Fourier transform of tAt_A evaluated at νx=x2/λf\nu_x=x_2/\lambda f, νy=y2/λf\nu_y=y_2/\lambda f, and explain why a detector placed in that plane cannot see the quadratic phase factor. ((b)) Apply the result to a single slit of width a=0.20 mma=0.20\,\mathrm{mm}, tA(x1,y1)=rect(x1/a)t_A(x_1,y_1)=\mathrm{rect}(x_1/a), held directly against a lens of focal length f=200 mmf=200\,\mathrm{mm} at λ=633 nm\lambda=633\,\mathrm{nm}. Find the distance from the axis to the first zero of irradiance in the back focal plane.

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