# Let’s combine two ideas we’ve learned previously. Remember that if you illuminate a transmissive… 1 answer below »

Let’s combine two ideas we’ve learned previously. Remember that if you illuminate a transmissive object with coherent light, the diffraction pattern at infinity is a Fourier transform of the transmissivity. (Morin Ch 9, eq 42 or Georgi 13.14, which is 2D but you can ignore the y-terms). Remember that parallel rays that leave the focal plane of a converging lens at an angle 0 from the axis converge at the other focus of the lens, a distance f0 from the axis. Let’s use a coordinate system where x is the distance along the, and y is the distance perpendicular to the axis. Imagine that there is a lens with focal length f, located at x = 0 a) Imagine two coherent sources are located at the front focus of the lens (x = -f), one on the axis = 0) and the other at y. Imagine two plane waves in phase leave these sources at an angle 0. They will intersect and interfere in the focal plane of the other lens (at f, f9). Use the principle of least time and the fact that the wave fronts are perpendicular to the rays to calculate the phase difference between the two waves at the point of intersection. b) From the previous result, if an object with 1lansmissivity T(y) is illuminated with coherent light of wavelength A and placed at the front focus of a thin lens with focal length f, find a formula for the relative (you can ignore overall constants) intensity of light at the opposite focus of the lens. Explain how “lenses make Fourier transforms.”

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