1. As we showed on Example Sheet 4, the Fourier transform of the function (d-a)/2

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1 As We Showed On Example Sheet 4 The Fourier Transform Of The Function D A 2 R D A 2 Otherwise F X 0 Is A 1 (79.97 KiB) Viewed 89 times

1. As we showed on Example Sheet 4, the Fourier transform of the function (d-a)/2 <r<(d+a)/2 otherwise f(x) = { : 0 is a = g(k) VIT e-ikd/2 sinc(ka/2). In wave optics you will see that this describes the diffraction pattern produced by a single slit of width a centred on r = +d/2. (a) Use this result to write down the Fourier transform corresponding to a slit centred on r = -d/2. (b) Find the Fourier transform corresponding to light from two slits, one centred on r = +d/2 and the other on r = -d/2. (c) Show that your result for two slits can be written as a product of two functions of k. What does this imply about the form of the original function of r? [Hint: Consider the Convolution Theorem.] (d) Interpret the significance of the two functions of k in part (c) using the result for a single slit from part (a) and the shape of the two-slit result in the limit of small a.