By J. F. James
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Extra info for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering, Third edition
These are subtleties which can safely be left to post-graduate students. 1 Fraunhofer diffraction 43 Fig. 3. Oblique incidence from a source not on the z-axis. to x is sin θ/λ and it is θ which matters. Diffraction theory alone says nothing about the size of the pattern: that depends on geometry. Very often, in practice, the diffracting aperture is followed by a lens, and the pattern is observed at the focal plane of this lens. The approximation that r 0 D r0 x sin θ is now exact, since the image of the focal plane, seen from the diffracting aperture, is at infinity.
7 Useful functions 17 Similarly, we find δ(a/b c/d) D acδ(ad bc) D bdδ(ad bc) δ(ax) D (1/a)δ(x). Another important consequence of the shift theorem is that 1 e2πipx δ(x a)dx D e2πipa 1 so that we can write δ(x δ(mx a) • e2πipa , a) • (1/m)e2πipa/m and a formula which we shall need in Chapter 7: 1 p δ n l r n Dδ pn l r •e 2πi ( pn l r ). 6 A pair of δ-functions If two δ-functions are equally disposed on either side of the origin, the Fourier transform is a cosine wave: δ(x a) C δ(x C a) • e2πipa C e 2πipa D 2 cos(2πpa).
A pair of δ-functions and its transform. Represent the spectral power density (‘the spectrum’, see Fig. 4) of the source by Sδ(λ λ0 ). The spectrometer will plot the graph as kSδ(λ λ0 ), where k is a factor which depends on the throughput of the spectrometer, its geometry and its detector sensitivity. No spectrometer is perfect in practice, and what a real instrument will plot in response to a monochromatic input is a continuous curve kSI (λ λ0 ), where 1 I (λ) is called the ‘instrumental function’ and 1 I (λ)dλ D 1.
A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering, Third edition by J. F. James