By J. F. James
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Additional resources for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering, Third edition
2 Alternating pulse-heights In a rectangular waveform with pulses of length a/4 separated by spaces of length a/4 and with alternate rectangles twice the height of their neighbours, the amplitude of the second harmonic is greater than the fundamental amplitude. The waveform can be represented by F (t) D h a/4 (t) [Шa (t) C Шa/2 (t)]. The Fourier transform is (ν) D (ah/4)sinc(π νa/4) [(1/a)Ш1/a (ν) C (2/a)Ш2/a (ν)] and the teeth of this Dirac comb are at ν D 1/a, 2/a, . . , with heights (h/4)sinc(π/4), (3h/4)sinc(π/2), (h/4)sinc(3π/4) .
4 Three parallel slits, each of width a, with centres separated by a distance b To simplify the algebra, put sin θ/λ D p: A(x) D 0 (x) [δ(x A(p) D k sinc(πpa)[e b) C δ(x) C δ(x C b)], 2πibp C1Ce 2πipb ] D k sinc(πpa)[2 cos(2πpb) C 1] and the intensity diffracted at angle θ is I (p) D k 2 sinc2 (πpa)[2 cos(4πpb) C 4 cos(2πpb) C 3] D k 2 sinc2 (π a sin θ/λ)[2 cos(4π b sin θ/λ) C 4 cos(2π b sin θ/λ) C 3]. See Fig. 7. 5 The transmission diffraction grating There are two obvious ways of representing the aperture function.
This can be written symbolically as F (x) δ(x a) D F (x a). Applying the convolution theorem to this is instructive since it yields the shift theorem: F (x) • (p); δ(x a) • e 2πipa so that F (x a) D F (x) δ(x a) • (p)e 2πipa . More interesting is the convolution of a pair of δ-functions with another function: [δ(x a) C δ(x C a)] • 2 cos(2πpa). 7) and this is illustrated in Fig. 5. 2 2 The Fourier transform of a Gaussian g(x) D e x /a is, from Chapter 1, p 2 2 2 2 2 a π e π p a . The convolution of two unequal Gaussian curves, e x /a 2 2 e x /b , can then be done, either as a tiresome exercise in elementary calculus, or by application of the convolution theorem: e x 2 /a 2 e x 2 /b2 • abπ e π 2 p2 (a 2 Cb2 ) 28 Useful properties and theorems Fig.