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Unformatted text preview: of only sine functions, as
cosine function are even whilst sine functions are odd. Some of the algebra needed above would thus be avoided by starting
out with a fourier decomposition in terms of 1 plus a sum over all possible sine functions. 4 3
3.1 Fourier transform, square pulse
Outright calculation and plot of transform ˜
E0 (kx ) = E
E0 (x) =
+∞ −∞ −ikx x dx · e · E0 (x) = −L/2 < x < +L/2 (18) (19) otherwise +L/2 −L/2 dx · e−ikx x · E0 = E0 Lsinc(kx L/2) Figure 2: p3.jpg 5 4 Shifting a Fourier transform 4.1 inverse Fourier transform of delta functions
f (x) =
2π +∞ ˜
dkx F (kx )e+ikx x −∞
dkx δ (kx − (−a/2)) + δ (kx − (+a/2)) e+ikx x
(22) “Makes sense” for the reason we have two plane waves in real space, each with a distinct wavenumber, so in a wavenumber
representation we would expect contributions at only those two wavenumbers. 4.2 k -space shifting
f (x) → f (x)eik0 x
F (kx ) = δ (−kx +...
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This document was uploaded on 03/11/2014 for the course PHYS 408 at University of British Columbia.
- Fall '11