PHYS 408 HOMEWORK 6 SOLUTIONS

PHYS 408 HOMEWORK 6 SOLUTIONS - solution set 6 Contents 1 2...

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solution set 6 November 15, 2009 Contents 1 Ronchi Ruling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 calculate Fourier decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 plot Fourier decomposition coe cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Another Fourier decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Outright calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Identify expansion in list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Fourier transform, square pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1 Outright calculation and plot of transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Shifting a Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.1 inverse Fourier transform of delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 k -space shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 Fourier transform of a gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.1 the transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 direct and inverse space widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 matching functions and their fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.2 reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 Convolution of rectangular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 8 copy and paste convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8.1 function, single delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8.2 function, two delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8.3 function, infinity of delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 Convolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 fourier optics, transfer functions, plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.1 single-wavevector wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.2 multi-wavevector wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.3 explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11 free-space propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11.1 three plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11.2 single plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11.3 delta-function slit, x -axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11.4 two delta-function slits, relative phase di ff erence . . . . . . . . . . . . . . . . . . . . . . . . . 12 11.5 delta-function slit along y = α x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1
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1 Ronchi Ruling 1.1 calculate Fourier decomposition Let E 0 ( x ) = E 0 f ( x ) with f ( x ) = 1 (4 n 1) b 2 < x < (4 n + 1) b 2 0 (4 n + 1) b 2 < x < (4 n 3) b 2 (1) Fourier decompose f ( x ): f ( x ) = + n = −∞ c n e ik n x k n = 2 π n 2 b = π n b (2) Integrate over one period using the piecewise form of f ( x ): + b b dx e ik m x f ( x ) = + b/ 2 b/ 2 dx e ik m x · 1 = b · sinc( π m/ 2) (3) Integrate over one period using the Fourier decomposition of f ( x ): + b b dx e ik m x f ( x ) = c m · 2 b (4) Thus c m = sinc( π m/ 2) 2 (5) and E 0 ( x ) = E 0 + n = −∞ sinc( π m/ 2) 2 e ik n x (6) = E 0 1 2 + + n =1 sinc( π m/ 2) cos( ik n x ) k n = π n b (7) 2
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1.2 plot Fourier decomposition coe cients Figure 1: p1.jpg 3
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2 Another Fourier decomposition 2.1 Outright calculation Assume period of 2 T and later adjust T as needed. f ( x ) = 2 (2 n ) T < x < (2 n + 1) T 0 (2 n + 1) T < x < (2 n + 2) T (8) Fourier decompose f ( x ): f ( x ) = + n = −∞ c n e ik n x k n = 2 π n 2 T = π n T (9) Integrate over one period using the piecewise form of f ( x ): 2 T 0 dx e ik m x f ( x ) = T 0 dx e ik m x · 2 = 2 T · e ik m T/ 2 sinc( k m T/ 2) (10) Integrate over one period using the Fourier decomposition of f ( x ): 2 T 0 dx e ik m x f ( x ) = c m · 2 T (11) Thus c m = e ik m T/ 2 sinc( k m T/ 2) (12) and f ( x ) = + n = −∞ e ik n T/ 2 sinc( k n T/ 2) e ik n x (13) = 1 + | n | odd e ik n T/ 2 sinc( k n T/ 2) e ik n x (14) = 1 + 2 n odd cos( k n x k n T/ 2)sinc( k n T/ 2) (15) = 1 + 4 π n odd sin( k n x ) n k n = π n T (16) For T = 1, f ( x ) = 1 + 4 π n odd sin( n π x ) n (17) 2.2 Identify expansion in list (c) 2.3 Shortcuts f ( x ) takes the form of 1 plus an odd function, so we would expect the odd function part to consist 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.
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