Fourier Transforms D. S. Sivia
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1 An Introduction to Fourier Transforms D. S. Sivia St. John s College Oxford, England June 28, 2013
2 Outline Approximating functions Taylor series Fourier series transform ISIS Neutron Training Course 2 / 38
3 Outline Approximating functions Taylor series Fourier series transform Some formal properties Symmetry Convolution theorem Auto-correlation function ISIS Neutron Training Course 2 / 38
4 Outline Approximating functions Taylor series Fourier series transform Some formal properties Symmetry Convolution theorem Auto-correlation function Physical insight Fourier optics ISIS Neutron Training Course 2 / 38
5 Taylor Series ISIS Neutron Training Course 3 / 38
6 Taylor Series (0) f(x) a 0 ISIS Neutron Training Course 4 / 38
7 Taylor Series (1) f(x) a 0 + a 1 (x x o ) ISIS Neutron Training Course 5 / 38
8 Taylor Series (2) f(x) a 0 + a 1 (x x o ) + a 2 (x x o ) 2 ISIS Neutron Training Course 6 / 38
9 Taylor Series (3) f(x) a 0 + a 1 (x x o ) + a 2 (x x o ) 2 + a 3 (x x o ) 3 ISIS Neutron Training Course 7 / 38
10 Taylor Series (4) f(x) a 0 + a 1 (x x o ) + a 2 (x x o ) 2 + a 3 (x x o ) 3 + a 4 (x x o ) 4 ISIS Neutron Training Course 8 / 38
11 Fourier Series Periodic: f(x) = f(x+λ) k = 2π λ (wavenumber) ISIS Neutron Training Course 9 / 38
12 Fourier Series (0) f(x) a 0 2 ISIS Neutron Training Course 10 / 38
13 Fourier Series (1) f(x) a A 1sin(kx+φ 1 ) ISIS Neutron Training Course 11 / 38
14 Fourier Series (1) f(x) a a 1cos(kx) + b 1 sin(kx) ISIS Neutron Training Course 12 / 38
15 Fourier Series (2) f(x) a a 1cos(kx) + a 2 cos(2kx) + b 1 sin(kx) + b 2 sin(2kx) ISIS Neutron Training Course 13 / 38
16 Fourier Series (3) f(x) a a 1cos(kx) + a 2 cos(2kx) + a 3 cos(3kx) + b 1 sin(kx) + b 2 sin(2kx) + b 3 sin(3kx) ISIS Neutron Training Course 14 / 38
17 Fourier Series (4) f(x) a a 1cos(kx) + a 2 cos(2kx) + a 3 cos(3kx) + a 4 cos(4kx) + b 1 sin(kx) + b 2 sin(2kx) + b 3 sin(3kx) + b 4 sin(4kx) ISIS Neutron Training Course 15 / 38
18 Taylor Versus Fourier Series Taylor: f(x) = a n = 1 n! d n f dx n n=0 xo a n (x x o ) n x x o <R ISIS Neutron Training Course 16 / 38
19 Taylor Versus Fourier Series Taylor: f(x) = a n = 1 n! d n f dx n n=0 xo a n (x x o ) n x x o <R Fourier: f(x) = a n=1 a n cos(nkx) + b n sin(nkx) k = 2π λ λ λ a n = 2 λ f(x) cos(nkx) dx and b n = 2 λ f(x) sin(nkx) dx 0 0 ISIS Neutron Training Course 16 / 38
20 Complex Fourier Series e iθ = cos θ + i sinθ, where i 2 = 1 ISIS Neutron Training Course 17 / 38
21 Complex Fourier Series e iθ = cos θ + i sinθ, where i 2 = 1 Fourier: f(x) = n= c n e inkx c n = 1 λ λ/2 f(x) e inkx dx λ/2 ISIS Neutron Training Course 17 / 38
22 Complex Fourier Series e iθ = cos θ + i sinθ, where i 2 = 1 Fourier: f(x) = n= c n e inkx c n = 1 λ λ/2 f(x) e inkx dx λ/2 c ±n = 1 2 (a n ib n ) for n 1 c 0 = a 0 ISIS Neutron Training Course 17 / 38
23 Fourier Transform As λ, so that k 0 and f(x) is non-periodic, c n e inkx n= c(q)e iqx dq ISIS Neutron Training Course 18 / 38
24 Fourier Transform As λ, so that k 0 and f(x) is non-periodic, c n e inkx n= c(q)e iqx dq In the continuum limit, Fourier sum (series) Fourier integral (transform) f(x) = F(q) e iqx dq F(q) = 1 2π f(x) e iqx dx ISIS Neutron Training Course 18 / 38
25 Some Symmetry Properties Even: f(x) = f( x) F(q) = F( q) Odd: f(x) = f( x) F(q) = F( q) ISIS Neutron Training Course 19 / 38
26 Some Symmetry Properties Even: f(x) = f( x) F(q) = F( q) Odd: f(x) = f( x) F(q) = F( q) Real: f(x) = f(x) F(q) = F( q) (Friedel pairs) ISIS Neutron Training Course 19 / 38
27 Convolution f(x) = g(x) h(x) = g(t) h(x t) dt ISIS Neutron Training Course 20 / 38
28 Convolution f(x) = g(x) h(x) = g(t) h(x t) dt ISIS Neutron Training Course 20 / 38
29 Convolution Theorem f(x) = g(x) h(x) F(q) = 2π G(q) H(q) ISIS Neutron Training Course 21 / 38
30 Convolution Theorem f(x) = g(x) h(x) F(q) = 2π G(q) H(q) f(x) = g(x) h(x) F(q) = 1 2π G(q) H(q) ISIS Neutron Training Course 21 / 38
31 Auto-correlation Function F(q) e iqx dq = f(x) ISIS Neutron Training Course 22 / 38
32 Auto-correlation Function F(q) e iqx dq = f(x) F(q) 2 e iqx dq = f(t) f(x+t) dt = ACF(x) Patterson map ISIS Neutron Training Course 22 / 38
33 Auto-correlation Function (1) ISIS Neutron Training Course 23 / 38
34 Auto-correlation Function (2) ISIS Neutron Training Course 24 / 38
35 Fourier Optics I(q) = ψ(q) 2 Fraunhofer: ψ(q) = ψ o A(x) e iqx dx where q = 2π sinθ λ ISIS Neutron Training Course 25 / 38
36 Young s Double Slits ISIS Neutron Training Course 26 / 38
37 Young s Double Slits ISIS Neutron Training Course 26 / 38
38 Young s Double Slits ISIS Neutron Training Course 26 / 38
39 Single Wide Slit ISIS Neutron Training Course 27 / 38
40 Single Wide Slit ISIS Neutron Training Course 27 / 38
41 Single Wide Slit ISIS Neutron Training Course 27 / 38
42 Two Wide Slits (0) ISIS Neutron Training Course 28 / 38
43 Two Wide Slits (1) ISIS Neutron Training Course 29 / 38
44 Two Wide Slits (2) ISIS Neutron Training Course 30 / 38
45 Two Wide Slits (3) ISIS Neutron Training Course 31 / 38
46 Finite Grating (0) ISIS Neutron Training Course 32 / 38
47 Finite Grating (1) ISIS Neutron Training Course 33 / 38
48 Finite Grating (2) ISIS Neutron Training Course 34 / 38
49 Finite Grating (3) ISIS Neutron Training Course 35 / 38
50 Write up of this Talk! Foundations of Science Mathematics (Chapter 15) Oxford Chemistry Primers Series, vol. 77 D. S. Sivia and S. G. Rawlings (1999), Oxford University Press Elementary Scattering Theory for X-ray and Neutron Users (Chapter 2) D. S. Sivia (January 2011), Oxford University Press Foundations of Science Mathematics: Worked Problems (Chapter 15) Oxford Chemistry Primers Series, vol. 82 D. S. Sivia and S. G. Rawlings (1999), Oxford University Press ISIS Neutron Training Course 36 / 38
51 The phaseless Fourier problem ISIS Neutron Training Course 37 / 38
52 The phaseless Fourier problem ISIS Neutron Training Course 38 / 38
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