An Introduction to Fourier Transforms D. S. Sivia St. John’s College Oxford, England June 28, 2013
An Introduction to
Fourier Transforms
D. S. Sivia
St. John’s College
Oxford, England
June 28, 2013
Outline
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■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
Outline
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■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
■ Some formal properties
◆ Symmetry
◆ Convolution theorem
◆ Auto-correlation function
Outline
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■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
■ Some formal properties
◆ Symmetry
◆ Convolution theorem
◆ Auto-correlation function
■ Physical insight
◆ Fourier optics
Taylor Series
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Taylor Series (0)
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■ f(x) ≈ a0
Taylor Series (1)
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■ f(x) ≈ a0 + a1(x−xo)
Taylor Series (2)
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■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2
Taylor Series (3)
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■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)
3
Taylor Series (4)
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■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)
3 + a4(x−xo)4
Fourier Series
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■ Periodic: f(x) = f(x+λ) k =2π
λ(wavenumber)
Fourier Series (0)
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■ f(x) ≈a0
2
Fourier Series (1)
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■ f(x) ≈a0
2+A1sin(kx+φ1)
Fourier Series (1)
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■ f(x) ≈a0
2+ a1cos(kx)
+ b1sin(kx)
Fourier Series (2)
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■ f(x) ≈a0
2+ a1cos(kx) + a2 cos(2kx)
+ b1sin(kx) + b2 sin(2kx)
Fourier Series (3)
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■ f(x) ≈a0
2+ a1cos(kx) + a2 cos(2kx) + a3 cos(3kx)
+ b1sin(kx) + b2 sin(2kx) + b3 sin(3kx)
Fourier Series (4)
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■ f(x) ≈a0
2+ a1cos(kx) + a2 cos(2kx) + a3 cos(3kx) + a4 cos(4kx)
+ b1sin(kx) + b2 sin(2kx) + b3 sin(3kx) + b4 sin(4kx)
Taylor Versus Fourier Series
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■ Taylor: f(x) =∞
∑
n =0
an(x−xo)n |x−xo|<R
◆ an =1
n!
dn f
dxn
∣
∣
∣
∣
xo
Taylor Versus Fourier Series
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■ Taylor: f(x) =∞
∑
n =0
an(x−xo)n |x−xo|<R
◆ an =1
n!
dn f
dxn
∣
∣
∣
∣
xo
■ Fourier: f(x) =a0
2+
∞∑
n =1
an cos(nkx) + bn sin(nkx) k =2π
λ
◆ an = 2
λ
λ∫
0
f(x) cos(nkx) dx and bn = 2
λ
λ∫
0
f(x) sin(nkx) dx
Complex Fourier Series
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eiθ = cos θ + i sin θ , where i2 = −1
Complex Fourier Series
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eiθ = cos θ + i sin θ , where i2 = −1
■ Fourier: f(x) =∞
∑
n =−∞cn einkx
◆ cn = 1
λ
λ/2∫
−λ/2
f(x) e−inkx dx
Complex Fourier Series
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eiθ = cos θ + i sin θ , where i2 = −1
■ Fourier: f(x) =∞
∑
n =−∞cn einkx
◆ cn = 1
λ
λ/2∫
−λ/2
f(x) e−inkx dx
■ c±n = 1
2(an ∓ ibn) for n>1
■ c0 = a0
Fourier Transform
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■ As λ→∞, so that k→0 and f(x) is non-periodic,
◆
∞∑
n =−∞cn einkx −→
∞∫
−∞
c(q) eiqx dq
Fourier Transform
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■ As λ→∞, so that k→0 and f(x) is non-periodic,
◆
∞∑
n =−∞cn einkx −→
∞∫
−∞
c(q) eiqx dq
■ In the continuum limit,
◆ Fourier sum (series) −→ Fourier integral (transform)
◆ f(x) =
∞∫
−∞
F(q) eiqx dq
■ F(q) = 1
2π
∞∫
−∞
f(x) e−iqx dx
Some Symmetry Properties
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■ Even: f(x) = f(−x) ⇐⇒ F(q) = F(−q)
■ Odd: f(x) = − f(−x) ⇐⇒ F(q) = −F(−q)
Some Symmetry Properties
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■ 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)
Convolution
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f(x) = g(x) ⊗ h(x) =
∞∫
−∞
g(t) h(x−t) dt
Convolution
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f(x) = g(x) ⊗ h(x) =
∞∫
−∞
g(t) h(x−t) dt
Convolution Theorem
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f(x) = g(x) ⊗ h(x) ⇐⇒ F(q) =√
2π G(q)×H(q)
Convolution Theorem
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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)
Auto-correlation Function
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∞∫
−∞
F(q) eiqx dq = f(x)
Auto-correlation Function
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∞∫
−∞
F(q) eiqx dq = f(x)
■
∞∫
−∞
∣
∣F(q)∣
∣
2eiqx dq =
∞∫
−∞
f(t)∗ f(x+t) dt = ACF(x)
◆ Patterson map
Auto-correlation Function (1)
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Auto-correlation Function (2)
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Fourier Optics
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I(q) =∣
∣ψ(q)∣
∣
2
■ Fraunhofer: ψ(q) = ψo
∞∫
−∞
A(x) eiqx dx where q =2π sin θ
λ
Young’s Double Slits
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Young’s Double Slits
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Young’s Double Slits
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Single Wide Slit
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Single Wide Slit
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Single Wide Slit
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Two Wide Slits (0)
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Two Wide Slits (1)
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Two Wide Slits (2)
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Two Wide Slits (3)
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Finite Grating (0)
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Finite Grating (1)
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Finite Grating (2)
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Finite Grating (3)
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Write up of this Talk!
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■ 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
The phaseless Fourier problem
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The phaseless Fourier problem
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