fourier - neutron scattering · taylor versus fourier series isis neutron training course 16 / 38...

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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