An Introduction to

Fourier Transforms

D. S. Sivia

St. John’s College

Oxford, England

August 12, 2015

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Taylor Series (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Taylor Series (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Taylor Series (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Taylor Series (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Taylor Series (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Fourier Series (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Fourier Series (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Fourier Series (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Fourier Series (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Fourier Series (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Fourier Series (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Taylor Versus Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Some Symmetry Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Auto-correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Auto-correlation Function (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Auto-correlation Function (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Fourier Optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Young’s Double Slits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Single Wide Slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Two Wide Slits (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Two Wide Slits (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Two Wide Slits (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Two Wide Slits (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Finite Grating (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Finite Grating (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Finite Grating (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1

Finite Grating (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Write up of this Talk!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

The phaseless Fourier problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

The phaseless Fourier problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2

Outline

■ Approximating functions

◆ Taylor series

◆ Fourier series → transform

■ Some formal properties

◆ Symmetry

◆ Convolution theorem

◆ Auto-correlation function

■ Physical insight

◆ Fourier optics

Oxford School on Neutron Scattering 2 / 38

Taylor Series

Oxford School on Neutron Scattering 3 / 38

3

Taylor Series (0)

■ f(x) ≈ a0

Oxford School on Neutron Scattering 4 / 38

Taylor Series (1)

■ f(x) ≈ a0 + a1(x−xo)

Oxford School on Neutron Scattering 5 / 38

4

Taylor Series (2)

■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2

Oxford School on Neutron Scattering 6 / 38

Taylor Series (3)

■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)

3

Oxford School on Neutron Scattering 7 / 38

5

Taylor Series (4)

■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)

3 + a4(x−xo)4

Oxford School on Neutron Scattering 8 / 38

Fourier Series

■ Periodic: f(x) = f(x+λ) k =2π

λ(wavenumber)

Oxford School on Neutron Scattering 9 / 38

6

Fourier Series (0)

■ f(x) ≈a0

2

Oxford School on Neutron Scattering 10 / 38

Fourier Series (1)

■ f(x) ≈a0

2+A1sin(kx+φ1)

Oxford School on Neutron Scattering 11 / 38

7

Fourier Series (1)

■ f(x) ≈a0

2+ a1cos(kx)

+ b1sin(kx)

Oxford School on Neutron Scattering 12 / 38

Fourier Series (2)

■ f(x) ≈a0

2+ a1cos(kx) + a2 cos(2kx)

+ b1sin(kx) + b2 sin(2kx)

Oxford School on Neutron Scattering 13 / 38

8

Fourier Series (3)

■ f(x) ≈a0

2+ a1cos(kx) + a2 cos(2kx) + a3 cos(3kx)

+ b1sin(kx) + b2 sin(2kx) + b3 sin(3kx)

Oxford School on Neutron Scattering 14 / 38

Fourier Series (4)

■ 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)

Oxford School on Neutron Scattering 15 / 38

9

Taylor Versus Fourier Series

■ Taylor: f(x) =

∞∑

n=0

an(x−xo)n |x−xo|<R

◆ an =1

n!

dnf

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

Oxford School on Neutron Scattering 16 / 38

Complex Fourier Series

eiθ = cos θ + i sin θ , where i2 = −1

■ Fourier: f(x) =

∞∑

n=−∞cn e

inkx

◆ cn = 1

λ

λ/2∫

−λ/2

f(x) e−inkx dx

■ c±n = 1

2(an∓ ibn) for n>1

■ c0 = a0

Oxford School on Neutron Scattering 17 / 38

10

Fourier Transform

■ As λ→∞, so that k→0 and f(x) is non-periodic,

◆

∞∑

n=−∞cn e

inkx −→

∞∫

−∞

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

Oxford School on Neutron Scattering 18 / 38

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)

Oxford School on Neutron Scattering 19 / 38

11

Convolution

f(x) = g(x)⊗ h(x) =

∞∫

−∞

g(t) h(x−t) dt

Oxford School on Neutron Scattering 20 / 38

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)

Oxford School on Neutron Scattering 21 / 38

12

Auto-correlation Function

∞∫

−∞

F(q) eiqx dq = f(x)

■

∞∫

−∞

∣

∣F(q)∣

∣

2eiqx dq =

∞∫

−∞

f(t)∗ f(x+t) dt = ACF(x)

◆ Patterson map

Oxford School on Neutron Scattering 22 / 38

Auto-correlation Function (1)

Oxford School on Neutron Scattering 23 / 38

13

Auto-correlation Function (2)

Oxford School on Neutron Scattering 24 / 38

Fourier Optics

I(q) =∣

∣ψ(q)∣

∣

2

■ Fraunhofer: ψ(q) = ψo

∞∫

−∞

A(x) eiqx dx where q =2π sin θ

λ

Oxford School on Neutron Scattering 25 / 38

14

Young’s Double Slits

Oxford School on Neutron Scattering 26 / 38

Single Wide Slit

Oxford School on Neutron Scattering 27 / 38

15

Two Wide Slits (0)

Oxford School on Neutron Scattering 28 / 38

Two Wide Slits (1)

Oxford School on Neutron Scattering 29 / 38

16

Two Wide Slits (2)

Oxford School on Neutron Scattering 30 / 38

Two Wide Slits (3)

Oxford School on Neutron Scattering 31 / 38

17

Finite Grating (0)

Oxford School on Neutron Scattering 32 / 38

Finite Grating (1)

Oxford School on Neutron Scattering 33 / 38

18

Finite Grating (2)

Oxford School on Neutron Scattering 34 / 38

Finite Grating (3)

Oxford School on Neutron Scattering 35 / 38

19

Write up of this Talk!

■ Elementary Scattering Theory for X-ray and Neutron Users (Chapter 2)

D. S. Sivia (2011), Oxford University Press

■ Foundations of Science Mathematics (Chapter 15)

Oxford Chemistry Primers Series, vol. 77 (and 82)

D. S. Sivia and S. G. Rawlings (1999), Oxford University Press

Oxford School on Neutron Scattering 36 / 38

The phaseless Fourier problem

Oxford School on Neutron Scattering 37 / 38

20

The phaseless Fourier problem

Oxford School on Neutron Scattering 38 / 38

21