Fourier - Neutron Scattering · Taylor Versus Fourier Series ISIS Neutron Training Course 16 / 38...
Transcript of 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
ISIS Neutron Training Course 2 / 38
■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
Outline
ISIS Neutron Training Course 2 / 38
■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
■ Some formal properties
◆ Symmetry
◆ Convolution theorem
◆ Auto-correlation function
Outline
ISIS Neutron Training Course 2 / 38
■ Approximating functions
◆ Taylor series
◆ Fourier series → transform
■ Some formal properties
◆ Symmetry
◆ Convolution theorem
◆ Auto-correlation function
■ Physical insight
◆ Fourier optics
Taylor Series
ISIS Neutron Training Course 3 / 38
Taylor Series (0)
ISIS Neutron Training Course 4 / 38
■ f(x) ≈ a0
Taylor Series (1)
ISIS Neutron Training Course 5 / 38
■ f(x) ≈ a0 + a1(x−xo)
Taylor Series (2)
ISIS Neutron Training Course 6 / 38
■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2
Taylor Series (3)
ISIS Neutron Training Course 7 / 38
■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)
3
Taylor Series (4)
ISIS Neutron Training Course 8 / 38
■ f(x) ≈ a0 + a1(x−xo) + a2(x−xo)2 + a3(x−xo)
3 + a4(x−xo)4
Fourier Series
ISIS Neutron Training Course 9 / 38
■ Periodic: f(x) = f(x+λ) k =2π
λ(wavenumber)
Fourier Series (0)
ISIS Neutron Training Course 10 / 38
■ f(x) ≈a0
2
Fourier Series (1)
ISIS Neutron Training Course 11 / 38
■ f(x) ≈a0
2+A1sin(kx+φ1)
Fourier Series (1)
ISIS Neutron Training Course 12 / 38
■ f(x) ≈a0
2+ a1cos(kx)
+ b1sin(kx)
Fourier Series (2)
ISIS Neutron Training Course 13 / 38
■ f(x) ≈a0
2+ a1cos(kx) + a2 cos(2kx)
+ b1sin(kx) + b2 sin(2kx)
Fourier Series (3)
ISIS Neutron Training Course 14 / 38
■ f(x) ≈a0
2+ a1cos(kx) + a2 cos(2kx) + a3 cos(3kx)
+ b1sin(kx) + b2 sin(2kx) + b3 sin(3kx)
Fourier Series (4)
ISIS Neutron Training Course 15 / 38
■ 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
ISIS Neutron Training Course 16 / 38
■ Taylor: f(x) =∞
∑
n =0
an(x−xo)n |x−xo|<R
◆ an =1
n!
dn f
dxn
∣
∣
∣
∣
xo
Taylor Versus Fourier Series
ISIS Neutron Training Course 16 / 38
■ 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
ISIS Neutron Training Course 17 / 38
eiθ = cos θ + i sin θ , where i2 = −1
Complex Fourier Series
ISIS Neutron Training Course 17 / 38
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
ISIS Neutron Training Course 17 / 38
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
ISIS Neutron Training Course 18 / 38
■ As λ→∞, so that k→0 and f(x) is non-periodic,
◆
∞∑
n =−∞cn einkx −→
∞∫
−∞
c(q) eiqx dq
Fourier Transform
ISIS Neutron Training Course 18 / 38
■ 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
ISIS Neutron Training Course 19 / 38
■ Even: f(x) = f(−x) ⇐⇒ F(q) = F(−q)
■ Odd: f(x) = − f(−x) ⇐⇒ F(q) = −F(−q)
Some Symmetry Properties
ISIS Neutron Training Course 19 / 38
■ 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
ISIS Neutron Training Course 20 / 38
f(x) = g(x) ⊗ h(x) =
∞∫
−∞
g(t) h(x−t) dt
Convolution
ISIS Neutron Training Course 20 / 38
f(x) = g(x) ⊗ h(x) =
∞∫
−∞
g(t) h(x−t) dt
Convolution Theorem
ISIS Neutron Training Course 21 / 38
f(x) = g(x) ⊗ h(x) ⇐⇒ F(q) =√
2π G(q)×H(q)
Convolution Theorem
ISIS Neutron Training Course 21 / 38
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
ISIS Neutron Training Course 22 / 38
∞∫
−∞
F(q) eiqx dq = f(x)
Auto-correlation Function
ISIS Neutron Training Course 22 / 38
∞∫
−∞
F(q) eiqx dq = f(x)
■
∞∫
−∞
∣
∣F(q)∣
∣
2eiqx dq =
∞∫
−∞
f(t)∗ f(x+t) dt = ACF(x)
◆ Patterson map
Auto-correlation Function (1)
ISIS Neutron Training Course 23 / 38
Auto-correlation Function (2)
ISIS Neutron Training Course 24 / 38
Fourier Optics
ISIS Neutron Training Course 25 / 38
I(q) =∣
∣ψ(q)∣
∣
2
■ Fraunhofer: ψ(q) = ψo
∞∫
−∞
A(x) eiqx dx where q =2π sin θ
λ
Young’s Double Slits
ISIS Neutron Training Course 26 / 38
Young’s Double Slits
ISIS Neutron Training Course 26 / 38
Young’s Double Slits
ISIS Neutron Training Course 26 / 38
Single Wide Slit
ISIS Neutron Training Course 27 / 38
Single Wide Slit
ISIS Neutron Training Course 27 / 38
Single Wide Slit
ISIS Neutron Training Course 27 / 38
Two Wide Slits (0)
ISIS Neutron Training Course 28 / 38
Two Wide Slits (1)
ISIS Neutron Training Course 29 / 38
Two Wide Slits (2)
ISIS Neutron Training Course 30 / 38
Two Wide Slits (3)
ISIS Neutron Training Course 31 / 38
Finite Grating (0)
ISIS Neutron Training Course 32 / 38
Finite Grating (1)
ISIS Neutron Training Course 33 / 38
Finite Grating (2)
ISIS Neutron Training Course 34 / 38
Finite Grating (3)
ISIS Neutron Training Course 35 / 38
Write up of this Talk!
ISIS Neutron Training Course 36 / 38
■ 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
ISIS Neutron Training Course 37 / 38
The phaseless Fourier problem
ISIS Neutron Training Course 38 / 38