Fourier Methods
Fraunhofer diffraction = Fourier transformConvolution theorem
easy solution to difficult diffraction problems(double slit of finite slit width, diffraction grating)
Fourier Methods
up = � i
⇤
��(⇥i, ⇥o)
us(x, y)r
eikrdS
Fresnel-Kirchhoff diffraction integral
Fraunhofer diffraction in 1D ➙simplifies to
� = k sin ⇥with
Note: Us(β) is the Fourier Transform of us(x)The Fraunhofer diffraction pattern is the Fourier transform
of the amplitude function leaving the diffracting aperture
up � Us(�) =�
us(x)ei�xdx
us(x)
Fourier Transform
time t and angular frequency ω
U(⇥) =� ⇥
�⇥u(t)ei�tdt
u(t) =12�
� ⇥
�⇥U(⇥)e�i�td⇥
Fourier transform
inverse transform
coordinate x and spatial frequency β:
U(�) =� ⇥
�⇥u(x)ei�xdx
u(x) =12⇥
� ⇥
�⇥U(�)e�i�xd�
Fourier transform
inverse transform
(ω,t)→(β,x)
Fourier Methods
Extension to two dimensions
spatial frequencies
�x = k sin⇤
�y = k sin ⇥
[β] = rad / m
up � U(�x, �y) =�
us(x, y)ei(�xx+�yy)dxdy
MonochromaticWave
T
Fourier Transforms
u(t)
u(t) = e�i�0t
⇥0 = 2�/T
FourierTransform
U(⇤) =2⇥ · �(⇤ � ⇤0)
��0
U(�)
δ-function V
β
Fourier Transforms
u(x)
Re[U(β)]
Fourier transform
Power spectrum
|U(�)|2 = const.
U(�) = ei�x0
u(x) = �(x� x0)
Comb of δ-functions
Diffraction Grating
u(x)
|U(β)|2Fourier transform
Power spectrum
|U(�)|2 =�
sin(N�d/2)sin(�d/2)
⇥2
U(�) =�
n
ein�d
u(x) =�
n
�(x� nd)
Comb of δ-functions
Diffraction Grating
u(x)
|U(β)|2
Plane waves
x’
� = k sin ⇥ � k x�/f
Fourier transform
Power spectrum
|U(�)|2 =�
sin(N�d/2)sin(�d/2)
⇥2
U(�) =�
n
ein�d
u(x) =�
n
�(x� nd)
Fraunhofer diffraction as Fourier transformFourier synthesis and analysisFourier transformsConvolution theorem:
Double slit of finite slit width, diffraction grating
Abbé theory of imagingResolution of microscopesOptical image processingDiffraction limited imaging
Fourier Methods
TF (f) =�
f(x)ei�xdx
Convolution Methods
h(x) = f(x)⇥ g(x) :=� ⇤
�⇤f(x⇥)g(x� x⇥)dx⇥
Convolution function
Convolution theorem TF (f ⇥ g) = TF (f) · TF (g)
TF (f · g) = TF (f)⇥ TF (g)
Fourier transform of the convolution h(x)=f(x)⊗g(x) is the product of the individual Fourier transforms (and vice versa)
g(x-x’ )f(x)
h(x)
Double Slit by Convolution
g(x-x’ )f(x)
h(x)
Double Slit by Convolution
f(x)
h(x)
g(x-x’ )
Convolution of Top-Hats →Triangle
f(x)
h(x)
g(x-x’ )
This is a self-convolution or Autocorrelation function
Convolution of Top-Hats →Triangle
Abbé theory of imaging
• spatial frequencies (image period d)
u(x) � u0 + u1 cos(2�
dx)
⇥S :=2�
d
• Fraunhofer diffraction
U(�) = 0 except for � = 0,±⇥S
diffraction angles⇥ =
⇤
2⌅� = 0,±⇤
d
Fourier Planes
Abbé theory of imaging
Objective magnification = v/u Eyepiece magnifies real image of object
The Compound Microscope
Abbé theory of imaging
Diffracted orders from high spatial frequencies miss the lens
High spatial frequencies are missing from the image.
θmax defines the numerical aperture… and resolution
Limited Resolution
Fourierplane
Image plane
Optical Image Processing
a b
a’ b’
(a) and (b) show objects: double helix
at different angle of view
Diffraction patterns of (a) and (b) observed in
Fourier plane
Computer performs Inverse Fourier transform
To find object “shape”
Simulation of X-Ray Diffraction
Resolution Limit → Image Brightness
Brightness: Energy falling on unit area of the image in unit time
D
��
D
f
⇥2
� D2
non-diffraction limited
Resolution Limit → Image Brightness
Brightness: Energy falling on unit area of the image in unit time
��
D
f
⇥2
� D2
non-diffraction limited
masking of source → image diffraction-limited
� D4diffraction-limited brightness
amplitude object
Spatial Filtering
measuring the local phase
phase object
Spatial Filtering
measuring the local phase
FourierPlane
Spatial Filtering: Schlieren Photography
phase → amplitude modulation
Summary of Lecture 9
Division of wavefront
Spatial frequencies and spatial filtering
Resolution limit
Image brightness
⇥S
k=
�
d< �max =
D
2x
d
x= ⇥ > 1.22
�
D
in the diffraction limit(source @ u=∞; image @ v=f)
dimg = 1.22�
Df
�
��
D2 resolvedD4 di�r. limited
φmin
dimg
point source
Interference → Division of Amplitude
Divide and (re)combine amplitudesMach-Zehnder interferometerLocalisation of fringes
Extended light sourcesMichelson interferometerFabry-Perot interferometer
∆φ
Mach-Zehnder Interferometer
u0 · eikx u1 · eikx
u2 · eiky
u2 · eikL
u0 · cos��
u0 · sin��
u1 · eikL · ei��
Interference fringes as a function of Δϕ in the output of the 2nd BS
... the ideal world
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