• Fresnel integral ! Fraunhofer diffraction
• Fraunhofer diffraction as Fourier transform
• Convolution theorem:
solving difficult diffraction problems
(double slit of finite slit width, diffraction grating)
lecture 7
Fourier Methods
Fourier Methods
up = ! i
λ
!η(θi, θo)
us(x, y)r
eikrdS
Fresnel-Kirchhoff diffraction integral
Fraunhofer diffraction in 1D !simplifies to
! = k sin "with
Note: Us(!) 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 !
U(!) =! !
"!u(t)ei!tdt
u(t) =12"
! !
"!U(!)e"i!td!
Fourier transform
inverse transform
coordinate x and spatial frequency ":
U(β) =! !
"!u(x)ei!xdx
u(x) =12π
! !
"!U(β)e"i!xdβ
Fourier transform
inverse transform
(",t)!(!,x)
Fourier Methods
Extension to two dimensions
spatial frequencies
!x = k sin"
!y = k sin #
[!] = rad / m
up ! U(!x, !y) =!
us(x, y)ei(!xx+!yy)dxdy
Monochromatic
WaveT
Fourier Transforms
u(t)
u(t) = e!i!0t
ω0 = 2π/T
Fourier
Transform
U(!) =2" · #(! ! !0)
!!0
U(!)
!-function V
"
Fourier Transforms
u(x)
Re[U(!)]
Fourier transform
Power spectrum
|U(!)|2 = const.
U(!) = ei!x0
u(x) = !(x− x0)
Comb of #-functions
Diffraction Grating
u(x)
|U(!)|2
Fourier transform
Power spectrum
|U(!)|2 =!
sin(N!d/2)sin(!d/2)
"2
U(!) =!
n
ein!d
u(x) =!
n
!(x! nd)
Comb of #-functions
Diffraction Grating
u(x)
|U(!)|2
Plane
waves
! = k sin "
Fourier transform
Power spectrum
|U(!)|2 =!
sin(N!d/2)sin(!d/2)
"2
U(!) =!
n
ein!d
u(x) =!
n
!(x! nd)
Comb of #-functions
Diffraction Grating
u(x)
|U(!)|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 transform
Fourier synthesis and analysis
Fourier transforms
Convolution theorem:
Double slit of finite slit width, diffraction grating
Abbé theory of imaging
Resolution of microscopes
Optical image processing
Diffraction limited imaginglecture 8
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/uEyepiece 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.
#max defines the numerical aperture… and resolution
Limited Resolution
Fourier
plane
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
Summary of MT 2008
Geometrical optics
Fraunhofer and Fresnel diffraction
Fresnel-Kirchhoff diffraction integral
Fourier transform methods
Convolution theorem:
Double slit of finite slit width, diffraction grating
Abbé theory of imaging
Resolution of microscopes, image processing
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