• 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