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