# Fresnel Kirchhoff

1988 ON THE BOUNDARIES OF APPLICABILITY OF KIRCHHOFF AND FRESNEL APPROXIMATION IN THE INVERSE PROBLEMS OF PLANE OPTICS SYNTHESIS. Kirchhoff's diffraction formula (also Fresnelâ€“Kirchhoff diffraction formula) can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave.

1. We had previously seen Mechanics and Theory of Relativity, and Electricity and Magnetism by A. In this post we will see another book, Optics by this great author. From the preface The.
3. . Fresnel integral! Fraunhofer diffraction. Fraunhofer diffraction as Fourier transform. Convolution theorem: solving difficult diffraction problems (double slit.

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## Fresnel Kirchhoff Beugungstheorie

• 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

(i, o)

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)eixdx

us(x)

• Fourier Transform

time t and angular frequency !

U() =

u(t)eitdt

u(t) =12

U()eitd

Fourier transform

inverse transform

coordinate x and spatial frequency ':

U() =

u(x)eixdx

u(x) =12

U()eixd

Fourier transform

inverse transform

(',t)!(!,x)

Fourier Methods

Extension to two dimensions

spatial frequencies

x = k siny = k sin

up U(x, y) =

us(x, y)ei(xx+yy)dxdy

• Monochromatic

WaveT

Fourier Transforms

u(t)

u(t) = ei0t

0 = 2/T

Fourier

Transform

U() =2 ( 0)

0

U()

!-function V

'

Fourier Transforms

u(x)

Re[U(!)]

Fourier transform

Power spectrum

U() 2 = const.

U() = eix0

u(x) = (x x0)

• Comb of #-functions

Diffraction Grating

u(x)

U(!) 2

Fourier transform

Power spectrum

U() 2 =(

sin(Nd/2)sin(d/2)

)2

U() =

n

eind

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(Nd/2)sin(d/2)

)2

U() =

n

eind

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(Nd/2)sin(d/2)

)2

U() =

n

eind

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

Abb theory of imaging

Resolution of microscopes

Optical image processing

Diffraction limited imaginglecture 8

Fourier Methods

TF (f) =

f(x)eixdx

• 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

Abb theory of imaging

spatial frequencies (image period d)

u(x) u0 + u1 cos(2d

x)S :=

2d

Fraunhofer diffraction

U() = 0 except for = 0,S

diffraction angles =

2 = 0,

d

• Fourier Planes

Abb theory of imaging

Objective magnification = v/uEyepiece magnifies real image of object

The Compound Microscope

Abb 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

Abb theory of imaging

Resolution of microscopes, image processing