Fresnel Kirchhoff

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1988 ON THE BOUNDARIES OF APPLICABILITY OF KIRCHHOFF AND FRESNEL APPROXIMATION IN THE INVERSE PROBLEMS OF PLANE OPTICS SYNTHESIS. Kirchhoff's diffraction formula (also Fresnel–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. Fresnel-kirchhoff Diffraction Theory
  2. Fresnel Kirchhoff Formula
  3. Fresnel Kirchhoff Beugungstheorie
  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.
  2. Introduction to Fourier optics Joseph W. Goodman download Z-Library. Download books for free.
  3. . Fresnel integral! Fraunhofer diffraction. Fraunhofer diffraction as Fourier transform. Convolution theorem: solving difficult diffraction problems (double slit.

Fresnel-kirchhoff Diffraction Theory

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Text of Fourier Methods - University of .Fraunhofer diffraction as Fourier ... Diffraction limited imaging

Fresnel Kirchhoff Formula

Integral

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

    [!] = rad / m

    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