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  • Diffraction

    Fraunhofer and Fresnel Diffraction

    Reports due this week

  • Two light sources have a frequency difference of 4000 GHz. Can

    we resolve these two light sources?

    A sodium atom has a doublet emission with a frequency difference

    of 0.1 GHz. Can we resolve the fine structure?

    A Fabry-Perot wave meter. Free spectral range: 3000 GHz.

    Finesse 100,000.

  • Hecht, Pg 312, Fig. 7.34

    The square pulse and its transform

    Fourier Integrals and Fourier Transforms

    0

    sin( / 2)( )

    / 2

    kLA k E L

    kL

    The sinc function.

    Inverse relationship between widths

    FT in single-slit Fraunhofer diffraction

    0 0

    1( ) ( )cos( ) ( )sin( )f x A k kx dk B k kx dk

    ( ) ( )cos( )A k f x kx dx

    ( ) ( )sin( )B k f x kx dx

  • aperture

    incident wave

    diffracted wave

    Diffraction

    Huygens-Fresnel Principle:Every unobstructed point of a wavefront acts

    as a source of spherical secondary wavelets.

    The field at any point is the superposition of all

    of the secondary wavelets.

    A

    B

    P Manifestation of the wave

    nature of light light bends

    around corners.

    When l ~ smallest dim

    max AP BP AB

    ABl Wave spreads in large angle

    ABl Multitude of wavelets

    emitted from the aperture

    interfere

  • Analytical Approach

    (of aperture)

    ikr ikr

    o o

    S S

    e eE E S E dS

    r r

    Individual Huygens wavelet generated from surface S

    Superposition of Huygens wavelets at r.

    Time dependence is same for all (same frequency)

    Observer

    (Detector)

    Source

    Co-ordinate

    System

    Eo

  • Analytical Approach - II

    'z z d

    The distance between source

    plane and detector plane.

  • Criterion for Fraunhofer

    Diffraction

    Can ignore quadratic terms if

    2 2

    12

    x yk

    d

    2wd

    l

    w is largest dim

    of aperture

    We assume the incoming wave is homogeneous

    Fraunhofer Diffraction: Far field diffraction

  • 1-D Fraunhofer Diffraction Single Slit

    Huygens Recipe

    Fraunhofer limit

    In general, related to

    Fourier Transform

    of aperture function

    SINC sq.

    function

    Hecht page 452-469

  • Can determine width w

    from minima location

    Single Slit Diffraction Pattern

    Smaller w, stronger diffraction

  • Fraunhofer Diffraction Rectangular Aperture

    ( ' ')ik xx yy

    o dE

    E e dsd

    Hecht page 452-469

  • Fraunhofer Diffraction Rectangular Aperture

    2

    1, ( 1/ 2)

    (0)

    mi

    i

    In

    I

    Hecht page 452-469

  • Fresnel Diffraction

    Occurs when w2/dl >> 1

    Near Field Effect

    Wavefront curvature

    is important.

  • Fresnel Diffraction

    rs rd

  • Fresnel Diffraction

    Occurs when w2/dl >> 1

    Near Field Effect

    Wavefront curvature

    is important.

    (keep quadratic terms)

    rsrd

  • Fresnel Diffraction

    Occurs when w2/dl >> 1

    Near Field Effect

    Wavefront curvature

    is important.

    Either source or

    Detector in near field

  • Fresnel Integrals

    C

    S

    I=|P2P1|2

  • The Cornu Spiral

    C

    S

    I=|P2P1|2

    Once width is fixed,

    the length of the

    curve is fixed

  • Fresnel Diffraction

    All the equations are derived for X = 0.

    P

    -w/2

    w/2

    u1

    u2P

    -w/2+d

    w/2+d

    u1

    u2

    P

  • The Cornu Spiral

  • Fraunhofer

    Diffraction

    on Cornu

    Spiral

    P

    -0.5

    0.5

    u1

    u2

    u=0

  • Fraunhofer

    Diffraction

    on Cornu

    Spiral

    P

    -0.5

    0.5

    u1 =0

    u2 = 1

    u=-0.5

  • Fraunhofer

    Diffraction

    on Cornu

    Spiral

    P

    -0.5

    0.5

    u1 =0.5

    u2 =1.5

    u=-1

  • Detector is at the u=0.

    Start with slit width =1. We continuously increase the slit width.

  • Detector is at the u=0.

    Start with slit width =1. We continuously increase the slit width.

  • Detector is at the u=0.

    Start with normalized slit width =1. We continuously increase the slit

    width.

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