Fraunhofer Diffraction

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Fraunhofer Diffraction

• Single Double Slit
• Rectangular Circular Apertures

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Huygens construction
Each point on the wavefront is the source of new
Huygen wavelets.
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Single Slit case in Fraunhofer limit.
Screen point so distant that all rays
are approximately parallel.
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The Hand waving version
Each Huygen wavelet contributes to the electric
field at the screen point.
ri cannot differ from r0 by more than the
distance w
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A
Each contribution to the sum is a phasor of the
same magnitude and frequency. Since the source
points for the wavelets are evenly spaced in y,
the angle between consecutive phasors is constant.
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The vibration curve it results from adding all
the phasors tip to tail
R
s
So the curve is a circle.
The angle ? is the angle between phasor 0 and
phasor 8
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R
s
Since
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Now via integration
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Double slit
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Rectangular Aperture
In 3-d we will use spherical Huygen wavelets
s
rA
Subscript A for aperture coordinate, S for screen
coordinate
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Neglect this part
How to justify that?
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ys
xs
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2a
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Recast
in polar coordinates
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?
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First root of J1(? ) is at ?3.832
gives location of first minimum
Solves to
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Rayleighs criterion A telescope is limited by
diffraction, each point of light is imaged as a
bulls eye pattern (Airy disk). If any two
point sources are close together so that their
1st minima overlap, they are said to be
unresolved.
D
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Rectangular Aperture
s
rA
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Arbitrary Aperture
Fourier Transform of Aperture Function
There could be variations of transparency, A does
not need to be constant. There could be
variations of thickness e.g. of emulsion in
hologram. A can be complex!
Computer generated holograms Pattern
recognition,.
Fourier Optics
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Fast Fourier Transform
Kinoform
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