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Fourier relations in Optics Near fieldFar field FrequencyPulse duration FrequencyCoherence length Beam waist Beam divergence Focal plane of lensThe other focal plane Spatial dimensionAngular dimension
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Huygens’ Principle E(r) E(R)
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Fourier theorem A complex function f(t) may be decomposed as a superposition integral of harmonic function of all frequencies and complex amplitude (inverse Fourier transform) The component with frequency has a complex amplitude F( ), given by (Fourier transform)
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Useful Fourier relations in optics between t and, and between x and .
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Useful Fourier relations in optics between t and, and between x and .
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Position or timeAngle or frequency
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Position or time Angle or frequency
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Single- slit diffractionApplication of Fourier relation: a
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-Spatial harmonics and angles of propagation The applications of the Fourier relation:
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?
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Frequency, time, or position
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N Time Frequency
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N Time Frequency Mode-locking
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N xx x0x0 Angle Position Diffraction grating, radio antenna array
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(8) The applications of the Fourier relation: Finite number of elements
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-Graded grating for focusing -Fresnel lens
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Fourier transform between two focal planes of a lens
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The use of spatial harmonics for analyses of arbitrary field pattern Consider a two-dimensional complex electric field at z=0 given by where the ’s are the spatial frequencies in the x and y directions. The spatial frequencies are the inverse of the periods.
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Thus by decomposing a spatial distribution of electric field into spatial harmonics, each component can be treated separately.
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Define a transfer function (multiplication factor) in free space for the spatial harmonics of spatial frequency x and y to travel from z=0 to z=d as
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Source z=0 EE
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To generalize: “Grating momentum”
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Stationary gratings vs. Moving gratings Deflection Deflection + Frequency shift
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The small angle approximation (1/ << ) for the H function A correction factor for the transfer function for the plane waves =
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D F(x) H( x )F(x) z=0
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Express F(x,z) in =x/z
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The effect of lenses A lens is to introduce a quadratic phase shift to the wavefront given by.
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Fourier transform using a lens
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Huygens’ Principle E(r) E(R)
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: Recording of full information of an optical image, including the amplitude and phase. Holography Amplitude only: Amplitude and phase
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k1k1 k2k2 A simple example of recording and reconstruction:
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k1k1 k2k2
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/2 ? k2k2 k1k1
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k1k1 k2k2 Another example: Volume hologram
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Volume grating
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k1k1
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k1k1 k1k1 k2k2
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d D C B A Bragg condition
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d D C B A
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Another example: Image reconstruction of a point illuminated by a plane wave. Writing
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Reading
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ErEr E(x,y) Recorded pattern
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Diffracted beam when illuminated by E R
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