# wk9a - Overview from last week Optical systems act as...

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Overview from last week Optical systems act as linear shift-invariant (LSI) filters (we have not yet seen why) Analysis tool for LSI filters: Fourier transform – decompose arbitrary 2D functions into superpositions of 2D sinusoids (Fourier transform) – use the transfer function to determine what happens to each 2D sinusoid as it is transmitted through the system (filtering) – recompose the filtered 2D sinusoids to determine the output 2D function (Fourier integral, aka inverse Fourier transform) MIT 2.71/2.710 Optics 11/01/04 wk9-a-1

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Today Wave description of optical systems Diffraction very short distances: near field, we skip – intermediate distances: Fresnel diffraction expressed as a convolution – long distances ( ): Fraunhofer diffraction expressed as a Fourier transform MIT 2.71/2.710 Optics 11/01/04 wk9-a-2
Space and spatial frequency representations g ( x,y) SPACE DOMAIN G ( u,v) () () ( ) y x y x g v u G vy ux i d d e , , 2 + = π 2D 2D Fourier transform Fourier transform 2D 2D Fourier integral Fourier integral aka inverse inverse 2D 2D Fourier transform Fourier transform SPATIAL FREQUENCY DOMAIN ( ) ( ) ( ) v u v u G y x g vy ux i d d e , , 2 + + = MIT 2.71/2.710 Optics 11/01/04 wk9-a-3

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2D 2D linear shift invariant systems input output convolution with convolution with impulse response impulse response g i ( x,y ) g o ( x , y ) () ( ) ( ) y x y y x x h y x g y x g d d , , , i o = inverse Fourier transform Fourier LSI LSI multiplication with multiplication with transfer function transfer function G i ( u,v ) G o ( u,v) ( ) ( )( ) v u H v u G v u G , , , i o = MIT 2.71/2.710 Optics 11/01/04 wk9-a-4
Wave description of optical imaging systems MIT 2.71/2.710 Optics 11/01/04 wk9-a-5

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Thin transparencies coherent illumination: plane wave Transmission function: ( ) { } , exp ) , ( ) , ( in y x i y x t y x g φ = >~ λ Field before transparency: = λ π z i z y x a 2 exp ) , , ( =0 Field after transparency: = + z i y x g z y x a 2 exp ) , ( ) , , ( in =0 assumptions: transparency at z=0 transparency thickness can be ignored <~50 λ MIT 2.71/2.710 Optics 11/01/04 wk9-a-6
Diffraction: Huygens principle MIT 2.71/2.710 Optics 11/01/04 wk9-a-7 ) , ( ) , , ( in y x g z y x a = + Field after transparency: Field at distance d : contains contributions from all spherical waves emitted at the transparency, summed coherently incident plane wave d

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Huygens principle: one point source incoming plane wave opaque screen spherical wave x=x 0 l MIT 2.71/2.710 Optics 11/01/04 wk9-a-8
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wk9a - Overview from last week Optical systems act as...

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