P. Piot, PHYS 630 – Fall 2008Fourier Optics• Provides a description of the propagation of light based on anharmonic analysis• It is in essence a signal processing description of light propagation• Generally Fourier transform is of the form (time-frequency)• Or (space-spatial frequency)P. Piot, PHYS 630 – Fall 2008Propagation of light through a system• We now deal with spatial frequency Fourier transforms and introducethe two-dimensional Fourier transform [associated to (x,y) plane]• The Fourier Optics allows easy description of a linear systemFourier transform! G("x,"y)spaceFourierP. Piot, PHYS 630 – Fall 2008• Consider the plane wave• This is a wave propagating with angle• At z=0• sincePlane waveSpatial harmonicwith freq. νx, νyWave period in x,y directionsxyzkθxθyP. Piot, PHYS 630 – Fall 2008• In the paraxial approximation we have• The inclinations of the wave vector is directly proportional to thespatial frequencies• There is a one-to-one between and the harmonic• If is given then• If is given thenPlane waveP. Piot, PHYS 630 – Fall 2008Propagation through an harmonic element• Consider a thin optical element with transmittance• The wave is modulated by anharmonic and thus• So the incident wave can be convertedinto a wave propagating with angleλλP. Piot, PHYS 630 – Fall 2008Propagation through a thin element• Consider a thin optical element with transmittance• Then• Plane waves are going to bedispersed along the directiondefine by the spatial frequencycontents of the elementλλP. Piot, PHYS 630 – Fall 2008amplitude modulation• Consider a thin optical element with transmittance• 1st term deflects wave at angles• Fourier transform of f(x,y)• So system deflects incoming wave atP. Piot, PHYS 630 – Fall 2008phase modulation• Consider a thin optical element with transmittance• Can Taylor-expand the argument around• So f(x,y) proportional to• This element introduced a position dependent deflectionSlowly varying functionof x and yP. Piot, PHYS 630 – Fall 2008Transfer function of the free space• An interesting properties is the determinant of the covariance matrix:• So the transfer functionis given bywherez0dP. Piot, PHYS 630 – Fall 2008Transfer function of the free space II• Transfer function modulus is unity for λνρ<1 bandwidth of propagation in freespace is 1/λ• The module decreases for larger νρ (this correspond to evanescent waves --waves that do not propagate)modulusargumentλνρλνρλ/(2d2)! "#$"x2+"y2P. Piot, PHYS 630 – Fall 2008Fresnel approximation• Assume then• So the transfer function takes the form• This is valid if• Introducing the conditionbecomesThis is the Fresnel approximationMaximum apertureP. Piot, PHYS 630 – Fall 2008• Given the transfer function, the impulsional response can becomputed• So each point in the input plane generates a paraboloidal wave andwe also have• Which is consistent with Huygens-Fresnel principleImpulsional
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