Fourier Optics • Provides a description of the propagation of light based on an harmonic 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 2008 Propagation of light through a system • We now deal with spatial frequency Fourier transforms and introduce the two-dimensional Fourier transform [associated to (x,y) plane] Fourier transform • The Fourier Optics allows easy description of a linear system space Fourier G(" x," y ) P. Piot, PHYS 630 – Fall 2008 ! Plane wave • Consider the plane wave y • This is a wave propagating with angle k z • At z=0 θy x θx • since Spatial harmonic with freq. νx, νy Wave period in x,y directions P. Piot, PHYS 630 – Fall 2008 Plane wave • In the paraxial approximation we have • The inclinations of the wave vector is directly proportional to the spatial frequencies • There is a one-to-one between and the harmonic • If is given then • If is given then P. Piot, PHYS 630 – Fall 2008 Propagation through an harmonic element • Consider a thin optical element with transmittance λ λ • The wave is modulated by an harmonic and thus • So the incident wave can be converted into a wave propagating with angle P. Piot, PHYS 630 – Fall 2008 Propagation through a thin element • Consider a thin optical element with transmittance • Then λ • Plane waves are going to be λ dispersed along the direction define by the spatial frequency contents of the element P. Piot, PHYS 630 – Fall 2008 amplitude 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 at P. Piot, PHYS 630 – Fall 2008 phase modulation • Consider a thin optical element with transmittance Slowly varying function of x and y • Can Taylor-expand the argument around • So f(x,y) proportional to • This element introduced a position dependent deflection P. Piot, PHYS 630 – Fall 2008 Transfer function of the free space • An interesting properties is the determinant of the covariance matrix: 0 d z • So the transfer function is given by where P. Piot, PHYS 630 – Fall 2008 Transfer function of the free space II modulus argument λ/(2d2) λνρ λνρ • Transfer function modulus is unity for λνρ<1 bandwidth of propagation in free space is 1/λ • The module decreases for larger νρ (this correspond to evanescent waves -- waves that do not propagate) 2 2 " # $ " x + " y P. Piot, PHYS 630 – Fall 2008 ! Fresnel approximation • Assume then • So the transfer function takes the form This is the Fresnel • This is valid if approximation Maximum aperture • Introducing the condition becomes P. Piot, PHYS 630 – Fall 2008 Impulsional response • Given the transfer function, the impulsional response can be computed • So each point in the input plane generates a paraboloidal wave and we also have • Which is consistent with Huygens-Fresnel principle P. Piot, PHYS 630 – Fall 2008.