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)

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

P. Piot, PHYS 630 – Fall 2008 Propagation through an harmonic element

• Consider a thin optical element with transmittance

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

• 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