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

Huygens’ Principle E(r) E(R)

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)

Useful Fourier relations in optics between t and, and between x and .

Useful Fourier relations in optics between t and, and between x and .

Position or timeAngle or frequency

Position or time Angle or frequency

Single- slit diffractionApplication of Fourier relation: a

-Spatial harmonics and angles of propagation The applications of the Fourier relation:

?

Frequency, time, or position

N   Time Frequency

N   Time Frequency Mode-locking

N xx x0x0 Angle Position Diffraction grating, radio antenna array

(8) The applications of the Fourier relation: Finite number of elements

-Graded grating for focusing -Fresnel lens

Fourier transform between two focal planes of a lens

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.

Thus by decomposing a spatial distribution of electric field into spatial harmonics, each component can be treated separately.

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

Source z=0 EE

To generalize: “Grating momentum”

Stationary gratings vs. Moving gratings Deflection Deflection + Frequency shift

The small angle approximation (1/ << ) for the H function A correction factor for the transfer function for the plane waves  =

D F(x) H( x )F(x) z=0

Express F(x,z) in  =x/z

The effect of lenses A lens is to introduce a quadratic phase shift to the wavefront given by.

Fourier transform using a lens

Huygens’ Principle E(r) E(R)

: Recording of full information of an optical image, including the amplitude and phase. Holography Amplitude only: Amplitude and phase

k1k1 k2k2  A simple example of recording and reconstruction:

k1k1 k2k2 

 /2 ? k2k2 k1k1

k1k1 k2k2  Another example: Volume hologram

Volume grating

k1k1

k1k1 k1k1 k2k2

 d D C B A Bragg condition

 d D C B A

Another example: Image reconstruction of a point illuminated by a plane wave. Writing

Reading

ErEr E(x,y) Recorded pattern

Diffracted beam when illuminated by E R