Fourier relations in Optics Near field Frequency Pulse

Fourier relations in Optics Near field Frequency Pulse duration Frequency Coherence length Beam waist Beam divergence Spatial dimension Angular dimension Focal plane of lens The other focal plane

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 time Angle or frequency

Position or time Angle or frequency

Application of Fourier relation: a Single- slit diffraction

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

Frequency, time, or position

N w 0 Frequency Dw Time

N w 0 Frequency Time Dw Mode-locking

N x 0 Position Dx Angle Diffraction grating, radio antenna array

The applications of the Fourier relation: (8) 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

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 E z=0

To generalize: “Grating momentum”

Stationary gratings vs. Moving gratings Deflection + Frequency shift

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

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

Express F(x, z) in =x/z

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)

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

A simple example of recording and reconstruction: k 1 k 2

A simple example of recording and reconstruction: k 1 k 2

k 1 k 2 /2 ?

Another example: Volume hologram k 1 k 2

Volume grating

k 1

k 1 k 2 k 1

D A d C B Bragg condition

D A d C B

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

Reading

E(x, y) Er Recorded pattern

Recorded pattern Diffracted beam when illuminated by ER