An optical correlator is an optical computer for comparing two signals by utilising the Fourier transforming properties of a lens. It is commonly used in

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

for target tracking and identification.

Introduction

The correlator has an input signal which is multiplied by some filter in the Fourier domain. An example filter is the matched filter which uses the cross correlation of the two signals. The cross correlation or correlation plane,

c(x,y)

of a 2D signal

i(x,y)

with

h(x,y)

is :

c(x,y)=i(x,y) \otimes h^(-x,-y)

This can be re-expressed in Fourier space as :

C(\xi,\eta)=I(\xi,\eta) H^(-\xi,-\eta)

where the capital letters denote the Fourier transform of what the lower case letter denotes. So the correlation can then be calculated by inverse Fourier transforming the result.

Implementation

According to

Fresnel Diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff's diffraction formula, Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near and far field, near fi ...

theory a convex lens of

focal length The focal length of an Optics, optical system is a measure of how strongly the system converges or diverges light; it is the Multiplicative inverse, inverse of the system's optical power. A positive focal length indicates that a system Converge ...

f

will produce the exact Fourier transform at a distance

f

behind the lens of an object placed

f

distance in front of the lens. So that complex amplitudes are multiplied, the light source must be coherent and is typically from a

laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...

. The input signal and filter are commonly written onto a spatial light modulator (SLM). A typical arrangement is the 4f correlator. The input signal is written to an SLM which is illuminated with a laser. This is Fourier transformed with a lens and this is then modulated with a second SLM containing the filter. The resultant is again Fourier transformed with a second lens and the correlation result is captured on a camera.

Filter design

Many filters have been designed to be used with an optical correlator. Some have been proposed to address hardware limitations, others were developed to optimize a merit function or to be invariant under a certain transformation.

Matched filter

The matched filter maximizes the signal-to-noise ratio and is simply obtained by using as a filter the Fourier transform of the reference signal

r(x,y)

. :

H(\xi,\eta) = R(\xi,\eta)

Phase-only filter

The phase-only filterJ. L. Horner and P. D. Gianino, ''Phase-only matched filtering'', Appl. Opt. 23, 1984, 812–816 is easier to implement due to limitation of many SLMs and has been shown to be more discriminant than the matched filter. :

H(\xi,\eta) = \frac

References

{{DEFAULTSORT:Optical Correlator Optical instruments Fourier analysis