An optical correlator is an

optical computer Optical computing or photonic computing uses light waves produced by lasers or incoherent sources for data processing, data storage or data communication for computing. For decades, photons have shown promise to enable a higher bandwidth than the ...

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 instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...

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 In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal wi ...

which uses the

cross correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used fo ...

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–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern ...

theory a

convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simp ...

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

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 In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...

are multiplied, the light source must be

coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...

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" is an acronym for "light amplification by stimulated emission of radiation". The fir ...

. The input signal and filter are commonly written onto a

spatial light modulator A spatial light modulator (SLM) is an object that imposes some form of spatially varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the term SLM is used, it means that the transparency can ...

(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