$$\newcommand{\FT}[1] {\mathcal{F}\left(#1\right)}

\newcommand{\iFT}[1] {\mathcal{F}^{-1}\left(#1\right)}$$

# Optical Correlators

The optical correlator is perhaps the most common use of optical

processing. It has many uses in signal identification and tracking – although

it has generated many more academic papers than real world applications due

to some practical problems that will be discussed later.

The correlator is related to the convolution

operator and the correlation of two signals $latex f$ and $latex g$ is often

written

$$c(x) = f(x) \odot g(x) = \int_{-\infty}^{\infty}

f(\xi)g(\xi-x) d \xi$$

where $latex \xi$ is a dummy variable.

When the signal is complex the complex conjugate of $latex g$ is used.

$$c(x) = f(x) \odot g^{\ast}(x) =\int_{-\infty}^{\infty} f(\xi)g^{\ast}(\xi-x) d \xi$$

This can be expressed in terms of convolution by

$$f(x) \odot g^{\ast}(x) = f(x) \otimes g^{\ast}(-x)$$

This can be re-expressed using Fourier transforms (and in 2D)

$$c(x,y) = \iFT{F(u,v)G^{\ast}(-u,-v)}$$

The correlation gives us a measure of the similarity of the two signals.

If they match or are similar there will be a peak in $latex c(x,y)$. If the two

signal are identical we have the auto-correlation of the signal.

## The 4-f correlator

The 4-*f* correlator is an optical device that implements the above

equation. It makes use of the fact that an exact Fourier transform of an

object is performed by a lens of focal length *f*, in the plane a

distance *f* behind the lens if the object is placed a distance

*f* in front and illuminated by coherent light.

If we look at the diagram above we have some input image illuminated by a

coherent source such as a laser (not shown), this is then Fourier transformed

by the first lens. \(G\) represents a filter, in practice this could be

a piece of film or a liquid crystal spatial light modulator. If the input

signal is \(f(x,y)\), and the signal placed on the filter is \(G^{\ast}(-u,-v)\),

the light field just after the filter is \(F(u,v)G^{\ast}(-u,-v)\). This is again

Fourier transformed by the second lens and we almost end up with the above

correlation equation. The only difference is that the lens can only perform a

forward Fourier transform, and not an inverse. This results in a constant

term, which is normally neglected, and the result has negative

co-ordinates.

Of course, this 4-f correlator uses a matched filter, in reality it could

be another type of filter.

The physical realisation of the filter is usually a problem. We required

full complex (amplitude and phase) modulation to fully implement it. Normally

this can not be done. The filter is limited to either amplitude or phase

modulation, not both together, and the filter attenuates the light, i.e., the

transmission goes from 1 to 0. It’s not possible to have this greater than

one.

### Spatial Invariance

If we consider the case when the input signal is shifted by some amount

\(\alpha\), is it is now \((x-\alpha)\). Inserting this into the Fourier

correlator we have

$$F(u)e^{-i 2 \pi \alpha u}G^{\ast}(-u)$$

we note that

$$C(u)=F(u)G^{\ast}(-u)$$

So our shifted function must equal \(C(u)e^{-i 2 \pi \alpha u}$ therefore $c(x-\alpha)

= f(x-\alpha) \odot g^{\ast}(x)\). The correlation moves as the input signal moves.

This is one of the most powerful features of this type of correlator. Not

only can the signal be identified, but its position can be determined as well

in one single operation.

The optical correlator is useful in the fact that is can provide target

identification and tracking. In practice something better than a matched

filter would be used. The main problems with a matched filter are the peak it

produced is too broad, spreading out the power; and it tends to be a very

indiscriminate filter. Its lack of ability to discriminate means it would

struggle to find the difference between the letter O and the letter C. More

about this here.