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

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 \( f\) and \( g\) is often
written

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

where \( \xi\) is a dummy variable.

When the signal is complex, the complex conjugate of \( g\) 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 \( c(x,y)\). 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.