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# Explanation of 4f optical systems

A 4f correlator optical system is a system architecture that uses  Fourier Optics.  In this article, we will explain why we need them and how to design a simple 4f correlator. We will leave out most of the mathematical derivations, but if you are interested in digging more in depth, we highly recommend “Introduction to Fourier Optics, by J. Goodman”

Imagine a scene in an action movie in which intelligence intercepts a radio communication from a hostile cell.  As you can imagine, the transmission is full of interference, noise, and it’s hard to understand the actual message.  This is where the tech-savvy character presses a bunch of buttons and twists some knobs while saying something like: “we need to clean the signal and filter out the voices…” and voila, a second later, the message is clear of noise and we can understand whatever was said.

Well, the process of “cleaning” up a signal using filters is quite common in signal processing.  Now here’s the catch: most of this filtering is done in the frequency domain.  So we first need to convert our time-based signal into the frequency domain, filter out the unwanted components, and convert back to the time domain so we can hear it.  That transformation between time-frequency is called Fourier Transformation, and as we apply it to electrical or voice signals, we can also apply it to images, but instead of time-frequency, the transformation is a space-frequency one.

Before we continue this discussion, let’s take a look at a typical 4f correlator system Figure 1. Basic structure of a 4f optical system.

The 4f correlator is in essence an optical relay that usually consists of two lenses.  The input plane is one focal length in front of Lens 1 while the output plane is located one focal length  after Lens 2.  In between the two lenses, we have the Fourier plane.  Here is where we have the Fourier transformation of the object placed at the output plane.  The magnification is found to be equal to −f2∕f1. A 1:1 relay (−1 magnification) and can be achieved if the two lenses have the same focal length.

At the Fourier plane, we can place masks of different shapes and opacities that can filter-out unwanted components from our original image.  The Fourier transformation of an image is very similar to a diffraction pattern, where low frequency components are located close to the optical axis and higher frequency ones are placed farther away from the origin. The shape of the mask varies depending on your application.  If you want to remove high frequency components, the mask can be as simple as a spherical aperture on a back plate.  Only low components close to the optical axis will pass, in essence removing the high components.

One interesting application of a 4f correlator is target identification.  Let’s say that you want to identify a target  in a non-controlled environment.  The target can be as diverse as an enemy tank, a bacterium, or a security mark on a credit card. Your mask will be the Fourier transformation of that target.  You can then create that mask using a lithographic method or a mathematical matrix.  On your input plane, you can have your real-life scenes. It can be a flow of liquids where you want to detect your bacteria,  the satellite image on enemy territory, or a user scanning an access card.   By passing your real-life system through a 4f system, the Fourier transformation of your image will interact with your target-mask at the Fourier plane and create a specific response where there is a match.

We most recently used a 4F system in a custom eye aberrometer.  An eye aberrometer measures the wavefront of light as it goes through the eye. In an ideal eye, the wavefront would be completely flat. An aberrometer uses the changes on the wavefront to calculate Zermike polynomials and a topographic image of the distorted wavefront.  This information can then be used to create custom corrected lenses, or corrective surgery (e.g LASIK). Our basic design is shown in Fig 2 Figure 2. Two aspherical lenses are placed so that their back focal points coincide. The first object placed  at infinity is imaged into an infinite image through the lenses (blue rays). The second finite object is placed at the forward focal point of the first lens. The image is conjugated with the back focal point of the second lens (green rays)