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We know that a simple lens is a Fourier transform engine, which does complicated ‘Fourier calculations’ of any object faster than any computer available today! An object placed at a distance d from the lens forms its Fourier image at the back focal length (f) of the lens. As we know, Fourier image is always formed at the focal plane of a lens, irrespective of the position of the object; you can keep the object wherever you want in front of a lens (d<< f, or d>>f).
The Fourier plane, thus formed, is mainly employed to carry out a spatial filtering process of an image. Image quality can be extensively enhanced after filtering the desired components of spatial frequencies that form the original image. A 4f configuration is widely employed to achieve this type of filtering. During the process of spatial filtering, some part of the Fourier plane is blocked such that the image is only formed due to the unblocked spatial frequencies. This is a common method adopted to efficiently remove noise from an image. Another application of the Fourier plane filtering that is not well explored is to realize angle-resolved measurements. In this tutorial, we will see the basics on how to employ angle-resolved measurements using the Fourier filtering technique. Here, we make use of the fact that all the light rays coming from an image, at a certain angle with respect to the object plane, will always converge at a particular point on the Fourier plane. By selectively passing the light rays from that point on the Fourier plane, it is possible to record the information (such as the energy of the light) about the rays coming at a certain angle from the object. This technique can be used to experimentally determine the dispersion relation (energy band diagram) of an unknown chemical material (e.g., Graphene, Metal-Organic Frameworks, etc.).
But when you work with materials of micrometre size (e.g., a small crystal you can barely see!), you can’t use ordinary lenses to do such angle-resolved measurements because of its poor resolution. So instead of an ordinary lens, we adopt an objective lens for better resolution. An objective lens can drastically enhance the resolution of the filtered image because the numerical aperture (NA) is usually very large for the objective lenses. But usually, the Fourier plane of an objective lens lies inside its barrel; therefore, the Fourier plane is not accessible to do spatial filtering. However, it is possible to take this Fourier plane out of an objective and make it available for the filtering process. The following figure is the configuration we require to take the Fourier plane out of an objective.
The above figure is simulated using: https://ricktu288.github.io/ray-optics/. Here the object is at the focal length (f = 100 units) of the lens 1, aka an objective lens, (In fact, a distance less than f is preferable, because the image formed from the object is not our concern. For an objective lens, this distance is called working distance whose magnitude usually is less than its focal length). The Fourier plane forms at the back focal length (at 300 units) of the lens 1 and the corresponding image is formed at infinity. For an objective, this Fourier plane lies within the barrel so that it remains inaccessible. So, we introduce a new lens (lens 2, f = 100 units) at a length of 3f (at 500 units) from the lens 1. This forms an image (not needed for us) and a Fourier plane (now it is out of the objective lens) at f (at 600 units) and 2f (at 700 units) of the lens 2 respectively. Here, the Fourier transform plane is formed at 2f of lens 2 because the first Fourier transform plane (formed at 300 units on the scale) acts as an ‘object’ for lens 2. It is to be noted that the Fourier transform formed this way is inverted but neither magnified nor diminished.
Now, we have re-constructed the Fourier plane outside of an objective lens barrel. Such a Fourier plane is readily available for the filtering process!
This simple technique is also beneficial in angle-resolved reflection or photoluminescence measurements of micro-sized samples.