# Taking Fourier plane out of an objective.

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We know that a lens is a Fourier transform engine, which does complicated Fourier calculations of a complex object, faster than any computer available today! An object placed at a distance d from the lens will form 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 the lens, irrespective of the position of the object; it can be very close to the lens (d<< f) or very far away from the lens (d>>f).

The Fourier plane, thus formed, is used mainly to do a spatial filtering process. Image quality can be extensively higher after filtering only desired spatial frequencies. The 4f configuration is widely employed to achieve this. Spatial filtering process includes blocking some part of the Fourier plane such that the image is formed only from the unblocked spatial frequencies. This is an efficient method to remove noise from an image. Another application of Fourier plane filtering is to realize angle-resolved measurements. All the light rays coming at an angle with respect to the object plane will converge at a certain point on the Fourier plane. By blocking all other points on the Fourier plane, it is possible to filter out the information 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 sample.

But when you work with objects of micrometre size, you can’t use ordinary lenses as the resolution is usually very poor. Employing an objective drastically increases the resolution of the image as the numerical aperture (NA) can be very high for objectives. But practically, it is not possible to access the Fourier plane of an objective to do spatial filtering, because this lies inside the objective barrel. However, it is possible to take this Fourier plane out of an objective and make it accessible for the filtering process. The following figure is the configuration we require to take 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 (In fact, a distance less than f is preferable, because the formation of the image formation of the object is not our concern. For an objective this called distance is working distance whose magnitude usually is less than its focal length). Fourier plane forms at the back focal length of the lens 1 (objective) and the 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 from the lens 1. This forms the image (not needed for us) and the Fourier transform plane (now it is out of the objective lens) at f and 2f of the lens 2 respectively. This 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.

This simple technique is beneficial in angle-resolved reflection or photoluminescence measurements of micro-sized samples.

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