# Diffraction from a set of concentric rings with random phase

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• Gezstarski
In summary, a Diffractive Optical Element (DOE) made up of concentric rings with random phase shifts acts as a diffraction limited lens with low efficiency. The resulting Point Source Response Function (PSF) has a core similar to an Airy pattern but with reduced amplitude and stronger side lobes. The majority of the flux is scattered at wide angles, but there is a central spot that is significantly brighter than the background.
Gezstarski
TL;DR Summary
Looking for a treatment of the focussing properties of a set of concentric rings with random phase
I have been considering the properties of a Diffractive Optical Element (DOE) consisting of a very large number of concentric rings of equal (small) width, where the thicknessses of the rings are such as to produce random phase shifts in the range 0 to 2pi. I think I understand the behaviour of such a DOE - it acts as a diffraction limited lens that is achromatic but of exceedingly low efficiency. I am sure that this problem must have been tackled before but I have not been able to find a reference or a textbook treatment. Can anyone help?

Each ring emits radiation of random phase. So there is no preferred direction of radiation. It is behaving like a white surface. It looks as if the incoming energy is re-radiated isotropically.

Its not as simple as that. Each ring individually would produce a diffraction pattern with a wave amplitude that is a J_0 Bessel function, with a peak on the axis and lower elsewhere. At any point in the image plane the J_0 contributions from all the rings combine with random phases, but just as the steps of a random walk will almost always add up to a net displacement, the expectation value for the intensity is non-zero. And on-axis, where all the contributions are largest, the resultant intensity will be highest - a random walk of N large steps ends up further away than one with N small steps.

For a random walk with steps of equal length L the mean square displacement after N steps is N L but the present case is a not quite the classical random walk one because at a given point in the image plane the contributions from the different rings are not equal. Its a random walk with a distribution of step lengths.

The net effect is a PSF (point source response function) that has a core rather like an Airy pattern in shape but very much reduced in amplitude and with stronger side lobes. In the figure the PSF has been normalised by a factor 3N/4. Most of the flux does indeed get scattered to wide angles but there is a central spot far brighter than the background.

tech99

## 1. What is diffraction from a set of concentric rings with random phase?

Diffraction from a set of concentric rings with random phase refers to the phenomenon where light waves passing through a medium with randomly distributed phase shifts create a pattern of concentric rings when observed on a screen. This is a result of the interference of the light waves, causing some areas to be brighter and others to be darker.

## 2. How does diffraction from a set of concentric rings with random phase occur?

Diffraction from a set of concentric rings with random phase occurs when light waves pass through a medium with varying refractive indices, causing the waves to bend and interfere with each other. The random phase shifts in the medium cause the light waves to constructively and destructively interfere, creating the pattern of concentric rings.

## 3. What factors affect the pattern of concentric rings in diffraction?

The pattern of concentric rings in diffraction is affected by the wavelength of the light, the distance between the light source and the screen, and the properties of the medium through which the light passes. The size and shape of the particles in the medium, as well as the distribution of their refractive indices, also play a role in the pattern formation.

## 4. What are some practical applications of diffraction from a set of concentric rings with random phase?

Diffraction from a set of concentric rings with random phase has various applications in fields such as microscopy, astronomy, and material science. It is used to study the properties of materials, such as their refractive indices and particle size distributions. In microscopy, it can be used to enhance image contrast and resolution. In astronomy, it is used to study the composition and structure of celestial objects.

## 5. Can diffraction from a set of concentric rings with random phase be controlled or manipulated?

Yes, diffraction from a set of concentric rings with random phase can be controlled and manipulated by changing the properties of the medium through which the light passes. For example, by adjusting the refractive indices or particle size distribution of the medium, the pattern of concentric rings can be altered. This allows for the customization of diffraction patterns for specific applications.

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