Software Application for Diffraction Patterns Visualization

What is diffraction?

Diffraction is a physical phenomenon consisting of the emergence of some geometric structures during the propagation of waves encountering obstacles in their path. A must condition for the appearance of diffraction is that the dimensions of the obstacle, whether it is a barrier or an aperture, should be comparable to the wavelength. This phenomenon is usually observable as a wave movement in the area of the so-called geometric shadow, where under other conditions, there would be no wave process.

This phenomenon has been known for centuries. It was first described in 1660 by the Italian Jesuit priest and physicist Francesco Grimaldi. In the light of classical physics, it is explained by the so-called Huygens-Fresnel principle, formulated in 1678 by the Dutch astronomer and mathematician Christiaan Huygens and quantitatively expanded by the French physicist Augustin-Jean Fresnel in 1818. To Fresnel's theoretical and experimental discoveries in the field of optics, we owe the triumph of the wave theory of light, which reached its peak in the second half of the 19th century.

The principle states that each point of the medium the wavefront has reached becomes a source of secondary spherical wavelets that interfere (interact) with each other. The envelope of their superposition (overlay) represents the new wavefront at the next moment in time.

The exact calculation of the diffraction patterns, proceeding from the principle itself, is a complex mathematical task. Therefore, an analytical solution has been obtained only in the cases of obstacles with simple geometry such as circular or rectangular apertures, rectilinear slits and edges, small spheres, etc.

The Diffractor application

In order to calculate diffraction patterns in an area near the focus of the optical system and their visualization, a GUI .NET application Diffractor is developed. The original version was written back in 2006 and later updated. The program calculates the diffraction of an optical system with a round entrance pupil without aberrations or with third-order monochromatic aberrations: spherical aberration, coma, astigmatism, field curvature and distortion. We decompose the aberration function as a sum of circular Zernike polynomials, orthonormal on a unit circle, to find a numerical solution of the diffraction integral as an infinite series of Bessel functions of the first kind [1]. The resulting two-dimensional map of the intensity distribution in the area around the focus is normalized to the maximum intensity in the geometric focus point in the case with no aberrations.

The program accepts the following input parameters:

• Size in pixels. The size of the resulting 2D image in pixels.
• Pupil diameter. The diameter in mm of the entrance pupil. For example, for a telescope, this is the diameter of its objective.
• Gauss radius. The radius in mm of the wavefront converging to the focus point. For a telescope, this is the focal length of its objective.
• Scale. The size of the side of the resulting two-dimensional picture, expressed in wavelengths λ.
• Abberation coefficient. Represents the magnitude of aberration.
• Focal distance. The distance from the geometric focus point along the axis of the system expressed in wavelengths λ. It can be a positive or negative number.
• Integration tolerance. Determines the accuracy of the numerical integration. A Romberg integration method for a two-dimensional complex function is used.
• Integral sums. The number of terms in the series expansion of the diffraction integral in the presence of aberrations.

The calculated intensity distribution is visualized on the screen and can be saved as a text file from the File → Save menu.

The following figures demonstrate two-dimensional diffraction patterns at various optical aberrations obtained with the Diffractor.

System requirements

*Windows XP requires .NET Framework 2.0 or higher to be installed.

Installation

You can download Diffractor from here. Then, unpack the zip archive somewhere on your system, for example, in C:\Program Files\Diffractor and run the .exe file. Additionally, you can create a shortcut on your Desktop or Start menu.

NOTE: The archive contains an executable file. Some antivirus software may restrict it from the download. Please, check the settings of your antivirus program.

NOTE: Do not change the structure and content of the unpacked zip archive.

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Reference

1. Born M., Wolf E. Principles of optics. Electromagnetic theory of propagation, interference and diffraction of light, Pergamon Press, 4th Edition, 1968.