# Diffraction pattern and Fourier Transform

• transform
In summary, the diffraction pattern after Laserlight passing through a grating pattern can be determined by the spatial frequencies of the grating pattern and the wavelength of the laser, and mathematically it involves the Fourier transform of the grating pattern. The far-field diffraction pattern is the complex Fourier transform of the aperture, with phase factors added when moving planes. Detectors are sensitive to the intensity only, but for coherent detection the transfer function is the field at the entrance pupil and for incoherent detection it is the autocorrelation of the entrance pupil. The cutoff frequency for aberration-free imaging is twice that of coherent detection.
transform
hello
I wonder if the diffraction pattern after Laserlight going thru a grating pattern (say 10 slots) has anything to do with the Fourier transform of the grating pattern.
I am not a physicist, but have some knowledge of Fourier math.
I think the spatial frequencies of the grating pattern plus the wavelength of the laser determin the diffraction pattern, and mathematically it has something to do with the FT of the grating pattern, but I am not sure!?
can anybody elaborate on this a little bit?
here is an example of the kind of diffraction pattern i mean
http://demonstrations.wolfram.com/DiffractionGratingIntensities/

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the book seems to cover all the aspects of FT and holography which I am interested in and want to know!
I just continued to look in the web for more information on diffraction pattern and FT. In one paper I found the notion that the actual (Fraunhofer) diffraction pattern gives back the intensity wave of the FT function, not directly the FT.
So can one say that the diffraction pattern gives a kind of picture what the FT of the grating pattern looks like?

The far-field diffraction pattern of an aperture is the (complex) Fourier transform of the aperture. Placing a lens after the aperture slightly alters this, but *if* the aperture is located at the entrance pupil of the lens, *then* the field at the back focal plane is the complex Fourier transform of the aperture. Moving planes around adds phase factors, essentially. Laue patterns, x-ray crystallography, Bragg scattering etc. all use this phenomonon.

Note it's the *field* that is transformed, not the intensity, and it is indeed a complex transform. Detectors (in the visible region) are sensitive to the intensity only; that has an effect but not a conceptual change. For coherent detection (phase-sensitive detectors), the transfer function is the field at the entrance pupil; for incoherent detection (intensity detection), the transfer function is the autocorrelation of the entrance pupil. For aberration-free imaging, the cutoff frequency in incoherent detection is twice that of coherent detection.

## 1. What is a diffraction pattern?

A diffraction pattern is a series of bright and dark spots that appear when a wave, such as light or sound, passes through a narrow opening or around an obstacle. This pattern is a result of the wave bending or diffracting as it passes through the opening or around the object.

## 2. How is a diffraction pattern related to Fourier Transform?

A Fourier Transform is a mathematical tool used to analyze the frequency components of a signal or wave. In the case of a diffraction pattern, the Fourier Transform can be used to break down the complex pattern into its individual frequency components, providing insight into the structure of the diffracting object.

## 3. Can diffraction patterns be seen in everyday life?

Yes, diffraction patterns can be observed in everyday life. For example, the colorful patterns seen on the surface of a CD or DVD are a result of diffraction of light. Diffraction patterns can also be seen in natural phenomena such as the iridescent colors of a soap bubble or the rainbow-like patterns seen on the surface of a CD.

## 4. What factors affect the appearance of a diffraction pattern?

The appearance of a diffraction pattern is affected by several factors including the wavelength of the wave, the size of the diffracting object, and the distance between the wave source and the object. The shape and size of the opening or obstacle also play a role in determining the pattern.

## 5. How is a Fourier Transform used in practical applications?

The Fourier Transform has a wide range of practical applications in various fields such as signal processing, image analysis, and data compression. It is used to analyze and manipulate signals and waves in fields such as telecommunications, medical imaging, and audio and video processing. It is also used in scientific research to analyze diffraction patterns and obtain information about the structure of materials.

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