Diffraction pattern and Fourier Transform

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Discussion Overview

The discussion revolves around the relationship between diffraction patterns produced by laser light passing through a grating and the Fourier transform of the grating pattern. Participants explore the mathematical connections and implications of these phenomena, touching on concepts from optics and Fourier analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the diffraction pattern from a grating is related to the Fourier transform of the grating pattern, suggesting that spatial frequencies and laser wavelength play a role.
  • Another participant mentions a paper indicating that the Fraunhofer diffraction pattern represents the intensity wave of the Fourier transform function, rather than the transform itself, raising the idea that the diffraction pattern may visually represent the Fourier transform of the grating.
  • A further contribution clarifies that the far-field diffraction pattern of an aperture is the complex Fourier transform of that aperture, with additional notes on how lenses can alter this relationship and the distinction between intensity and field in the context of detection.
  • It is noted that coherent and incoherent detection methods affect the interpretation of the transfer function related to the aperture's Fourier transform.

Areas of Agreement / Disagreement

Participants express differing views on the exact nature of the relationship between diffraction patterns and Fourier transforms, with some proposing that the diffraction pattern reflects aspects of the Fourier transform while others clarify the complexities involved. No consensus is reached on the precise connections.

Contextual Notes

Participants highlight the importance of distinguishing between intensity and field in the context of diffraction and Fourier transforms, as well as the impact of detection methods on the interpretation of results. There are unresolved aspects regarding the mathematical details and assumptions underlying these relationships.

Who May Find This Useful

This discussion may be of interest to those studying optics, Fourier analysis, and related fields, particularly in understanding the interplay between diffraction patterns and mathematical transforms.

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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/
 
Science news on Phys.org
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.
 

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