Light Diffraction: Explaining Airy Equation

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Discussion Overview

The discussion revolves around the theoretical description of the Airy equation in the context of light diffraction, specifically seeking a derivation from the wave equation to the Airy equation as it applies to diffraction patterns produced by circular apertures.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the theoretical description of the Airy equation in light diffraction.
  • Another participant describes the Airy disc as the diffraction pattern resulting from a circular aperture and provides a link to further information.
  • A participant expresses a desire for a derivation of the Airy equation from the wave equation, indicating that existing resources do not address this need.
  • One participant suggests applying the theory of Fraunhofer diffraction and provides an aperture function in cylindrical coordinates as a starting point for deriving the diffraction pattern.
  • A later reply acknowledges the challenge of deriving the Airy equation from the two-dimensional Fourier integral equation and expresses intent to explore this further.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple viewpoints regarding the derivation of the Airy equation and the resources available for understanding it.

Contextual Notes

The discussion highlights the dependence on specific mathematical frameworks and the potential complexity of deriving the Airy equation, which may not be fully addressed in the provided references.

a-lbi
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What describes Airy equation in theoretical description of light diffraction?
 
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Thank you for your response. I've read this article. But there are no answers for my question.

Writing more precisly. I would like to see derivation from (for instance) wave equation to get Airy equation in a light diffraction problem.
 
To find a diffraction pattern for a given aperture, you need to apply the theory of Fraunhofer diffraction.

To do this, define an aperture function in cylindrical coordinates as follows;

a(r) = 1 for r less than or equal to a.
a(r) = 0 for r greater than a.

Where a is the radius of the aperture.

The resultant diffraction pattern is a 2D Fourier transform of this function (i.e. the Airy function). I'm sure you can find the gory details of this calculation somewhere (or maybe post in the math forums).

Claude.
 
Thank you for your response. I can see that the problem is to get Airy equation from the two dimensional Fourier integral equation. If I have some time I will try to follow the way.
 

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