I How Can One Derive the Diffraction Pattern Formula from a 1D Aperture?

AI Thread Summary
The discussion focuses on deriving the diffraction pattern formula for a plane wave passing through a 1D aperture, described by an opacity function. The Huygens-Fresnel principle is applied to model the wave's amplitude as it reaches the screen, leading to an integral expression for the resulting diffraction. Concerns are raised about the need to normalize the integral by the total length of the aperture, similar to averaging a function over an interval. The complexity of the integral is noted, with challenges in finding a closed analytical solution, prompting suggestions for numerical integration methods. The conversation also touches on the desire to relate the diffraction pattern to concepts akin to the Fourier transform.
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I've been trying to derive a formula for diffraction pattern formed by casting a plane-wave through a generic 1D aperture onto a screen distanced ##L## from the aperture. The aperture is described by an opacity function ##f:\mathbb{R} \rightarrow [0,1]## so it can be a single slit, multiple slits, shaded glass with varying opacity. By the Huygens-Fresnel principle every point on that aperture is a spherical wave and we weigh them by infinitesimal ##du## so that it can be integrated.

Let ##u## be the position on the aperture relative to some chosen axis which also goes through the screen. Let the position on the screen relative to that axis be ##x##. The opacity function is a function of ##u##, i.e. ##f=f(u)##.

The wave that goes through point ##u## on reaching the screen has amplitude

$$A f(u) \cos\left(\frac{2\pi}{\lambda}\sqrt{(x-u)^2+L^2 }\right) du,$$

and the resulting diffraction will be

$$A \int_{\mathbb{R}} f(u) \cos\left(\frac{2\pi}{\lambda}\sqrt{(x-u)^2+L^2 }\right) du.$$

It's a function of ##x## and we would square it to get intensity. I'm not sure how to proceed or what I did wrong. This approach is the only approach I am interested in. I'm trying to obtaining something similar to Fourier transform. What's missing here?

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I would like delete this post I am convinced nobody cares. How does one delete?
 
I care but I am not so good in this type of problems that's why I hesitate to type my thoughts. However I liked the generalization of this problem (function ##f(u)##) and the way you use Huygens-Fresnel principle to integrate over all possible sources. The only problem I see with that integral expression is that :
  • you have to divide by the total length of the aperture, pretty much the same way you divide by ##b-a## when you calculate the average value of a function ##f(x)## over the interval ##[a,b]##, $$\mu=\frac{1}{b-a}\int_a^b f(x)dx$$
  • your integral though it doesn't seem very complicated, yet it doesn't seem to have a closed analytical antiderivative, at least wolfram alpha can't find it (I tried it with ##f(u)=1## the constant function).
Have you try to do numerical integration using some math software (mathematica, MATLAB e.t.c) for the case f(u) is the function that corresponds to a single slit to see what diffraction pattern you get?

What do you mean when you say you want to get something similar to Fourier transform? Fourier transform of ##f(u)##?
 
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