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

• I
• struggling_student
In summary, the conversation discusses the derivation of a formula for the diffraction pattern formed by a plane-wave passing through a 1D aperture onto a screen. The aperture is described by an opacity function and the Huygens-Fresnel principle is used to integrate over all possible sources on the aperture. The resulting diffraction is a function of the position on the screen and can be squared to obtain the intensity. The only issue with the proposed formula is the lack of division by the total length of the aperture. It is suggested to try numerical integration for a single slit to see the resulting diffraction pattern. The speaker also mentions wanting something similar to a Fourier transform, but it is not specified what exactly they mean by this.
struggling_student
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?

Delta2
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)##?

## 1. What is a diffraction pattern?

A diffraction pattern is a specific pattern that is produced when light waves or other waves pass through a narrow opening or slit, causing them to spread out and interfere with each other. This pattern is characterized by alternating bright and dark fringes or bands.

## 2. How is a diffraction pattern formed?

A diffraction pattern is formed when a wavefront of light or other waves encounters an obstacle or slit that is comparable in size to the wavelength of the waves. The waves are diffracted or bent as they pass through the opening, causing them to interfere and produce the distinct pattern.

## 3. What factors affect the shape of a diffraction pattern?

The shape of a diffraction pattern is affected by several factors, including the size of the opening or obstacle, the wavelength of the waves, and the distance between the source of the waves and the screen or detector where the pattern is observed. Additionally, the type of wave (e.g. light, sound, etc.) and the properties of the medium through which it travels can also impact the diffraction pattern.

## 4. What is the significance of a diffraction pattern in science?

Diffraction patterns are important in many areas of science, including physics, chemistry, and biology. They provide valuable information about the properties of waves, such as their wavelength and the size of the obstacles they encounter. Diffraction patterns are also used in various technologies, such as X-ray crystallography, which is used to determine the structure of molecules.

## 5. How is a diffraction pattern used in real-life applications?

Diffraction patterns have numerous practical applications, such as in the design of optical instruments like microscopes and telescopes. They are also used in various imaging techniques, such as electron microscopy and diffraction-enhanced imaging, which can provide detailed images of biological tissues and structures. Additionally, diffraction patterns are used in spectroscopy to analyze the composition of materials and in the production of holograms for security and entertainment purposes.

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