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

[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?

 

[struggling_student]

I would like delete this post I am convinced nobody cares. How does one delete?

 

[Delta2]

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 $f(u)$?