I have been studying the intensity patterns of single slit, double slit and triple slits. Each intensity pattern depending on how you add the individual electric field components at a point on the screen and using the relationship $I=kE^2$ where intensity is proportional to the electric field squared.
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide14.pdf
The entire methodology for computing the intensity really lies in the idea of finding the total electric field added up together then squaring it to find intensity. For instance in this website by MIT, the mathematical expressions for single slit, double slit and triple slit is given by:
Single Slit ~ $I=\frac{4I_0}{\beta^2_0}\sin^2\frac{\beta}{2}\space\space\text{page 16-18}$
Double Slit ~ $I=I_0\cos^2\frac{\phi}{2}\space\space\text{page 8-10}$
Triple Slit ~ $I=I_0\frac{(1+2\cos\phi)^2}{9} \space\space\text{page 11-12}$
The trigonometric approach to deriving the triple slit and double slit as well as single slit (although little different from double slit and triple slit) I understand well matheamtically as it inrpoates some trigonometric identites only.
However when I try to generalize for $N$ slits I tried to see the patterns by dividing the cases up in to odd and even number of slits. For instance when computing intensity patterns for $N=4$, I firstly add the following four expressions:
$$E_T=\sum^4_{i=1}E_i=\sin(\omega t)+\sin(\omega t+\phi)+\sin(\omega t+2\phi)+\sin(\omega t+3\phi))$$
The whole idea is to simply add the sine functions and reducing it in to products as clearly done in the MIT pdf attached on this post. However as I get to more number of slits like $N=5$, or $N=8$, it is possible to reduce the trigonometric sum, however I do not get like a neat result which can help me generalize an expression for any $N=$ even or any $N=$ odd slits.
My question is are there any sources which show the generalized $N$ slits equation for intensity? I am just curious as I believe it can be done, but I am having trouble doing so.
