On this website (http://www.ualberta.ca/~pogosyan/teaching/PHYS_130/FALL_2010/lectures/lect35/lecture35.html) it says that the formula $\sin(\theta_{max}) = (m+\frac{1}{2})\lambda$ derived for the maxima of a single slit interference is only approximate, while the formula for the minima $\sin(\theta_{max}) = m\lambda$ is exact (if we neglect other approximations like interference of parallel light waves). Why are the maxima not just in the middle of two minima?

We can clearly divide a light ray in three parts from which two cancel and thus get the mentioned formula for the maxima. Why is this formula not exact?

## Best Answer

The function describing the interference pattern is $f(x)=(\sin (x)/x)^2$, where $x=\frac{\pi a}{\lambda} \sin(\theta).$ To find the extremes you need to differentiate and equate to zero. For the minima it is easy, you find the condition $\sin x=0$, but for the maxima, you get $\tan x=x$ which has to be solved numerically.