In rectangular aperture or single slit diffraction, how does one find the location of the secondary MAXIMA of the diffraction pattern and their amplitude for a normalized central amplitude of unity?

Question

Once you determine the transcendental equation that determines the maxima, use a graphing calculator (free online version) to find the exact solutions without trying to be consumed by numerical analysis (should take 5-10 minutes instead of hours using numerical analysis). The phase maxima occurs midway between the minima which are easily found with m*lambda = d*sin(theta), but the I = A*(sinx / x) dependency pulls the maxima toward the origin so 3/2 pi, 5/2 pi, 7/2pi, etc. is a crude approximation.

Answer 1

If you differentiate (sin x) / x and set it to zero, where x = pi d / lambda * sin(theta), you get:

[(cos x) / x - (sin x) / (x^2)] * dx / d(theta) = 0

The solution is not 3/2 pi, 5/2 pi, 7/2 pi, 9/2 pi, etc even though dx/d(theta) is zero at those angles. The actual solution is where the quantity inside the square brackets is zero, namely at tan x = x. This gives x = 4.49, 7.73, 10.90, 14.07, ... all values that are slightly smaller than the respective half integer multiples of pi. You can then use those values of x to solve for theta.