Intensity in Single-Slit Diffraction, Quantitatively — Question 3

Question 3

Secondary maxima satisfy a transcendental stationarity condition

Consider $f(\beta)=(\sin\beta/\beta)^2$. The first two positive secondary-maximum roots are approximately $\beta_1=4.493409458$ and $\beta_2=7.725251837$.

1. Differentiate f and derive the equation obeyed by nonzero, non-dark stationary points.

2. Use that equation to express the maximum intensity in terms of its root and evaluate both side-lobe heights.

Question 3 - Solution

Secondary maxima satisfy a transcendental stationarity condition

(a) Method and calculation

$$f^\prime(\beta)=\frac{2\sin\beta(\beta\cos\beta-\sin\beta)}{\beta^3}.$$$$\sin\beta\ne0:\quad\boxed{\tan\beta=\beta}.$$

(b) Interpretation and further calculation

$$\sin\beta=\beta\cos\beta\Longrightarrow f=\cos^2\beta=\boxed{1/(1+\beta^2)}.$$$$\boxed{I_1/I_c=0.0471904,\quad I_2/I_c=0.0164800}.$$

Physical check

The side maxima do not lie exactly halfway between neighboring zeros; their locations follow the derivative equation.