Intensity in Single-Slit Diffraction, Quantitatively — Question 6

Question 6

Most of the one-dimensional diffraction power lies between the first zeros

In the paraxial one-dimensional model, power per unit $\beta$ is proportional to $(\sin\beta/\beta)^2$. Use $\int_{-\infty}^{\infty}(\sin\beta/\beta)^2d\beta=\pi$ and $\int_0^b(\sin\beta/\beta)^2d\beta=\operatorname{Si}(2b)+(\cos2b-1)/(2b)$, where $\operatorname{Si}(2\pi)=1.418151576$.

1. Calculate the fraction of transmitted power in the central lobe.

2. For total transmitted power $2.00\,\mathrm{mW}$, find the central-lobe power and total power in all other lobes.

Question 6 - Solution

Most of the one-dimensional diffraction power lies between the first zeros

(a) Method and calculation

$$\eta_c=\frac1\pi\int_{-\pi}^{\pi}(\sin\beta/\beta)^2d\beta.$$$$\boxed{\eta_c=2\operatorname{Si}(2\pi)/\pi=0.902823}.$$

(b) Interpretation and further calculation

$$\boxed{P_c=\eta_cP=1.80565\,\mathrm{mW}}.$$$$\boxed{P_{\mathrm{side}}=(1-\eta_c)P=0.194353\,\mathrm{mW}}.$$

Physical check

This is a one-dimensional slit result; the central-disk energy fraction of a circular aperture is different.