Intensity in Single-Slit Diffraction, Quantitatively — Question 8

Question 8

A finite pixel records a nonzero average at a perfect diffraction zero

A pixel of width w is centered on the first minimum at $y_1=\lambda f/a$ in a paraxial focal plane. Let $\epsilon=\pi(y-y_1)/y_1$, so $\beta=\pi+\epsilon$. Assume $w\ll y_1$ and a uniform pixel response.

1. Expand the intensity near the minimum and average it over the pixel.

2. Find the residual recorded intensity for $w=0.10y_1$, as a fraction of $I_c$.

Question 8 - Solution

A finite pixel records a nonzero average at a perfect diffraction zero

(a) Method and calculation

$$\frac{I}{I_c}=\frac{\sin^2(\pi+\epsilon)}{(\pi+\epsilon)^2}\simeq\frac{\epsilon^2}{\pi^2}=\frac{(y-y_1)^2}{y_1^2}.$$$$\frac{\overline I}{I_c}\simeq\frac1w\int_{-w/2}^{w/2}\frac{z^2}{y_1^2}dz=\boxed{\frac{w^2}{12y_1^2}}.$$

(b) Interpretation and further calculation

$$w=0.10y_1\Longrightarrow\boxed{\overline I/I_c\simeq1/1200=8.3333\times10^{-4}}.$$$$\boxed{\overline I\to0\ \text{as}\ w\to0,\quad\overline I\propto w^2\ \text{to leading order}}.$$

Physical check

A filled recorded minimum can result from spatial averaging alone, even when the optical intensity is exactly zero at the pixel center.