Diffraction by a Single Slit: Locating the Minima — Question 6

Question 6

A rectangular opening diffracts more strongly along its narrower dimension

A uniformly illuminated rectangular aperture has horizontal width $a_x$ and vertical height $a_y$. In a paraxial focal plane, its first dark lines are $x=\pm f\lambda/a_x$ and $y=\pm f\lambda/a_y$. Let $f=0.50\,\mathrm m$, $\lambda=600\,\mathrm{nm}$, and central-lobe dimensions $W_x=6.0\,\mathrm{mm}$, $W_y=2.0\,\mathrm{mm}$.

1. Infer both aperture dimensions.

2. Predict the central-lobe dimensions if the aperture is rotated by $90^\circ$ about the optical axis.

Question 6 - Solution

A rectangular opening diffracts more strongly along its narrower dimension

(a) Method and calculation

$$W_x=2f\lambda/a_x,\quad W_y=2f\lambda/a_y.$$$$\boxed{a_x=0.100\,\mathrm{mm},\quad a_y=0.300\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$a_x^\prime=a_y,\quad a_y^\prime=a_x.$$$$\boxed{W_x^\prime=2.0\,\mathrm{mm},\quad W_y^\prime=6.0\,\mathrm{mm}}.$$

Physical check

The elongated diffraction pattern is perpendicular to the long dimension of the opening.