Any physics sophomore who had taken Optics would probably know the Airy formula for diffraction over a circular aperture. The image of a point source, taken by a telescope for example, is formed by a central bright fringe (about 84% of the light) surrounded by concentric rings that get dimmer radially. The angular size $\Delta\theta$ of this pattern is first derived by George Airy in 1835,

\[ \Delta\theta = 1.22\dfrac{\lambda}{D} \]

This poses a fundamental limit for imaging techniques: for a fixed wavelength, the resolution is inversely proportional to the diameter of the aperture $D$. Interestingly enough, it can also be derived from Heisenberg’s uncertainty principle.

Qualitatively speaking, when a photon passes through an aperture, we gain some information regardings its position $x$, and as a consequence we are less sure about its momentum $p$ on the axis which its motion is constrained. The Heisenberg’s uncertainty principle states that,

\[ \Delta x \Delta p \geq \dfrac{\hbar}{2} \]

It is the uncertainty in momentum that gives the photon more degrees of freedom in choosing its future direction. If we were to consider a collective of photons–a beam, then this beam is effectively spread out as passing the aperture, hence the Airy pattern. The smaller the aperture $d\to 0$, the more information we gain about the photon’s position, and thus the larger the Airy rings $\delta\theta\to \infty$.

To obtain the Airy formula, we first consider a photon of momentum $h/\lambda$ along the $x$-axis passing a slot aperture $D$ along the $y$-axis. Since the photon passes through the slit of width $D$,

\[ \Delta p_y D \geq \dfrac{\hbar}{2} \]

Note that the $x$-momentum is $p_x = 2\pi\hbar/\lambda$ we can substitute $\hbar$ by

\[\begin{align*} \Delta p_y D \geq \lambda p_x/4\pi \newline \Delta p_y/p_x \geq \lambda/4\pi D \end{align*} \]

Let us pause for a moment and think about the meaning of the expression on the left hand side. Remember that $p_x$ itself is a vector, pointing along the $x$-axis; $\Delta p_y$ itself is also a vector, pointing along the circumferential axis. The ratio $\Delta p_y/p_x$ thus denote the angular uncertainty, $\Delta\theta$ of the photon passing through the aperture; hence,

\[ \Delta\theta \propto \dfrac{\lambda}{D} \]

Note that the photon’s motion here is 1-dimensional. For an aperture disk of radius $D$, we expect it to be two-dimensional: hence two axes along which the position of the photon is constrained.