Show that when a circular aperture is illuminated by a plane wave whose amplitude varies with radius as \(T(\rho)\), from (9.4.19), the diffraction pattern is given by $$ \psi(u)=2 \pi \breve{C}^{\prime} \int_{0}^{a} T(\rho) J_{c}\left(\frac{u \rho}{a}\right) \rho d \rho, $$ which reduces to \((10.5 .4)\) when \(T(\rho)=1\).

Bessel functions play a central role in wave propagation problems, especially when dealing with circular symmetry—a common scenario in optical and acoustical systems.

Bessel functions, denoted by \(J_n(x)\), where \(n\) represents the order of the function, are solutions to Bessel's differential equation. This equation arises in problems exhibiting cylindrical or spherical symmetry. The solutions \(J_n(x)\) oscillate in a manner similar to sine and cosine functions but are not periodic. An important aspect of these functions for physical applications is the behavior near the origin, where they start at a finite value for \(n = 0\) and are zero for any other \(n\).

When analyzing the diffraction patterns produced by circular apertures, Bessel functions become essential as they describe how the intensity pattern varies with distance from the center of the diffraction pattern. Understanding Bessel functions can be enhanced by tracing their graphs, which exhibit a central peak (for the zero-order Bessel function) and subsequent decrease with increasing values of \(x\), showing a series of maxima and minima.

Plane wave diffraction occurs when a wave encounters an obstacle that disrupts its propagation, resulting in the bending and spreading of the waves around the edges of the obstacle.

In the context of our exercise, we examine the diffraction of a planar wavefront passing through a circular aperture. The term 'plane wave' implies that the wave's amplitude is uniform across wavefronts, and these wavefronts are perpendicular to the direction of wave propagation. When such a wave diffracts through a circular opening, the resulting pattern cannot be described using elementary functions, and that is where Bessel functions come into play.

The pattern formed due to diffraction is the culmination of the interference of many waves emanating from different points on the aperture. These patterns are generally complex, but they can be predicted and understood mathematically using the principles of wave superposition and the particular behavior of Bessel functions in such geometrical configurations.

The amplitude function, \(T(\rho)\), in the context of diffraction, describes how the amplitude of the illuminating plane wave varies across the aperture. For a uniform plane wave, \(T(\rho) = 1\), the amplitude is the same at all points on the aperture.

In practice, the amplitude might change due to various factors such as obstruction, variation in light source intensity, and the lens' or aperture's material properties. The exercise demonstrates that when \(T(\rho)\) is uniform, the integral simplifies, and we see a relationship that is independent of \(\rho\). However, when \(T(\rho)\) varies with \(\rho\), the resulting diffraction pattern will also vary, and the integral will need to account for this variation. This complexity is dealt with by integrating the product of the amplitude function and the Bessel function over the area of the aperture.

The key take-away is that the amplitude function allows us to model a wide range of physical configurations and predict the corresponding diffraction pattern. It is an essential component for understanding how light behaves when it passes through apertures with different illumination profiles.