Show that the fraction of the total energy in the circular-aperture diffraction pattern out to the radius specified by \(u_{\operatorname{msx}}\) is $$ \frac{1}{2} \int_{0}^{u_{\max }}\left[\frac{2 J_{1}(u)}{u}\right]^{2} u d u=1-J_{0}^{2}\left(u_{\max }\right)-J_{1}^{2}\left(u_{\max }\right) . $$ Hence, verify the final column of Table \(10.2\).

Bessel functions are essential in various areas such as physics, engineering, and mathematics, particularly when dealing with problems which have radial symmetry. They appear in solutions to wave equations in cylindrical coordinates and are therefore pivotal in analyzing wave behaviors, such as those in circular aperture diffraction patterns.

For example, in optics, the Bessel function of the first kind, denoted as \(J_n(u)\), where \(n\) is the order, characterizes how light waves propagate through circular openings. These functions oscillate and have a decaying amplitude, much like how ripples behave on a water surface after a stone is dropped.

Specifically, in the context of a circular aperture diffraction problem, the term \(J_1(u)\), represents the first-order Bessel function. This mathematical representation is critical to predict the intensity variation in the resulting diffraction pattern. As light waves interfere with each other, they give rise to a series of bright and dark rings witnessed in the pattern, and understanding Bessel functions allows us to describe this behavior accurately.

Students struggling with Bessel functions should start by grasping their oscillatory nature and relation to sine and cosine functions. Moreover, they should become familiar with standard Bessel function properties, such as recurrence relations, which are incredibly useful in simplifying complex expressions in problems such as diffraction pattern calculations.

The intensity pattern of light obtained in a diffraction experiment is critical for understanding how energy is distributed across the diffraction pattern. To quantify the distribution, we integrate the intensity pattern over the relevant area.

In our exercise, the intensity pattern \(I(u)\) is expressed in terms of the Bessel function and the variable \(u\), which is a dimensionless quantity related to the radius in the diffraction pattern. This pattern is not uniform; instead, it varies with the radial distance, showing high intensity at the center which gradually decreases with increases in radius.

To determine the total energy within a certain radius \(u_{\text{max}}\), the intensity function \(I(u)\) is integrated with respect to \(u\), weighted by the radial distance \(u\). This process, known as intensity pattern integration, involves not only the integration of the simple square of the Bessel function but also the incorporation of the variable \(u\) which accounts for the circular geometry of the aperture.

When integrating the intensity pattern, students must remember that the integration bounds are crucial. They need to integrate up to \(u_{\text{max}}\) to find the energy fraction within the radius specified. The integration thus captures the energy distribution up to that point, reflecting how much of the light after passing through the aperture falls within a given radius.

Talking about the energy fraction in diffraction is essentially addressing how much of the incident light's energy is distributed within a specific area of the diffraction pattern. This concept is extremely valuable in applications where precise energy measurements matter, such as in designing optical instruments or analyzing the light-gathering capabilities of telescopes.

The exercise we are considering requires us to calculate the fraction of the total energy within a circular area of radius \(u_{\text{max}}\) of the diffraction pattern. The derived expression combines the mathematical framework provided by Bessel functions with physical insights about energy conservation in wave phenomena. By integrating the intensity, we account for all the light beam contributions up to the specified radius.

The result is intuitive; as we increase the radius \(u_{\text{max}}\), we expect more energy to be included in our calculation. The expression \(1 - J_0^2(u_{\text{max}}) - J_1^2(u_{\text{max}})\) indicates that there's a certain amount of energy outside the radius \(u_{\text{max}}\), which corresponds to the sum of the squares of the zeroth and first-order Bessel functions evaluated at that radius.

This part of the study is challenging, but students should understand that it is about dividing the diffraction pattern into infinitesimally small concentric rings, integrating the energy over those rings, and then summing up to get the total energy within \(u_{\text{max}}\). By comparing the results with established values (like those in Table 10.2), students can validate their understanding and the calculated expressions.