(a) Show that the on-axis intensity maxima and minima for a slit occur approximately at one-quarter of a zone less than integral zones, as stated in (11.3.12). * \((b)\) Show that the error involved in this approximation is $$ n_{\text {extremum }}-k+\frac{1}{4} \approx \frac{(-1)^{k}}{2 \pi^{2}\left(k-\frac{1}{4}\right)^{3 / 2}} $$ II \(i \boldsymbol{n} t:\) The asymptotic expansions of the Fresnel integrals \((11.2 .16)\) and \((11.2 .17)\) for \(u \gg 1\) are $$ \begin{aligned} &C(u)=\frac{1}{2}+\frac{\sin (\pi / 2) u^{2}}{\pi u}-\frac{\cos (\pi / 2) u^{2}}{\pi^{2} u^{3}}-\cdots \\ &S(u)=\frac{1}{2}-\frac{\cos (\pi / 2) u^{2}}{\pi u}-\frac{\sin (\pi / 2) u^{2}}{\pi^{2} u^{3}}+\cdots \end{aligned} $$

When light waves pass through a slit, they create a pattern of bright and dark regions on a screen. These regions are called intensity maxima and minima. The intensity maxima are bright spots where the light waves interfere constructively, meaning they add together to make a brighter light. Conversely, the minima occur when waves interfere destructively, cancelling each other out and producing dark spots.

Mathematically, these patterns are described by Fresnel integrals, which predict the distribution of light intensity across the screen. The positions of the maxima and minima can be found by setting the values of the Fresnel cosine and sine integrals, denoted as C(u) and S(u), equal to 1/2k where k is an integer, representing the order of the maxima or minima. For a slit, maxima and minima happen at positions which are a quarter-zone less than whole zones, leading to a quarter-zone phase shift in the pattern. This shift is pivotal in calculating the accurate positions of these bright and dark fringes and thus is crucial in the study of wave optics.

Asymptotic expansions are mathematical expressions that approximate more complex functions as a variable tends to infinity. In the context of wave optics and Fresnel integrals, these expansions become useful when analyzing the behavior of light at large distances from the slit, or for high values of the variable u. Basically, while the Fresnel integrals are not trivial to solve exactly, their asymptotic expansions allow for simpler, approximate solutions that can describe the intensity pattern quite accurately at far distances.

When u is much greater than 1, the expansions of C(u) and S(u) are expressed in a series involving trigonometric functions of u squared, divided by powers of u. These expansions provide an effective tool for physicists and engineers to predict optical phenomena without resorting to complex numerical calculations, particularly in scenarios where a high degree of precision is not necessary, but an overall understanding of the behavior is required.

Wave diffraction is a phenomenon that occurs when waves encounter an obstacle or a slit that is comparable in size to their wavelength. It refers to the bending and spreading out of waves as they pass through an opening or around edges. Diffraction is not exclusive to light waves; it can also be observed with sound, water, and other types of waves.

The diffracted waves give rise to interference patterns that can be described by Fresnel integrals. The intensity of light in these patterns is not uniform, leading to the formation of maxima and minima. Engineering applications of wave diffraction range from the design of optical instruments to the understanding and manipulation of radio wave propagation for communication technologies. Comprehending wave diffraction through mathematical formulas like Fresnel integrals is a fundamental part of physics and optics, allowing us to predict and explain a myriad of natural and technological phenomena.