(a) A plane wave (λ=6 ×10^-5 cm) is incident normally on a circular aperture of radius a. (i) A

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(a) A plane wave (λ=6 ×10^-5 cm) is incident normally on a...

(a) $A$ plane wave $\left(\lambda=6 \times 10^{-5} \mathrm{~cm}\right)$ is incident normally on a circular aperture of radius $a$. (i) Assume $a=1 \mathrm{~mm}$. Calculate the values of $z$ (on the axis) for which maximum intensity will occur. Plot the intensity as a function of $z$ and interpret physically. (ii) Assume $z=50 \mathrm{~cm}$. Calculate the values of $a$ for which minimum intensity will occur on the axial point. Plot the intensity variation as a function of $a$ and interpret physically. (b) Repeat the calculations for $\lambda=5 \times 10^{-5} \mathrm{~cm}$ and discuss chromatic aberration of a zone plate.

(a) $A$ plane wave $\left(\lambda=6 \times 10^{-5} \mathrm{~cm}\right)$ is incident normally on a circular aperture of radius $a$.
(i) Assume $a=1 \mathrm{~mm}$. Calculate the values of $z$ (on the axis) for which maximum intensity will occur. Plot the intensity as a function of $z$ and interpret physically.
(ii) Assume $z=50 \mathrm{~cm}$. Calculate the values of $a$ for which minimum intensity will occur on the axial point. Plot the intensity variation as a function of $a$ and interpret physically.
(b) Repeat the calculations for $\lambda=5 \times 10^{-5} \mathrm{~cm}$ and discuss chromatic aberration of a zone plate.

Key Concepts

Diffraction Through a Circular Aperture

This concept refers to the bending and spreading of waves when they pass through a circular opening. The phenomenon produces an intensity distribution that features bright and dark regions along the observation axis, which are analyzed using mathematical functions. When a plane wave is incident on a circular aperture, the resultant diffraction pattern depends on the geometry of the aperture and the wavelength of the light, and it is described by the theory of diffraction, often involving Bessel functions.

Axial Intensity Variation

Axial intensity variation describes how the intensity of light varies along the central axis (z-axis) after passing through an aperture. This variation results from interference between different zones of the diffracted wavefront, leading to positions of constructive interference (maxima) and destructive interference (minima). The calculation of these points involves understanding the phase differences accumulated over the propagation distance due to the geometry of the aperture and wavelength.

Bessel Functions and Their Zeros

Bessel functions play a crucial role in solving diffraction problems involving circular apertures because the radial part of the wave equation in cylindrical coordinates leads to these functions. The locations of the intensity maxima and minima along the axis are related to the zeros of certain combinations of Bessel functions. Identifying these zeros allows for the prediction of positions where maximum or minimum intensity occurs, which is essential in designing optical systems that rely on precise diffraction control.

Zone Plates and Chromatic Aberration

Zone plates are diffractive optical elements that focus light by using a series of concentric rings or zones, with each zone contributing to constructive interference at the focal point. Chromatic aberration in zone plates arises because the focal length is wavelength-dependent; different wavelengths are focused at different positions. This wavelength dependence must be carefully managed in applications requiring high precision, as it can lead to image blurring when a broad spectrum of light is used.

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