A circular aperture of radius 0.01 cm is placed in front of a convex lens of focal length 25 c

step 1 So students in this question we have a circular aperture of radius a equals to 0 .01 centimeter

A circular aperture of radius 0.01 cm is placed in front of...

A circular aperture of radius $0.01 \mathrm{~cm}$ is placed in front of a convex lens of focal length $25 \mathrm{~cm}$ and illuminated by a parallel beam of light of wavelength $5 \times 10^{-5} \mathrm{~cm}$. Calculate the radii of the first three dark rings.

Key Concepts

Aperture Diffraction

Aperture diffraction describes how the size and shape of an opening in an optical system constrain the propagation of light and lead to a diffraction pattern. For a circular aperture, the interplay of the aperture size, the wavelength, and the focal length of the lens determines the scale of the diffraction pattern, including the spacing of the dark and bright rings.

Bessel Function Zeros

In the analysis of the Airy pattern, the Bessel function of the first kind, specifically J1, naturally appears due to the circular symmetry of the aperture. The positions of the dark rings correspond to the zeros of J1. Knowing these zeros (for example, approximately 3.83, 7.02, and 10.17 for the first three zeros) allows one to calculate the angular positions of the dark fringes, which are then mapped into physical distances in the focal plane by the lens.

Fraunhofer Diffraction

Fraunhofer diffraction refers to the far?field diffraction of light, where the incident wavefronts are almost planar. In the context of a circular aperture and a convex lens, the lens produces a Fourier transform of the aperture’s transmission function in its focal plane, which manifests as a diffraction pattern that is easier to analyze and predict.

Airy Pattern

The Airy pattern is the characteristic intensity distribution produced by diffraction through a circular aperture. It features a central bright spot (the Airy disk) surrounded by concentric dark and bright rings. The radii of the dark rings are determined by the zeros of the Bessel function associated with the aperture, and the pattern is essential for understanding the limits of resolution in optical systems.

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