Consider a rectangular aperture of dimensions 0.2 mm ×0.3 mm. Obtain the positions of the firs

step 1 Hello students in this question we have a rectangular aperture of dimension 0 .2 m m m by 0 .3 m

Consider a rectangular aperture of dimensions 0.2 mm ×0.3...

Consider a rectangular aperture of dimensions $0.2 \mathrm{~mm}$ $\times 0.3 \mathrm{~mm}$. Obtain the positions of the first few maxima and minima in the Fraunhofer diffraction pattern along directions parallel to the length and breadth of the rectangle. Assume $\lambda=5 \times 10^{-5} \mathrm{~cm}$ and that the diffraction pattern is produced at the focal plane of a lens of focal length $20 \mathrm{~cm}$. [Ans: Along the $x$ -axis, minima will occur at $x \approx 0.05$, $0.10,0.15, \ldots \mathrm{cm} ;$ along the $y$ -axis, minima will occur at $$ y \approx 0.033,0.067,0.1, \ldots \mathrm{cm}] $$.

Consider a rectangular aperture of dimensions $0.2 \mathrm{~mm}$ $\times 0.3 \mathrm{~mm}$. Obtain the positions of the first few maxima and minima in the Fraunhofer diffraction pattern along directions parallel to the length and breadth of the rectangle. Assume $\lambda=5 \times 10^{-5} \mathrm{~cm}$ and that the diffraction pattern is produced at the focal plane of a lens of focal length $20 \mathrm{~cm}$. [Ans: Along the $x$ -axis, minima will occur at $x \approx 0.05$, $0.10,0.15, \ldots \mathrm{cm} ;$ along the $y$ -axis, minima will occur at
$$
y \approx 0.033,0.067,0.1, \ldots \mathrm{cm}]
$$.

Key Concepts

Fourier Transform of the Aperture Function

The Fourier transform is a mathematical tool used to convert a spatial function (like the aperture function) into its frequency domain representation. In the context of diffraction, the aperture function's Fourier transform directly gives the amplitude and phase distribution in the far field. This approach simplifies the analysis of complex aperture shapes and is essential for predicting the location of minima and maxima in the diffraction pattern.

Diffraction Minima and Maxima Conditions

The positions of the minima and maxima in a diffraction pattern are determined by interference conditions. For a rectangular aperture, minima occur at the zeros of the sinc function components, which are dictated by the dimensions of the aperture. The conditions for these minima are derived from setting the argument of the sine functions equal to multiples of ?, leading to positions where destructive interference causes the intensity to vanish.

Fraunhofer Diffraction

Fraunhofer diffraction refers to the far?field diffraction phenomenon where the observed diffraction pattern is formed at a large distance (or equivalently at the focal plane of a converging lens). In this regime, spherical wavefronts can be approximated as planar, and the diffraction pattern is given by the Fourier transform of the aperture function. This concept is fundamental in understanding how light behaves when it encounters obstacles like apertures and slits under far-field conditions.

Rectangular Aperture Diffraction Pattern

A rectangular aperture produces a diffraction pattern that is the product of two one-dimensional diffraction patterns, one for each dimension of the aperture. The pattern is typically described by two sinc functions (arising from the Fourier transform of the rectangular shape), with one sinc function corresponding to the width and the other to the length. As a result, the overall intensity distribution in the diffraction pattern is modulated independently along both the horizontal and vertical directions.

Mapping of Angular Coordinates to the Focal Plane

When a lens is used to produce the diffraction pattern, the angular deviations in the diffraction pattern are mapped onto linear positions at the focal plane of the lens. This mapping means that the angles at which the diffraction minima and maxima occur can be directly converted into positions on the screen using the focal length of the lens. This process is essential for experimental determination of the diffraction characteristics of an aperture.

Transcript

Please give Ace some feedback