A slit 0.360 mm wide is illuminated by parallel rays of...

A slit 0.360 mm wide is illuminated by parallel rays of light that have a wavelength of 540 nm. The diffraction pattern is observed on a screen that is 1.20 m from the slit. The intensity at the center of the central maximum $(\theta = 0^\circ)$ is $I_0$. (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the distance on the screen from the center of the central maximum to the point where the intensity has fallen to $I_0$/2?

Key Concepts

Small Angle Approximation

The small angle approximation is used when the angle ? is small enough that sin ? is approximately equal to tan ?, which in turn is approximately equal to ? in radians. This simplification allows for easier calculation of the positions of diffraction features on a screen, where the linear distance y can be approximated by y = L?, with L being the distance from the slit to the screen.

Intensity Distribution in Single-Slit Diffraction

The intensity distribution in a single-slit diffraction pattern is described by the equation I(?) = I? (sin ?/?)², where ? = (?a sin ?)/? and I? is the central maximum intensity. This formula explains how the brightness diminishes from the center, with specific conditions corresponding to the first minimum and other points where the intensity drops to a fraction of I?.

Diffraction

Diffraction is the bending and spreading of waves around obstacles or through openings. In optics, it explains how light waves spread out after passing through narrow apertures, leading to characteristic patterns of light and dark regions on a screen.

Single-Slit Diffraction

Single-slit diffraction refers to the phenomenon observed when coherent light passes through a narrow slit, producing a central bright fringe flanked by alternating dark and bright fringes. The positions of the dark (minimum) fringes are given by the condition a sin ? = m?, where a is the slit width, ? is the angular deviation, ? is the wavelength, and m is a non-zero integer.

Fraunhofer Diffraction

Fraunhofer diffraction describes the far-field diffraction pattern observed when the light source and the screen are sufficiently far from the diffracting aperture, or when lenses are used to simulate this condition. Under these circumstances, the diffraction pattern exhibits well-defined interference fringes that can be analyzed using simple trigonometric relationships.

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