Suppose we have 15 parallel long narrow slits in an opaque...

Name: Problem 20, Chapter 10: Optics
Uploaded: 2021-07-30T06:28:53-08:00
Duration: 3 min 20 s
Channel: Khoobchandra Agrawal
Description: Suppose we have 15 parallel long narrow slits in an opaque screen. Furthermore, suppose each slit is separated from the next by a center-to-center distance that is equal to 4 slit widths. Given that a Fraunhofer diffraction pattern appears on a screen, determine the ratio of the irradiance of the second-order principal maximum to that of the zeroth-order maximum.

Suppose we have 15 parallel long narrow slits in an opaque screen. Furthermore, suppose each slit is separated from the next by a center-to-center distance that is equal to 4 slit widths. Given that a Fraunhofer diffraction pattern appears on a screen, determine the ratio of the irradiance of the second-order principal maximum to that of the zeroth-order maximum.

Key Concepts

Irradiance Distribution in Diffraction Patterns

The irradiance distribution in diffraction and interference patterns is determined by both the constructive interference among light from multiple slits and the modulation provided by the diffraction envelope of each slit. The total intensity at any point on the screen is the product of the interference term (which scales with the square of the number of slits when fully constructive) and the envelope function that arises from the finite width of each slit. This explains how higher-order maxima, while still produced by constructive interference, often have lower irradiance than the zeroth-order maximum due to the decreasing amplitude of the envelope function.

Diffraction Envelope

In a multi-slit system, each slit also produces its own diffraction pattern (often called the single-slit diffraction envelope), which modulates the overall intensity distribution. This envelope is typically represented by a sinc-squared function and determines the amplitude of the interference fringes. The finite width of each slit means that even at angles where the interference condition is met, the intensity is reduced if the diffraction envelope is not at its maximum.

Principal Maxima

Principal maxima refer to the bright fringes in an interference pattern where the contributions from all the slits add constructively with maximum intensity. In an array of slits, the condition for the occurrence of these maxima is that the phase difference between light waves from adjacent slits is an integer multiple of 2?. However, the actual observed intensity of each principal maximum is further affected by the single-slit diffraction envelope, which may reduce the irradiance at higher-order maxima compared to the central maximum.

Fraunhofer Diffraction

Fraunhofer diffraction is the approximation used for analyzing diffraction patterns when both the light source and the observation screen are effectively at infinite distances from the diffracting aperture. In this regime, the waves reaching the screen are essentially plane waves, and the resulting pattern can be derived using Fourier transform techniques, which simplifies the analysis of interference and diffraction effects in the far-field.

Multiple-Slit Interference

Multiple-slit interference occurs when coherent light passes through an array of slits, causing the waves emanating from each slit to superpose and produce an interference pattern on a distant screen. This pattern features a series of bright (constructive interference) and dark (destructive interference) regions. The position of the bright fringes (principal maxima) is determined by the condition that the path difference between light from adjacent slits is an integer multiple of the wavelength.

Please give Ace some feedback