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Chapter 36: Problem 37

Monochromatic light with wavelength 620 nm passes through a circular aperture with diameter 7.4 \(\mu\)m. The resulting diffraction pattern is observed on a screen that is 4.5 m from the aperture. What is the diameter of the Airy disk on the screen?

Short Answer

Expert verified
The diameter of the Airy disk is approximately 0.9216 m.

Step by step solution

01

Understanding the Problem

We need to find the diameter of the Airy disk produced by light diffracting through a circular aperture. The key formula to use is for the angular width of the central maximum in the Airy disk: \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength and \( D \) is the aperture diameter.
02

Calculate Angular Width

Use the formula for the angular width \( \theta = 1.22 \frac{\lambda}{D} \). Here, \( \lambda = 620 \times 10^{-9} \) m and \( D = 7.4 \times 10^{-6} \) m. Substitute these values:\[ \theta = 1.22 \times \frac{620 \times 10^{-9}}{7.4 \times 10^{-6}} \approx 1.024 \times 10^{-1} \text{ radians} \]
03

Determine the Radius on the Screen

The radius \( r \) of the Airy disk on the screen is given by \( r = L \cdot \theta \), where \( L \) is the distance to the screen (4.5 m). Thus:\[ r = 4.5 \cdot 1.024 \times 10^{-1} \approx 0.4608 \text{ m} \]
04

Calculate the Diameter of the Airy Disk

The diameter of the Airy disk is twice the radius:\[ \text{Diameter} = 2 \cdot r = 2 \times 0.4608 = 0.9216 \text{ m} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Airy disk
The Airy disk is a fundamental concept in optics, representing the central bright spot seen in a diffraction pattern. When light waves interfere, the pattern forms due to diffraction through an aperture, like a circular opening. The Airy disk is the region at the center of this pattern, appearing bright because light waves constructively interfere there. It is named after Sir George Biddell Airy, who first described this phenomena.
Here are key points about the Airy disk:
  • It forms the central maximum of a diffraction pattern.
  • The size of the Airy disk is dependent on the aperture size and the wavelength of light used.
  • It plays a crucial role in determining the resolution of optical instruments.
The diameter of this disk is crucial for understanding the resolution limits of devices like telescopes and microscopes, as it sets the smallest detail that can be observed clearly.
circular aperture
A circular aperture is simply a round opening through which light passes. This shape is significant in diffraction because it affects how light waves are distributed after passing through. Thus, when monochromatic light enters a circular aperture, it spreads out and forms a distinct diffraction pattern characterized by concentric rings, with the Airy disk at its center.
Key characteristics of circular apertures include:
  • The shape influences diffraction patterns profoundly, producing circular patterns due to symmetry.
  • Aperture size ( D ) is inversely related to the size of the diffraction pattern on a screen.
In optical devices like cameras and telescopes, the aperture size can significantly affect image sharpness and clarity due to this diffraction pattern, reinforcing its importance in designing lenses and other components.
monochromatic light
Monochromatic light refers to light of a single wavelength or color. It is crucial in many scientific experiments because it offers consistency and simplicity in studying wave behaviors like diffraction.
Let's highlight a few points about monochromatic light:
  • It does not have variations in wavelength, providing a stable pattern of interference and diffraction.
  • Common sources include lasers and particular specialized light filters.
  • Its wavelength (denoted as \( \lambda \)) significantly affects the visual outcomes, such as the size of the Airy disk.
Using monochromatic light helps us accurately apply equations in optical physics, like calculating diffraction effects, because there are no complications from mixed wavelengths.
wavelength
Wavelength, denoted by \( \lambda \), represents the distance between two consecutive peaks of a wave. In optical physics, it is measured in nanometers (nm) or micrometers (碌m) and is crucial in determining how light behaves when interacting with objects.In the context of diffraction through apertures, the wavelength of light directly affects the diffraction pattern's features, such as the size of the Airy disk.
Consider these points about wavelength:
  • Longer wavelengths produce larger diffraction patterns, meaning a bigger Airy disk.
  • The interaction of light of different wavelengths with apertures can result in colorful patterns if not monochromatic.
  • In the problem scenario, the wavelength given is 620 nm, which serves as a key parameter in calculations involving the diffraction pattern.
Thus, understanding wavelength allows predictions on how light will diffract and how patterns will manifest, making it an essential concept in optics.

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Most popular questions from this chapter

The Hubble Space Telescope has an aperture of 2.4 m and focuses visible light (380-750 nm). The Arecibo radio telescope in Puerto Rico is 305 m (1000 ft) in diameter (it is built in a mountain valley) and focuses radio waves of wavelength 75 cm. (a) Under optimal viewing conditions, what is the smallest crater that each of these telescopes could resolve on our moon? (b) If the Hubble Space Telescope were to be converted to surveillance use, what is the highest orbit above the surface of the earth it could have and still be able to resolve the license plate (not the letters, just the plate) of a car on the ground? Assume optimal viewing conditions, so that the resolution is diffraction limited.

Red light of wavelength 633 nm from a helium-neon laser passes through a slit 0.350 mm wide. The diffraction pattern is observed on a screen 3.00 m away. Define the width of a bright fringe as the distance between the minima on either side. (a) What is the width of the central bright fringe? (b) What is the width of the first bright fringe on either side of the central one?

A slit \(0.240 \mathrm{~mm}\) wide is illuminated by parallel light rays of wavelength \(540 \mathrm{nm} .\) The diffraction pattern is observed on a screen that is \(3.00 \mathrm{~m}\) from the slit. The intensity at the center of the central maximum \(\left(\theta=0^{\circ}\right)\) is \(6.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\). (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

If you can read the bottom row of your doctor's eye chart, your eye has a resolving power of 1 arcminute, equal to \(1\over{60}\) degree. If this resolving power is diffraction limited, to what effective diameter of your eye's optical system does this correspond? Use Rayleigh's criterion and assume \(\lambda\) = 550 nm.

Monochromatic light with wavelength 490 nm passes through a circular aperture, and a diffraction pattern is observed on a screen that is 1.20 m from the aperture. If the distance on the screen between the first and second dark rings is 1.65 mm, what is the diameter of the aperture?
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