Diffraction Patterns Due to a ‘Single-Slit’ and a ‘Circular Aperture

  • Diffraction is the bending and spreading of waves around obstacles or through narrow openings.
  • It occurs when the size of the obstacle or opening is on the same order of magnitude as the wavelength of the wave.
  • Two common examples of diffraction patterns are those produced by a single-slit and a circular aperture.
  • In this lecture, we will explore the diffraction patterns formed by these two types of apertures.
  • We will analyze their characteristics and understand the factors that affect them.

Diffraction patterns from a single-slit

  • A single-slit diffraction pattern is formed when a wave passes through a narrow opening.
  • Light waves passing through the slit interfere with one another, resulting in a pattern of dark and bright fringes.
  • The central bright fringe is wider and more intense, while the outer fringes gradually become fainter and narrower.
  • The width of the central bright fringe is twice that of the other bright fringes.
  • The pattern can be observed on a screen placed behind the single-slit opening.

Equation for the single-slit diffraction pattern

  • The equation for the location of the minima (dark fringes) in the single-slit diffraction pattern is given by:
    • asin(θ) = mλ
  • Where:
    • a is the width of the slit
    • θ is the angle between the central maximum and the minima
    • m is the order of the minima
    • λ is the wavelength of the incident wave

Example: Single-slit diffraction pattern

  • Let’s consider a single-slit with a width of 0.1 mm and an incident light of wavelength 600 nm.
  • Find the angle at which the 2nd order minima occurs. Using the equation asin(θ) = mλ:

0.1 mm * sin(θ) = 2 * 600 nm sin(θ) = 2 * 600 nm / 0.1 mm sin(θ) = 0.012 θ ≈ 0.7° Therefore, the 2nd order minima occurs at an angle of approximately 0.7°.

Diffraction patterns from a circular aperture

  • Diffraction patterns can also be observed when a wave passes through a circular aperture.
  • These patterns are characterized by concentric rings of alternating dark and bright regions.
  • The central bright spot is called the Airy disk, and it is surrounded by concentric bright and dark rings.
  • The size and intensity of the Airy disk depend on the diameter of the aperture and the wavelength of the incident wave.

Equation for the circular aperture diffraction pattern

  • The equation for the radius of the n-th dark ring in the circular aperture diffraction pattern is given by:
    • r = (n * λ * D) / (2 * a)
  • Where:
    • r is the radius of the dark ring
    • n is the order of the ring
    • λ is the wavelength of the incident wave
    • D is the distance between the aperture and the screen
    • a is the diameter of the circular aperture

Example: Circular aperture diffraction pattern

  • Consider a circular aperture with a diameter of 2 mm and an incident light of wavelength 500 nm.
  • The distance between the aperture and the screen is 1 m.
  • Find the radius of the 3rd order dark ring. Using the equation r = (n * λ * D) / (2 * a): r = (3 * 500 nm * 1 m) / (2 * 2 mm) r = 0.375 mm Therefore, the radius of the 3rd order dark ring is 0.375 mm.

Factors affecting diffraction patterns

Diffraction patterns can be affected by various factors, including:

  1. Wavelength of the incident wave: Longer wavelengths result in wider and more spread out diffraction patterns.
  1. Size of the aperture or slit: Smaller apertures or slits produce wider and more intense central maxima.
  1. Distance between the aperture or slit and the screen: Increasing the distance narrows and brightens the diffraction pattern.
  1. Order of the minima or ring: Higher-order minima or rings become narrower and less intense.

Applications of diffraction patterns

Diffraction patterns have several practical applications, such as:

  1. Analyzing the structure of crystals using X-ray diffraction.
  1. Studying the properties of waves and their behavior around obstacles.
  1. Designing optical instruments like microscopes, telescopes, and spectrometers.
  1. Understanding the nature of light and its wave-particle duality.

Summary

  • Diffraction occurs when waves pass through narrow openings or around obstacles.
  • The single-slit diffraction pattern consists of a central bright fringe and alternating dark and bright fringes.
  • The circular aperture diffraction pattern forms concentric dark and bright rings, with a central bright spot called the Airy disk.
  • The diffraction patterns are affected by factors such as wavelength, aperture size, distance, and order of minima or rings.
  • Diffraction has practical applications in crystallography, optics, and fundamental understanding of light.
  1. Diffraction Patterns Due to a ‘Single-Slit’ and a ‘Circular Aperture - Example 2
  • Let’s consider another example to understand the diffraction patterns.
  • In this example, we will analyze the diffraction patterns produced by a single-slit and a circular aperture using different parameters.
  • Through this example, we will deepen our understanding of the factors affecting the diffraction patterns.
  1. Single-slit diffraction pattern - Example 2
  • Consider a single-slit with a width of 0.05 mm and an incident light of wavelength 550 nm.
  • Find the angle at which the 1st order minima occurs. Using the equation asin(θ) = mλ:

0.05 mm * sin(θ) = 1 * 550 nm sin(θ) = 550 nm / 0.05 mm sin(θ) = 0.11 θ ≈ 6.33° Therefore, the 1st order minima occurs at an angle of approximately 6.33°.

  1. Circular aperture diffraction pattern - Example 2
  • Consider a circular aperture with a diameter of 2.5 mm and an incident light of wavelength 600 nm.
  • The distance between the aperture and the screen is 1.5 m.
  • Find the radius of the 4th order dark ring. Using the equation r = (n * λ * D) / (2 * a): r = (4 * 600 nm * 1.5 m) / (2 * 2.5 mm) r = 0.72 mm Therefore, the radius of the 4th order dark ring is 0.72 mm.
  1. Factors affecting diffraction patterns - continued
  • Polarization of the incident wave: The polarization of the wave affects the intensity and shape of the diffraction pattern.
  • Nature of the obstacle or opening: Different shapes and sizes of the obstacle or opening can lead to different diffraction patterns.
  • Interference of multiple waves: When multiple waves interfere, they can amplify or cancel out certain regions in the diffraction pattern.
  1. Applications of diffraction patterns - continued
  1. Holography: Diffraction patterns are used to create holograms, which are three-dimensional images.
  1. Particle size analysis: Diffraction patterns are utilized to determine the size and distribution of particles in various materials.
  1. Astronomy and space research: Diffraction patterns help understand the properties of electromagnetic waves emitted by celestial objects.
  1. Medical imaging techniques: Diffraction patterns are employed in techniques such as X-ray diffraction for imaging and diagnosing medical conditions.
  1. Summary - continued
  • Diffraction patterns occur when waves encounter narrow openings or obstacles.
  • The single-slit diffraction pattern consists of a central bright fringe and alternating dark and bright fringes that become narrower as the order increases.
  • The circular aperture diffraction pattern forms concentric dark and bright rings, with the Airy disk at the center.
  • Factors affecting diffraction patterns include wavelength, aperture size, distance, polarization, nature of the obstacle, and interference of multiple waves.
  • Diffraction patterns find applications in various fields, such as crystallography, optics, holography, particle analysis, astronomy, and medical imaging.
  1. Conclusion
  • Diffraction patterns exhibit fascinating wave behavior and are essential to understanding the properties of waves and light.
  • By analyzing these patterns, we can gain insights into the wave-particle duality of light and its behavior around obstacles.
  • Learning about diffraction patterns enables us to comprehend the principles behind many optical devices and techniques used in various scientific disciplines.
  • With a thorough understanding of diffraction patterns, we can explore and contribute further to the field of wave optics.
  1. Thank you!
  • Thank you for joining this lecture on diffraction patterns.
  • By mastering the concepts and equations related to diffraction, you are well-prepared for the 12th Boards Physics exam.
  • If you have any questions or need further clarification, feel free to ask. Good luck with your exam preparations!
  1. References
  • Insert references here as required.
  1. Questions?
  • Now is the time to ask any questions regarding the topic of diffraction patterns.
  • Feel free to ask for any additional examples, explanations, or clarifications.
  • Let’s enhance our understanding together!
  1. Diffraction Patterns Due to a ‘Single-Slit’ and a ‘Circular Aperture - Example 2
  • In this example, we will explore the diffraction patterns produced by a single-slit and a circular aperture using different parameters.
  • The goal is to deepen our understanding of the factors influencing diffraction patterns.
  • Through these examples, we will reinforce our knowledge of the equations and concepts related to diffraction.
  1. Single-slit diffraction pattern - Example 2
  • Consider a single-slit with a width of 0.05 mm and an incident light of wavelength 550 nm.
  • Find the angle at which the 1st order minima occurs. Using the equation asin(θ) = mλ:

0.05 mm * sin(θ) = 1 * 550 nm sin(θ) = 550 nm / 0.05 mm sin(θ) = 0.11 θ ≈ 6.33° Therefore, the 1st order minima occurs at an angle of approximately 6.33°.

  1. Circular aperture diffraction pattern - Example 2
  • Consider a circular aperture with a diameter of 2.5 mm and an incident light of wavelength 600 nm.
  • The distance between the aperture and the screen is 1.5 m.
  • Find the radius of the 4th order dark ring. Using the equation r = (n * λ * D) / (2 * a): r = (4 * 600 nm * 1.5 m) / (2 * 2.5 mm) r = 0.72 mm Therefore, the radius of the 4th order dark ring is 0.72 mm.
  1. Factors affecting diffraction patterns - continued
  • Polarization of the incident wave: The polarization of the wave affects the intensity and shape of the diffraction pattern.
  • Nature of the obstacle or opening: Different shapes and sizes of the obstacle or opening can lead to different diffraction patterns.
  • Interference of multiple waves: When multiple waves interfere, they can amplify or cancel out certain regions in the diffraction pattern.
  • Distance between the screen and the observer: Changing the distance alters the visibility and sharpness of the diffraction pattern.
  • Coherence of the wavefront: A coherent wavefront produces a cleaner and more distinct diffraction pattern.
  1. Applications of diffraction patterns - continued
  • Holography: Diffraction patterns are used to create holograms, which are three-dimensional images.
  • Particle size analysis: Diffraction patterns are utilized to determine the size and distribution of particles in various materials.
  • Astronomy and space research: Diffraction patterns help understand the properties of electromagnetic waves emitted by celestial objects.
  • Medical imaging techniques: Diffraction patterns are employed in techniques such as X-ray diffraction for imaging and diagnosing medical conditions.
  • Laser applications: Diffraction patterns assist in laser beam shaping, optical data storage, and laser-based communication systems.
  1. Summary - continued
  • Diffraction patterns occur when waves encounter narrow openings or obstacles.
  • The single-slit diffraction pattern consists of a central bright fringe and alternating dark and bright fringes that become narrower as the order increases.
  • The circular aperture diffraction pattern forms concentric dark and bright rings, with the Airy disk at the center.
  • Factors affecting diffraction patterns include wavelength, aperture size, distance, polarization, nature of the obstacle, interference, distance from the screen, and coherence of the wavefront.
  • Diffraction patterns find applications in various fields, such as crystallography, optics, holography, particle analysis, astronomy, medicine, and laser technology.
  1. Conclusion
  • Diffraction patterns provide valuable insights into the behavior of waves and play a crucial role in numerous scientific and technological advancements.
  • By studying and understanding diffraction patterns, we can unravel the nature of light, explore its wave-particle duality, and apply this knowledge in various disciplines.
  • As you prepare for the 12th Boards Physics exam, remember to revise and practice diffraction patterns using different examples and scenarios.
  • If you have any questions or need further assistance, please don’t hesitate to ask.
  • Best of luck with your exam preparations!
  1. References
  • Insert references here as required.
  1. Questions?
  • Now is the time to ask any questions regarding the topic of diffraction patterns.
  • Feel free to seek clarification on any concept, example, or equation discussed today.
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  • Let’s thrive together in Physics!
  1. Thank you!
  • Thank you for joining today’s lecture on diffraction patterns.
  • We hope it provided you with valuable insights and enhanced your understanding of this topic.
  • If you have any additional questions, feel free to reach out.
  • Good luck with your exam preparations and have a great day!