Glossary

27.6 Limits of Resolution: The Rayleigh Criterion

3 min read•june 18, 2024

The sets the bar for optical resolution, determining how well we can distinguish tiny details. It's all about the dance between light waves and apertures, with and size playing key roles in sharpening our view.

Diffraction is the party crasher of perfect images, spreading light and blurring fine points. But we're not helpless – shorter wavelengths and bigger lenses can push back against diffraction's limits, helping us see the universe more clearly.

Limits of Resolution and the Rayleigh Criterion

Rayleigh criterion for resolution limits

Top images from around the web for Rayleigh criterion for resolution limits

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

1 of 3

Top images from around the web for Rayleigh criterion for resolution limits

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

  • Limits of Resolution: The Rayleigh Criterion | Physics View original

    Is this image relevant?

1 of 3
  • Defines minimum between two distinguished as separate entities by an optical system
  • Based on produced by point sources when light passes through aperture or lens
  • Two point sources considered just resolved when central maximum of one source's diffraction pattern falls on first minimum of other source's diffraction pattern
    • Noticeable dip in intensity between two sources allows them to be perceived as separate
  • Sets fundamental limit on of optical systems (telescopes, microscopes, cameras)
    • Determines smallest angular separation between two objects clearly distinguished by system
  • Factors affecting resolution limit according to :
    • Wavelength of light used (λ\lambda): shorter wavelengths allow better resolution (visible light, X-rays)
    • Diameter of aperture or lens (DD): larger apertures or lenses improve resolution (astronomical telescopes, wide-aperture cameras)
  • : measure of an optical system's ability to distinguish small details of an object

Minimum angular separation calculations

  • : θ=1.22λD\theta = 1.22 \frac{\lambda}{D}
    • θ\theta: minimum angular separation between two point sources resolved (radians)
    • λ\lambda: wavelength of light used (meters)
    • DD: diameter of aperture or lens (meters)
  • Calculation steps:
    1. Identify wavelength of light (λ\lambda) and diameter of aperture or lens (DD)
    2. Substitute values into Rayleigh criterion formula: θ=1.22λD\theta = 1.22 \frac{\lambda}{D}
    3. Calculate minimum angular separation in radians
    4. If required, convert angular separation from radians to degrees by multiplying result by 180π\frac{180}{\pi}
  • Example: with lens diameter of 10 cm observing light with wavelength of 550 nm
    • θ=1.22550×10−9m0.1m=6.71×10−6\theta = 1.22 \frac{550 \times 10^{-9} m}{0.1 m} = 6.71 \times 10^{-6} radians (0.00038 degrees)

Diffraction effects on image resolution

  • Diffraction: light waves bending and spreading out when encountering obstacle or aperture
  • In optical instruments, diffraction occurs when light passes through aperture or lens
    • Forms diffraction patterns: central bright spot () surrounded by alternating dark and bright rings
  • Size of Airy disk and spacing between rings determined by wavelength of light and diameter of aperture or lens
    • Smaller apertures or lenses result in larger Airy disks and more spread-out diffraction patterns
  • Diffraction limits ability of optical instruments to resolve fine details ()
    • Images of point sources spread out into Airy disks
    • When two point sources too close together, Airy disks overlap, making it difficult to distinguish as separate entities
  • Rayleigh criterion based on diffraction-limited nature of optical systems
    • Provides quantitative measure of minimum angular separation needed to resolve two point sources
  • Improving resolution and minimizing diffraction effects:
    • Use shorter wavelengths of light: produce smaller Airy disks, allow better resolution (ultraviolet, X-rays)
    • Increase diameter of aperture or lens: reduces size of Airy disk, improves resolution (large telescopes, objectives)
    • Employ advanced techniques: adaptive optics, post-processing algorithms to compensate for diffraction effects

Advanced Resolution Concepts

  • : measure of how rapidly image intensity changes over distance, affecting resolution capabilities
  • : alternative to Rayleigh criterion, defines resolution limit when intensity between two point sources is flat
  • : describes how different spatial frequencies are transmitted through an optical system, impacting overall image quality and resolution

Key Terms to Review (24)

Acceleration due to gravity: Acceleration due to gravity is the rate at which an object accelerates when falling freely under the influence of Earth's gravitational pull. Its standard value on Earth's surface is approximately $9.81 \text{ m/s}^2$.
Airy Disk: The Airy disk is the diffraction pattern created by the interference of light waves passing through a circular aperture, such as a telescope lens or pupil. It is a fundamental concept in the study of the limits of optical resolution and the Rayleigh criterion.
Angular Resolution: Angular resolution refers to the ability of an optical instrument, such as a telescope or microscope, to distinguish between two closely spaced objects or features. It is a measure of the smallest angular separation that can be detected, allowing the instrument to resolve fine details in an image.
Angular Separation: Angular separation is the measure of the angle between two objects as viewed from a specific point, often expressed in degrees or radians. This concept is crucial in understanding how closely spaced two points of light, such as stars or other celestial bodies, can be before they appear as one due to the limitations of resolution in optical systems. It connects to the limits of resolution, particularly in regards to the Rayleigh criterion, which sets the threshold for distinguishing between two closely located objects.
Aperture: Aperture refers to the opening or diameter of a lens or mirror in an optical instrument, such as a telescope or camera. It is a critical parameter that determines the amount of light that can enter the system and affects the instrument's resolution and light-gathering capabilities.
Camera: A camera is an optical instrument used to capture and record visual images. It is a device that collects and focuses light to create a photograph or digital image. Cameras are an essential tool in the context of the Rayleigh Criterion, which is a fundamental concept in the study of the limits of resolution in optical systems.
De Broglie wavelength: The de Broglie wavelength is the wavelength associated with a particle and is inversely proportional to its momentum. It highlights the wave-particle duality of matter.
Diffraction Limit: The diffraction limit is a fundamental constraint that sets the maximum resolution or smallest distinguishable detail that can be achieved by an optical system, such as a telescope or microscope. It arises from the wave-like nature of light and its interaction with the aperture or lens of the optical device.
Diffraction Patterns: Diffraction patterns are the distinctive interference patterns created when waves, such as light or sound, encounter an obstacle or aperture. These patterns arise from the bending and spreading of waves as they pass through or around the edges of a barrier, revealing important information about the nature and properties of the waves.
Interference: Interference is the phenomenon that occurs when two or more waves interact with each other, resulting in the creation of a new wave pattern. This concept is central to understanding the behavior of light and other forms of electromagnetic radiation in the context of Young's Double Slit Experiment and the Rayleigh Criterion for the limits of resolution.
Lord Rayleigh: Lord Rayleigh, also known as John William Strutt, was a renowned British physicist who made significant contributions to the field of optics, particularly in the understanding of the limits of resolution. His work on the Rayleigh criterion, which establishes the minimum angular separation required for two point sources to be distinguished as separate, is a fundamental concept in the study of wave interference and diffraction.
Microscope: A microscope is an optical instrument used to magnify and observe small objects or details that are not visible to the naked eye. It is a crucial tool in various scientific fields, including biology, materials science, and medicine, enabling the detailed examination of microscopic structures and phenomena.
Optical Transfer Function: The optical transfer function (OTF) is a mathematical representation that describes how an optical system transfers input spatial frequencies to output spatial frequencies. It provides crucial insights into the resolution and contrast capabilities of an optical system, linking the performance of imaging devices to their ability to reproduce details in an object. The OTF is composed of two parts: the modulation transfer function (MTF) and the phase transfer function (PTF), both of which are essential for understanding limits of resolution in imaging systems.
Point Sources: A point source is a localized and discrete source of radiation, light, or other energy that can be treated as originating from a single point in space. In the context of the Rayleigh criterion, point sources are the basis for understanding the limits of resolution in optical systems.
Rayleigh criterion: The Rayleigh criterion defines the minimum angular separation at which two point light sources can be resolved as distinct. It is determined by the diffraction limit of an optical system.
Rayleigh Criterion: The Rayleigh criterion is a fundamental principle in optics that defines the limit of resolution for optical instruments, such as telescopes and microscopes. It establishes the minimum angular separation required for two point sources to be distinguished as separate entities by an optical system.
Rayleigh Criterion Formula: The Rayleigh criterion is a formula used to determine the minimum angular separation between two point sources of light, such as stars, for which they can still be visually distinguished as separate objects. It is a fundamental concept in the study of the limits of optical resolution.
Resolving Power: Resolving power refers to the ability of an optical system, like a telescope, to distinguish between two closely spaced objects. It is a crucial feature because it determines the clarity and detail of the images produced by the optical system. Higher resolving power means finer details can be seen, which is essential in astronomical observations where distant celestial bodies need to be differentiated from each other.
Sparrow Criterion: The Sparrow criterion, also known as the Rayleigh criterion, is a fundamental principle in optics that defines the limit of resolution for an optical system. It establishes the minimum angular separation required between two point sources for them to be distinguished as separate entities by the optical system.
Spatial Frequency: Spatial frequency refers to the level of detail present in a visual image, typically measured in cycles per unit distance. It is a crucial concept in understanding how resolution limits are determined, particularly in optics and imaging systems. Higher spatial frequencies indicate finer details, while lower frequencies correspond to coarser structures, affecting the ability to resolve two closely spaced objects.
Telescope: A telescope is an optical instrument that uses lenses, mirrors, or a combination of both to gather and focus light from distant objects, allowing for their observation and study. Telescopes are essential tools in the field of astronomy, enabling the exploration of the universe beyond our immediate surroundings.
Wavelength: Wavelength is a fundamental characteristic of waves, representing the distance between consecutive peaks or troughs in a wave. It is a crucial parameter that describes the spatial extent of a wave and is closely related to other wave properties such as frequency and speed.
θ (Theta): Theta (θ) is a fundamental mathematical symbol used to represent an angle in various contexts, including rotation, wave interference, and resolution limits. It is a Greek letter that serves as a variable or parameter to quantify and analyze angular relationships and phenomena.
λ: Lambda (λ) is a Greek letter that represents the wavelength of a wave, which is the distance between two consecutive peaks or troughs of the wave. This term is particularly important in the context of Young's Double Slit Experiment and the Rayleigh Criterion, as the wavelength of light is a crucial factor in understanding the behavior of light and the limits of optical resolution.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide