Topic 9: Diffraction; Diffraction by a Circular Aperture; Airy’s Disc
Introduction
Diffraction is a fundamental phenomenon of wave optics that describes the bending and spreading of waves when they encounter obstacles or apertures. Unlike reflection and refraction, diffraction reveals the wave nature of light most explicitly, affecting image formation, resolution of optical instruments, and many practical optical systems.
This detailed article covers the principles of diffraction, special consideration of diffraction by circular apertures, and the concept of Airy’s disc—the fundamental diffraction pattern formed by circular openings, crucial to understanding the limits of resolution in optics.
1. What is Diffraction?
Definition
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| Diffraction of light through wide and narrow gap |
Diffraction is the phenomenon in which waves bend around obstacles or spread after passing through small apertures, causing deviations from geometric optics predictions. This bending leads to characteristic patterns of light and dark regions due to interference between different parts of the wavefront.
Historical Context
Diffraction was first systematically studied by Francesco Maria Grimaldi in the 17th century, who coined the term "diffraction." Thomas Young and Augustin-Jean Fresnel later developed quantitative wave theories that explained diffraction patterns through interference of secondary wavelets.
Huygens-Fresnel Principle
Diffraction is explained by the Huygens-Fresnel principle, which states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets. When parts of the wavefront are obstructed or limited by an aperture, the secondary wavelets interfere, producing diffraction patterns.
---2. Types of Diffraction
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| Fraunhofer and Fresnel Diffraction |
- Fresnel Diffraction (Near-Field Diffraction): Observed when the source or the screen is at a finite distance from the aperture. The wavefront curvature is important, and the pattern changes with distance.
- Fraunhofer Diffraction (Far-Field Diffraction): Occurs when both source and observation screen are effectively at infinite distance or when lenses are used to create parallel incident and observation beams. The pattern becomes simpler and stable, described by Fourier transforms of aperture shapes.
3. Diffraction by a Single Slit (Brief Overview)
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| Diffraction by a Single Slit |
When light passes through a narrow slit of width a, it spreads out rather than continuing in a straight line. The Fraunhofer diffraction pattern consists of a central bright maximum with successive dark and bright fringes on either side.
The central maximum is much wider and brighter than the secondary maxima, defining the diffraction envelope for more complex systems.
---4. Diffraction by a Circular Aperture
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| Diffraction by a Circular Aperture |
Physical Description
Circular apertures are common in optical instruments like telescopes, microscopes, and cameras, where light passes through a circular opening. Diffraction through such apertures leads to unique patterns governed by the geometry of the circle.
Formation of Diffraction Pattern
When monochromatic light passes through a circular aperture of diameter D, the diffraction pattern observed on a distant screen consists of a bright central spot surrounded by concentric dark and bright rings called the Airy pattern.
Mathematical Treatment
The intensity distribution I(θ) is expressed as:
I(θ) = I0 × [ (2 J1(k a sinθ)) / (k a sinθ) ]2
- I0: Intensity at the center (θ = 0)
- J1: First-order Bessel function of the first kind
- k = (2π / λ): Wave number
- a = D / 2: Radius of aperture
Angular Radius of Airy Disc
The radius of the first dark ring surrounding the central maximum is given by:
sinθ = 1.22 × (λ / D)
For small angles:
θ ≈ 1.22 × (λ / D)
The amplitude distribution of the diffracted light is given by the Fraunhofer diffraction integral, which involves Bessel functions for circular apertures. The intensity distribution I(θ) is expressed as:
---5. Airy’s Disc
Definition
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| Airy's Disc |
The bright central spot in the diffraction pattern produced by a circular aperture is known as the Airy disc. It is named after George Biddell Airy, who first described this pattern mathematically.
Angular Radius of Airy Disc
The radius of the first dark ring surrounding the central maximum (Airy disc radius) is given by:
sin θ = 1.22 × (λ / D)
For small angles:
θ ≈ 1.22 × (λ / D)
This angle defines the size of the Airy disc.
Physical Interpretation
The Airy disc represents the fundamental limit to the resolution of any optical system with a circular aperture. Two point sources closer than the Airy disc radius cannot be distinctly resolved, as their diffraction patterns overlap significantly.
Intensity Distribution of Airy Pattern
The intensity drops rapidly from the center, with the first minimum defining the boundary of the central bright spot. The intensity of the rings decreases with increasing radius.
---6. Applications and Significance of Airy’s Disc
Limit of Resolution
The size of the Airy disc limits the resolving power of optical instruments such as telescopes and microscopes. It defines how close two objects can be while still appearing as separate.
Rayleigh Criterion
According to Rayleigh’s criterion, two point sources are just resolved when the principal maximum of one Airy pattern coincides with the first minimum of the other. This is given by:
θmin = 1.22 × (λ / D)
Optical System Design
Understanding Airy disc size helps optical engineers optimize aperture size, focal length, and wavelength to improve image quality and resolution.
---7. Factors Affecting Diffraction Patterns
- Aperture Size (D): Smaller apertures produce larger Airy discs, reducing resolution.
- Wavelength (λ): Longer wavelengths increase diffraction effects, enlarging Airy discs.
- Distance to Screen (Fraunhofer): Far-field diffraction pattern stability depends on distance.
- Coherence: Higher coherence improves fringe visibility and pattern contrast.
8. Experimental Observations
Diffraction by circular apertures can be observed using pinholes illuminated by lasers. The resulting Airy pattern is seen on a screen as a bright central spot with faint rings. By measuring the diameter of the Airy disc and varying aperture size or wavelength, the theoretical predictions can be confirmed.
---9. Practical Importance in Optometry
Diffraction limits the resolving power of the human eye and optical instruments like slit lamps and fundus cameras. Understanding diffraction and Airy discs helps optometrists and optical engineers interpret the limits of detail perception and devise better optical aids.
---10. Summary
Diffraction is a core wave optics phenomenon characterized by the bending and spreading of waves near apertures and obstacles. Diffraction by a circular aperture produces the Airy pattern, whose central bright spot—the Airy disc—defines the fundamental resolution limit of optical systems. Mastery of these concepts is essential for understanding image formation and resolution constraints in optics and optometry.
This extensive article covers diffraction fundamentals, circular aperture diffraction, Airy’s disc, and their significance for optical resolution and instrument design, forming a crucial part of physical optics knowledge.
Topic 10: Resolution of an Instrument (Telescope, for example); Rayleigh’s Criterion
Introduction
Resolution is a fundamental property of optical instruments that determines their ability to distinguish between two closely spaced objects as separate entities. Whether it is a telescope observing distant stars, a microscope viewing microscopic structures, or the human eye discerning fine details, resolution sets the ultimate limit of clarity and detail. This article explores the concept of resolution, with a special focus on telescopes, and thoroughly explains Rayleigh’s criterion—the widely accepted standard for defining resolution limits in optics.
1. What is Resolution?
Definition
Resolution refers to the minimum angular or spatial separation between two point sources such that they can still be seen as distinct rather than merged into one blurred image. In simpler terms, it is the measure of an optical system’s ability to reveal fine detail.
Types of Resolution
- Angular Resolution: The smallest angle between two objects seen as separate. Common in telescopes and the human eye.
- Spatial Resolution: The smallest distance between two points that can be resolved, relevant in microscopy and imaging.
2. Factors Affecting Resolution
- Aperture Size (D): Larger apertures improve resolution by reducing diffraction effects.
- Wavelength (λ): Shorter wavelengths yield better resolution.
- Quality of Optics: Imperfections and aberrations degrade resolution.
- Atmospheric Conditions: For telescopes, atmospheric turbulence (seeing) affects resolution.
3. Diffraction Limit and Resolution
No optical system can achieve infinite resolution due to diffraction, the bending of light waves around apertures and edges. Diffraction causes point sources to be imaged as blurred spots rather than perfect points. The shape of these spots depends on the aperture shape, typically producing an Airy pattern in circular apertures.
The size of the diffraction spot limits the minimum separation at which two points can be distinguished.
---Rayleigh Criterion for Resolution
Definition
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| Rayleigh Criterion |
Lord Rayleigh proposed a criterion to define when two point sources are just resolvable. According to Rayleigh’s criterion, two sources are considered just resolved when the principal maximum (brightest spot) of one Airy pattern coincides with the first minimum (dark ring) of the other.
Mathematical Expression
For a circular aperture, the minimum resolvable angular separation θ_min is:
θmin = 1.22 × (λ / D)
- λ: Wavelength of light
- D: Diameter of the aperture
Interpretation
The factor 1.22 arises from the first zero of the Bessel function that describes the Airy pattern. This formula defines the diffraction limit—the fundamental physical limit on resolution for any circular aperture.
---5. Resolution in Telescopes
Angular Resolution of a Telescope
Using Rayleigh’s criterion:
θmin = 1.22 × (λ / D)
For example, a telescope with a 100 mm aperture observing light of wavelength 550 nm (green light) has:
θmin = 1.22 × (550 × 10-9 / 0.1) = 6.7 × 10-6 radians ≈ 1.38 arc seconds
Practical Implications
- The telescope cannot distinguish two stars closer than 1.38 arcseconds under ideal conditions.
- Atmospheric turbulence often limits ground-based telescopes to worse resolution (seeing limited).
- Space telescopes overcome atmospheric effects to achieve near-diffraction-limited performance.
Factors Limiting Telescope Resolution
- Diffraction Limit: Fundamental physical limit as described.
- Atmospheric Seeing: Turbulence blurs images beyond diffraction limit.
- Optical Aberrations: Imperfect lens or mirror shapes distort images.
6. Resolution in Microscopes
Resolution in Microscopes
Microscopes also face diffraction-limited resolution determined by the wavelength and numerical aperture (NA) of the objective lens. The spatial resolution δ is given by:
The spatial resolution δ is given by:
δ = 0.61 × (λ / NA)
Where:
- λ = Wavelength of light used
- NA = Numerical Aperture of the objective lens
Here, the numerical aperture depends on the refractive index and acceptance angle of the objective. This sets the smallest distance between two points that can be distinguished in the specimen.
---7. Other Resolution Criteria
Abbe’s Criterion
Ernst Abbe developed a criterion similar to Rayleigh’s, often used in microscopy. It relates resolution to the wavelength and numerical aperture, emphasizing the role of light collection angle.
Sparrow’s Criterion
Sparrow proposed a criterion based on the disappearance of the dip between two peaks in the intensity distribution, leading to a slightly more stringent resolution limit than Rayleigh’s.
Practical Choice of Criterion
Rayleigh’s criterion remains widely used due to its simplicity and physical meaning, but in some fields, other criteria are preferred based on specific imaging goals.
---8. Improving Resolution
Increasing Aperture Size
Larger apertures reduce the diffraction limit, improving resolution. This is why astronomical telescopes have enormous mirrors.
Using Shorter Wavelengths
Since resolution scales inversely with wavelength, using ultraviolet or electron beams (in electron microscopes) achieves much higher resolution.
Aperture Synthesis and Adaptive Optics
Techniques like interferometry combine light from multiple telescopes to simulate a large aperture. Adaptive optics corrects atmospheric distortion in real-time, pushing ground telescopes toward their diffraction limit.
---9. Mathematical Derivation of Rayleigh’s Criterion
Starting from the Airy pattern intensity distribution given by:
I(θ) = I0 × [ (2 J1(k a sin θ)) / (k a sin θ) ]2
The first minimum of this function occurs where the argument of the Bessel function satisfies:
k a |sin θ| = 3.832
Since k = 2π / λ and a = D / 2, we get:
(2Ï€/λ) × (D/2) × |sin θ| = 3.832 &Rightarrow |sin θ| = 1.22 × (λ / D)
This gives the angular radius of the Airy disc’s first dark ring and sets the minimum resolvable angle.
---10. Examples and Numerical Calculations
Example 1: Resolving Two Stars
Given a telescope aperture of 200 mm and wavelength 600 nm, calculate the minimum angular separation of two stars that can be resolved.
θmin = 1.22 × (600 × 10-9 / 0.2) = 3.66 × 10-6 radians ≈ 0.75 arc seconds
Example 2: Resolving Features in a Microscope
Using light of wavelength 500 nm and an objective with numerical aperture 1.4, the resolution limit is:
δ = (0.61 × 500 × 10-9) / 1.4 = 2.18 × 10-7 meters = 218 nm
---11. Summary
Resolution is a key characteristic of any optical instrument, defining its ability to distinguish fine detail. Diffraction imposes a fundamental limit on resolution, expressed quantitatively by Rayleigh’s criterion, which relates resolution to wavelength and aperture size. Understanding and applying these concepts is essential in designing and interpreting the performance of telescopes, microscopes, and other optical devices.
Modern techniques like adaptive optics and aperture synthesis continue to push the boundaries set by diffraction, enabling remarkable imaging capabilities in astronomy and microscopy.
This article provides a thorough understanding of resolution limits and Rayleigh’s criterion, complete with mathematical derivations and practical examples for students of optics and optometry.
Topic 11: Scattering; Rayleigh Scattering; Tyndall Effect
Introduction
Scattering of light is a vital phenomenon in optics, explaining a wide variety of natural and technological effects such as the blue color of the sky, the red hues at sunset, and the visibility of beams in fog. It occurs when light encounters particles or inhomogeneities smaller or comparable to its wavelength, causing redirection of light in different directions.
This article explores the nature of light scattering with detailed emphasis on Rayleigh scattering and the Tyndall effect, their underlying physics, mathematical formulation, and applications relevant to optics and optometry.
---1. What is Scattering?
Definition
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| Scattering |
Scattering refers to the process by which a light wave or other electromagnetic radiation is forced to deviate from a straight trajectory due to non-uniformities or particles in the medium. This causes the light to spread out in various directions rather than simply passing through or being absorbed.
Types of Scattering Based on Particle Size
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| Rayleigh, Mie Scattering and Mie Scattering with larger particles |
- Rayleigh Scattering: Scattering by particles much smaller than the wavelength of light (typically d << λ).
- Mie Scattering: Scattering by particles comparable to or larger than the wavelength (such as water droplets, dust).
- Non-selective Scattering: Scattering by particles much larger than the wavelength, leading to wavelength-independent scattering.
2. Rayleigh Scattering
Physical Explanation
Rayleigh scattering arises when light interacts with particles significantly smaller than its wavelength. These small particles act as oscillating dipoles induced by the incident electromagnetic wave, reradiating light in different directions. Because the particle size is much smaller than the wavelength, the scattered intensity depends strongly on the wavelength.
Mathematical Formulation
The intensity of scattered light I at angle θ is given by Rayleigh’s scattering formula:
I(θ) = I0 × (1 + cos2θ) / (2 R2) × [ (2Ï€ / λ)4 × ((n2 - 1) / (n2 + 2))2 × (d / 2)6 ]
- I0: Intensity of incident light
- R: Distance from the scattering particle
- λ: Wavelength of light
- n: Refractive index of the particle relative to the medium
- d: Diameter of the particle
Wavelength Dependence
The scattered intensity is proportional to 1/λ^4. This means shorter wavelengths (blue light) scatter much more than longer wavelengths (red light). This explains many natural phenomena such as:
- The blue color of the clear daytime sky.
- Red and orange colors during sunrise and sunset (due to scattering of shorter wavelengths out of the direct path).
3. Polarization in Rayleigh Scattering
Rayleigh scattered light is partially polarized. At 90 degrees to the incident beam, scattered light is strongly polarized perpendicular to the scattering plane. This effect is used in atmospheric optics and remote sensing.
---4. Applications of Rayleigh Scattering
- Atmospheric Optics: Understanding sky color, twilight, and atmospheric visibility.
- Optical Fiber Communication: Rayleigh scattering causes signal attenuation and noise in fibers.
- Biomedical Optics: Used in tissue imaging and characterization.
- Remote Sensing: Polarization measurements aid in identifying atmospheric particles.
5. Tyndall Effect
Definition
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| Phenomenon of Tyndall Effect |
The Tyndall effect is the scattering of visible light by particles in a colloidal suspension or very fine suspensions, making the path of the light beam visible.
Physical Basis
The particles in colloids are larger than those responsible for Rayleigh scattering but still small enough to scatter light effectively. The scattered light is visible as a beam when viewed from the side, unlike pure solutions where the beam is invisible.
Differences Between Rayleigh Scattering and Tyndall Effect
| Aspect | Rayleigh Scattering | Tyndall Effect |
|---|---|---|
| Particle Size | Much smaller than wavelength (< λ/10) | Comparable to wavelength |
| Type of Medium | Gases, very fine particles | Colloidal suspensions |
| Visibility of Light Beam | Usually not visible | Visible as scattered beam |
| Wavelength Dependence | Strong (1/λ⁴) | Less wavelength dependent |
6. Examples of Tyndall Effect
- Light beam visible in fog or smoke.
- Blue color of smoke from burning organic material.
- Visibility of beams in dusty or smoky rooms.
- Appearance of milk or other colloidal suspensions.
7. Importance in Optometry and Vision Science
Scattering phenomena affect ocular media transparency and visual quality. For example:
- Corneal and Lens Clarity: Scattering by opacities or cataracts causes glare and reduced vision.
- Blue Sclera: Resulting from altered scattering properties of the sclera.
- Retinal Imaging: Scattering affects image contrast in fundus photography and optical coherence tomography (OCT).
8. Controlling and Measuring Scattering
In optical systems, scattering can cause noise and degrade image quality. Methods to control scattering include:
- Using purer materials and cleaner environments.
- Anti-reflection coatings.
- Optical filters and polarizers to reduce scattered light effects.
Scattering measurements help characterize particle size and concentration in medical and environmental applications.
---9. Summary
Scattering is a key optical phenomenon explaining many natural and practical observations. Rayleigh scattering governs the color of the sky and polarization effects, while the Tyndall effect reveals light scattering in colloids, making beams visible. Both phenomena are integral to optics, atmospheric science, and vision science, impacting optical instrument design and clinical diagnostics.
This comprehensive article covers the theory, mathematical basis, differences, and applications of scattering, Rayleigh scattering, and the Tyndall effect for a clear understanding suited to exam preparation and practical optics.
Topic 12: Fluorescence and Phosphorescence
Introduction
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| Fluorescence and Phosphresence |
Fluorescence and phosphorescence are two important types of photoluminescence—processes by which certain materials absorb light energy and then re-emit it as visible light. Both phenomena are widely observed in nature and find applications in optics, medicine, and various industries. Understanding their physical mechanisms, differences, and practical implications is essential in optics and optometry.
---1. What is Photoluminescence?
Photoluminescence is the emission of light from a material after it absorbs photons (light energy). When photons of sufficient energy excite electrons in a substance, the electrons jump to higher energy states. As they return to their original states, they release energy as light. The timing and mechanism of this emission classify photoluminescence into fluorescence or phosphorescence.
---2. Fluorescence
Definition
Fluorescence is the rapid emission of light by a substance that has absorbed electromagnetic radiation, occurring almost instantaneously (within about 10-8 seconds). The emitted light usually has a longer wavelength (lower energy) than the absorbed light.
Physical Mechanism
- Absorption: A molecule absorbs a photon, exciting an electron from the ground state (S0) to a higher singlet excited state (Sn).
- Internal Conversion: The excited electron quickly relaxes to the lowest vibrational level of the first excited singlet state (S1) without photon emission.
- Fluorescence Emission: The electron returns from S1 to the ground state S0, emitting a photon.
Jablonski Diagram
The Jablonski diagram visually represents these electronic transitions including absorption, fluorescence, and other processes.
Characteristics of Fluorescence
- Emission occurs rapidly (nanoseconds).
- Emission wavelength is longer than excitation wavelength (Stokes shift).
- Stops immediately when excitation source is removed.
- Highly sensitive to environmental factors like temperature and oxygen.
3. Phosphorescence
Definition
Phosphorescence is a delayed emission of light from a material after it absorbs radiation, continuing from microseconds to several minutes or even hours after the excitation source is removed.
Physical Mechanism
- Absorption: As in fluorescence, electrons are excited to higher singlet states.
- Intersystem Crossing: Electrons undergo a spin change to reach a triplet excited state (T1), which is metastable.
- Phosphorescence Emission: Electrons slowly return from T1 to the ground singlet state (S0), emitting light over an extended time.
Jablonski Diagram
The triplet state and intersystem crossing explain the prolonged emission, distinguishing phosphorescence from fluorescence.
Characteristics of Phosphorescence
- Emission persists after excitation stops.
- Longer emission wavelength than excitation.
- Emission time can vary from milliseconds to hours.
- More sensitive to quenching by oxygen and temperature.
4. Differences Between Fluorescence and Phosphorescence
| Feature | Fluorescence | Phosphorescence |
|---|---|---|
| Emission Time | Nanoseconds (fast) | Milliseconds to hours (slow) |
| Electronic States Involved | Singlet to singlet transition | Triplet to singlet transition (intersystem crossing) |
| Persistence | Stops immediately after excitation ceases | Continues after excitation stops |
| Spin Forbiddenness | No | Yes (slower transition) |
| Typical Applications | Fluorescent dyes, sensors, microscopy | Glow-in-the-dark materials, safety signs |
5. Applications of Fluorescence and Phosphorescence
Fluorescence Applications
- Fluorescence Microscopy: Enables visualization of biological specimens tagged with fluorescent dyes.
- Fluorescent Lamps: Efficient light sources using fluorescence in mercury vapor.
- Optometry: Fluorescein dye helps in diagnosing corneal abrasions and tear film abnormalities.
- Security Printing: Fluorescent inks used in banknotes and documents.
Phosphorescence Applications
- Glow-in-the-Dark Materials: Used in safety signs, watches, and toys.
- Time-Resolved Fluorescence Studies: Differentiate phosphorescence from fluorescence in chemical analysis.
6. Factors Affecting Fluorescence and Phosphorescence
- Temperature: Higher temperatures often quench emission.
- Oxygen Concentration: Oxygen quenches phosphorescence more than fluorescence.
- Matrix or Environment: Rigidity and polarity affect emission properties.
- Concentration: High concentration can lead to quenching (concentration quenching).
7. Quantum Yield and Lifetime
Quantum yield is the efficiency of photon emission, defined as the ratio of emitted photons to absorbed photons. Lifetimes characterize how long excited states last before emission.
- Fluorescence lifetime: Typically nanoseconds.
- Phosphorescence lifetime: Much longer, up to hours in some materials.
8. Summary
Fluorescence and phosphorescence are two types of photoluminescence with distinct mechanisms and temporal characteristics. Fluorescence is fast, involving singlet-singlet transitions, while phosphorescence is slow, involving forbidden triplet-singlet transitions. These properties enable diverse applications from biological imaging to safety materials. Understanding these phenomena is important for optics, photonics, and optometric diagnostics.
This article offers a detailed overview of fluorescence and phosphorescence, suitable for exam preparation and practical understanding in optics and vision sciences.
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