Topic 13: Basics of Lasers – Coherence; Population Inversion; Spontaneous Emission; Einstein’s Theory of Lasers
Introduction
Lasers (Light Amplification by Stimulated Emission of Radiation) represent a revolutionary technology in optics, enabling the generation of intense, highly coherent, and monochromatic light beams. Since their invention in the 1960s, lasers have become indispensable in fields ranging from medicine and communications to research and industry.
This article explores the fundamental principles underlying laser operation, including coherence, population inversion, spontaneous and stimulated emission, and Einstein’s theoretical contributions that laid the groundwork for laser physics.
---1. What is a Laser?
 |
| LASER Beam |
A laser is a device that produces a concentrated beam of light with unique properties: high intensity, directionality, monochromaticity (single wavelength), and coherence (fixed phase relationship). Unlike ordinary light sources, which emit incoherent light in multiple directions and wavelengths, lasers generate light through a controlled process called stimulated emission.
---2. Coherence
Definition
 |
| Coherence |
Coherence refers to a fixed relationship between the phases of waves at different points in space (spatial coherence) or time (temporal coherence). In lasers, coherence enables the emitted light waves to maintain a constant phase difference, leading to constructive interference and a narrow, intense beam.
Types of Coherence
- Spatial Coherence: Correlation between the phases of a wave at different points in space perpendicular to propagation direction. High spatial coherence means the beam is well collimated.
- Temporal Coherence: Correlation of the wave phase at different points along the direction of propagation over time. Temporal coherence is related to the monochromaticity and spectral linewidth of the laser.
Coherence Length and Time
Temporal coherence length is the distance over which the wave maintains a predictable phase relationship and is inversely proportional to spectral linewidth. Coherence time is the time interval over which the wave phase is predictable.
---3. Atomic Energy Levels and Population
Atoms have discrete energy levels. Electrons can move between these levels by absorbing or emitting photons with energies matching the energy difference between levels:
E2 − E1 = h ν
- E2 and E1: Energy levels
- h: Planck’s constant
- ν: Frequency of the emitted or absorbed photon
4. Spontaneous Emission
 |
| Spontaneous and Stimulated Emission |
Process Description
When an electron in an excited state (E2) spontaneously returns to a lower energy state (E1), it emits a photon randomly in time and direction. This is spontaneous emission and is the source of incoherent light in conventional lamps.
Characteristics
- Random phase and direction
- Emission rate governed by the lifetime of the excited state
- Not sufficient for laser action alone
5. Stimulated Emission
Concept
Proposed by Einstein in 1917, stimulated emission occurs when an incoming photon of energy hν interacts with an excited electron, causing it to drop to a lower energy state and emit a second photon identical in phase, frequency, polarization, and direction to the incoming photon.
Significance
- Fundamental mechanism behind laser operation
- Produces coherent, monochromatic light amplification
6. Einstein’s Coefficients and Theory
Introduction
Einstein introduced three coefficients to describe the interaction of light with matter:
- A21: Probability per unit time of spontaneous emission from level 2 to 1.
- B12: Probability per unit time of absorption from level 1 to 2 induced by radiation.
- B21: Probability per unit time of stimulated emission from level 2 to 1 induced by radiation.
Equilibrium Conditions
Einstein showed that in thermal equilibrium, the rate of absorption equals the total rate of emissions (spontaneous + stimulated), leading to the Planck radiation law. This theory laid the groundwork for understanding laser amplification.
---7. Population Inversion
Definition
 |
| Population Inversion of Laser |
Population inversion is the condition where more atoms are in the excited state than in the ground state. It is essential for net light amplification via stimulated emission because normally more atoms occupy the ground state.
Achieving Population Inversion
Since thermal equilibrium favors the ground state, external energy (pumping) must be applied, such as optical pumping, electrical discharge, or chemical reactions, to raise enough atoms to excited states.
---8. Laser Amplification and Resonance Cavity
Amplification
In the presence of population inversion, an incident photon can stimulate emission of more photons, resulting in exponential amplification of light intensity.
Resonator Cavity
A pair of mirrors form a resonant optical cavity that reflects the light back and forth through the gain medium, amplifying it further. One mirror is partially transparent to allow some light to escape as the laser beam.
---9. Characteristics of Laser Light
- Monochromatic: Very narrow spectral linewidth.
- Coherent: High temporal and spatial coherence.
- Directional: Low divergence beam.
- High Intensity: High power concentrated in a small area.
10. Types of Lasers
- Gas Lasers: Helium-Neon (He-Ne), CO2.
- Solid-State Lasers: Ruby, Nd:YAG.
- Semiconductor Lasers: Laser diodes.
- Liquid Lasers: Dye lasers.
11. Applications of Lasers in Optometry and Medicine
- Refractive Surgery: LASIK uses excimer lasers for precise corneal reshaping.
- Retinal Therapy: Laser photocoagulation treats retinal diseases.
- Diagnostic Imaging: OCT uses low-coherence lasers for retinal cross-section imaging.
12. Summary
Lasers operate on the principles of stimulated emission and population inversion, producing coherent, monochromatic, and intense light beams. Einstein’s theoretical framework of spontaneous and stimulated emission remains foundational. Understanding coherence and laser mechanisms is crucial for optics and optometry applications, enabling advances in imaging, therapy, and communication.
This comprehensive article covers laser basics, including coherence, population inversion, emission processes, and Einstein’s theory, designed to support detailed exam preparation and practical understanding.
Topic 14: Radiometry; Solid Angle; Radiometric Units; Photopic and Scotopic Luminous Efficiency and Efficacy Curves; Photometric Units
Introduction
Radiometry and photometry are branches of optical science concerned with the measurement of electromagnetic radiation and visible light, respectively. While radiometry deals with all wavelengths of electromagnetic radiation, photometry focuses specifically on light as perceived by the human eye. Understanding these measurement principles is critical for designing and evaluating optical instruments, lighting systems, and vision-related applications in optometry.
---1. Radiometry: Definition and Importance
Radiometry is the science of measuring electromagnetic radiation in terms of absolute power, independent of human visual response. Radiometric quantities describe the physical energy flow of electromagnetic waves across all wavelengths, from gamma rays through ultraviolet, visible, and infrared to radio waves.
Radiometric measurements are fundamental for characterizing light sources, optical materials, and environmental radiation.
---2. Solid Angle
Definition
A solid angle is a three-dimensional analog of a two-dimensional planar angle, representing the portion of space that an object subtends at a point. It quantifies how large an object appears to an observer from that point.
Units
The SI unit of solid angle is the steradian (sr). A sphere encompasses a total solid angle of 4Ï€ steradians.
Mathematical Expression
The solid angle Ω subtended by a surface area A on a sphere of radius r is:
Ω = A / r2 (steradians)
- Ω: Solid angle
- A: Surface area subtended
- r: Radius of the sphere
3. Radiometric Quantities and Units
 |
| Radiometric Units |
| Quantity | Symbol | Definition | SI Unit | Description |
|---|---|---|---|---|
| Radiant Energy | Q or We | Total energy emitted, transferred, or received | Joule (J) | Energy carried by radiation |
| Radiant Flux (Power) | Φe | Energy per unit time | Watt (W = J/s) | Power of electromagnetic radiation |
| Irradiance (Radiant Flux Density) | Ee | Radiant flux incident per unit area | W/m2 | Light power received per unit area |
| Radiant Intensity | Ie | Radiant flux per unit solid angle | W/sr | Power emitted per solid angle |
| Radiance | Le | Radiant flux per unit area per unit solid angle | W/(m2·sr) | Power per area and direction |
4. Photometry: Human Visual Response
Photometry measures light in terms of its perceived brightness to the average human eye. Since the eye's sensitivity varies with wavelength, photometric quantities weight radiometric measures by the eye’s spectral sensitivity.
---5. Luminous Efficiency Functions
 |
| Photopic and Scotopic Luminous Efficiency Curve |
Photopic Luminous Efficiency Curve (V(λ))
Photopic vision is mediated by cone cells and dominates under well-lit conditions. The photopic luminous efficiency curve represents the relative sensitivity of the human eye to different wavelengths of light under bright conditions. It peaks around 555 nm (green-yellow light), where the eye is most sensitive.
Scotopic Luminous Efficiency Curve (V’(λ))
Scotopic vision is mediated by rod cells and dominates under low-light (night) conditions. The scotopic luminous efficiency curve peaks around 507 nm (blue-green light) and differs from photopic sensitivity.
---6. Luminous Flux and Radiant Flux
Definitions
- Radiant Flux (Φe): Total power of electromagnetic radiation emitted, measured in watts.
- Luminous Flux (Φv): Power of light weighted by the eye's sensitivity, measured in lumens (lm).
Conversion between Radiant and Luminous Flux
The luminous flux is obtained by integrating the spectral radiant flux weighted by the luminous efficiency function:
Φv = 683 × ∫0∞ Φe,λ V(λ) dλ
Here, 683 lm/W is the maximum luminous efficacy at 555 nm.
---7. Photometric Units and Their Radiometric Counterparts
| Photometric Quantity | Symbol | Radiometric Equivalent | Unit |
|---|---|---|---|
| Luminous Energy | Qv | Radiant Energy | lumen second (lm·s) |
| Luminous Flux | Φv | Radiant Flux | lumen (lm) |
| Illuminance | Ev | Irradiance | lux (lx) = lm/m2 |
| Luminous Intensity | Iv | Radiant Intensity | candela (cd) = lm/sr |
| Luminance | Lv | Radiance | cd/m2 |
8. Luminous Efficacy
Definition
Luminous efficacy is the ratio of luminous flux to radiant flux, expressing how efficiently a light source produces visible light:
η = Φv / Φe (lm/W)
Importance
It helps compare light sources in terms of their perceived brightness per unit power consumed. The theoretical maximum efficacy is 683 lm/W at 555 nm.
---9. Spectral Power Distribution (SPD)
The SPD of a light source describes the power emitted at each wavelength. It is essential for calculating photometric quantities since eye sensitivity varies with wavelength.
---10. Practical Examples
Example 1: Calculating Illuminance
A lamp emits 1000 lumens uniformly over a surface area of 10 m2. Calculate illuminance:
Ev = Φv / A = 1000 lm / 10 m2 = 100 lux
Example 2: Difference Between Photopic and Scotopic Vision
Under low light, the human eye sensitivity shifts toward blue-green light (scotopic), affecting perceived brightness and color. This impacts lighting design and vision testing.
---11. Summary
Radiometry and photometry form the backbone of measuring light energy and its perception. Solid angle provides a spatial framework, while radiometric and photometric units quantify power and perceived brightness respectively. Understanding the luminous efficiency curves and unit conversions is essential for optical instrument design, lighting engineering, and vision science.
This article provides an extensive understanding of radiometric and photometric principles, crucial for students and professionals in optics and optometry.
Topic 15: Inverse Square Law of Photometry; Lambert’s Law
Introduction
Understanding how light intensity changes with distance and how surfaces reflect or emit light is fundamental in optics and photometry. The Inverse Square Law explains the variation of illuminance with distance from a point light source, while Lambert’s Law describes the directional dependence of light intensity from ideal diffuse surfaces. These principles have broad applications in lighting design, vision science, and optical instrumentation.
---1. Inverse Square Law of Photometry
Definition
The Inverse Square Law states that the illuminance (E) produced by a point source of light is inversely proportional to the square of the distance (r) from the source:
E ∝ 1 / r2
Mathematical Expression
If I is the luminous intensity of the point source (in candela), then the illuminance at distance r is given by:
E = I / r2 (where E is in lux, r in meters)
Physical Meaning
As light radiates spherically from a point source, it spreads over the surface area of a sphere (4Ï€r²). Since the total luminous flux remains constant, illuminance decreases as the area increases with distance squared.
Assumptions and Limitations
- Source is a perfect point light source.
- Light propagates in free space without absorption or scattering.
- Surface receiving light is perpendicular to the incident beam.
Applications
- Design of lighting systems and estimation of required lamp power.
- Understanding retinal illumination in optometry.
- Distance estimation in optical instruments.
2. Lambert’s Law (Cosine Law of Illumination)
Definition
Lambert’s Law describes how the intensity of light reflected or emitted by an ideal diffusely reflecting surface (Lambertian surface) varies with the angle of observation. The observed brightness is proportional to the cosine of the angle (θ) between the surface normal and the observer's line of sight:
I(θ) = I0 × cos θ
Physical Explanation
A Lambertian surface emits or reflects light uniformly in all directions, but due to projection effects, the apparent brightness decreases with increasing angle. This cosine dependence ensures that the surface appears equally bright from all viewing angles.
Implications
- Real surfaces approximate Lambertian behavior, simplifying modeling in computer graphics and vision science.
- Explains why a surface appears dimmer when viewed at a steep angle.
- Important for understanding luminance and radiance distributions.
3. Relationship Between Inverse Square Law and Lambert’s Law
Both laws describe different aspects of light propagation and reflection. The Inverse Square Law governs how light intensity decreases with distance from a point source, while Lambert’s Law governs how light intensity varies with viewing angle on diffuse surfaces. Together, they help predict illumination and brightness in complex lighting environments.
---4. Mathematical Derivations
Inverse Square Law (Photometry/Radiometry)
The inverse square law states that the illuminance or intensity per unit area from a point source decreases in proportion to the square of the distance from the source. As light spreads spherically, the same total power is distributed over a surface area that grows as 4Ï€r².
Why it holds
A point source emits uniformly in all directions. At distance r, light power is spread over the surface of a sphere of area A = 4Ï€r². Doubling r quadruples the area, so illuminance drops to one-fourth: E ∝ 1/r².
| Symbol | Quantity | SI Unit |
|---|---|---|
| I | Luminous intensity | candela (cd) |
| E | Illuminance | lux (lx) = lm·m−2 |
| r | Distance from source | metre (m) |
Worked Example
A lamp behaves as a point source with luminous intensity I = 500 cd. Find the illuminance on a screen at:
- (a) r = 2 m
- (b) r = 5 m
Formula: E = I / r² (lux)
(a) E = 500 / (2)² = 500 / 4 = 125 lx
(b) E = 500 / (5)² = 500 / 25 = 20 lx
Moving the screen from 2 m to 5 m multiplies distance by 2.5; illuminance is divided by (2.5)² ≈ 6.25, dropping from 125 lx to 20 lx.
Practical Notes (Limitations)
- Valid for point-like sources (far field). Large/extended sources or near-field conditions deviate.
- Assumes no absorption or scattering by the medium (use in air over short distances).
- Holds for isotropic emission. Directional emitters (e.g., LEDs with optics) require angular intensity patterns.
Lambert’s Law
Lambert's Law states that the luminous intensity observed from a perfectly diffusing surface is directly proportional to the cosine of the angle (θ) between the incident light direction and the surface normal.
Formula:
Iθ = I0 × cos θ
- Iθ = Luminous intensity at angle θ
- I0 = Luminous intensity when θ = 0° (normal incidence)
- θ = Angle between incident light and surface normal
Explanation: The law explains how brightness appears reduced at oblique viewing angles, and it is important in surface photometry and visual ergonomics.
Combined Lambert–Beer Law
The Lambert–Beer Law combines Lambert’s Law of absorption and Beer’s Law of concentration. It states that the absorbance of light passing through a medium is directly proportional to the path length and the concentration of the absorbing substance.
Formula:
A = ε × c × l
- A = Absorbance (no units)
- ε = Molar absorptivity (L·mol⁻¹·cm⁻¹)
- c = Concentration of the solution (mol·L⁻¹)
- l = Path length of the light through the sample (cm)
Explanation: As concentration or path length increases, absorbance increases proportionally. This principle is widely used in spectrophotometry to determine the concentration of unknown solutions.
If I_0 is the intensity normal to the surface, the intensity at angle θ is:
I(θ) = I_0 \cos θ
This arises from the projected area of the surface element decreasing by cos θ when viewed at an angle.
---5. Experimental Verification
- Illuminance measurements at varying distances from a point source confirm the inverse square dependence.
- Diffuse reflectance measurements with goniometers demonstrate the cosine dependence of intensity with angle.
6. Applications in Optometry and Vision Science
Illumination of the Retina
Understanding how light intensity varies with distance and angle helps calculate retinal illumination, crucial for vision testing and diagnostics.
Design of Optical Devices
Lighting instruments and vision aids use these laws to optimize brightness, contrast, and viewing angles.
Visual Perception
The human eye’s perception of brightness relates to Lambertian reflection properties of surfaces, influencing contrast and object recognition.
---7. Limitations and Extensions
Real light sources are not perfect points, and surfaces often deviate from ideal Lambertian behavior. Corrections and more complex models account for anisotropic emission, surface texture, and atmospheric effects.
---8. Summary
The Inverse Square Law and Lambert’s Law are fundamental to understanding how light intensity varies with distance and angle. They underpin the measurement and application of illumination in optics, vision science, and lighting design, providing essential tools for professionals in optometry and related fields.
This article offers a detailed explanation of the Inverse Square Law and Lambert’s Law, with derivations, examples, and applications, supporting thorough exam preparation and practical understanding.
Topic 16: Other Units of Light Measurement; Retinal Illumination; Trolands
Introduction
In the field of optics and vision science, accurate measurement of light and its effects on the human eye is essential. Beyond fundamental photometric and radiometric units, specialized concepts such as retinal illumination and Trolands provide deeper insight into how light interacts with the eye's anatomy and physiology. This topic explores additional units of light measurement, the concept of retinal illumination, and the important unit known as the Troland, which bridges physical light intensity and the eye's optical response.
---1. Recap of Basic Photometric and Radiometric Units
Before diving into specialized units, it is important to briefly revisit fundamental units:
- Radiometric units: Measure absolute physical energy (e.g., watts, W).
- Photometric units: Weight radiometric units by human eye sensitivity (e.g., lumens, candela, lux).
These basic units quantify light emission, intensity, and illumination but do not directly account for the eye's optical characteristics or retinal response.
---2. Limitations of Basic Units for Vision Science
Photometric units like illuminance (lux) describe the amount of light falling on a surface, such as the cornea or a sheet of paper, but do not capture how much light actually reaches or stimulates the retina. Factors like pupil size, ocular media absorption, and the eye's optics influence retinal illumination, necessitating more refined measures.
---3. Retinal Illumination: Concept and Significance
Definition
Retinal illumination refers to the amount of light that effectively reaches and stimulates the retina, measured as the luminous flux per unit area on the retinal surface. It accounts for the eye's optics and pupil size, providing a physiologically relevant measure of light input.
Importance
- Critical in visual perception studies.
- Helps quantify light exposure in diagnostic and therapeutic procedures.
- Facilitates understanding of visual adaptation to different lighting conditions.
4. Calculation of Retinal Illumination
Retinal illumination is not directly measurable but is calculated from measurable quantities such as corneal illuminance and pupil diameter. The relationship is based on the geometry of the eye and optics principles.
Formula for Retinal Illuminance
Retinal illuminance (in Trolands, Td) is given by:
Retinal Illuminance (Td) = Illuminance on the cornea (lux) × Pupil area (mm²)
Since pupil area is proportional to the square of the pupil diameter (d), it can be expressed as:
Td = E × Ï€ (d/2)² = E × (Ï€ d²)/4
---5. The Troland (Td): Definition and Origin
Definition
The Troland is a unit of retinal illuminance that combines corneal illuminance with pupil size, reflecting the actual luminous flux incident on the retina.
Historical Background
Named after Leonard T. Troland, an American physicist and psychologist, the unit was introduced to better quantify retinal stimulation beyond basic photometric measures.
---6. Practical Examples of Trolands
Example 1: Calculating Retinal Illuminance
Suppose corneal illuminance is 1000 lux, and the pupil diameter is 4 mm. Then retinal illuminance is:
Td = 1000 × Ï€ × (4/2)² = 1000 × Ï€ × 2² = 1000 × Ï€ × 4 ≈ 12566 Td
Example 2: Effect of Pupil Size
For the same corneal illuminance, if the pupil constricts to 2 mm diameter (due to bright light), retinal illuminance reduces to:
Td = 1000 × Ï€ × (1)² = 1000 × Ï€ ≈ 3142 Td
This shows how pupil size regulates retinal light exposure.
---7. Applications of Retinal Illumination and Trolands
- Vision Testing: Standardizing luminance conditions considering pupil size.
- Visual Adaptation Studies: Analyzing how retinal illumination affects rod and cone activity.
- Clinical Procedures: Managing light exposure during eye examinations and surgeries.
- Lighting Design: Optimizing environments for human visual comfort and performance.
8. Other Specialized Units of Light Measurement
Retinal Illuminance Units in Research
Beyond Trolands, researchers use units such as Lambert and Stilb for luminance, and candela per square meter (cd/m²) for brightness measurements, each with specific contexts and conversions.
Lambert (L)
The Lambert is a unit of luminance defined as 1 lumen per square centimeter per steradian, equivalent to 3183 cd/m². It is primarily used in photographic and vision research.
Stilb (sb)
The Stilb is another unit of luminance, equal to 1 candela per square centimeter. It is not widely used but appears in some scientific literature.
---9. Relationship Between Retinal Illuminance and Visual Performance
Studies show a close link between retinal illuminance levels and visual acuity, contrast sensitivity, and color perception. Higher retinal illuminance generally improves visual performance up to a saturation point, beyond which photoreceptor bleaching or discomfort glare occurs.
---10. Measurement Techniques for Retinal Illuminance
Indirect Methods
Direct measurement is difficult; instead, retinal illuminance is estimated using corneal illuminance measurements (lux meters) combined with pupil size measurements (pupillometry).
Instruments
- Lux Meter: Measures illuminance at the cornea.
- Pupillometer: Measures pupil diameter under test conditions.
- Retinoscope/Infrared Cameras: Assess retinal illumination indirectly.
11. Factors Affecting Retinal Illumination
- Pupil Size: Most significant factor; controlled by ambient light, accommodation, and autonomic input.
- Ocular Media Transmission: Light absorption by cornea, lens, vitreous can reduce retinal illuminance.
- Wavelength: Eye’s spectral sensitivity affects effective retinal stimulation.
- Age: Aging changes pupil size and media clarity, impacting retinal illumination.
12. Clinical Relevance
Light Adaptation and Testing
Retinal illuminance must be controlled during visual field testing, electrophysiological exams, and low vision rehabilitation to ensure accurate results.
Phototoxicity Considerations
Excessive retinal illuminance can cause photochemical damage; understanding and managing Trolands helps prevent iatrogenic injury during diagnostic or therapeutic exposure.
---13. Summary
Retinal illumination and the Troland unit provide vital links between external light conditions and the eye’s internal visual processing. These specialized measurements account for physiological and optical factors, enhancing the precision of vision science and clinical optics. Mastery of these concepts is essential for optometry students, researchers, and practitioners.
This comprehensive article covers advanced units of light measurement, retinal illumination, and Trolands, supporting detailed understanding and practical application in optics and vision science.
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