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General Concept of Laser

LASER stands for Light Amplification by Stimulated Emission of Radiation, producing a highly directional, coherent, and monochromatic beam of light. Its operation relies on population inversion and optical or electrical pumping within an active medium.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic principles of absorption, spontaneous emission, stimulated emission, Einstein's coefficients, and general characteristics of laser light.

PGT extension

Detailed analysis of population inversion, three-level and four-level laser systems, threshold pumping conditions, spatial and temporal coherence, and cavity resonance modes.

Prerequisites

Bohr's atomic model and energy levels
Planck's radiation law and thermal equilibrium
Wave optics, interference, and spatial/temporal coherence

Stimulated Emission and Einstein A & B Coefficients

Light interaction with matter involves three processes: induced absorption, spontaneous emission, and stimulated emission.
Stimulated emission occurs when an incident photon of energy hν triggers an excited electron to drop to a lower energy state, releasing a photon identical in phase, frequency, polarization, and direction.
Einstein's coefficients A₂₁, B₁₂, and B₂₁ quantify the transition rates for spontaneous emission, absorption, and stimulated emission respectively.

Population Inversion and Pumping Mechanisms

Under thermal equilibrium, the population of lower energy states exceeds higher states according to the Boltzmann distribution.
Population inversion is a non-equilibrium state where the population of a higher energy level (N₂) exceeds that of a lower level (N₁), i.e., N₂ > N₁.
Pumping is the method of exciting atoms from the ground state to higher energy states using optical, electrical, or chemical energy sources to achieve population inversion.

Optical Resonator and Laser Cavity

An optical cavity consists of two mirrors placed facing each other, one fully reflective and the other partially transmitting, enclosing the active medium.
The cavity provides optical feedback, forcing photons to pass back and forth through the active medium, causing cumulative stimulated emission.
Standing waves are established in the cavity corresponding to specific resonant frequencies whose wavelengths fit an integral multiple of half the cavity length.

Characteristics of Laser Radiation

Monochromaticity: The emitted light possesses an extremely narrow spectral width due to stimulated transitions between specific atomic energy levels.
Coherence: Photons are in phase with each other, showing high temporal and spatial coherence.
Directionality: The beam exhibits extremely low angular divergence, allowing it to propagate over long distances without significant spreading.

Derivations

Relation Between Einstein's Coefficients

Conditions: System in thermal equilibrium with radiation field of energy density U(ν)

Write the rate of absorption: R₁₂ = N₁ U(ν) B₁₂.
Write the total rate of emission: R₂₁ = N₂ [A₂₁ + U(ν) B₂₁].
Equate absorption rate and emission rate for thermal equilibrium: N₁ U(ν) B₁₂ = N₂ A₂₁ + N₂ U(ν) B₂₁.
Express energy density U(ν) from the resulting equation: U(ν) = \frac{A_{21}/B_{21}}{\frac{N_{1}}{N_{2}}\frac{B_{12}}{B_{21}} - 1}.
Substitute the Boltzmann distribution ratio \frac{N_{1}}{N_{2}} = \frac{g_{1}}{g_{2}} e^{h\nu/kT} and compare with Planck's radiation law U(\nu) = \frac{8pi h\nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1}.

Result: $B_{12} = B_{21} (for non-degenerate levels)$ and \frac{A_{21}}{B_{21}} = \frac{8pi h\nu^3}{c^3}

Threshold Condition for Population Inversion

Conditions: Steady-state operation in a laser cavity of length L with mirror reflectivities R₁ and R₂

Consider a light wave of intensity I making a round trip in the cavity of length L with active medium gain coefficient g_th.
Account for cavity losses per round trip due to mirror transmission, diffraction, and scattering, represented by absorption coefficient α.
Write the threshold condition for steady-state amplification: R_{1} R_{2} e^{2(g_{th} - alpha)L} = 1.
Take the natural logarithm of both sides to solve for threshold gain g_{th}.

Result: $g_{th} = \alpha + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Rate of Induced Absorption	$$R_{12} = N_{1} U(\nu) B_{12}$$	$N_1 = lower level population$, U(\nu) = radiation energy density, $B_{12} = Einstein B coefficient$ for absorption m^{-3} s^{-1}	Valid for isotropic radiation field
Rate of Spontaneous Emission	$$R_{spont} = N_{2} A_{21}$$	$N_2 = upper level population$, $A_{21} = Einstein A coefficient$ m^{-3} s^{-1}	Independent of external radiation field
Rate of Stimulated Emission	$$R_{stim} = N_{2} U(\nu) B_{21}$$	$N_2 = upper level population$, U(\nu) = energy density, $B_{21} = Einstein B coefficient$ for stimulated emission m^{-3} s^{-1}	Triggered by incoming resonant photon
Ratio of Einstein Coefficients	$$\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$$	h = Planck constant, \nu = frequency, c = speed of light in vacuum J s (for ratio, kg m^2 s^{-2} Hz^3)	Derived using Planck's blackbody radiation law
Boltzmann Population Distribution	$$\frac{N_{2}}{N_{1}} = \frac{g_{2}}{g_{1}} e^{-(E_2 - E_1)/kT}$$	g_1, $g_2 = statistical weights$, k = Boltzmann constant, T = absolute temperature Dimensionless	Thermal equilibrium
Threshold Gain Coefficient	$$g_{th} = \alpha + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$$	$\alpha = internal cavity loss$, L = cavity length, R_1, $R_2 = mirror reflectivities$ m^{-1}	Steady-state laser oscillation
Cavity Resonant Frequency Condition	$$\nu = \frac{qc}{2nL}$$	q = integer mode number, c = speed of light, n = refractive index of medium, L = cavity length Hz	Fabry-Perot resonator cavity
Laser Beam Divergence	$$\theta \approx \frac{\lambda}{\pi w_0}$$	$\theta = beam divergence half-angle$, $\lambda = wavelength$, $w_0 = beam waist radius$ radians	Gaussian beam propagation

Graphs and experiments

Energy Level Diagram for Three-Level and Four-Level Lasers

Draw: Schematic energy levels E_0, E_1, E_2, E_3 showing pumping transitions, fast non-radiative decay, and laser transition.

Interpret: Illustrates how population inversion is maintained more efficiently in four-level systems due to a lower threshold compared to three-level systems.

Exam use: Frequently asked in conceptual questions comparing laser efficiency and pumping requirements.

Laser Output Power vs Pumping Current/Power

Draw: A curve showing near-zero output power up to a critical pumping threshold, followed by a linear rise in output power beyond the threshold.

Interpret: Identifies the threshold pumping rate required to overcome cavity losses and initiate laser action.

Exam use: Tested via graphical interpretation questions in PGT examinations.

Fabry-Perot Resonator Modes

Draw: Frequency spectrum showing discrete longitudinal modes separated by frequency interval Δ$\nu = c / (2nL)$.

Interpret: Demonstrates that cavity resonance allows multiple standing wave modes within the gain bandwidth.

Exam use: Numerical problems calculating longitudinal mode separation.

Solved numericals

TGT core

Calculate the ratio of the rate of spontaneous emission to the rate of stimulated emission for a system in thermal equilibrium at temperature T = 300 K, emitting radiation of wavelength $\lambda = 600 nm$.

Recall the ratio of emission rates: R_{spont} / $R_{stim} = A_{21} / [U(\nu) B_{21}] = e^{(h\nu/kT)} - 1$.
Calculate the photon energy h\nu: h$\nu = hc / \lambda = (6.63 \times 10^{-34} \text{ J s} \times 3 \times 10^8 \text{ m/s}) / (600 \times 10^{-9} \text{ m}) = 3.315 \times 10^{-19} \text{ J}$.
Calculate thermal energy kT: k$T = (1.38 \times 10^{-23} \text{ J/K}) \times 300 \text{ K} = 4.14 \times 10^{-21} \text{ J}$.
Evaluate the exponent h\nu/kT: (3.315 \times 10^{-19}) / (4.14 \times 10^{-21}) \approx 80.07.
Calculate ratio: R_{spont} / $R_{stim} = e^{80.07} - 1 \approx 2.45 \times 10^{34}$.

Answer: 2.45 \times 10^{34}

PGT extension

A laser optical cavity has a length L = 30 cm and mirror reflectivities $R_1 = 1.0$ and $R_2 = 0.95$. The internal scattering loss coefficient $\alpha = 0.02 cm^{-1}$. Find the threshold gain coefficient g_{th}.

Convert cavity length to meters or keep in cm consistent with α: L = 30 cm.
Use the threshold gain formula: $g_{th} = \alpha + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$.
Substitute values: $\alpha = 0.02 cm^{-1}$, $R_1 = 1.0$, $R_2 = 0.95$.
Calculate loss term: \frac{1}{2(30)} \ln\left(\frac{1}{(1.0)(0.95)}\right) = \frac{1}{60} \ln(1.0526) = \frac{0.05129}{60} = 0.000855 \text{ cm}^{-1}.
Sum the terms: $g_{th} = 0.02 + 0.000855 = 0.020855 \text{ cm}^{-1}$.

Answer: 0.0209 cm^{-1}

PGT extension

Find the longitudinal mode separation (frequency interval Δν) for a He-Ne laser operating at $\lambda = 632.8 nm with a cavity length L = 40 cm$ and refractive index n = 1.

Recall formula for longitudinal mode spacing: \Delta\nu = \frac{c}{2nL}.
Identify given values: $c = 3 \times 10^8 \text{ m/s}$, n = 1, L = 0.4 \text{ m}.
Substitute values into the formula: \Delta$\nu = \frac{3 \times 10^8}{2 \times 1 \times 0.4} = \frac{3 \times 10^8}{0.8}$.
Calculate final value: \Delta$\nu = 3.75 \times 10^8 \text{ Hz} = 375 \text{ MHz}$.

Answer: 375 MHz

Assumptions and limitations

Einstein's relations assume a simple two-level or three-level atomic system interacting with isotropic blackbody radiation.
Laser threshold equations assume steady-state conditions and uniform gain distribution across the optical cavity mode volume.

Common misconceptions

Common error: Spontaneous emission produces coherent light.

Correct: Spontaneous emission produces incoherent light with random phase.

Independent atoms emit photons independently without phase coordination in spontaneous emission.

Common error: Population inversion can be achieved in a two-level system using optical pumping.

Correct: Population inversion is impossible in a two-level system under steady-state optical pumping.

Pumping rates cannot exceed stimulated emission rates in a two-level system, capping the population ratio at 1:1.

Quick revision

LASER requires population inversion, stimulated emission, and an optical feedback cavity.
Einstein's relation states A_{21}/$B_{21} = 8\pih\nu^3 / c^3$.
Four-level lasers are more efficient than three-level lasers because the lower laser level empties rapidly via non-radiative decay.

Must remember

Stimulated emission produces photons identical in phase, frequency, polarization, and direction to the incident photon.
Population inversion ($N_2 > N_1$) is a non-equilibrium state essential for laser action and is impossible to sustain in a simple two-level system.
The ratio of Einstein coefficients $A_{21}$ and $B_{21}$ is given by $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$.
Laser characteristics include high monochromaticity, spatial and temporal coherence, and extremely low angular divergence.
An optical cavity provides positive feedback using mirrors (one fully reflective, one partially transmitting) to sustain laser oscillation.

Glossary

Population InversionA non-thermal equilibrium state where the population of atoms in a higher energy level ($N_2$) exceeds that of a lower energy level ($N_1$).

PumpingThe process of exciting atoms from a lower energy state to a higher energy state using external energy to achieve population inversion.

Einstein A CoefficientQuantifies the transition rate of spontaneous emission from an excited state to a lower state.

Einstein B CoefficientQuantifies the transition rates for induced absorption ($B_{12}$) and stimulated emission ($B_{21}$).

Optical ResonatorAn arrangement of mirrors that reflects light back and forth through the gain medium to build up laser oscillation.

Spontaneous vs Stimulated Emission

Parameter	Spontaneous Emission	Stimulated Emission
Trigger	Untriggered (spontaneous drop)	Triggered by incident photon
Phase & Direction	Random phase, isotropic direction	In-phase, highly directional
Coherence	Incoherent	Coherent
Governing Coefficient	Einstein A coefficient ($A_{21}$)	Einstein B coefficient ($B_{21}$)

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following processes involves the release of a photon identical in phase, frequency, and direction to the incident photon?

Explanation: Stimulated emission occurs when an incident photon triggers an excited electron to drop, releasing a photon with identical phase, frequency, polarization, and direction.

2. What is the condition for population inversion between two energy levels with populations $N_1$ and $N_2$?

Explanation: Population inversion is defined as a non-equilibrium state where the population of the higher energy level exceeds that of the lower energy level, i.e., $N_2 > N_1$.

3. Which Einstein coefficient represents the rate of spontaneous emission?

Explanation: $A_{21}$ is the Einstein coefficient quantifying the rate of spontaneous emission from upper level 2 to lower level 1.

4. The ratio of Einstein coefficients $\frac{A_{21}}{B_{21}}$ is proportional to which power of frequency $\nu$?

Explanation: From Planck's law derivation, the ratio $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$, which shows proportionality to $\nu^3$.

5. Why is population inversion impossible to achieve in a steady-state two-level energy system?

Explanation: In a two-level system, the upward optical pumping rate equals the downward stimulated emission rate once populations become equal, preventing $N_2 > N_1$.

6. What is the primary function of an optical cavity in a laser system?

Explanation: The optical cavity uses mirrors to provide optical feedback, forcing photons to pass back and forth through the gain medium to amplify stimulated emission.

7. Which characteristic of laser light refers to the extreme narrowness of its spectral width?

Explanation: Monochromaticity denotes that the emitted laser light possesses an extremely narrow spectral width due to transitions between specific discrete energy levels.

8. The rate of induced absorption $R_{12}$ is given by which expression?

Explanation: The rate of induced absorption is proportional to the lower level population $N_1$, radiation energy density $U(\nu)$, and Einstein coefficient $B_{12}$.

9. What does the Fabry-Perot resonant frequency condition $\nu = \frac{qc}{2nL}$ imply for allowed cavity frequencies?

Explanation: Standing waves are established in the cavity corresponding to specific resonant frequencies where an integral multiple of half wavelengths fits the cavity length.

10. Which of the following correctly describes the nature of photons produced in spontaneous emission?

Explanation: Spontaneous emission produces incoherent light with random phase because independent atoms emit photons without phase coordination.

11. In the threshold gain condition $g_{th} = \alpha + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$, what do $R_1$ and $R_2$ represent?

Explanation: $R_1$ and $R_2$ denote the reflectivities of the two end mirrors enclosing the laser optical cavity.

12. Beam divergence half-angle $\theta$ for a Gaussian laser beam is inversely proportional to which parameter?

Explanation: The formula $\theta \approx \frac{\lambda}{\pi w_0}$ shows that beam divergence is inversely proportional to the beam waist radius $w_0$.

13. Under thermal equilibrium, the ratio of populations $N_2 / N_1$ follows which distribution?

Explanation: Under thermal equilibrium, populations follow the Boltzmann distribution: $\frac{N_2}{N_1} = \frac{g_2}{g_1} e^{-(E_2 - E_1)/kT}$.

14. What is the primary role of 'pumping' in a laser system?

Explanation: Pumping provides optical, electrical, or chemical energy to excite atoms to higher levels, overcoming thermal equilibrium to establish population inversion.

15. Which parameter has units of $\text{m}^{-1}$ in the context of laser cavities?

Explanation: Gain coefficients, internal losses, and threshold gain $g_{th}$ have units of inverse length ($\text{m}^{-1}$).

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The ratio of the Einstein A coefficient to the B coefficient ($A_{21}/B_{21}$) for a transition of frequency $\nu$ is given by:

Explanation: Using Planck's blackbody radiation law and Einstein's relations at thermal equilibrium, the ratio is derived as $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$.

2. Why is population inversion not possible in a two-level laser system under optical pumping?

Explanation: In a two-level system, as pumping intensity increases, the rate of stimulated emission matches the absorption rate, restricting the upper level population to at most equal the lower level population.

3. What condition must be satisfied for steady-state laser oscillation in a cavity of length $L$ and mirror reflectivities $R_1, R_2$ with internal loss $\alpha$?

Explanation: Laser oscillation builds up when the gain provided by the active medium overcomes internal cavity losses and mirror transmission losses, i.e., $g \ge g_{th}$.

4. The resonant frequencies of a Fabry-Perot laser cavity of length $L$ and refractive index $n$ are given by:

Explanation: Fabry-Perot cavity resonant frequencies correspond to standing waves where an integer multiple of half wavelengths equals the optical path length, yielding $\nu = \frac{qc}{2nL}$.

5. Which of the following characteristics distinguishes laser light from ordinary thermal light?

Explanation: Laser light is characterized by exceptional spatial and temporal coherence, high monochromaticity, and minimal angular divergence compared to thermal light sources.

6. If the energy density of radiation is $U(\nu)$, the rate of stimulated emission $R_{stim}$ is proportional to:

Explanation: Stimulated emission depends on the upper state population $N_2$, radiation energy density $U(\nu)$, and the Einstein B coefficient for stimulated emission $B_{21}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which process is directly responsible for the amplification of light in a laser medium?

Explanation: Stimulated emission produces coherent photons identical to the triggering photon, leading to light amplification (LASER = Light Amplification by Stimulated Emission of Radiation).

2. Under thermal equilibrium at temperature $T$, what does the Boltzmann distribution predict about the populations $N_1$ and $N_2$ for $E_2 > E_1$?

Explanation: Under thermal equilibrium, lower energy states are always more populated than higher energy states ($N_1 > N_2$) due to the negative exponent in the Boltzmann factor.

3. What is the unit of the Einstein A coefficient ($A_{21}$)?

Explanation: The Einstein A coefficient represents the probability per unit time of spontaneous transition, so its unit is reciprocal seconds ($\text{s}^{-1}$).

4. How does the beam divergence half-angle $\theta$ change if the beam waist radius $w_0$ of a laser is doubled?

Explanation: Since $\theta \approx \frac{\lambda}{\pi w_0}$, doubling $w_0$ results in halving the beam divergence half-angle $\theta$.

5. In a laser cavity, what is the purpose of making one mirror partially transmitting instead of fully reflective?

Explanation: A partially transmitting mirror allows a fraction of the circulating optical power to escape the cavity, forming the useful external laser beam output.

6. Which expression represents the rate of spontaneous emission?

Explanation: Spontaneous emission occurs independently of external radiation and its rate is simply $R_{spont} = N_2 A_{21}$.

7. What causes the extremely narrow spectral width (high monochromaticity) of laser radiation?

Explanation: Laser emission originates from transitions between discrete energy levels filtered by the resonant modes of the optical cavity, resulting in high monochromaticity.

8. If the temperature $T$ of a system approaches infinity, what does the Boltzmann population ratio $N_2/N_1$ approach (assuming equal statistical weights $g_1 = g_2$)?

Explanation: As $T \to \infty$, the exponent $-(E_2 - E_1)/kT \to 0$, so $\frac{N_2}{N_1} \to e^0 = 1$.

9. What is the physical significance of internal cavity loss $\alpha$ in the threshold gain equation?

Explanation: Internal cavity loss $\alpha$ quantifies losses due to scattering, absorption, and diffraction within the laser gain medium per unit length.

10. Which of the following statements is correct regarding stimulated emission?

Explanation: Stimulated emission is triggered by an incoming resonant photon matching the transition energy, leading to an amplified coherent photon.