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Stimulated Emission

Stimulated emission is a quantum process in which an incoming photon of specific energy interacts with an excited atom, triggering it to drop to a lower energy state and emit a second photon identical in phase, frequency, direction, and polarization. It forms the foundational physical principle behind laser action and light amplification.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic understanding of absorption, spontaneous emission, and stimulated emission, along with Einstein's A and B coefficients and their qualitative relations.

PGT extension

Detailed mathematical formulation of Einstein's relations, thermal equilibrium derivation of population inversion requirements, and threshold conditions for laser operation.

Prerequisites

Bohr's atomic model and discrete energy states
Planck's radiation law and thermal equilibrium
Maxwell-Boltzmann distribution of atomic populations

Fundamental Interaction Processes

Absorption involves an electron in a lower energy state E₁ absorbing an incident photon of energy h$\nu = E_{2} - E_{1}$ and transitioning to the excited state E₂.
Spontaneous emission is the random decay of an electron from the excited state E₂ to E₁ without external stimulation, emitting an incoherent photon with random phase and direction.
Stimulated emission requires an incoming photon of exact energy h$\nu = E_{2} - E_{1}$ to induce an excited electron to drop to E₁, releasing a clone photon.

Characteristics of Stimulated Photons

Coherence: The emitted photon is exactly in phase with the stimulating photon, maintaining spatial and temporal coherence.
Monochromaticity: The frequency of the emitted photon strictly matches the frequency of the incident photon, $\nu = (E_{2} - E_{1})/h$.
Directionality: The secondary photon travels in the precise geometrical direction of the primary stimulating photon, yielding a highly collimated beam.

Einstein's Coefficients

A₂₁ represents the Einstein coefficient for spontaneous emission, signifying the probability per unit time that an atom in state 2 decays to state 1.
B₂₁ represents the Einstein coefficient for stimulated emission, governing the transition rate from state 2 to state 1 induced by radiation density ρ(ν).
B₁₂ represents the Einstein coefficient for induced absorption, governing the upward transition rate from state 1 to state 2.

Condition for Amplification and Population Inversion

Under normal thermal equilibrium, the lower state population N₁ exceeds the upper state population N₂ according to the Boltzmann factor.
For light amplification via stimulated emission to dominate over absorption, the condition N₂ > N₁ must be satisfied, known as population inversion.
An external energy source called a 'pump' is required to achieve and maintain population inversion against natural decay rates.

Derivations

Derivation of Einstein's Relations between A and B Coefficients

Conditions: System of atoms in thermal equilibrium with radiation of energy density ρ(ν) at temperature T.

Write the rate of upward transitions (absorption): R₁₂ = N_1 B_{12} ρ(ν).
Write the total rate of downward transitions (spontaneous + stimulated): R₂₁ = N_2 [A_{21} + B_{21} ρ(ν)].
Equate upward and downward transition rates in thermal equilibrium: N_1 B_{12} ρ(ν) = N_2 [A_{21} + B_{21} ρ(ν)].
Express energy density from this balance: ρ(ν) = \frac{A_{21} / B_{21}}{(N_1 / N_2)(B_{12} / B_{21}) - 1}.
Substitute the Maxwell-Boltzmann ratio N_1 / $N_2 = \frac{g_1}{g_2} \exp\left(\frac{h\nu}{k_B T}\right)$. Assuming $g_1 = g_2 = 1$, N_1 / $N_2 = \exp\left(\frac{h\nu}{k_B T}\right)$.
Compare the resulting expression with Planck's Radiation Law: ρ(ν) = \frac{8\pi h \nu^3}{c^3} \frac{1}{\exp(h\nu / k_B T) - 1}.
Equating coefficients yields $B_{12} = B_{21}$ and \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}.

Result: $B_{12} = B_{21}$ and A_{21} / $B_{21} = 8\pi h \nu^3 / c^3$

Ratio of Spontaneous to Stimulated Emission Probabilities

Conditions: Harmonic radiation field in an optical cavity at frequency ν.

Define the ratio of spontaneous emission rate to stimulated emission rate as $R = \frac{A_{21}}{B_{21} \rho(\nu)}$.
Substitute Planck's expression for energy density ρ(ν) = \frac{8\pi h \nu^3}{c^3} \frac{1}{\exp(h\nu / k_B T) - 1}.
Substitute the relation A_{21} / $B_{21} = \frac{8\pi h \nu^3}{c^3}$.
Simplify the resulting algebraic fraction to isolate the exponential temperature dependence term.

Result: \frac{\text{Spontaneous Emission Rate}}{\text{Stimulated Emission Rate}} = \exp\left(\frac{h\nu}{k_B T}\right) - 1

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Rate of Induced Absorption	$$R_{12} = N_1 B_{12} \rho(\nu)$$	$N_1 = population of lower state$, $B_{12} = Einstein coefficient$ for absorption, \rho(\nu) = radiation energy density m^{-3} s^{-1}	Valid for non-degenerate atomic levels
Rate of Spontaneous Emission	$$R_{spont} = N_2 A_{21}$$	$N_2 = population of upper state$, $A_{21} = Einstein coefficient$ for spontaneous emission m^{-3} s^{-1}	Independent of external radiation field
Rate of Stimulated Emission	$$R_{stim} = N_2 B_{21} \rho(\nu)$$	$N_2 = population of upper state$, $B_{21} = Einstein coefficient$ for stimulated emission, \rho(\nu) = energy density m^{-3} s^{-1}	Requires incident photon matching transition energy
Einstein B Coefficient Equality	$$B_{12} = B_{21}$$	B_{12}, $B_{21} = Einstein coefficients$ for absorption and stimulated emission respectively m^3 J^{-1} s^{-2}	Assumes equal statistical weights for states 1 and 2
Ratio of Einstein A to B Coefficients	$$\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$$	A_{21}, $B_{21} = Einstein coefficients$, h = Planck's constant, \nu = frequency, c = speed of light J m^{-3}	Derived under thermal equilibrium using Planck's law
Boltzmann Population Ratio	$$\frac{N_2}{N_1} = \exp\left(-\frac{h\nu}{k_B T}\right)$$	N_1, $N_2 = level populations$, h = Planck's constant, \nu = frequency, $k_B = Boltzmann constant$, T = absolute temperature Dimensionless	Valid at thermal equilibrium
Transition Frequency Condition	$$h\nu = E_2 - E_1$$	h = Planck's constant, \nu = frequency, E_1, $E_2 = energy levels$ Joules (J)	Bohr frequency condition
High-Frequency Dominance of Spontaneous Emission	$$R_{spont} \gg R_{stim} \text{ at large } \nu$$	R_{spont}, $R_{stim} = emission rates$, \nu = frequency Dimensionless ratio context	Optical and UV frequencies where h\nu \gg k_B T

Graphs and experiments

Three-Level Laser Pumping Scheme Energy Diagram

Draw: Three horizontal lines representing E_1 (ground state), E_2 (metastable state), and E_3 (pump band), with upward arrows for optical pumping, non-radiative decay from E_3 to E_2, and downward laser transition from E_2 to E_1.

Interpret: Illustrates how atoms are pumped to a short-lived upper band from where they decay rapidly to a metastable intermediate state, facilitating population inversion relative to the ground state.

Exam use: Tested via conceptual labeling questions in PGT physics regarding ruby laser mechanics.

Wavelength Dependence of Spontaneous vs Stimulated Emission

Draw: A log-scale plot showing the ratio of spontaneous to stimulated emission rates increasing sharply as frequency ν increases in the optical domain.

Interpret: Demonstrates why achieving stimulated emission and laser action is significantly more difficult at shorter wavelengths (X-rays/UV) compared to microwaves and infrared (masers).

Exam use: Explains frequency limits of lasers and conceptual constraints in high-energy photon amplification.

Boltzmann Population Distribution Curve across Thermal States

Draw: Exponential decay curve of population N vs Energy E for different absolute temperatures T_1 and T_2.

Interpret: Shows that under thermal equilibrium, higher energy states are always less populated than lower energy states, proving that external pumping is mandatory to invert populations.

Exam use: Direct application in numericals calculating temperature or population ratios.

Solved numericals

TGT core

Calculate the ratio of the rate of spontaneous emission to the rate of stimulated emission for a system operating at wavelength $\lambda = 500 nm at room temperature T = 300 K$. (Take $h = 6.63 \times 10^{-34} J s$, $c = 3 \times 10^8 m/s$, $k_B = 1.38 \times 10^{-23} J/K)$

Find the frequency $\nu = c / \lambda = (3 \times 10^8) / (500 \times 10^{-9}) = 6 \times 10^{14} Hz$.
Calculate the exponent term hν / (k_B T) = \frac{6.63 \times 10^{-34} \times 6 \times 10^{14}}{1.38 \times 10^{-23} \times 300} = \frac{3.978 \times 10^{-19}}{4.14 \times 10^{-21}} \approx 96.08.
Evaluate the ratio R_spont / $R_stim = \exp(h\nu / k_B T) - 1 = \exp(96.08) - 1$.

Answer: \approx \exp(96.08) (extremely large, showing spontaneous emission completely dominates at optical wavelengths at room temperature).

PGT extension

Given that the Einstein A coefficient for a certain atomic transition is 2.5 × 10^7 s^{-1} and the transition wavelength is 6000 Å, calculate the Einstein B coefficient for stimulated emission. ($h = 6.63 \times 10^{-34} J s$, $c = 3 \times 10^8 m/s)$

Identify the relation between coefficients: \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{$c^3 = \frac{8\pi h c}{\lambda^3}$.
Substitute the given values: $\lambda = 6000 Å = 6 \times 10^{-7} m$, $A_{21} = 2.5 \times 10^7 s^{-1}$.
Calculate $B_{21} = \frac{A_{21} \lambda^3}{8 \pi h c}$.
$B_{21} = \frac{(2.5 \times 10^7) \times (6 \times 10^{-7})^3}{8 \times 3.1416 \times (6.63 \times 10^{-34}) \times (3 \times 10^8)}$.
Numerato$r = 2.5 \times 10^7 \times 2.16 \times 10^{-19} = 5.4 \times 10^{-12}$.
Denominato$r = 25.1327 \times 6.63 \times 10^{-34} \times 3 \times 10^8 = 499.9 \times 10^{-26} = 4.999 \times 10^{-24}$.
$B_{21} = \frac{5.4 \times 10^{-12}}{4.999 \times 10^{-24}} \approx 1.08 \times 10^{12} m^3 J^{-1} s^{-2}$.

Answer: 1.08 \times 10^{12} m^3 J^{-1} s^{-2}

PGT extension

At what temperature would the rate of spontaneous emission equal the rate of stimulated emission for a radiation wavelength of 1.0 μm? ($h = 6.63 \times 10^{-34} J s$, $c = 3 \times 10^8 m/s$, $k_B = 1.38 \times 10^{-23} J/K)$

Set the ratio equal to 1: \exp(h\nu / k_B T) - 1 = 1, which means \exp(h\nu / k_B T) = 2.
Take natural logarithm on both sides: \frac{h\nu}{k_B T} = \ln 2 \approx 0.693.
Express frequency $\nu = c / \lambda = (3 \times 10^8) / (1.0 \times 10^{-6}) = 3 \times 10^{14} Hz$.
Isolate temperature T: $T = \frac{h c}{\lambda k_B \ln 2}$.
Substitute values: $T = \frac{(6.63 \times 10^{-34}) \times (3 \times 10^8)}{(1.0 \times 10^{-6}) \times (1.38 \times 10^{-23}) \times 0.693}$.
Numerato$r = 1.989 \times 10^{-25}$.
Denominato$r = 1.0 \times 10^{-6} \times 1.38 \times 10^{-23} \times 0.693 = 9.5634 \times 10^{-30}$.
T = \frac{1.989 \times 10^{-25}}{9.5634 \times 10^{-30}} \approx 20,797 K.

Answer: 2.08 \times 10^4 K

Assumptions and limitations

Assumes two-level or simple multi-level atomic systems interacting with isotropic blackbody radiation fields in thermal equilibrium derivations.
Neglects Doppler broadening and collision broadening effects when defining monochromatic transition frequencies in ideal derivations.

Common misconceptions

Common error: Stimulated emission produces photons of higher energy than the incident photon.

Correct: Stimulated emission produces a photon with exactly the same energy (frequency) as the incident photon.

Energy conservation dictates that the emitted photon energy equals the exact energy difference E_2 - E_1 between the discrete atomic states.

Common error: Einstein's coefficient B_{12} is greater than B_{21} for non-degenerate systems.

Correct: For non-degenerate energy levels, B_{12} equals B_{21}.

Microscopic reversibility and thermal equilibrium proofs establish identical transition probability magnitudes for absorption and stimulated emission.

Quick revision

Stimulated emission yields coherent, monochromatic, and directionally aligned photons triggered by an incident photon of matching energy h$\nu = E_2 - E_1$.
Rate of Induced Absorption: $R_{12} = N_1 B_{12} \rho(\nu)$
Rate of Spontaneous Emission: $R_{spont} = N_2 A_{21}$

Must remember

Stimulated emission requires an incoming photon of exact energy $h\nu = E_2 - E_1$ to induce an excited electron to drop to $E_1$, releasing a clone photon.
The emitted photon in stimulated emission is strictly in phase, monochromatic, and travels in the same direction as the incident photon.
Under thermal equilibrium, the Boltzmann population ratio is given by $\frac{N_2}{N_1} = \exp\left(-\frac{h\nu}{k_B T}\right)$.
Population inversion ($N_2 > N_1$) is a mandatory condition for light amplification via stimulated emission to dominate over absorption.
For non-degenerate energy states, the Einstein B coefficients for absorption and stimulated emission are equal: $B_{12} = B_{21}$.
The ratio of Einstein coefficients is given by $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$, showing strong high-frequency dominance of spontaneous emission.

Glossary

Stimulated EmissionProcess where an incoming photon of energy $h\nu = E_2 - E_1$ triggers an excited atom to drop to a lower energy state, releasing an identical secondary photon.

Population InversionA non-equilibrium state in which the population of an upper energy level exceeds that of a lower energy level ($N_2 > N_1$).

Einstein A Coefficient ($A_{21}$)Probability per unit time for spontaneous decay of an atom from an excited state $2$ to a lower state $1$.

Einstein B Coefficient ($B_{21}$)Coefficient governing the rate of stimulated transition from state $2$ to state $1$ induced by radiation energy density.

Boltzmann FactorThe relative probability of finding a system in a state of higher energy compared to a lower energy state under thermal equilibrium.

Comparison of Emission Processes

Parameter	Spontaneous Emission	Stimulated Emission
External Stimulus	Not required (occurs naturally)	Required (incident photon of exact $h\nu$)
Phase & Direction	Random phase, incoherent, omnidirectional	Coherent, same phase, highly directional
Governing Rate Parameter	Einstein A coefficient ($A_{21}$)	Einstein B coefficient ($B_{21}\rho(\nu)$)

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following conditions must be satisfied for stimulated emission to occur when a photon interacts with an excited atom?

Explanation: Stimulated emission requires an incoming photon of exact energy matching the Bohr frequency condition $h\nu = E_2 - E_1$.

2. What is the relationship between the Einstein coefficients $B_{12}$ and $B_{21}$ for non-degenerate atomic energy levels?

Explanation: Under microscopic reversibility and thermal equilibrium proofs, the transition probabilities for absorption and stimulated emission are equal for non-degenerate levels, giving $B_{12} = B_{21}$.

3. Which characteristic is unique to the photon produced by stimulated emission compared to spontaneous emission?

Explanation: Stimulated emission produces a clone photon that is strictly in phase, monochromatic, and coherent with the primary stimulating photon.

4. What does the ratio $\frac{A_{21}}{B_{21}}$ depend upon according to Einstein's relations?

Explanation: Derived via Planck's law, $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$, showing it explicitly depends on the frequency $\nu$.

5. Under normal thermal equilibrium, which energy level population dominates according to the Boltzmann factor?

Explanation: Under thermal equilibrium, lower energy states are more populated than upper states, meaning $N_1 > N_2$ via the Boltzmann distribution $\frac{N_2}{N_1} = \exp\left(-\frac{h\nu}{k_B T}\right)$.

6. What is the primary purpose of an external 'pump' in a laser system?

Explanation: An external pumping source is strictly required to force atoms into the excited state, achieving population inversion against natural decay rates.

7. What is the unit of the Einstein coefficient $B_{21}$ for stimulated emission?

Explanation: Since $R_{stim} = N_2 B_{21} \rho(\nu)$ and $\rho(\nu)$ has units of $\text{J m}^{-3} \text{Hz}^{-1}$ or $\text{J m}^{-3} \text{s}$, $B_{21}$ has units of $\text{m}^3 \text{J}^{-1} \text{s}^{-2}$.

8. At very high frequencies (such as in the UV or X-ray region), which process dominates over stimulated emission?

Explanation: Since $\frac{A_{21}}{B_{21}} \propto \nu^3$, spontaneous emission dominates heavily at high frequencies where $h\nu \gg k_B T$.

9. What is the expression for the rate of induced absorption $R_{12}$?

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

10. Which transition process is entirely independent of external radiation fields?

Explanation: Spontaneous emission occurs randomly without external stimulation, governed solely by the internal characteristics of the excited state.

11. If the temperature $T \to \infty$ in a two-level atomic system, what does the population ratio $\frac{N_2}{N_1}$ approach?

Explanation: As $T \to \infty$, $\exp\left(-\frac{h\nu}{k_B T}\right) \to \exp(0) = 1$, meaning $N_2 = N_1$.

12. What condition represents population inversion in a two-level system?

Explanation: Population inversion is defined as the state where the upper energy level population exceeds the lower level population ($N_2 > N_1$).

13. What is the mathematical expression for the Bohr frequency condition governing atomic transitions?

Explanation: The energy difference between two discrete states is related to the photon frequency via $h\nu = E_2 - E_1$.

14. Which of the following statements about stimulated emission is incorrect?

Explanation: Stimulated emission produces coherent photons with precise directional and phase alignment, not random ones.

15. In the derivation of Einstein's relations under thermal equilibrium, what radiation law is utilized for energy density $\rho(\nu)$?

Explanation: Planck's radiation law is equated to the thermal equilibrium condition involving absorption and emission rates to derive the ratio of Einstein coefficients.

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 coefficient for spontaneous emission ($A_{21}$) to that of stimulated emission ($B_{21}$) varies with frequency $\nu$ as:

Explanation: From Einstein's relations, $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$, showing a cubic dependence on frequency $\nu$.

2. For light amplification via stimulated emission to take place in a medium, the condition to be satisfied is:

Explanation: Amplification requires stimulated emission to dominate over absorption, which necessitates population inversion ($N_2 > N_1$).

3. In a non-degenerate two-level atomic system in thermal equilibrium, the ratio of population of the upper state to the lower state is given by:

Explanation: The Boltzmann population ratio is $\frac{N_2}{N_1} = \exp\left(-\frac{h\nu}{k_B T}\right)$.

4. Which of the following properties is shared by photons emitted in stimulated emission?

Explanation: Stimulated emission yields clone photons that match the incident photon in frequency, phase, and direction of propagation.

5. If $B_{12}$ and $B_{21}$ are the Einstein coefficients for absorption and stimulated emission for non-degenerate energy states, then:

Explanation: Microscopic reversibility proves that the probability coefficients for absorption and stimulated emission are equal: $B_{12} = B_{21}$.

6. Why is it difficult to construct an X-ray laser compared to a visible light laser?

Explanation: Since $\frac{A_{21}}{B_{21}} \propto \nu^3$, spontaneous decay dominates overwhelmingly at X-ray frequencies, making population inversion extremely hard to sustain.

Timed Topic Test

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

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

Explanation: $A_{21}$ represents the probability per unit time of spontaneous decay, giving it the unit of reciprocal seconds ($\text{s}^{-1}$).

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

Explanation: Stimulated emission produces coherent clone photons, resulting in optical gain and light amplification.

3. In a two-level system under thermal equilibrium at absolute temperature $T$, if $T \to 0$, what happens to the population ratio $N_2 / N_1$?

Explanation: As $T \to 0$, $\exp\left(-\frac{h\nu}{k_B T}\right) \to \exp(-\infty) = 0$, meaning virtually all atoms reside in the lower ground state.

4. What does the radiation energy density $\rho(\nu)$ represent in the rate equation for stimulated emission?

Explanation: Energy density $\rho(\nu)$ represents spectral energy density with units of energy per unit volume per unit frequency.

5. Which of the following statements correctly contrasts spontaneous and stimulated emission?

Explanation: Spontaneous emission radiates isotropically in random directions, whereas stimulated emission follows the exact trajectory of the stimulating photon.

6. If the energy difference between two levels is $1.eV$, what is the approximate frequency of the photon involved in stimulated emission?

Explanation: Using $\nu = \frac{E_2 - E_1}{h} = \frac{1.6 \times 10^{-19} \text{ J}}{6.626 \times 10^{-34} \text{ J s}} \approx 2.4 \times 10^{14} \text{ Hz}$.

7. Under what condition can population inversion be achieved in a practical laser system?

Explanation: A two-level system cannot maintain population inversion under continuous pumping because absorption and stimulated emission rates equalize. Three or four-level systems are required.

8. In terms of Einstein coefficients, what is the rate equation for spontaneous emission?

Explanation: The spontaneous emission rate is given by $R_{spont} = N_2 A_{21}$.

9. What is the primary reason why stimulated emission alone cannot sustain population inversion in a two-level atomic system without pumping?

Explanation: Since $B_{12} = B_{21}$, an incident radiation field tends to equalize the populations of the two levels ($N_1 = N_2$), destroying any inversion.

10. Which of the following optical phenomena relies directly on stimulated emission?

Explanation: Lasers (Light Amplification by Stimulated Emission of Radiation) operate fundamentally on stimulated emission.