Derivation

Lasers: Stimulated Emission, Population Inversion and Gain

D-435 Home PU-306 Threads light · energy · chance Depends on Einstein Coefficients and Spontaneous Emission, The Boltzmann Distribution

Statement

For a two-level atomic system with lower level 1 (degeneracy \(g_1\)) and upper level 2 (degeneracy \(g_2\)) separated by frequency \(f\), the Einstein coefficients satisfy \(g_1 B_{12} = g_2 B_{21}\) and \(A_{21}/B_{21} = 8\pi h f^3/c^3\). Consequently, the net stimulated gain coefficient is proportional to the population inversion \(N_2 - (g_2/g_1)N_1\), which is strictly negative in thermal equilibrium at any finite temperature \(T\), making optical amplification impossible without a non-thermal pumping mechanism that drives the system out of equilibrium.

Why it matters

Laser operation relies on the dominance of stimulated emission over absorption. The Einstein relations provide the fundamental link between the microscopic quantum transition rates and the macroscopic thermodynamic properties of the radiation field. By deriving these relations from the requirement of consistency with Planck's law of blackbody radiation, we establish why population inversion is a necessary condition for gain and why it cannot be achieved in a two-level system under thermal equilibrium.

This derivation is the cornerstone of laser physics. It explains the design of three-level and four-level laser media, the threshold condition for lasing, and the fundamental limits of optical amplification. Understanding the interplay between spontaneous emission, stimulated emission, and absorption is essential for grasping how coherent light is generated and amplified in devices ranging from He-Ne lasers to fiber amplifiers.

Assumptions

The radiation field is in thermal equilibrium with the atoms.This allows the use of Planck's law for the spectral energy density \(\rho(f)\) and the Boltzmann distribution for the atomic populations. If the field is not thermal (e.g., a laser beam), the detailed balance argument fails, but the Einstein coefficients themselves remain valid constants of the atomic system.

The atomic levels are non-degenerate or degeneracies are accounted for.If \(g_1 \neq g_2\), the degeneracy factors must be included in the Einstein relations. Ignoring them leads to incorrect predictions for the required inversion level. For simplicity, many introductory treatments assume \(g_1 = g_2 = 1\), but the general case is presented here.

The transition frequency \(f\) is well-defined and narrow.We assume the atomic line shape is much narrower than the bandwidth of the radiation field, or that we are considering a monochromatic field at the line center. This justifies using the delta-function approximation for the transition rates.

The system is in the weak-field limit.The Einstein coefficients \(A, B\) are defined for weak fields where the transition rates are linear in the field intensity. At very high intensities, saturation effects modify the rates, and the simple linear gain model breaks down. This is relevant for the "Breaks when" section.

Derivation

\[ \frac{dN_2}{dt} = \underbrace{B_{12}\,\rho(f)\,N_1}_{\text{absorption}} - \underbrace{B_{21}\,\rho(f)\,N_2}_{\text{stimulated emission}} - \underbrace{A_{21}\,N_2}_{\text{spontaneous emission}} \]

Einstein’s rate equation for the upper-level population \(N_2\). Absorption lifts atoms from level 1 at a rate proportional to the spectral energy density \(\rho(f)\); stimulated emission returns them at a rate also proportional to \(\rho(f)\); spontaneous emission happens whether or not radiation is present. A

\[ B_{12}\,\rho\,N_1 = B_{21}\,\rho\,N_2 + A_{21}\,N_2 \quad\Longrightarrow\quad \rho(f) = \frac{A_{21}}{B_{12}\,(N_1/N_2) - B_{21}} \]

In thermal equilibrium \(dN_2/dt = 0\) (in fact detailed balance holds transition by transition). Solve for \(\rho(f)\). A

\[ \frac{N_2}{N_1} = \frac{g_2}{g_1}\,e^{-hf/kT} \]

Boltzmann distribution between the two levels, each state weighted by \(e^{-E/kT}\) and each level containing \(g_i\) states; \(E_2 - E_1 = hf\). A

\[ \rho(f) = \frac{A_{21}}{B_{12}\,\dfrac{g_1}{g_2}\,e^{hf/kT} - B_{21}} \]

Insert Step 3 into Step 2. A

\[ \rho(f) = \frac{8\pi h f^3}{c^3}\,\frac{1}{e^{hf/kT} - 1} \]

Planck’s law for the spectral energy density of black-body radiation (energy per unit volume per unit frequency). Atoms and radiation in a cavity at temperature \(T\) must reproduce it. A

\[ \frac{A_{21}/B_{21}}{\dfrac{g_1B_{12}}{g_2B_{21}}\,e^{hf/kT} - 1} = \frac{8\pi h f^3/c^3}{e^{hf/kT} - 1} \quad\text{for all } T \]

Divide numerator and denominator of Step 4 by \(B_{21}\) and equate to Step 5. The Einstein coefficients are properties of the atom, so they cannot depend on \(T\); the identity must hold at every temperature. B

\[ g_1 B_{12} = g_2 B_{21} \]

As \(T\to\infty\), \(\rho\) must grow without bound (Rayleigh–Jeans, \(\rho \approx 8\pi f^2 kT/c^3\)). The left side stays finite unless its denominator vanishes when \(e^{hf/kT}\to 1\), which forces \(g_1B_{12}/(g_2B_{21}) = 1\). C

\[ \frac{A_{21}}{B_{21}} = \frac{8\pi h f^3}{c^3} \]

With Step 7 the two denominators in Step 6 are identical, so the numerators must agree. Spontaneous and stimulated emission are therefore not independent: knowing one fixes the other. B

\[ \frac{\text{spontaneous rate}}{\text{stimulated rate}} = \frac{A_{21}}{B_{21}\,\rho(f)} = e^{hf/kT} - 1 \]

Divide Step 8 by Planck’s \(\rho\). For visible light from a thermal source (\(hf \approx 2\ \mathrm{eV}\), \(kT \approx 0.5\ \mathrm{eV}\) even at \(6000\ \mathrm{K}\)) the ratio is about \(e^{4}\approx 50\): thermal radiation is overwhelmingly spontaneous, which is why stimulated emission had to be engineered. B

\[ \frac{dI}{dz} = h f\,\frac{B_{21}}{c}\,\phi(f)\,\Big(N_2 - \frac{g_2}{g_1}N_1\Big)\,I \equiv g\,I \]

A beam of intensity \(I\) crossing the medium gains energy from stimulated emission and loses it to absorption; with Step 7, \(B_{12}N_1 = (g_2/g_1)B_{21}N_1\). Spontaneous photons go into all directions and do not add to the beam. \(\phi(f)\) is the normalised line shape. Writing \(g = \sigma\,(N_2 - (g_2/g_1)N_1)\) defines the stimulated-emission cross-section \(\sigma\). B

\[ g > 0 \iff N_2 > \frac{g_2}{g_1}N_1 \iff \frac{N_2}{g_2} > \frac{N_1}{g_1} \]

Amplification requires population inversion: more atoms per state in the upper level than in the lower. A

\[ \text{equilibrium: } \frac{N_2/g_2}{N_1/g_1} = e^{-hf/kT} < 1 \quad\text{for every } T>0 \]

From Step 3. A medium in thermal equilibrium at any positive temperature always absorbs; inversion is a non-equilibrium state that must be pumped. A

\[ g_1 = g_2:\quad 0 = W(N_1 - N_2) - A_{21}N_2 \;\Longrightarrow\; \frac{N_2}{N_1} = \frac{W}{W + A_{21}} < 1 \]

Pump a two-level system optically at its own transition with rate \(W = B\rho\) per atom. The pump stimulates emission exactly as efficiently as it causes absorption, so even an infinitely strong pump only equalises the populations. Inversion needs a third (and preferably fourth) level: pump into an auxiliary level that decays quickly into the upper laser level, and (four-level) let the lower laser level empty quickly into the ground state. C

\[ R_1R_2\,e^{2g_{\text{th}}L} = 1 \quad\Longrightarrow\quad g_{\text{th}} = \frac{1}{2L}\ln\frac{1}{R_1R_2} \]

In a cavity of length \(L\) with mirror reflectivities \(R_1, R_2\), light oscillates when one round trip (gain \(e^{2gL}\), mirror losses \(R_1R_2\)) returns at least as much power as it started with; other losses are neglected. A

Result

\[ g_1 B_{12} = g_2 B_{21}, \quad \frac{A_{21}}{B_{21}} = \frac{8 \pi h f^3}{c^3}, \quad \text{Gain} \propto N_2 - \frac{g_2}{g_1} N_1 \]

Reading. The Einstein relations establish the fundamental constants governing atomic transitions. The gain coefficient is positive only if the upper-level population exceeds the lower-level population by a factor of the degeneracy ratio. In thermal equilibrium, \(N_2/N_1 = (g_2/g_1) e^{-h f / k T} < g_2/g_1\), so the gain is always negative (absorption dominates). Population inversion requires non-thermal pumping.

Units check. \(A_{21}\) is a rate, \(\mathrm{s^{-1}}\). \(\rho(f)\) is energy per volume per frequency, \(\mathrm{J\,m^{-3}\,Hz^{-1} = J\,s\,m^{-3}}\), so \(B_{21}\rho\) is a rate only if \([B_{21}] = \mathrm{m^3\,J^{-1}\,s^{-2}}\); then \([A_{21}/B_{21}] = \mathrm{J\,s\,m^{-3}}\). The right side: \([hf^3/c^3] = \mathrm{J\,s \cdot s^{-3}/(m^3\,s^{-3}) = J\,s\,m^{-3}}\). Consistent.

Limiting cases

High-temperature limit (\(T \to \infty\)). The Boltzmann factor \(e^{-h f / k T} \to 1\), so \(N_2/N_1 \to g_2/g_1\). The population inversion approaches zero, and the gain vanishes. The radiation field approaches the Rayleigh-Jeans law.
Low-temperature limit (\(T \to 0\)). The Boltzmann factor \(e^{-h f / k T} \to 0\), so \(N_2/N_1 \to 0\). All atoms are in the ground state, and absorption dominates completely. The radiation field vanishes.
Equal degeneracies (\(g_1 = g_2\)). The Einstein relations simplify to \(B_{12} = B_{21}\) and \(A_{21}/B_{21} = 8 \pi h f^3/c^3\). The condition for gain is simply \(N_2 > N_1\).
Spontaneous emission dominance. At low temperatures or low field intensities, the spontaneous emission term \(A_{21} N_2\) dominates the rate equation. This is the source of noise in lasers (spontaneous emission into the lasing mode).

Breaks when

Gain saturation. At high intensities, the stimulated emission rate becomes comparable to the spontaneous emission rate, and the population inversion is depleted. The simple linear gain model \(dI/dz = g I\) fails, and saturation effects must be included.
Inhomogeneous broadening. If the atomic transition frequency varies across the ensemble (e.g., due to Doppler shift or crystal field variations), the simple two-level model with a single frequency \(f\) is insufficient. The gain spectrum becomes broadened, and the inversion condition must be considered at each frequency component.
Three-level systems. In a three-level laser (e.g., ruby), the lower laser level is the ground state. Achieving inversion requires pumping more than half the atoms to the upper level, which is energetically inefficient. The simple two-level equilibrium argument still holds, but the pumping mechanism is different.
Non-equilibrium pumping. The derivation assumes thermal equilibrium to derive the Einstein relations. However, lasers operate far from equilibrium. The relations themselves are valid, but the population inversion is maintained by external pumping, not by thermal distribution.

Failure modes

Assuming \(B_{12} = B_{21}\) without degeneracy factors. If \(g_1 \neq g_2\), the relation is \(g_1 B_{12} = g_2 B_{21}\). Ignoring degeneracies leads to incorrect predictions for the required inversion level.
Confusing spectral energy density \(\rho(f)\) with intensity \(I\). \(\rho(f)\) is energy per unit volume per unit frequency (\(\mathrm{J m^{-3} Hz^{-1}}\)), while \(I\) is power per unit area (\(\mathrm{W m^{-2}}\)). They are related by \(I = c \rho(f) \Delta f\) for a narrow band, but they are not the same quantity.
Believing that thermal equilibrium can produce inversion. The Boltzmann distribution always favors the lower energy level at finite temperature. Inversion is strictly a non-equilibrium phenomenon.
Ignoring spontaneous emission in the rate equations. Spontaneous emission is crucial for the initial seed of laser light and for the noise properties of the laser. It cannot be neglected in a complete description.

Discussion

The Einstein relations are a profound result because they connect microscopic quantum transition rates to macroscopic thermodynamic laws. The requirement that the atomic system and radiation field be in thermal equilibrium forces the coefficients \(A, B_{12}, B_{21}\) to satisfy specific constraints. This consistency check is a powerful tool in quantum optics, ensuring that any model of light-matter interaction is compatible with the second law of thermodynamics.

The key insight for laser physics is that stimulated emission is the only process that can produce coherent light. Absorption and spontaneous emission are incoherent processes. Stimulated emission produces a photon that is identical to the stimulating photon in frequency, phase, direction, and polarization. This is the mechanism by which a laser amplifies light. However, for net gain, the rate of stimulated emission must exceed the rate of absorption. This requires a population inversion, where the upper-level population exceeds the lower-level population (adjusted for degeneracy).

In a two-level system, optical pumping cannot produce inversion at all (Step 13): the pump light that lifts atoms up stimulates them back down just as efficiently. This is why most practical lasers use three-level or four-level schemes. In a four-level system (e.g., Nd:YAG, He-Ne), the lower laser level is rapidly depopulated to a lower energy state, allowing the upper laser level to accumulate a large population. This makes inversion easier to achieve and maintain.

The threshold condition for lasing arises from the balance between gain and losses in the optical cavity. The round-trip gain must exceed the round-trip losses. For a cavity with mirror reflectivities \(R_1, R_2\) and length \(L\), the threshold gain coefficient is \(g_{\text{th}} = \frac{1}{2L} \ln \frac{1}{R_1 R_2}\). This condition ensures that the light intensity grows with each round trip, leading to sustained oscillation. Below threshold, the losses dominate, and the cavity emits only spontaneous radiation (fluorescence).

Common misconceptions. A common misconception is that lasers emit light because of spontaneous emission. While spontaneous emission provides the initial seed, the coherent laser light is produced by stimulated emission. Another misconception is that population inversion means \(N_2 > N_1\) always. If the levels are degenerate, this is true, but if \(g_2 > g_1\), the condition is \(N_2 > (g_2/g_1) N_1\). Finally, some students believe that the Einstein relations are derived from quantum mechanics alone. In fact, they are derived from the requirement of consistency with thermodynamics, which is a classical constraint on the quantum rates.

Worked examples

Example 1. A two-level atomic system has degeneracies \(g_1 = 2\) and \(g_2 = 3\). The transition frequency is \(f = 5.0 \times 10^{14}\ \mathrm{Hz}\). Calculate the ratio \(B_{12}/B_{21}\) and the value of \(A_{21}/B_{21}\). Use \(h = 6.626 \times 10^{-34}\ \mathrm{J \cdot s}\) and \(c = 3.0 \times 10^8\ \mathrm{m/s}\).

\[ \frac{B_{12}}{B_{21}} = \frac{g_2}{g_1} = \frac{3}{2} = 1.5 \]

Use the first Einstein relation \(g_1 B_{12} = g_2 B_{21}\) to find the ratio of the B coefficients. A

\[ \frac{A_{21}}{B_{21}} = \frac{8 \pi h f^3}{c^3} \]

Use the second Einstein relation to express the ratio \(A_{21}/B_{21}\) in terms of fundamental constants and the frequency. A

\[ \frac{A_{21}}{B_{21}} = \frac{8 \pi (6.626 \times 10^{-34}) (5.0 \times 10^{14})^3}{(3.0 \times 10^8)^3} \]

Substitute the given values into the formula. A

\[ \frac{A_{21}}{B_{21}} = \frac{8 \pi (6.626 \times 10^{-34}) (1.25 \times 10^{44})}{27 \times 10^{24}} = \frac{8 \pi (8.2825 \times 10^{10})}{27 \times 10^{24}} \]

Calculate the numerator and denominator separately. A

\[ \frac{A_{21}}{B_{21}} = \frac{2.078 \times 10^{12}}{2.7 \times 10^{25}} \approx 7.7 \times 10^{-14}\ \mathrm{J\,s\,m^{-3}} \]

Perform the division to find the final value. A

\[ \frac{B_{12}}{B_{21}} = 1.5, \quad \frac{A_{21}}{B_{21}} \approx 7.7 \times 10^{-14}\ \mathrm{J\,s\,m^{-3}} \]

Reading. The B coefficients are related by the degeneracy ratio. The A/B ratio is the spectral energy density at which stimulated emission equals spontaneous emission; because it grows as \(f^3\), spontaneous emission becomes ever harder to beat at short wavelengths, which is why X-ray lasers are so hard to build.

Units check. \(A_{21}/B_{21}\) has units \(\mathrm{J\,s\,m^{-3}}\), those of a spectral energy density \(\rho(f)\).

Example 2. A laser cavity has length \(L = 1.0\ \mathrm{m}\) and mirror reflectivities \(R_1 = 0.95\) and \(R_2 = 0.90\). Calculate the threshold gain coefficient \(g_{\text{th}}\). Assume no other losses.

\[ g_{\text{th}} = \frac{1}{2L} \ln \frac{1}{R_1 R_2} \]

Use the threshold condition for lasing, where the round-trip gain equals the round-trip losses. A

\[ R_1 R_2 = 0.95 \times 0.90 = 0.855 \]

Calculate the product of the mirror reflectivities. A

\[ \ln \frac{1}{0.855} = \ln (1.1696) \approx 0.1568 \]

Calculate the natural logarithm of the inverse of the reflectivity product. A

\[ g_{\text{th}} = \frac{1}{2(1.0)} (0.1568) = 0.0784\ \mathrm{m^{-1}} \]

Substitute the values into the threshold gain formula. A

\[ g_{\text{th}} \approx 0.078\ \mathrm{m^{-1}} \]

Reading. The gain coefficient must exceed this value for lasing to occur. This is a relatively low threshold, typical for high-reflectivity cavities.

Units check. \(g_{\text{th}}\) has units of \(\mathrm{m^{-1}}\), which is consistent with a gain coefficient (inverse length).

Problems

A two-level system has \(g_1 = 1\) and \(g_2 = 1\). If the transition frequency is \(f = 1.0 \times 10^{15}\ \mathrm{Hz}\), calculate \(A_{21}/B_{21}\). Use \(h = 6.626 \times 10^{-34}\ \mathrm{J \cdot s}\) and \(c = 3.0 \times 10^8\ \mathrm{m/s}\).
Solution
\[ \frac{A_{21}}{B_{21}} = \frac{8 \pi h f^3}{c^3} = \frac{8 \pi (6.626 \times 10^{-34}) (1.0 \times 10^{15})^3}{(3.0 \times 10^8)^3} \] \[ = \frac{8 \pi (6.626 \times 10^{-34}) (1.0 \times 10^{45})}{27 \times 10^{24}} = \frac{1.665 \times 10^{13}}{2.7 \times 10^{25}} \approx 6.2 \times 10^{-13}\ \mathrm{J\,s\,m^{-3}} \]
A laser medium has a population inversion \(N_2 - N_1 = 1.0 \times 10^{20}\ \mathrm{m^{-3}}\) and a stimulated emission cross-section \(\sigma = 1.0 \times 10^{-20}\ \mathrm{m^2}\). Calculate the gain coefficient \(g\).
Solution
With equal degeneracies the gain coefficient is \(g = \sigma (N_2 - N_1)\) (Step 10). \[ g = (1.0 \times 10^{-20}\ \mathrm{m^2}) (1.0 \times 10^{20}\ \mathrm{m^{-3}}) = 1.0\ \mathrm{m^{-1}} \]
A three-level laser has a lower laser level that is the ground state. If the total number of atoms is \(N = 1.0 \times 10^{22}\ \mathrm{m^{-3}}\), what is the minimum population of the upper level \(N_2\) required for inversion?
Solution
For inversion, \(N_2 > N_1\). Since \(N_1 + N_2 = N\) (assuming no atoms in the third level), we have \(N_2 > N - N_2\), so \(2 N_2 > N\), or \(N_2 > N/2\). \[ N_2 > \frac{1.0 \times 10^{22}}{2} = 5.0 \times 10^{21}\ \mathrm{m^{-3}} \]
A laser cavity has length \(L = 0.5\ \mathrm{m}\) and mirror reflectivities \(R_1 = 0.99\) and \(R_2 = 0.98\). Calculate the threshold gain coefficient \(g_{\text{th}}\).
Solution
\[ g_{\text{th}} = \frac{1}{2L} \ln \frac{1}{R_1 R_2} = \frac{1}{2(0.5)} \ln \frac{1}{0.99 \times 0.98} \] \[ R_1 R_2 = 0.9702 \] \[ \ln \frac{1}{0.9702} = \ln (1.0307) \approx 0.0302 \] \[ g_{\text{th}} = 1.0 \times 0.0302 = 0.0302\ \mathrm{m^{-1}} \]
A two-level system is in thermal equilibrium at \(T = 300\ \mathrm{K}\). The transition frequency is \(f = 5.0 \times 10^{14}\ \mathrm{Hz}\). Calculate the ratio \(N_2/N_1\) and determine if population inversion is possible. Use \(k = 1.38 \times 10^{-23}\ \mathrm{J/K}\) and \(h = 6.626 \times 10^{-34}\ \mathrm{J \cdot s}\).
Solution
\[ \frac{N_2}{N_1} = e^{-h f / k T} = e^{-\frac{(6.626 \times 10^{-34})(5.0 \times 10^{14})}{(1.38 \times 10^{-23})(300)}} \] \[ \frac{h f}{k T} = \frac{3.313 \times 10^{-19}}{4.14 \times 10^{-21}} \approx 80.0 \] \[ \frac{N_2}{N_1} = e^{-80.0} \approx 1.8 \times 10^{-35} \] Since \(N_2/N_1 \ll 1\), the upper level is almost empty, and population inversion is impossible in thermal equilibrium.