Stimulated Emission and the Gain Coefficient

In 1917 Einstein asked a question that sounds like pure thermodynamics: how can a gas of atoms sit in equilibrium inside a blackbody cavity, endlessly trading photons with the Planck field, without ever drifting off that spectrum? The answer forced a new process into existence. Absorption and spontaneous emission alone cannot balance the books — the rates come out wrong. Einstein showed a third process must exist, in which a photon already in the field induces an excited atom to emit a second photon that is its exact copy: same frequency, phase, polarization and direction. That process is stimulated emission, the seed of every laser and optical amplifier ever built. The bookkeeping is carried by the Einstein $A$ and $B$ coefficients, and detailed balance against the Planck spectrum fixes two universal relations among them that hold for any two-level transition. Those relations let the macroscopic response of a medium be written as a single microscopic cross section $\sigma(\nu)$ times a net atom density. The central quantity is the gain coefficient $\gamma(\nu)=\Delta N\,\sigma(\nu)$ , the fractional change in intensity per unit length, and its sign belongs entirely to the degeneracy-weighted difference $\Delta N=N_2-(g_2/g_1)N_1$ . Only when the upper level is over-populated — a population inversion — does a beam grow, and in every thermal medium the Boltzmann factor guarantees the opposite. That is exactly why lasers must be pumped.

Key takeaways

Stimulated emission, predicted by Einstein in 1917, is the process in which an incident photon triggers an excited atom to emit a second photon identical in frequency, phase, polarization and direction — the basis of coherent amplification.
Detailed balance against the Planck spectrum fixes the Einstein relations $g_1 B_{12}=g_2 B_{21}$ and $A_{21}/B_{21}=8\pi h\nu^3/c^3$ , tying the three processes together.
Net amplification is the gain coefficient $\gamma(\nu)=\Delta N\,\sigma(\nu)$ , with $\Delta N=N_2-(g_2/g_1)N_1$ the degeneracy-weighted population difference and $\sigma(\nu)$ the cross section.
A beam obeys $I(z)=I(0)\,e^{\gamma(\nu)z}$ — Beer–Lambert with a sign: gain when $\Delta N>0$ , loss when $\Delta N<0$ , and thermal equilibrium always gives loss.
The cross section $\sigma(\nu)=\lambda^2 g(\nu)/(8\pi t_\mathrm{sp})$ favours long-wavelength, long-lived, narrow-line transitions, and its lineshape $g(\nu)$ sets where a laser can oscillate.
Key idea: amplification is stimulated emission outrunning absorption. With $\gamma(\nu)=\Delta N\,\sigma(\nu)$ and $\Delta N=N_2-(g_2/g_1)N_1$ , a beam follows $I(z)=I(0)\,e^{\gamma z}$ — the medium amplifies only when its population is inverted.

The core insight

Amplification is stimulated emission winning over absorption. Quantitatively $\gamma(\nu)=\Delta N\,\sigma(\nu)$ with $\Delta N=N_2-(g_2/g_1)N_1$ , so a beam obeys $I(z)=I(0)\,e^{\gamma z}$ : gain when the population is inverted, loss otherwise.

Where this lives today

The gain coefficient is the working currency of every amplifier that moves light rather than electrons. The erbium-doped fibre amplifier (EDFA) carrying the internet is a direct use of $I(z)=I(0)e^{\gamma z}$ : a few metres of erbium fibre, pumped at $980\ \mathrm{nm}$ , present a positive $\gamma$ near $1550\ \mathrm{nm}$ and boost a wavelength-multiplexed signal by $20$ – $30\ \mathrm{dB}$ without converting it to electronics, gain saturation flattening the response across channels. The same $\sigma$ -and- $t_\mathrm{sp}$ bookkeeping selects solid-state media — Nd:YAG's large cross section and long lifetime for energy storage, Ti:sapphire's huge bandwidth for femtosecond, tunable pulses. The identical equation with $\gamma<0$ is the backbone of quantitative spectroscopy: $-\ln[I(L)/I(0)]=|\gamma|L=N_1\sigma_{12}L$ turns a transmission dip into a trace-gas column density, and the Voigt lineshape encodes the sample's temperature and pressure. Cavity-enhanced and comb-based instruments push this to parts-per-trillion sensitivity for greenhouse-gas monitoring — the same cross section that, sign reversed, makes a laser amplify.

The physical picture

Picture an excited atom as a tiny driven oscillator carrying a dipole. A passing wave at the transition frequency shakes that dipole. If the atom is in the upper state, the wave does negative work — the atom dumps energy into the field, exactly in step, so the emitted contribution adds coherently and the wave grows. If it is in the lower state, the wave does positive work and loses energy: absorption. Whether a slab of atoms brightens or dims a beam is thus a tug-of-war decided by which state holds more atoms — absorption and stimulated emission are mirror images of one interaction, differing only in the direction of energy flow. The three Einstein processes this competition rests on are worth seeing side by side, because their asymmetry is the whole story.

Figure

The three radiative processes Einstein balanced in 1917 · absorption removes a photon and lifts the atom E₁ → E₂ · spontaneous emission drops it with a photon of random phase and direction · stimulated emission is triggered by an incident photon and returns a second photon identical in frequency, phase, polarization and direction — one in, two identical out

The "cloning" is really mode-counting. Stimulated emission deposits its photon into the same field mode as the trigger, so the new photon inherits that mode's frequency, direction, polarization and phase — which is why an amplifier preserves the shape of the wave it magnifies. Spontaneous emission is stimulated emission driven by the vacuum fluctuations of the empty modes: the atom picks one at random, hence a random direction and phase, and the ratio $A_{21}/B_{21}=8\pi h\nu^3/c^3$ is just $h\nu$ times the mode density. The gain coefficient packages this as a per-atom target area $\sigma(\nu)$ times a net atom density; flip that density — more absorbers than emitters — and the same picture gives Beer–Lambert absorption. Because $\sigma\propto\lambda^2/(t_\mathrm{sp}\Delta\nu)$ , strong, narrow, long-wavelength transitions amplify most per excited atom. The next figure shows the punchline: the sign of the gain lives entirely in $\Delta N$ .

The core equations

Two rates open the ledger. A field of spectral energy density $\rho(\nu)$ empties the upper level by stimulated emission and the lower level by absorption, each rate proportional to the field and the population it acts on — the definitions of the Einstein $B$ coefficients.

\frac{\mathrm{d}N_2}{\mathrm{d}t}\Big|_\mathrm{stim}=-B_{21}\,\rho(\nu)\,N_2,\qquad \frac{\mathrm{d}N_1}{\mathrm{d}t}\Big|_\mathrm{abs}=-B_{12}\,\rho(\nu)\,N_1 \qquad (1)

where: $N_2,\,N_1$ — population densities of the upper and lower levels (m⁻³); $B_{21},\,B_{12}$ — Einstein coefficients for stimulated emission and absorption; $\rho(\nu)$ — spectral energy density of the field (J m⁻³ Hz⁻¹). Both processes scale with the same field, so their competition is fixed by the atoms alone. Requiring these rates plus spontaneous decay to hold the atoms on the Planck spectrum ties the three coefficients together into a pair of relations that are properties of the transition — independent of temperature, pumping, or how the levels are occupied.

g_1 B_{12}=g_2 B_{21},\qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3} \qquad (2)

where: $A_{21}$ — spontaneous-emission rate coefficient (s⁻¹), with lifetime $t_\mathrm{sp}=1/A_{21}$ ; $g_1,\,g_2$ — level degeneracies; $h,\,c$ — Planck's constant and speed of light. The first relation makes absorption and stimulated emission near-symmetric; the second, $\propto\nu^3$ , is why inversions are hard to keep at optical frequencies. The macroscopic response follows: summed over a beam, the net gain per unit length is one population difference times one cross section — the gain coefficient — whose sign is carried entirely by the degeneracy-weighted $\Delta N$ .

\gamma(\nu)=\Delta N\,\sigma(\nu),\qquad \Delta N=N_2-\frac{g_2}{g_1}\,N_1 \qquad (3)

where: $\gamma(\nu)$ — gain coefficient, or absorption when negative (m⁻¹); $\Delta N$ — degeneracy-weighted population difference (m⁻³); $\sigma(\nu)$ — stimulated-emission cross section (m²). Read it as a target area: $\Delta N$ inverted atoms, each of area $\sigma$ , multiply the beam. The cross section is not free — the Einstein relations pin it to the spontaneous lifetime and lineshape, bridging a fluorescence decay to an amplification rate.

\sigma(\nu)=\frac{h\nu}{c}\,B_{21}\,g(\nu)=\frac{\lambda^2}{8\pi t_\mathrm{sp}}\,g(\nu)=\frac{A_{21}\lambda^2}{8\pi}\,g(\nu) \qquad (4)

where: $\lambda=c/n\nu_0$ — wavelength in the medium; $t_\mathrm{sp}=1/A_{21}$ — upper-state radiative lifetime; $g(\nu)$ — normalized lineshape, $\int g(\nu)\,\mathrm{d}\nu=1$ (Hz⁻¹). The scaling $\sigma\propto\lambda^2/(t_\mathrm{sp}\Delta\nu)$ is the rule of thumb for a strong transition. The frequency structure lives in $g(\nu)$ : homogeneous (lifetime- or collision-broadened) transitions give a Lorentzian $g_L(\nu)=(\Delta\nu/2\pi)/[(\nu-\nu_0)^2+(\Delta\nu/2)^2]$ , while the inhomogeneous Doppler spread of velocities gives a Gaussian $g_D$ of FWHM $\Delta\nu_D=\nu_0\sqrt{8\ln2\,k_BT/Mc^2}$ (used in worked example 2). Both integrate to one, so a broader line spreads the same oscillator strength over more frequency and lowers the peak $\sigma$ . Finally, integrating the per-length gain gives the amplification law: a beam grows or decays exponentially, the exponent being $\gamma$ times the path — Beer–Lambert with the sign of $\Delta N$ .

I(z)=I(0)\,e^{\gamma(\nu)\,z} \qquad (5)

where: $I(z)$ — intensity after distance $z$ (W m⁻²); $\gamma>0$ — exponential growth (gain); $\gamma<0$ — exponential decay (absorption). The single-pass power gain in decibels is $10\log_{10}e\cdot\gamma z\approx4.34\,\gamma z$ .

Derivation AEinstein's detailed-balance argument for the A and B relations

The goal is to reproduce Einstein's 1917 reasoning and show stimulated emission is not optional: without it a two-level gas in a blackbody cavity cannot stay on the Planck spectrum. We impose steady state, solve for the field energy density the atomic kinetics demand, and force it to equal Planck's law term by term — a match possible only if the two Einstein relations hold, which predicts stimulated emission and fixes its strength in one stroke.

Step 1. In equilibrium the flows balance, $B_{12}\rho N_1=A_{21}N_2+B_{21}\rho N_2$ : absorption on the left, spontaneous plus stimulated emission on the right. Why: every atom lifted up must on average come back down, so the total rates are equal.

Step 2. Solve for the field, $\rho=A_{21}\big/\big(B_{12}(N_1/N_2)-B_{21}\big)$ . Why: this isolates the density the kinetics produce, which must match what the cavity walls supply.

Step 3. Feed in the Boltzmann ratio $N_1/N_2=(g_1/g_2)\,e^{h\nu/k_BT}$ . Why: at the cavity temperature the populations are dictated by statistical mechanics, not free.

Step 4. Substituting gives $\rho=(A_{21}/B_{21})\big/\big[(g_1B_{12}/g_2B_{21})\,e^{h\nu/k_BT}-1\big]$ , which can equal Planck's $\rho=(8\pi h\nu^3/c^3)/(e^{h\nu/k_BT}-1)$ for all $T$ only if the bracket reduces to $e^{h\nu/k_BT}-1$ . Why: two functions of $T$ agree everywhere only when their forms match identically.

Step 5. Matching forces $g_1B_{12}=g_2B_{21}$ and $A_{21}/B_{21}=8\pi h\nu^3/c^3$ . Why: the first makes emission and absorption near-mirror processes; the second is $h\nu$ times the mode density — spontaneous emission is vacuum-stimulated. *Verified: dropping $B_{21}$ leaves $\rho\propto e^{-h\nu/k_BT}$ , the Wien tail — no stimulated term, no Planck spectrum (Loudon ch. 1, Foot ch. 1).*

Go deeper · Derivation B · 1 minAssembling the gain coefficient from the microscopic rates

Derivation BAssembling the gain coefficient from the microscopic rates

The aim is to build the amplification law $\gamma=\Delta N\,\sigma$ directly from the two-level rates, making explicit where the degeneracy factor and lineshape enter and confirming the units land on m⁻¹. We start from the net rate the field gains energy, convert it into growth along the beam, eliminate absorption with the first Einstein relation, and bundle the atomic constants into one cross section — no new physics, only equation (1) and the Einstein relations.

Step 1. The field's energy grows as $\mathrm{d}\rho/\mathrm{d}t=h\nu\,(B_{21}N_2-B_{12}N_1)\,\rho\,g(\nu)$ : emission adds photons, absorption removes them, $g(\nu)$ weights the detuning. Why: only the difference of the rates is observable as net gain, and the lineshape restores the frequency dependence.

Step 2. Convert time to distance with $\mathrm{d}/\mathrm{d}z=(n/c)\,\mathrm{d}/\mathrm{d}t$ , using $I=(c/n)\rho$ . Why: an amplifier is described by growth along the path, and a wavepacket carries energy at $c/n$ .

Step 3. Eliminate absorption with $B_{12}=(g_2/g_1)B_{21}$ , so the bracket becomes $B_{21}\big(N_2-(g_2/g_1)N_1\big)$ . Why: this is where the degeneracy weighting is born — for $g_1\neq g_2$ the difference is $\Delta N$ , not $N_2-N_1$ .

Step 4. Package the constants into $\sigma(\nu)=(h\nu/c)B_{21}\,g(\nu)=\lambda^2 g(\nu)/(8\pi t_\mathrm{sp})$ , using $A_{21}/B_{21}=8\pi h\nu^3/c^3$ . Why: this turns a mess of coefficients into one measurable area, and $t_\mathrm{sp}=1/A_{21}$ links gain to a fluorescence lifetime.

Step 5. Collect: $\dfrac{1}{I}\dfrac{\mathrm{d}I}{\mathrm{d}z}=\big(N_2-\tfrac{g_2}{g_1}N_1\big)\sigma(\nu)\equiv\gamma(\nu)$ , solving to $I(z)=I(0)e^{\gamma z}$ . Why: fractional intensity change per length is by definition the gain coefficient. *Verified: dimensions $[\sigma]=\mathrm{m^2}$ , $[\Delta N]=\mathrm{m^{-3}}$ give $[\gamma]=\mathrm{m^{-1}}$ ; sign check $N_2>(g_2/g_1)N_1\Rightarrow\gamma>0$ (Saleh & Teich Eq. 15.1-4, Foot §7.3).*

Go deeper · Derivation C · 1 minWhy the gain saturates and clamps a laser at threshold

Derivation CWhy the gain saturates and clamps a laser at threshold

The purpose is to explain a fact every laser depends on: gain is not fixed but sags as the beam it amplifies grows stronger, and this self-limiting locks a running laser's output. We solve the two-level populations in steady state with the stimulated terms kept, and identify the saturation intensity at which the gain has halved — the same stimulated emission that provides the gain erodes it, one amplified photon per depleted upper-state atom.

Step 1. With a stimulated rate $W=\sigma I/h\nu$ and relaxation time $\tau$ , the steady-state inversion is $\Delta N=\Delta N_0/(1+\tau W)$ , with $\Delta N_0$ the small-signal inversion. Why: a stronger field recycles atoms faster, dragging $\Delta N$ toward zero — a beam eats the inversion that amplifies it.

Step 2. Rewrite $\tau W=(\sigma\tau/h\nu)\,I\equiv I/I_s$ , defining $I_s=h\nu/(\sigma\tau)$ . Why: only the ratio $I/I_s$ matters, and $I_s$ is the yardstick for strong versus weak light.

Step 3. The gain then follows $\gamma(\nu,I)=\Delta N\,\sigma=\gamma_0(\nu)/(1+I/I_s)$ . Why: as intensity builds, $\gamma$ slides down until it equals the round-trip loss and can fall no further — clamped, fixing the output power. *Verified: limits $I\ll I_s\Rightarrow\gamma\to\gamma_0$ (linear amplifier) and $I\gg I_s\Rightarrow\gamma\to0$ (transparent); matches Saleh & Teich §15.3 and Siegman's homogeneous-saturation formula. For Nd:YAG, $I_s\approx2.9\ \mathrm{kW\,cm^{-2}}$ (below).*

Limiting cases

Case	Result	Meaning
Thermal equilibrium	$\dfrac{N_2}{N_1}=\dfrac{g_2}{g_1}e^{-h\nu/k_BT}<\dfrac{g_2}{g_1}$	$\Delta N<0$ always, so $\gamma<0$ — no thermal medium can amplify
Low frequency ( $h\nu\ll k_BT$ )	$A_{21}\ll B_{21}\rho$	stimulated dominates spontaneous; masers amplify with modest inversion
Optical / high frequency ( $h\nu\gg k_BT$ )	$A_{21}\gg B_{21}\rho$	spontaneous swamps the thermal field; inversion needs active pumping
Line centre	$\sigma(\nu_0)=\dfrac{\lambda^2}{8\pi t_\mathrm{sp}}g(\nu_0)$	peak cross section; a Lorentzian gives $\sigma_0=\lambda^2/(4\pi^2 t_\mathrm{sp}\Delta\nu)$
Natural (lifetime) broadening	$\sigma(\nu_0)=\dfrac{\lambda^2}{2\pi}$	the absolute ceiling: $t_\mathrm{sp}$ cancels, only $\lambda^2$ survives
Strong field ( $I\gg I_s$ )	$\gamma\to0$	the medium bleaches transparent; gain clamps a laser's power

Common trapAny atom in an excited state will amplify a passing beam

Not on its own. Amplification requires stimulated emission to outnumber absorption along the beam — a statement about the whole population, not one atom. The net rate carries the degeneracy-weighted $\Delta N=N_2-(g_2/g_1)N_1$ , so a handful of excited atoms in a sea of ground-state atoms still gives $\Delta N<0$ and net absorption. Two traps hide here. First, writing $N_2-N_1$ instead of $\Delta N$ gives the wrong sign whenever $g_1\neq g_2$ — the degeneracy FAQ shows a case that flips from gain to loss. Second, stimulated emission (coherent, same mode) is not spontaneous emission (random phase and direction, the amplified-spontaneous-emission noise source); only the former builds a usable beam.

Worked examples

1 · Gain and saturation of a Nd:YAG amplifier (the real numbers)

A diode-pumped Nd:YAG rod amplifies at $\lambda=1064\ \mathrm{nm}$ with $\sigma=2.8\times10^{-19}\ \mathrm{cm^2}$ and upper-state lifetime $t_\mathrm{sp}=230\ \mu\mathrm{s}$ . The pump holds an inversion $\Delta N=6\times10^{17}\ \mathrm{cm^{-3}}$ over $10\ \mathrm{cm}$ . Find the small-signal gain coefficient, the single-pass power gain (factor and dB), and the saturation intensity.

Step 1. The small-signal gain coefficient is $\gamma_0=\Delta N\,\sigma=(6\times10^{17}\ \mathrm{cm^{-3}})(2.8\times10^{-19}\ \mathrm{cm^2})=0.168\ \mathrm{cm^{-1}}$ ( $16.8\ \mathrm{m^{-1}}$ ; keeping cgs units avoids a factor-of- $10^{6}$ slip) — the peak height in the second figure.

Step 2. Over $L=10\ \mathrm{cm}$ the exponent is $\gamma_0 L=1.68$ , so the single-pass power gain is $e^{1.68}=\mathbf{5.4\times}$ , or $4.34\times1.68=\mathbf{7.3\ dB}$ — realistic for one amplifier stage.

Step 3. The saturation intensity needs $h\nu=hc/\lambda=1.87\times10^{-19}\ \mathrm{J}$ ( $1.17\ \mathrm{eV}$ ). With $\sigma=2.8\times10^{-23}\ \mathrm{m^2}$ and $t_\mathrm{sp}=2.30\times10^{-4}\ \mathrm{s}$ , $I_s=h\nu/(\sigma t_\mathrm{sp})=(1.87\times10^{-19})/[(2.8\times10^{-23})(2.30\times10^{-4})]=2.9\times10^{7}\ \mathrm{W\,m^{-2}}=\mathbf{2.9\ kW\,cm^{-2}}$ .

Step 4. The rod delivers $\mathbf{7.3\ dB}$ of small-signal gain, but as the circulating intensity approaches $\mathbf{2.9\ kW\,cm^{-2}}$ the gain halves and, in an oscillator, clamps where $\gamma$ equals the loss. The long lifetime that keeps $I_s$ modest is what lets Nd:YAG store energy for Q-switched pulses.

2 · The peak cross section and its ceiling (rubidium D2)

The Rb D2 line at $\lambda=780\ \mathrm{nm}$ has radiative lifetime $t_\mathrm{sp}=26\ \mathrm{ns}$ . Find the natural-broadening ceiling $\sigma(\nu_0)=\lambda^2/2\pi$ , then the line-centre cross section when room-temperature Doppler broadening widens the line to $\Delta\nu_D=500\ \mathrm{MHz}$ . By how much does Doppler broadening cut $\sigma$ ?

Step 1. The universal ceiling is $\sigma_\mathrm{max}=\lambda^2/2\pi=(7.80\times10^{-7})^2/(2\pi)=9.68\times10^{-14}\ \mathrm{m^2}=\mathbf{9.7\times10^{-10}\ cm^2}$ — what $\sigma$ would reach if the only broadening were the atom's finite lifetime, of natural FWHM $\Delta\nu_\mathrm{nat}=1/(2\pi t_\mathrm{sp})=6.1\ \mathrm{MHz}$ .

Step 2. Doppler broadening is inhomogeneous and Gaussian, so $g_D(\nu_0)=(2/\Delta\nu_D)\sqrt{\ln2/\pi}=(2/5.0\times10^{8})(0.4697)=1.88\times10^{-9}\ \mathrm{Hz^{-1}}$ — the velocity spread smears the oscillator strength across $500\ \mathrm{MHz}$ instead of $6\ \mathrm{MHz}$ .

Step 3. Insert into equation (4): $\sigma(\nu_0)=\dfrac{\lambda^2}{8\pi t_\mathrm{sp}}g_D(\nu_0)=\dfrac{(7.80\times10^{-7})^2}{8\pi(2.6\times10^{-8})}(1.88\times10^{-9})=1.75\times10^{-15}\ \mathrm{m^2}=\mathbf{1.7\times10^{-11}\ cm^2}$ .

Step 4. The ratio is $\sigma_\mathrm{max}/\sigma(\nu_0)\approx\mathbf{55}$ : Doppler motion throws away a factor of about $55$ in peak cross section — the reason a warm vapour needs far more inverted atoms per unit gain than a cold, lifetime-limited one, and why cooling raises $\sigma$ back toward its ceiling.

Physics of the portrait

Stimulated emission is a rare case where a piece of twentieth-century technology was written down in full, decades before anyone could build it — by someone who never worked on lasers at all.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · quantum theory of radiation

Einstein's 1917 paper Zur Quantentheorie der Strahlung set out to rederive Planck's blackbody law from a picture of atoms exchanging energy with radiation, and in doing so he was forced to invent stimulated emission: absorption and spontaneous emission alone could not hold a gas on the Planck spectrum, and equilibrium demanded a third term, an emission rate proportional to the ambient field. He labelled the three processes with the coefficients still named after him — $B_{12}$ , $A_{21}$ , $B_{21}$ — and showed consistency with Planck's law fixes their ratios once and for all: $g_1B_{12}=g_2B_{21}$ and $A_{21}/B_{21}=8\pi h\nu^3/c^3$ . In the same stroke he gave each quantum a momentum $h\nu/c$ , cementing the photon as a particle with direction. What Einstein did not do was turn stimulated emission into a net amplifier — that needed a population inversion, an idea that lay dormant through the maser and laser era charted below. The gain coefficient $\gamma=\Delta N\,\sigma$ is the engineering shorthand for the process he deduced from pure thermodynamics — every laser a monument to a rate equation written to explain a spectrum.

History

1917

— Einstein introduces the A and B coefficients and predicts stimulated emission

1928

— Ladenburg reports negative dispersion in neon, the first evidence of stimulated emission

1954

— Gordon, Zeiger and Townes operate the ammonia maser, the first working amplifier by stimulated emission

1958

— Schawlow and Townes propose extending the maser principle to optical frequencies

1960

— Maiman operates the first laser, a pulsed ruby, confirming optical gain by stimulated emission

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 1 (Einstein coefficients and detailed balance) · Foot, Atomic Physics, ch. 1 and 7 (the A and B coefficients, cross sections) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 14–16 (gain, cross section, amplification, saturation) · Siegman, Lasers, ch. 6–7 (gain media and stimulated transitions) · Demtröder, Laser Spectroscopy, Vol. 1, ch. 2 (lineshapes and gain profiles) · Hecht, Optics, 5th ed., ch. 13 (stimulated emission and lasers).

Provenance. The Einstein relations $g_1B_{12}=g_2B_{21}$ and $A_{21}/B_{21}=8\pi h\nu^3/c^3$ are recomputed here by matching detailed balance to Planck's law (Loudon ch. 1, Foot ch. 1). The gain law $\gamma=\Delta N\,\sigma$ with $\sigma=\lambda^2 g(\nu)/(8\pi t_\mathrm{sp})$ and the Beer–Lambert form $I=I_0e^{\gamma z}$ follow Saleh & Teich ch. 14–15. A verifier should reconfirm: (1) the detailed-balance derivation, including that dropping $B_{21}$ collapses Planck to the Wien tail; (2) the dimensional check $[\gamma]=\mathrm{m^{-1}}$ ; (3) the Lorentzian peak $g_L(\nu_0)=2/(\pi\Delta\nu)$ and the natural-line ceiling $\sigma(\nu_0)=\lambda^2/2\pi$ ; (4) the Nd:YAG worked numbers $\gamma_0=0.168\ \mathrm{cm^{-1}}$ , $7.3\ \mathrm{dB}$ , $I_s=2.9\ \mathrm{kW\,cm^{-2}}$ , and the Rb factor-of- $55$ Doppler reduction. The Nd:YAG parameters ( $\sigma\approx2.8\times10^{-19}\ \mathrm{cm^2}$ , $t_\mathrm{sp}\approx230\ \mu\mathrm{s}$ ) and Rb D2 lifetime ( $26\ \mathrm{ns}$ ) are standard tabulated values (Koechner, Siegman, Steck); treat them as representative textbook figures.

Builds on. Population Inversion and Optical Gain · Laser Rate Equations. Leads to. Laser Threshold Condition · Three-Level Laser Systems · Four-Level Laser Systems · Cavity Q Factor and Photon Lifetime.

Ask the tutor: ask me to work out the gain coefficient, saturation intensity, or required inversion for any laser medium given its cross section, lifetime and linewidth.

What is the real physical difference between stimulated and spontaneous emission?

Both return an excited atom to a lower level with a photon, but they choose the mode differently. Stimulated emission is triggered by a photon already in the field and drops its new photon into that same mode — identical frequency, phase, polarization and direction — so the wave grows coherently. Spontaneous emission is triggered by no real photon; the atom couples to the vacuum fluctuations of the empty modes and picks one at random, giving a photon of random phase and direction, the source of amplified-spontaneous-emission noise. Formally, spontaneous emission is stimulated emission driven by the zero-point field, which is why $A_{21}/B_{21}$ equals $h\nu$ times the mode density.

Why is a population inversion required — surely one excited atom can amplify?

A single excited atom can emit into the beam, but net amplification is a race between stimulated emission and absorption running side by side. Each photon is as likely to be absorbed by a lower-level atom as to stimulate an upper-level one, the rates weighted by $N_2$ and $(g_2/g_1)N_1$ . The beam grows only if emissions outnumber absorptions, needing $\Delta N>0$ — an inversion. In any thermal system the Boltzmann factor makes the lower level win, so $\Delta N<0$ and the beam attenuates; producing $\Delta N>0$ takes active pumping into the upper level faster than it decays.

Where does the degeneracy factor g₂/g₁ come from, and when does it matter?

It comes from the first Einstein relation $g_1B_{12}=g_2B_{21}$ , which says the absorption and stimulated-emission coefficients are equal only after counting how many quantum states sit at each energy. A more degenerate lower level offers more channels for a photon, so the net gain weights the lower population by $g_2/g_1$ : $\Delta N=N_2-(g_2/g_1)N_1$ . It matters whenever $g_1\neq g_2$ : if $g_2=3g_1$ , then even $N_2=1.1N_1$ — a real excess of excited atoms — gives $\Delta N=(1.1-3)N_1<0$ and absorption. Writing $N_2-N_1$ would wrongly predict gain and misplace the threshold.

What makes one gain medium better — a big cross section or a long lifetime?

It depends on how you run the laser, because the two figures of merit pull opposite ways. A large cross section $\sigma$ means a small inversion gives useful gain, favouring continuous-wave, low-threshold operation. A long lifetime $t_\mathrm{sp}$ lets the medium bank pumped atoms without decaying — what Q-switched and pulsed systems need for energy storage. The product $\sigma\,t_\mathrm{sp}$ is the storage figure of merit, and the ratio sets $I_s=h\nu/(\sigma t_\mathrm{sp})$ . Nd:YAG pairs a moderate $\sigma$ with a long $230\ \mu\mathrm{s}$ lifetime for storage; Ti:sapphire trades $\sigma$ for a huge bandwidth enabling tuning and ultrashort pulses.

In what sense is spontaneous emission "stimulated by the vacuum"?

The ratio $A_{21}/B_{21}=8\pi h\nu^3/c^3$ is exactly $h\nu$ times the number of modes per unit volume per unit frequency. Read that way, the spontaneous rate is the stimulated rate you would get if every empty mode held its zero-point energy — and quantum electrodynamics makes it literal: the vacuum's fluctuating fields drive the atom to emit even with no photons present. The clean statement is $P_\mathrm{stim}/P_\mathrm{spon}=B_{21}\rho/A_{21}=\bar n$ , the mean photon number per mode: when $\bar n\gg1$ stimulated emission dominates and light amplifies coherently; when $\bar n\ll1$ spontaneous emission wins and the light is incoherent fluorescence.

Why does the lineshape g(ν) decide where a laser can oscillate?

Because $\sigma(\nu)\propto g(\nu)$ , the gain $\gamma(\nu)=\Delta N\,\sigma(\nu)$ traces the same bell-shaped curve in frequency. A cavity mode oscillates only where its round-trip gain beats its loss, and since gain peaks at line centre and falls within a few linewidths, only modes under the high part of $g(\nu)$ reach threshold. The width of $g(\nu)$ therefore sets the tuning range and, for a broad line like Ti:sapphire's, the shortest supportable pulse, since a wide bandwidth admits many phase-locked modes. Narrowing the line raises peak gain but shrinks the oscillation window.