Lasers: Stimulated Emission & Population Inversion

The laser went from Maiman's bench curiosity to one of the most consequential inventions of the century, and every use rests on the properties stimulated emission confers: coherence, monochromaticity, directionality and brightness. Fibre-optic communication — the backbone of the internet — sends data as laser pulses down glass fibres, exploiting the beam's tight spectrum and directionality. Barcode scanners, laser printers and optical drives read and write with focused laser spots. Surgery uses lasers as bloodless scalpels and to weld detached retinas; lithography with ultraviolet lasers patterns every microchip. Metrology reached a new era when the laser's coherence let physicists measure distance by counting wavelengths, culminating in LIGO, whose kilometre-scale laser interferometers detected gravitational waves by sensing mirror motions a thousandth the width of a proton. And the same Einstein coefficients that predict lasing predict its inverse in astrophysics: interstellar clouds act as natural masers, amplifying microwaves by population inversions pumped by starlight — Einstein's 1917 bookkeeping written across the galaxy.

Every atom is a ladder of allowed energy levels — the same quantised levels the Bohr model gave for hydrogen and that line spectra reveal for every element. Light interacts with an atom only when a photon's energy matches the gap between two of those levels. Call the lower level $E_1$ and an upper level $E_2$; the resonant photon then carries energy

where: $h = 6.63\times10^{-34}\ \mathrm{J\,s}$ is Planck's constant, $f$ is the light's frequency, and $E_2 - E_1$ is the level spacing. The physical reading is a strict matching rule: only a photon tuned to the gap can be absorbed on the way up or emitted on the way down. This is why each element has its own sharp spectral lines, and it is why a given laser shines at one precise colour — its colour is a fingerprint of one particular level pair in its atoms. Now the crucial question a candle never forces you to ask: when an excited atom drops from $E_2$ to $E_1$, what decides when and in which direction it emits? There are two utterly different answers, and a laser lives on the second one. The first is spontaneous emission. Left alone, an excited atom is unstable; after a short, random wait — typically about $10^{-8}\ \mathrm{s}$ — it drops on its own and flings out a photon in a random direction with a random phase. A candle, a filament bulb and a neon sign all glow by spontaneous emission: countless atoms emitting independently, out of step, in all directions, over a spread of colours. The light is incoherent by construction. The second is stimulated emission, Einstein's 1917 addition. If a photon whose energy already equals $E_2 - E_1$ sweeps past an atom that is still sitting in the upper level, it can knock the atom down early — and the newly emitted photon is a perfect copy of the trigger: identical frequency, identical direction, identical phase, identical polarization. One photon in, two identical photons out. Those two can each stimulate another excited atom, giving four, then eight — a coherent avalanche of clones all marching together. That lockstep is exactly what makes laser light monochromatic, directional and coherent.

Einstein's insight was to count these processes as rates — how many atoms per second undergo each — because it is the competition between them that decides whether a beam grows or dies. Let $N_1$ be the number of atoms in the lower level and $N_2$ the number in the upper level, and let $u$ be the energy density of the light already present at the resonant frequency. Then, in words the three rates read:

where: the proportionality constants are Einstein's coefficients — $B_{12}$ for absorption, $A_{21}$ for spontaneous emission, $B_{21}$ for stimulated emission. The physical reading of eq (2) is that absorption and stimulated emission both need the light to be present (they scale with $u$) and are opposite processes: absorption removes a photon and needs a lower-level atom, stimulated emission adds a photon and needs an upper-level atom. Spontaneous emission needs no light at all — it happens in the dark — which is exactly why it emits at random. Send a beam through the medium and two of these processes fight over it. Every absorption event ($\propto N_1$) subtracts a photon from the beam; every stimulated emission event ($\propto N_2$) adds a photon that is a perfect copy and so joins the beam. Since the two are governed by the same coefficient ($B_{12} = B_{21}$, as the derivation below shows), the beam is amplified only when

that is, only when more atoms sit in the upper level than the lower one. This upside-down arrangement is a population inversion. Spontaneous emission, meanwhile, is the laser's enemy of coherence: it pours photons out in random directions and phases and drains the upper level uselessly, so a good laser is one in which stimulated emission is made to dominate spontaneous emission.

Here is the wall every would-be laser hits. In ordinary matter at temperature $T$, the number of atoms in a level of energy $E$ follows the Boltzmann distribution: the population falls off exponentially as you climb. For our two levels,

where: $k = 1.38\times10^{-23}\ \mathrm{J\,K^{-1}} = 8.62\times10^{-5}\ \mathrm{eV\,K^{-1}}$ is Boltzmann's constant and $\Delta E = E_2 - E_1$ is the level gap. Read eq (4) carefully: the exponent is negative, so the ratio $N_2/N_1$ is always less than one. The upper level is always less populated than the lower level in thermal equilibrium — for a visible-light gap it is emptier by an astronomical factor. No matter how hot you make the gas, the best you can approach is $N_2/N_1 \to 1$ as $T \to \infty$; you can never reach $N_2 > N_1$ by heating. A population inversion is not a hot state — it is a non-thermal state, and the only way to reach it is to drive energy in from outside faster than the atoms can relax. That driving is called pumping. The derivation puts a number on just how empty the upper level is, and the worked examples show that even a thirty-thousand-degree furnace would not get you halfway to inversion.

Pumping alone is not enough, because of a nasty feedback: the very light you want to amplify also drives absorption and spontaneous decay, and a normal excited level empties in about $10^{-8}\ \mathrm{s}$ — far too fast to stack atoms up. The trick is a metastable state: a special excited level from which the ordinary fast decay is forbidden by the atom's internal rules, so it lives thousands of times longer, of order $10^{-3}\ \mathrm{s}$. That is roughly $10^{5}$ times the ordinary lifetime — long enough for the pump to herd a majority of atoms into it before they leak away, so the population there builds past the lower level and the inversion is achieved. The classic scheme is the three-level laser of Maiman's ruby. A bright flash pumps atoms from the ground level $E_1$ up to a broad, short-lived band $E_3$. From there they tumble almost instantly, without radiating, down to a metastable level $E_2$, where they accumulate. Once more atoms sit in $E_2$ than in the ground state $E_1$, a single spontaneously emitted photon of energy $E_2 - E_1$ sweeping down the crystal stimulates a cascade, and the laser fires at the wavelength set by that gap. Placing the crystal between two mirrors — one fully reflecting, one slightly leaky — sends the growing beam back and forth through the pumped medium many times, amplifying it on each pass; the fraction that escapes the leaky mirror is the output beam. That mirror box is the optical cavity, and it is what turns a one-shot amplifier into a self-sustaining oscillator.

Board level — the ruby laser photon (ISC pattern)

A ruby laser emits deep-red light of wavelength $694.3\ \mathrm{nm}$. Take $hc = 1240\ \mathrm{eV\,nm}$, $e = 1.60\times10^{-19}\ \mathrm{C}$. Find (a) the energy of one photon in eV and in joules, (b) the energy gap between the two ruby levels involved, and (c) the number of photons in a single $1.0\ \mathrm{mJ}$ pulse.

Step 1. Photon energy from the wavelength: $E = 1240/\lambda = 1240/694.3 = \boxed{1.79\ \mathrm{eV}}$. In joules, $E = 1.79 \times 1.60\times10^{-19} = \boxed{2.86\times10^{-19}\ \mathrm{J}}$.

Step 2. The energy gap is the photon energy — that is the meaning of $hf = E_2 - E_1$. So $E_2 - E_1 = \boxed{1.79\ \mathrm{eV}}$; the laser's colour reads the level spacing directly.

Step 3. A pulse of energy $E_{\text{pulse}} = 1.0\ \mathrm{mJ} = 1.0\times10^{-3}\ \mathrm{J}$ is made of identical photons each carrying $2.86\times10^{-19}\ \mathrm{J}$, so the count is $N = E_{\text{pulse}}/E = (1.0\times10^{-3})/(2.86\times10^{-19}) = \boxed{3.5\times10^{15}\ \text{photons}}$. Every one of them is a near-perfect clone of the others — that is what "coherent" buys you.

Board level — identifying the process (ISC and A-Level pattern)

In each case name the radiative process: (a) an atom in the ground state absorbs a photon and jumps up; (b) an isolated excited atom drops after a random wait, emitting a photon in an arbitrary direction; (c) a photon of the resonant energy passes an excited atom and two identical photons emerge.

Step 1. Case (a): a photon is destroyed and the atom is raised — this is absorption, rate $\propto N_1 u$.

Step 2. Case (b): no incoming light is needed and the direction and phase are random — this is spontaneous emission, rate $\propto N_2$, the mechanism of ordinary lamps.

Step 3. Case (c): one photon in, two identical photons out, triggered by the passing photon — this is stimulated emission, rate $\propto N_2 u$, the amplifying process at the heart of every laser. Marks are lost here only by confusing (b) and (c); the tell is whether an incoming photon is required and whether the output is a copy.

Competitive level — could a furnace ever invert ruby? (A-Level / Olympiad-style)

For the ruby transition $\Delta E = 1.79\ \mathrm{eV}$, take $k = 8.62\times10^{-5}\ \mathrm{eV\,K^{-1}}$. (a) Find $N_2/N_1$ at $T = 300\ \mathrm{K}$. (b) Find the temperature at which $N_2/N_1$ would reach $\tfrac{1}{2}$, and comment on whether heating can ever invert the populations.

Step 1. Room temperature ratio. $kT = (8.62\times10^{-5})(300) = 0.0259\ \mathrm{eV}$, so the exponent is $\Delta E/kT = 1.79/0.0259 = 69.1$ and $N_2/N_1 = e^{-69.1} \approx \boxed{10^{-30}}$. The upper level is empty for all practical purposes.

Step 2. Temperature for $N_2/N_1 = \tfrac{1}{2}$. Set $e^{-\Delta E/kT} = \tfrac{1}{2}$, so $\Delta E/kT = \ln 2 = 0.693$ and $T = \Delta E/(k\ln 2) = 1.79/[(8.62\times10^{-5})(0.693)] = \boxed{3.0\times10^{4}\ \mathrm{K}}$.

Step 3. Interpret. Even at thirty thousand kelvin — hotter than the surface of most stars — you reach only $N_2/N_1 = \tfrac{1}{2}$, i.e. the upper level is still half as populated as the lower one. As $T$ climbs further the ratio creeps toward $1$ but never crosses it: heating cannot deliver $N_2 > N_1$. Inversion is a non-thermal, actively pumped state, full stop. Timing note: about two minutes. Part (a) is one product ($kT$), one division, one exponential; part (b) is the same equation solved for $T$ with $\ln 2$ in place of the exponent. The examiner's real target is the one-line comment in Step 3 — that no finite temperature inverts the populations — which is worth as much as the arithmetic.

"Stimulated" vs "spontaneous" — the definitional trap. Spontaneous emission needs no incoming light and sends its photon out in a random direction and phase; stimulated emission is triggered by a resonant photon and emits a copy of it. Any question describing "two identical photons" or "same phase and direction" is testing stimulated emission. Mixing these up is the single most common ISC/A-Level slip.

A population inversion is not high temperature. Do not write that a laser medium is "very hot" so that the upper level is full. $N_2/N_1 = e^{-\Delta E/kT} < 1$ for every finite $T$. Inversion requires pumping out of equilibrium, and formally corresponds to a negative temperature — say "pumped", never "heated".

The metastable state is essential, not decorative. Marks are lost by describing a two-level pump-and-lase scheme without a metastable level. Without a long-lived state ($\sim 10^{-3}\ \mathrm{s}$ versus the ordinary $\sim 10^{-8}\ \mathrm{s}$), atoms decay before they can be stacked and no inversion forms. Name the metastable level explicitly.

LASER is an acronym — expand it correctly. Light Amplification by Stimulated Emission of Radiation. Not "spontaneous", not "stimulated absorption". The examiner will accept full marks only for the exact expansion.

The photon energy equals the level gap, not $kT$. $hf = E_2 - E_1$ sets the laser's colour; $kT$ is only the thermal yardstick you compare the gap against. Don't conflate them: ruby's gap is $1.79\ \mathrm{eV}$ while $kT$ at room temperature is a mere $0.0259\ \mathrm{eV}$.

Board scope. Stimulated emission, population inversion and the laser are enrichment topics carried by ISC (CISCE Class 12) and A-Level (CIE 9702). They are not core CBSE or NEET material and are not in AP Physics 2 or JEE's required syllabus, and IGCSE has no atomic-spectra content at all — so treat this page as background depth, not exam drill, unless you sit ISC or A-Level.

Texts. ISC Physics (Class 12), the chapter on the laser and stimulated emission · A. K. Ghatak, Optics, the chapters on the Einstein coefficients and laser principles · Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, on the A and B coefficients · for CIE 9702, the "Quantum physics" topic on energy levels and lasers.

Provenance. HS constants: $h = 6.63\times10^{-34}\ \mathrm{J\,s}$, $c = 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$, $e = 1.60\times10^{-19}\ \mathrm{C}$, $k = 1.38\times10^{-23}\ \mathrm{J\,K^{-1}} = 8.62\times10^{-5}\ \mathrm{eV\,K^{-1}}$, $hc = 1240\ \mathrm{eV\,nm}$. Recompute: ruby $\Delta E = 1240/694.3 = 1.79\ \mathrm{eV} = 2.86\times10^{-19}\ \mathrm{J}$; $f = (3.00\times10^{8})/(694.3\times10^{-9}) = 4.32\times10^{14}\ \mathrm{Hz}$; $kT(300\ \mathrm{K}) = 0.0259\ \mathrm{eV}$; $\Delta E/kT = 69.1$; $N_2/N_1 = e^{-69.1} \approx 10^{-30}$; $T$ for $N_2/N_1 = \tfrac12$ is $1.79/(8.62\times10^{-5}\times0.693) = 3.0\times10^{4}\ \mathrm{K}$; He–Ne $\Delta E = 1240/632.8 = 1.96\ \mathrm{eV}$; photons in a $1.0\ \mathrm{mJ}$ ruby pulse $= (1.0\times10^{-3})/(2.86\times10^{-19}) = 3.5\times10^{15}$; ordinary-to-metastable lifetime ratio $\sim 10^{-3}/10^{-8} = 10^{5}$; Einstein 1917 to Maiman 1960 is 43 years.

Prerequisites. Energy Levels & the Hydrogen Spectrum — where the level ladder and $hf = E_2 - E_1$ are built · Atomic Spectra: Emission & Absorption Lines, for how transitions between levels make the sharp lines a laser exploits. Leads to. For the university treatment, the enrichment seams Stimulated Emission and the Laser and Einstein's A and B Coefficients carry the full rate-equation and cavity analysis (go-deeper pointers, never an exam dependency here).