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Stimulated emission adds a photon to an already occupied radiation mode; spontaneous emission can occur when that mode initially contains no photons. In the electric-dipole approximation the atom-field interaction is $H_{\mathrm{int}}=-\mathbf{d}\cdot \mathbf{E}$. Quantizing the radiation field makes the emission matrix element into a mode with occupation $n$ proportional toThis photon occupation enhancement of emission makes the emission rate into that mode proportional to $n+1$: the part proportional to $n$ is stimulated and the remaining part is spontaneous. Absorption is proportional to $|⟨n-1|a|n⟩{|}^2=n$. The photon energy is the atomic energy difference, up to any specified recoil or line broadening.

The stimulated contribution preferentially populates the stimulating mode, including its propagation direction and polarization; it provides optical gain. Spontaneous photons populate the available modes according to the transition's dipole matrix elements and radiation environment. They are not universally isotropic. An excited bare atom with the field vacuum is not an eigenstate of the fully coupled Hamiltonian, which resolves the apparent paradox of a nominally stationary excited state radiating.^[ref]

For lower and upper levels 1 and 2, with radiation spectral energy density $u_{\nu }$ per unit ordinary frequency, the rates areThermal detailed balance with the Planck spectrum givesThe statistical weights $g_i$ here are level degeneracies, not Landé g-factors. A definition of $B$ multiplying intensity or angular-frequency density would change the normalization factors.

For a specified free-space electric-dipole transition, summing radiation modes givesAppropriate sums over final magnetic substates and averages over initial substates are needed for an unresolved degenerate level. The radiative lifetime is $1/A_{21}$ only when this is the sole decay channel. In a laser, net stimulated gain requires $N_2/g_2>N_1/g_1$, as well as sufficient gain to overcome cavity loss; population inversion alone does not specify the lasing threshold.

Let $A_{21}$ be the spontaneous emission coefficient, and let $B_{12}\rho$ and $B_{21}\rho$ be the absorption and stimulated emission rates per atom under the radiation spectral energy density $\rho$ at resonance. Because the two levels are nondegenerate, the Einstein coefficients satisfy $B_{12}=B_{21}=B$.

The radiative balance in a nondegenerate two-level atom isThe total spontaneous and stimulated emission rates are $R_{\mathrm{sp}}=N_2{A}_{21}$ and $R_{\mathrm{st}}=N_{2}B\rho$. Dividing the balance equation by $N_{2}B\rho$ givesThis presumes nonzero drive and upper-level population, and no other population-transfer mechanism. It does not require a thermal Boltzmann distribution. In this closed driven two-level system, spontaneous decay keeps $N_1>N_2$; arbitrarily intense driving approaches equal populations, rather than producing a population inversion.

The specific intensity $I_{\nu }$ is energy transported per unit time, projected area, solid angle and frequency. The path coordinate $s$ increases along the ray. The emission coefficient $j_{\nu }$ is emitted energy per unit time, volume, solid angle and frequency, whereas the absorption coefficient ${\alpha }_{\nu }$ has units of inverse length. Thus $j_{\nu }ds$ adds specific intensity and ${\alpha }_{\nu }{I}_{\nu }ds$ removes it. Stimulated emission can be included as negative absorption.

Define the optical depth between two points by ${\tau }_{\nu }(s_1,s_2)={\int }_{s_1}^{s_2}{\alpha }_{\nu }(u)du$. Multiplying the radiative transfer equation by the integrating factor $\exp [{\tau }_{\nu }(s_0,s)]$ and integrating givesEach local contribution is attenuated only by the material between its emission point and the observer. Where ${\alpha }_{\nu }\ne 0$, the source function is $S_{\nu }=j_{\nu }/{\alpha }_{\nu }$ and the same formal solution of the radiative transfer equation is $I_{\nu }(\tau )=I_{\nu }(0)e^{-\tau }+{\int }_0^{\tau }{S}_{\nu }(t)e^{-(\tau -t)}dt$. A constant source function gives $I_{\nu }=I_{\nu }(0)e^{-\tau }+S_{\nu }(1-e^{-\tau })$.

Use the Einstein coefficients convention in which $B_{ij}$ multiplies spectral radiation energy density $u_{\nu }$, with $u_{\nu }=c^{-1}\int I_{\nu }d\Omega$. A normalized spectral line profile obeys $\int \phi (\nu )d\nu =1$. The spontaneous emission probability per upper-state atom per unit time is $A_{21}$; the stimulated emission and upward absorption rates per atom are respectively $B_{21}\int u_{\nu }\phi (\nu )d\nu$ and $B_{12}\int u_{\nu }\phi (\nu )d\nu$.

For isotropic spontaneous emission, $n_2{A}_{21}$ transitions occur per unit volume and time, each supplying photon energy $h{\nu }_{12}$. Dividing among $4\pi$ steradians givesFor a small incident beam, its contribution to $u_{\nu }$ is $I_{\nu }d\Omega /c$. Absorption removes and stimulated emission adds photons in the incident direction. Matching the net loss to ${\alpha }_{\nu }{I}_{\nu }d\Omega$ therefore givesReplacing ${\nu }_{12}$ by $\nu$ across the narrow spectral line profile gives the printed expressions. The negative sign is essential: an inverted population can amplify the incident specific intensity. Thermal equilibrium and the Planck function imply $g_1{B}_{12}=g_2{B}_{21}$ and $A_{21}/B_{21}=8\pi h{\nu }_{12}^3/c^3$, where $g_i$ is the state degeneracy. A convention using mean intensity instead changes both $B$ coefficients by $4\pi /c$.

Specify the radiation normalization first. Let $u_{\nu }$ be radiation energy density per ordinary frequency, with SI units $\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$. Then $A_{21}$ is the spontaneous emission rate per upper-state chemical atom, while $B_{12}{u}_{\nu }$ and $B_{21}{u}_{\nu }$ are the induced absorption and stimulated emission rates, evaluated at a sharp transition frequency. For a finite spectral line profile, replace $u_{\nu }$ by its profile-weighted average.

The Einstein coefficients obeyHere $g_i$ are the statistical weights of the two energy levels, and the field is unpolarized and isotropic. The units and physical dimensions areThe apparently time-independent dimension of $B$ follows because a per-hertz density includes an extra unit of time. If instead $B$ multiplies the mean intensity $J_{\nu }$, use $B^{(J)}=(4\pi /c)B$ and $A/B^{(J)}=2h{\nu }_0^3/c^2$. These distinct normalizations must not be mixed in the radiative transfer equation.

Define $r=g_2/g_1$ and let $\sigma ={\sigma }_{21}(\omega )$ be the stimulated-emission cross section, averaged over upper magnetic sublevels. The Einstein coefficients relation $g_1{B}_{12}=g_2{B}_{21}$ gives the absorption cross section ${\sigma }_{12}=r\sigma$. Assume equally populated magnetic sublevels within each energy level, a steady pump, and a monochromatic beam. The photon flux is $I/(\hslash \omega )$; define $W=\sigma I/(\hslash \omega )$ and the effective population inversion $D=N_2-r{N}_1$. Then the rate equations areThe $WD$ term includes both stimulated emission and absorption. In steady state, adding the two rate equations gives $N_1=R_2{\tau }_1$. The first equation then yieldsThe directed beam gains energy at rate $\hslash \omega WD$ per volume, giving $dI/dz=\sigma DI$ and the optical gain coefficient $\alpha =\sigma D$. At line centre, write ${\sigma }_0={\sigma }_{21}({\omega }_{21})$. ThereforeHere ${\alpha }_0$ is the small-signal optical gain coefficient and $I_s$ is the saturation intensity, at which the local optical gain is half its small-signal value. Positive optical gain requires $r{A}_{21}{\tau }_1<1$: the lower energy level must empty sufficiently quickly.

The cancellation of ${\tau }_1$ from $I_s$ is specific to this model's steady fixed pump and the fact that every upper spontaneous emission feeds the lower energy level. Adding an extra lower-state accumulation factor to $I_s$ would contradict these rate equations. Alternative conventions using the absorption cross section must replace ${\sigma }_0$ by ${\sigma }_{12}/r$ throughout. Homogeneous spectral broadening permits the whole selected ensemble to share this local saturation response.

A two-photon transition is a second-order atom-field process. For two-photon absorption, energy conservation requires $E_f-E_g=\hslash ({\omega }_1+{\omega }_2)$. A virtual intermediate state in two-photon absorption need not be populated as a real stationary state. In the electric dipole approximation, the two time orderings give an amplitude proportional toThe electric-field amplitudes multiply this expression. Far from intermediate resonance, the denominators are large relative to linewidths and one-photon couplings, keeping real intermediate occupation small. Near an intermediate resonance, sequential absorption, linewidths and saturation require a fuller treatment. For weak unsaturated absorption the rate scales as $I_1{I}_2$, or $I^2$ when both photons come from one field. Absorption followed by stimulated emission instead gives a difference-frequency Raman transition; it is not two-photon absorption at the sum frequency.

Two electric dipole interactions change parity twice, so initial and final states have the same parity. For a one-electron orbital, the usual changes are $\Delta \ell =0,\pm 2$, subject to angular-momentum and polarization constraints. For identical equal-frequency photons, exchange symmetry removes the antisymmetric rank-one tensor: ranks zero and two survive. Thus $J=0\to 0$ is possible, whereas $J=0\to 1$ is forbidden in this degenerate electric-dipole approximation. It would be too strong to prohibit every $\Delta J=1$ transition between nonzero angular momenta, since a rank-two tensor can connect some such pairs. Spin remains unchanged to leading electric-dipole order.

A central example is hydrogen $1S\to 2S$, forbidden as a one-photon electric dipole transition but accessible with two photons near $243\mathrm{nm}$. Absorption of one photon from each of equal-frequency counterpropagating beams makes the first-order Doppler shift $({\mathbf{k}}_1+{\mathbf{k}}_2)\cdot \mathbf{v}$ vanish and suppresses net recoil. The two same-direction absorption paths can still give a broad Doppler background. A primary demonstration used a frequency-doubled pulsed dye laser near this wavelength and compared with a simultaneously recorded Balmer reference.^[ref]

Experimentally, scan the sum frequency and detect the excited state, its subsequent fluorescence or its ionization. Counterpropagating and single-beam spectra distinguish the narrow Doppler-free contribution from its background. Varying intensity tests weak-field scaling and extrapolates AC Stark shifts and power broadening; frequency calibration, beam overlap, residual angle and second-order Doppler shifts still matter. “Doppler free” here cancels the leading velocity term, not every systematic shift.

A forward-biased p-n junction injects electrons from its n-type semiconductor and electron holes from its p-type semiconductor into an active region. A direct band gap allows radiative transitions without a compensating phonon, so stimulated emission can compete effectively with nonradiative processes. At thermal equilibrium there is one Fermi level; forward injection instead establishes separate electron and valence-electron quasi-Fermi levels $F_c$ and $F_v$. Sufficient injection makes conduction states more occupied than the corresponding valence states, producing optical gain.

Two parallel cleaved facets supply an optical resonator. At normal incidence their approximate intensity reflectance follows the Fresnel equations:A resonant normal mode grows when its round-trip amplification exceeds its losses. For length $L$, material optical gain $g$, overlap factor $\Gamma$, internal intensity loss ${\alpha }_i$ and facet reflectances $R_1,R_2$,The pump supplies the emitted energy; the facets provide feedback and let some output escape. Positive optical gain alone does not establish laser threshold.

A laser relaxation oscillation is the damped exchange between stored upper-level excitation and intracavity photon number after a small perturbation of a lasing steady state. Extra photons increase stimulated depletion of the upper level; the resulting smaller gain then reduces the photon population. Pumping subsequently replenishes the inversion, so the two populations overshoot their equilibrium values before damping settles them.

Laser spiking is the corresponding large-signal sequence of intense output pulses, often during switch-on. Excitation first accumulates while the optical field is small. Once gain exceeds loss, rapid stimulated emission releases this stored energy in a spike, depleting the inversion below threshold. Recovery can produce further spikes. This is distinct from an externally imposed Q-switching cycle, although both draw on stored excitation.

Many solid-state gain media have upper-state lifetimes far longer than the cavity photon lifetime. The optical polarization relaxes rapidly, leaving the two slow rate-equation variables of a class-B laser. A long-lived population reservoir and a much faster photon response make underdamped recovery and pronounced startup spiking common. The criterion and frequency follow from the linearization in part c; they are not universal consequences of the word “solid-state”.

Let $v_g\simeq c/n_r$ be the light's group velocity in the fibre, with unit mode overlap and a uniform transverse mode area. The fibre-laser gain-volume rate equations follow because the upper-level density is defined per doped volume, whereas the photon density is defined per full cavity volume. Stimulated emission therefore removes upper-level ions at rate $v_g{\sigma }_{21}{N}_{2}n$ per doped volume, but supplies cavity photons at the volume-averaged rate $f{v}_g{\sigma }_{21}{N}_{2}n$. ThusAt threshold the stimulated photon growth rate balances the photon loss rate, givingUsing this relation yields the stated equations, including the essential $1/f$ in the ion-depletion term:Above threshold, the nonzero stationary photon density requires $N_{2,\mathrm{ss}}=N_{\mathrm{th}}$. The remaining stationary equation then givesThe parameter $r$ is the pump rate in units of its lasing-threshold value. The above-threshold branch requires $r>1$; below threshold the ideal stationary solution is $n=0$, $N_2=S{\tau }_2$. Spontaneous emission into the lasing mode has been omitted in this deterministic model, so an exactly zero initial field would need a seed or noise to start, even when the zero-field branch is unstable.

The two $D$ states are metastable, so a quantum superposition can retain its relative phase much longer than one involving a rapidly decaying $P$ state. In the stipulated transition diagram they have no allowed downward electric-dipole transition; real weaker decay channels limit, rather than eliminate, this lifetime advantage. Their distinct energies permit resolved state-selective optical control.

Both $D$ levels couple to $P_{3/2}$. Apply coherent beams near 850 nm and 854 nm, detuned from that common intermediate level by a large Rabi detuning ${\Delta }_R$. Absorption from one beam and stimulated emission into the other transfer population between the two $D$ states without appreciably populating the short-lived intermediate level. Tune the frequency difference to their splitting, including differential AC Stark shifts. In the far-detuned limit,for an initially pure $D_{3/2}$ state on two-photon resonance. The controllable relative optical phase sets the rotation axis; the large Rabi detuning suppresses spontaneous-scattering errors. Suitable polarizations and a chosen pair of magnetic sublevels are required in an actual experiment.

In a direct band gap semiconductor, the small photon momentum makes optical transitions approximately vertical in wave vector. Their energy isThe joint density of states per unit volume per transition energy, including the supplied spin counting, isFor an upper electron state of occupation $f_c$ and a lower electron state of occupation $f_v$, stimulated emission has occupation factor $f_c(1-f_v)$; absorption has factor $f_v(1-f_c)$. Their difference is exactly $f_c-f_v$. Multiplication by the joint density of states therefore givesThe proportionality suppresses the optical transition matrix element and other smoothly varying optical prefactors. It assumes parabolic isotropic energy bands, an allowed direct transition, and negligible linewidth and many-body corrections. Here $f_v$ counts electrons, not electron holes.

A quasi-Fermi level is the chemical potential parameter of an approximately thermalized carrier population within one energy band. Rapid intraband scattering can establish a Fermi-Dirac distribution in each energy band before slower interband recombination equilibrates their populations. Thus separate $E_{\mathrm{Fc}}$ and $E_{\mathrm{Fv}}$ describe a pumped population inversion even though the entire semiconductor is far from thermal equilibrium.

At a common positive temperature, the Fermi-Dirac distribution is strictly decreasing in $E-E_F$. ConsequentlyThe joint density of states is nonzero only above the band gap, soIn particular the quasi-Fermi-level splitting must exceed $E_g$. The upper boundary is transparency, where stimulated emission balances absorption; the lower boundary has zero joint density of states in this ideal three-dimensional parabolic model. The strict occupation comparison uses $T>0$; at exactly zero temperature occupations become step functions, and their supports should be checked explicitly, as in the next part.

A double heterostructure laser sandwiches a narrow-band gap active layer between wider-band gap cladding layers. In a type-I band alignment, the conduction band edge is lower and the valence band edge higher in the active layer. Electrons therefore see an energy well, and electron holes also see a well because their excitation energy increases when the valence band edge decreases. Both carrier populations remain concentrated where stimulated emission occurs.

The active layer also normally has the higher refractive index, forming an optical waveguide. The optical normal mode overlaps the injected carriers strongly, and the wider-band gap cladding has less interband absorption at the emitted photon energy. Carrier confinement reduces leakage, while optical confinement reduces the required pump current and hence heating.

Writing $\Gamma$ for the fraction of the optical normal mode overlapping the active layer, the threshold condition is $\Gamma g={\alpha }_i+{\alpha }_m$, with internal and mirror losses ${\alpha }_i,{\alpha }_m$. A larger overlap and a higher carrier number density at a given current both lower the threshold current. The figure shows composition-induced offsets; a working junction also has band bending under bias, which is omitted to make the confinement mechanism visible.

In Q-switching, hold the optical resonator's quality factor low while pumping builds population inversion, then rapidly reduce the loss. The stored excitation produces a large optical gain, the intracavity light intensity grows, and stimulated emission depletes the population inversion until the optical gain falls below the laser threshold. The resulting optical pulse is a transient discharge of stored excitation.

Its duration depends on the high-quality factor photon lifetime, initial population inversion, saturation dynamics and switching time. For example, a simple laser rate equation model iswhere $S$ is intracavity photon number, $N$ is the available inversion in compatible units, $R_p$ is the pump rate and ${\tau }_c$ is the photon lifetime. During the short discharge, pumping and upper-state decay can often be neglected. The build-up and depletion rates determine the width; there is no universal equality between the width and ${\tau }_c$. The initially stored energy and the output coupling also matter. Typical Q-switching produces nanosecond-scale optical pulses.

Let $A=A_{21}$ and $r=g_2/g_1$. Before stimulated emission and stimulated absorption are added, the source and decay terms giveThe upper level is fed at a specified rate of addition to the number density, its spontaneous decay feeds the lower level, and lower-level removal leads to states outside this two-level system. This is therefore an open pumped system, not a closed two-level system with conserved population.

At steady state,The degeneracy-weighted population inversion compares population per magnetic sublevel. It is positive only if $rA{\tau }_1<1$; a positive pump rate alone cannot guarantee optical gain if lower-level removal is too slow. A short lower-state lifetime is consequently helpful even when the upper-state pumping is unchanged.

The equations assume finite positive $A$ and ${\tau }_1$, approximately constant pumping and no other specified sources, sinks or branching channels. A finite steady state would not exist for continuous pumping if the lower level had no escape route. The fixed source rate also presumes access to an external reservoir; it does not imply that the two populations can grow beyond all available active atoms in a real finite medium.

Stimulated emission depletion microscopy narrows the region allowed to emit detectable fluorescence. A focused excitation pulse or beam populates the excited state. A red-shifted depletion beam with a central intensity minimum drives stimulated emission from the surrounding excited fluorophores. The stimulated photons are rejected from the fluorescence detection channel, leaving spontaneous emission concentrated near the depletion zero. Scanning this small effective emission region builds the image.^[ref]

Near an ideal zero, take depletion intensity proportional to squared radius. The radius at which it reaches a saturation scale decreases as the depletion amplitude grows. A common approximate resolution law iswhere $d_0$ is the undepleted characteristic width and $I_{\mathrm{sat}}$ depends on fluorophore kinetics and illumination timing. It is an idealized scaling, not a universal instrument calibration. For example, a threefold width reduction needs $I_{\max }/I_{\mathrm{sat}}\approx 8$ in this model. The illumination beams remain diffraction limited; nonlinear control of the emitting population creates the smaller effective response.

The cost is high depletion intensity, with possible photobleaching, phototoxicity and heating, demanding fluorophore photophysics, precise beam alignment and maintenance of the central zero. Point scanning limits speed and trades off field size, dwell time and photon statistics. Scattering and aberrations can fill the depletion minimum and degrade performance in depth. A simple lateral doughnut mainly improves lateral resolution; improved axial confinement requires an appropriate three-dimensional depletion pattern. Resolution gain is therefore conditional on usable photons, sample integrity and the actual depletion geometry.

The creation-operator matrix element $⟨n+1|a^{†}|n⟩=\sqrt{n+1}$ makes emission into a radiation mode proportional to $n+1$. The $n$ part is stimulated emission and the remaining part is spontaneous emission. Absorption has the corresponding occupation factor $n$.

A population inversion puts more emitters in the upper than in the lower quantum state of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing photon absorption rate proportional to $N_1$, so net optical gain is proportional to $N_2-N_1$. For $N_2=6\times 10^{20}{\mathrm{m}}^{-3}$ and $N_1=4\times 10^{20}{\mathrm{m}}^{-3}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per statistical weight of an atomic level, rather than simply $N_2>N_1$.

Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both photon absorption and stimulated emission. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation isat a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain population inversion in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible laser gain mechanisms. ^[ref].

In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the energy: extra levels provide a route to population inversion, not free energy. ^[ref].

Q-switching creates a short high-energy optical pulse by first storing population inversion while an optical cavity has high loss, then increasing its quality factor so that stimulated emission rapidly extracts the stored excitation. Its duration depends on photon lifetime, excess optical gain, inversion depletion and switching time; it is not fixed by a universal time-bandwidth equality.