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A physicist.^[ref]

A physicist.^[ref]

The experimental distinction is concrete. A 1962 gallium arsenide p-n junction laser used top and bottom contacts, parallel polished faces, and current pulses with the device in liquid nitrogen. An infrared image converter recorded the radiation pattern; a spectrometer measured narrowing above laser threshold. Comparing the same device below and above laser threshold supplied a control. Pulsed, cooled operation did not demonstrate continuous operation at room temperature.^[ref]

Zhores I. Alferov's 1970 team photolithographically patterned stripe double heterostructure lasers on silver-covered copper plates. He reports sustained laser emission at room temperature; Fig. 4 shows the device structure. Independently, Izuo Hayashi, Morton B. Panish and collaborators demonstrated continuous double heterostructure laser operation with diamond heat sinks.^[ref]

The below-threshold comparison tests the onset of stimulated emission; continuous operation additionally requires sustained removal of waste heat. Neither observation alone measures the separate band offsets, optical overlap, or every loss mechanism. The schematic above therefore explains a design principle rather than reconstructing the historical apparatus.

For a vertical optical transition between parabolic energy bands with common carrier temperature, net stimulated emission has occupation factor $f_c(E_c)-f_v(E_v)$. The monotonic Fermi-Dirac distribution makes this positive exactly when the quasi-Fermi-level splitting exceeds the transition photon energy. An allowed transition must also lie above the band gap. Broadening, band degeneracy and a zero-temperature occupation window require their own qualifications; this condition alone does not establish laser threshold.

A double heterostructure laser electrically pumps its central layer for stimulated emission. Band offsets confine charge carriers; refractive index contrast confines the optical normal mode.^[ref]

For an original bookkeeping model, let $R$ be the injected pair rate per unit area, $d$ the active-layer thickness, $N$ its excess pair number density, $\tau$ a recombination lifetime, and ${\tau }_e$ an escape time. Before appreciable stimulated emission, supposeChoose two otherwise identical devices with ${\tau }_e=\tau$ and ${\tau }_e=9\tau$. The resulting steady state populations are $R\tau /(2d)$ and $9R\tau /(10d)$: improved confinement raises the population by a factor $1.8$ at fixed injection. This linear model assumes equal excess electron and electron hole densities, fixed lifetimes, and no laser gain clamping.

Optical confinement enters separately. For a chosen optical normal mode, define $0<\Gamma \le 1$ as its gain-overlap factor with the active layer. With material optical gain $g$, internal loss ${\alpha }_i$, and mirror loss ${\alpha }_m$, a simple laser threshold condition isFor stipulated losses ${\alpha }_i+{\alpha }_m=3$ in inverse-length units, $\Gamma =1/2$ requires $g=6$, whereas $\Gamma =3/4$ requires $g=4$. This calculation varies optical overlap only; the preceding population calculation varies carrier escape only. Their combination explains the usefulness of controlling both the energy band profile and the optical normal mode.

Herbert Kroemer proposed the double heterostructure laser in 1963; he did not participate in its 1970 experimental realization.^[ref] Zhores I. Alferov describes his group's independent 1963 proposal and later experimental program.^[ref]

The effective area converting incident photon flux into a stimulated emission rate for a specified transition, frequency, polarization and sublevel distribution. For statistically populated energy levels of statistical weights $g_1,g_2$, the Einstein coefficients imply $g_1{\sigma }_{12}=g_2{\sigma }_{21}$. The related absorption cross section therefore differs when the statistical weights differ.

An absorption equivalent width is $W_{\lambda }=\int [1-I_{\lambda }/I_{\lambda }^0]d\lambda$, the wavelength width of a completely black interval that removes the same continuum-normalized area. For a narrow spectral line profile, small optical depth and negligible stimulated emission, the energy-density convention for Einstein coefficients gives $W_{\lambda }=h{B}_{12}{N}_1/{\nu }_{12}$. This follows by replacing $1-e^{-{\tau }_{\nu }}$ by ${\tau }_{\nu }$ and using $|d\lambda /d\nu |=c/{\nu }^2$. At redshift $z$, the observed and rest-frame values satisfy $W_{\mathrm{obs}}=(1+z)W_{\mathrm{rest}}$.

A physicist.^[ref][ref]

A maser using a hydrogen atom transition.

A physicist.^[ref]

A laser generates amplified light through stimulated emission. Its name abbreviates “light amplification by stimulated emission of radiation”. Lasers supply the intense beams used in optical tweezers; optical coherence also makes them useful for wave interference measurements. ^[ref].

In the usual material picture, a gain medium stores energy supplied by pumping, for example by incident light or an electric current. Stimulated emission transfers that stored energy to a selected optical normal mode, while competing photon absorption removes light. Producing net optical gain requires a suitable population distribution, such as population inversion in the simple model below.

A familiar laser oscillator uses mirrors to return light through the gain medium. This optical feedback favours resonant normal modes; a partly transmitting mirror lets useful power escape. Feedback and mode selection have different implementations, including ring arrangements: every laser need not resemble the same two-mirror device. The laser threshold example treats just that device. ^{[ref] [ref]}.

The useful combination is concentrated power and controlled optical normal modes. It gives optical tweezers a localized source of force and wave interference measurements a source with useful optical coherence. A laser can nevertheless emit several frequencies; “laser” alone does not guarantee a single frequency. ^[ref].

An original experimental landmark is T. H. Maiman's 1960 report of stimulated optical radiation from optically pumped ruby at about $694.3\mathrm{nm}$. His paper describes both the pumping path and the observed emission, connecting stored material energy with amplified light. ^[ref].

The material in which pumping produces optical gain through stimulated emission. Its inversion density, mode-overlap volume and emission cross section determine whether the gain overcomes cavity losses.

A laser rate equation approximates coupled evolution of population inversion and intracavity photon number, including pumping, spontaneous emission, stimulated emission and optical cavity losses. A simple model is $\dot{S}=[g(N)-1/{\tau }_c]S$, $\dot{N}=R_p-N/{\tau }_u-g(N)S$, where compatible normalizations absorb mode-volume factors. Large excess optical gain drives rapid field growth until inversion depletion reduces it below threshold.

Laser spectroscopy uses a tunable laser to interrogate energy levels through photon absorption, stimulated emission or fluorescence. A narrow optical source is useful only if the sample and readout preserve the information: Doppler broadening can hide closely spaced spectral lines even when the source itself is narrow.

A maser uses stimulated emission to amplify microwaves. With population inversion and feedback it can sustain a coherent wave field as an oscillator. The name abbreviates "microwave amplification by stimulated emission of radiation".^[ref]

Why make molecules supply an oscillator? Their energy levels offer a reproducible frequency reference, while stimulated emission lets successive molecules reinforce one coherent wave field. In his 1964 lecture, Charles Hard Townes traced his motivation to microwave spectroscopy: resolving molecular structure required useful coherent sources.^[ref] The interesting balance is between molecular amplification and escaping energy: even a beam with population inversion fails to oscillate if cavity losses win. This is the microwave counterpart of laser threshold.

Optical gain is amplification of light by a medium. Here the small-signal intensity gain coefficient $g$ has units ${\mathrm{m}}^{-1}$: for fixed populations, a weak beam obeysin the nondegenerate resonant two-level model, with common transition cross-section $\sigma$. Thus $g$ includes the competition between stimulated emission and photon absorption on that transition. Separate passive intensity loss $\alpha$ gives $I(z)=I(0)e^{(g-\alpha )z}$. Both coefficients describe light intensity, not wave amplitude; the corresponding field-amplitude exponent is half as large. ^[ref].

For $g=0.30{\mathrm{m}}^{-1}$, $\alpha =0.10{\mathrm{m}}^{-1}$ and $z=0.50\mathrm{m}$, the single-pass intensity multiplier is $e^{0.10}\approx 1.10517$. This is amplification during one passage, before mirror losses. Population inversion can therefore yield positive optical gain while the complete oscillator remains below laser threshold.

A three-level laser uses a pump transition from ground level1 to a higher level3, rapid relaxation $3\to 2$, and a long-lived upper lasing level2. The laser transition is $2\to 1$, so its lower level is the ground state. Fast feeding and slow upper-state decay allow excitation to accumulate, unlike steady resonant pumping of a simple two-level transition.

Stimulated emission adds photons to the stimulating optical mode, but ground-state photon absorption competes with it. The population inversion condition is $N_2/g_2>N_1/g_1$ when the relevant transition strengths obey the usual Einstein relations. For equal degeneracies and negligible $N_3$, it reduces to $N_2>N_1$, requiring more than half the active population in level2.

For a simple pump rate R per ground-state emitter, fast $3\to 2$ relaxation and negligible laser intensity, the three-level laser rate equation reduces toThis three-level ground-state inversion threshold is only transparency; positive gain must also overcome optical losses. For an illustrative uniformly filled cavity of length $\ell$, equal absorption/emission cross-section $\sigma$, internal intensity-loss coefficient ${\alpha }_i$ and mirror power reflectivities $R_1,R_2$, the threshold condition isHere $n_i$ are number densities, unlike the total populations $N_i$. The optical resonator supplies feedback and selects modes; a partially transmitting mirror couples out light. Spontaneous emission seeds the field, and stimulated transitions deplete the inversion until saturated gain balances loss. A ruby laser is the standard example. Real level degeneracies, branching losses and pump saturation modify the simple equations. ^[ref]

Define $\hslash {\omega }_0=E_2-E_1$ and use radiation energy density $u(\omega )$ per unit angular frequency, excluding the zero-point contribution. The Angular-frequency Einstein coefficients give rates per atom: $B_{12}u({\omega }_0)$ for absorption, $B_{21}u({\omega }_0)$ for stimulated emission, and $A_{21}$ for spontaneous emission. Stimulated emission adds a photon to the stimulating mode; spontaneous emission persists with no incident photons.

In thermal equilibrium the Boltzmann distribution gives $N_2/N_1=e^{-\hslash {\omega }_0/(k_{\mathrm{B}}T)}$. Balancing up and down rates yieldsMatching the Planck law for every temperature fixesNo degeneracy factor is needed for these nondegenerate levels.^[ref]

For plates of large area $A$ and spacing $d$, the surviving low-frequency transverse electromagnetic mode has one polarization, $\mathbf{E}\parallel \hat{\mathbf{z}}$, and in-plane wavevector. Periodic in-plane counting givesMultiplying this single-polarization two-dimensional photon density of states by $\hslash \omega$ and the Bose-Einstein distribution produces the thermal radiation between parallel conducting plates:The same rate balance therefore givesThese $B$ coefficients refer to the radiation polarization in their respective environments: isotropic free-space $B$ must not automatically be equated to the coefficient for a specified $z$-polarized field.

For hydrogen, take the quantization axis along $z$, neglect spin-orbit coupling, and assume perfect plates, a wide in-plane region and an unmodified atomic wavefunction: $a_{\mathrm{B}}\ll d\ll c/{\omega }_0$, with negligible surface-induced mixing or shifts. The electric-dipole selection rules for hydrogen require $\Delta l=\pm 1$; the allowed $z$ field has $\Delta m_{\ell }=0$. Hence the polarization-selective spontaneous emission between conducting plates givesThe thin-cavity enhancement of a normal electric dipole applies: the $2p_0\to 1s$ dipole points along $z$ and couples to the enhanced mode density. For $m_{\ell }=\pm 1$, the transition dipoles lie in the $x$–$y$ plane and couple to none of these low-frequency modes: the electric-dipole decay rate is zero in the ideal model, so this channel's lifetime is infinite. Real finite, imperfect plates and additional decay or mixing channels limit the lifetime. The order-one orientation factor for the allowed transition does not change the requested lifetime scaling.

A laser combines pumping, an amplifying medium and optical feedback. Stimulated emission adds photons to an occupied optical mode; net amplification requires population inversion between the relevant levels after accounting for their degeneracies. Pumping through additional levels and rapid lower-level relaxation can sustain inversion; a closed two-level system under steady incoherent resonant pumping cannot sustain it.

An optical resonator selects modes and feeds light back through the gain medium. For intensity gain coefficient $g$, internal intensity loss ${\alpha }_i$, length $L$ and mirror intensity reflectivities $R_1,R_2$, the round-trip laser threshold condition isAbove threshold, saturation limits gain and a partially transmitting mirror provides output. Typical properties are strong directionality, substantial coherence and narrow bandwidth; multiple modes and pulsed operation are possible, so perfect monochromaticity is not universal.^[ref]