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Carrier confinement changes both the energies and the available optical transitions of a semiconductor. A quantum well confines one direction and leaves two continuous momentum coordinates; a wire confines two directions and leaves one; a quantum dot confines all three and has discrete ideal levels. Confinement becomes important when its level spacing is comparable with thermal, lifetime and disorder energy scales, and strong excitonic confinement involves comparison with the exciton Bohr radius.

Transition energies. In a simple infinite-well effective-mass model of width $L$, a same-index electron-hole transition has thresholdThus decreasing $L$ generally blue-shifts interband absorption and emission through the $L^{-2}$ confinement contribution. Finite barriers, band mixing, strain, nonparabolicity, dielectric environment and Coulomb interactions alter this elementary estimate. In dots, confinement shifts and exciton binding have different size dependences, so a single inverse-square law is not an exact nanocrystal spectrum.

Dimensional density of states. For a parabolic continuum the phase-space volume below kinetic energy $E$ scales as $E^{d/2}$, hence the density of states scales as $E^{d/2-1}$. Each subband threshold therefore contributes a square-root onset in three free dimensions, a constant step in two, and an inverse-square-root singularity in one. A zero-dimensional ideal dot has discrete lines rather than a continuum power law. Real temperature, disorder and lifetime broadening smooth singularities and lines. Optical absorption uses the joint electron-hole density of states multiplied by transition matrix elements and occupations; a bare electronic density of states is not the entire absorption spectrum.

Selection and Coulomb effects. Envelope-function overlap and Bloch-state dipole selection rules determine allowed transitions and their polarization. For equal ideal infinite-well envelopes, orthogonality favors matching transverse indices; finite barriers and asymmetric fields can relax that envelope rule. Reduced dimensionality can enhance electron-hole attraction and oscillator strength near an exciton resonance, but the binding energy depends on the actual confinement and dielectric screening. Intersubband transitions within a conduction band are different from interband recombination; in a quantum well they require an electric-field polarization component along the confinement direction.

Devices. Quantum wells, wires and dots allow wavelength engineering, carrier localization and altered optical gain for LEDs and semiconductor lasers. Concentrating states near selected transition energies can improve differential gain and reduce thresholds under appropriate injection and loss conditions. Dot ensembles can instead have substantial inhomogeneous broadening from size variation; nonradiative surface traps and carrier capture can limit an otherwise attractive design. Material confinement does not alone ensure high quantum efficiency.

An applied field across a quantum well shifts electron and hole envelopes apart, usually red-shifting the exciton transition and reducing overlap: the quantum-confined Stark effect. This supports electroabsorption modulation and tuneable optical transitions. Conduction-band intersubband energies can also be engineered for infrared detectors and cascade-laser designs. Optical waveguides and cavities confine photons and affect mode overlap or spontaneous-emission rates, but that photonic confinement is distinct from the carrier quantization derived above. Both must be specified when predicting a real device's threshold, spectrum and output.

In the ideal three-level laser, pumping excites atoms out of the ground state to a higher auxiliary level, rapid relaxation fills a relatively long-lived upper laser level, and the laser transition returns to the ground state. A ruby laser is a standard example of this classification. Since the lower laser state initially holds most atoms, substantial pumping is needed to reverse the population per sublevel. With upper and lower degeneracies $g_2,g_1$ and $N_1+N_2\simeq N_T$, the positive-gain condition isFor equal degeneracies more than half the active atoms must be excited before optical gain is even positive.

In a four-level laser the lower laser level is above the ground reservoir and empties rapidly after laser emission. Pumping again feeds a higher auxiliary level which relaxes into the upper laser state, but the lower laser-state population can remain small. The helium-neon laser provides a multilevel physical realization often represented by this scheme for a four-level laser. Positive population inversion then requires much less depletion of the ground reservoir, giving a much lower laser pump threshold for comparable lifetimes, emission cross sections and cavity losses.

The threshold of a four-level laser is not literally zero. Oscillation requires round-trip optical gain to overcome output coupling and other losses, and competing decay or incomplete lower-level removal raises the required pump. Real laser media contain more atomic states than ideal diagrams for a three-level laser or a four-level laser; the labels describe the relevant pumping and laser-transition cycle rather than the total number of states in the material.

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.

Define $\sigma (\omega )$ as the emission cross section for an upper-level atom, averaged over the participating sublevels, and let $W=\sigma I/(\hslash \omega )$. Detailed balance gives the upward stimulated rate per lower-level atom as $rW$, where $r=g_2/g_1$. With $D=N_2-r{N}_1$ the complete laser rate equations areAdding them eliminates the internal transitions. Therefore steady state has $N_1=R_2{\tau }_1$ even when the radiation is strong. Substitution into the upper equation givesOne net stimulated transition changes the beam energy by $\hslash \omega$, so the coefficient of optical gain for intensity is $\alpha (\omega ,I)=\sigma (\omega )D$. At line center, with ${\sigma }_0=\sigma ({\omega }_{21})$,Here ${\alpha }_0$ is the unsaturated intensity optical gain per length, which may be negative, and $I_s$ is the intensity at which its magnitude is halved in this model with homogeneous broadening. The cancellation of ${\tau }_1$ from $I_s$ follows from exclusive upper-to-lower decay and constant upper-state pumping. Keeping $N_1=A{\tau }_1{N}_2$ in the stimulated problem would discard the stimulated feeding term and give an incorrect saturation scale. Other decay branches or different pump constraints lead to different expressions.

For a traveling beam with uniform pump, no additional loss and local steady populations,If ${\alpha }_0>0$ and $I\gg I_s$, the increment becomes $dI/dz\simeq {\alpha }_0{I}_s=\hslash {\omega }_{21}{R}_2(1-rA{\tau }_1)$, approximately constant. Intensity then grows linearly, rather than exponentially, with distance while that strong-saturation regime persists. Negative ${\alpha }_0$ gives saturable absorption and a decreasing beam.

Keep the given emission cross section in square centimetres so that the density remains per cubic centimetre. For the first transition, $r=5/3$ andThusThe separate populations are $N_2^{(0)}=1.667\times 10^{13}$ cm${}^{-3}$ and $N_1^{(0)}=1.00\times 10^{12}$ cm${}^{-3}$. Subtracting $5N_1/3$, rather than $N_1$ alone, reproduces the population inversion used above.

Here $rA{\tau }_1=0.10<1$, so upper-state pumping and sufficiently rapid lower-level removal produce positive small-signal optical gain. This coefficient is an intensity optical gain; a weak field amplitude would grow with half this exponent. A positive single-pass coefficient also does not by itself establish laser oscillation without a cavity-loss budget. No transition wavelength was supplied for this numerical example, so a numerical saturation intensity is not needed or invented to calculate ${\alpha }_0$.

For the second transition, $r=3$ and the much longer lower-level lifetime changes the sign of the available population inversion:If its emission cross section equals the first transition's value, the estimate isThe word “comparable” does not specify exact equality, so the more general answer is ${\alpha }_0=-(0.1133\mathrm{c}{\mathrm{m}}^{-1})[{\sigma }_2/(4\times 10^{-15}\mathrm{c}{\mathrm{m}}^2)]$. The quoted numerical value is an explicitly equal-cross-section estimate, while the negative sign is robust for any positive comparable emission cross section.

In this case $rA{\tau }_1=2.7>1$. At fixed pump, $N_1=1.50\times 10^{13}$ cm${}^{-3}$, so its degeneracy-weighted contribution exceeds the upper population. The medium therefore absorbs a weak beam at this transition. Increasing the radiation saturates the photon absorption magnitude toward zero according to c; it does not convert this fixed-pump model into an optical amplifier with positive gain. Simply inserting the magnitude $|{\alpha }_0|$ into the propagation law for positive optical gain would discard the physically important population balance.

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.

An open pumped two-level system amplifier with fixed upper-state source $R_2$, upper spontaneous emission only into the lower energy level, and lower-state removal rate $N_1/{\tau }_1$. Let $r=g_2/g_1$, $W={\sigma }_{21}I/(\hslash \omega )$ and $D=N_2-r{N}_1$. Steady rate equations give $N_1=R_2{\tau }_1$ and $D=R_2(1-r{A}_{21}{\tau }_1)/(A_{21}+W)$. Consequently the optical gain is $\alpha ={\alpha }_0/(1+I/I_s)$ with ${\alpha }_0={\sigma }_{21}{R}_2(A_{21}^{-1}-r{\tau }_1)$. Lower-state removal controls whether population inversion is possible, while its lifetime cancels from this model's saturation intensity. Different upper-state branching or a nonconstant pump generally changes that conclusion.

Stimulated emission is emission induced by incident light, in contrast with the unprompted decay in spontaneous emission. It supplies the amplification mechanism of a laser. ^[ref].

For an emitter with upper and lower energy values $E_2>E_1$, the resonant photon has frequency $\nu$ satisfyingwhere $h$ is the Planck constant. In the elementary photon picture, an incident photon stimulates an emitter in the upper quantum state to emit an additional photon into the same optical normal mode, with the same frequency, propagation direction and photon polarization. The emitter supplies the extra energy; the stimulating photon is not consumed. This builds the existing field and its optical coherence. An emitter in the lower quantum state can instead undergo photon absorption, so stimulated emission alone does not establish net optical gain.

For example, a $2.0\mathrm{eV}$ gap corresponds to $\nu \approx 4.8360\times 10^{14}\mathrm{Hz}$ and a vacuum wavelength of about $620\mathrm{nm}$. One ideal stimulated emission event converts one incoming photon into two outgoing photons in the selected optical normal mode, adding $2.0\mathrm{eV}$ of light energy while lowering the emitter's energy by that amount.

Consider an original reduced model with a fixed number $N$ of active emitters. Let $u=N_2/N$ be the upper lasing-state fraction, $1-u=N_1/N$ the ground-state fraction, and $s=n/N$ the number of cavity photons per emitter. Assume the pump state empties rapidly, so its population can be neglected. Measure time by $x=t/\tau$, where $\tau$ is the upper state's spontaneous lifetime. Let $r=W\tau$ be the effective pump rate, $C>0$ the dimensionless stimulated-coupling coefficient, $K>0$ the photon-loss rate in these units, and $0\le \beta \le 1$ the fraction of spontaneous decays feeding this optical normal mode. The rate equations areEach term has a physical role. Pumping transfers emitters out of the ground state; spontaneous emission removes upper-state population. The net stimulated term includes emission from fraction $u$ and absorption by fraction $1-u$, giving their difference $2u-1$. A stimulated transition removes one stored excitation and creates one cavity photon, or does the reverse. The factor $\beta$ accounts for spontaneous emission into the selected mode without pretending that all spontaneous emission enters it.

For negligible cavity light, the upper-state equation is linear. A constant pump applied to $u(0)=0$ givesThus this model first permits population inversion at $r>1$. Positive optical gain is still insufficient for an oscillator: the gain must exceed the complete laser threshold, including photon loss. The small-signal photon growth coefficient changes sign whenIf $C\le K$, even complete inversion cannot exceed the assumed loss in this model. For chosen $C=100$ and $K=20$, the threshold population is $u_{\mathrm{th}}=0.6$ and the pump threshold is $r_{\mathrm{th}}=1.5$. At $r=2$, the unsaturated population first reaches $0.6$ at $x=(\log 10)/3\simeq 0.7675$. These numbers are stipulated examples, not ruby material parameters.