Nanolaser (Microdisk, Microring and Photonic Crystal)

1. Theory

• Definition: A nanolaser is a semiconductor laser whose cavity is only a few wavelengths across, so that the mode volume is of the order of one cubic wavelength in the material or less; the cavity then changes the spontaneous emission itself (the Purcell effect), and a large part of the spontaneous emission can enter the lasing mode.
• Cavities on this page: The microdisk and the microring, which are whispering-gallery cavities in which light circulates along the rim by total internal reflection, and the photonic-crystal defect cavity, in which a small defect in a periodic hole pattern traps light by the photonic bandgap.
• Microdisk: A thin disk of gain material; disks 1 $\mu$m in diameter with InGaN quantum dots, 120 or 200 nm thick, lased below 430 nm at room temperature under pulsed optical pumping with a measured $Q$ of up to about 6,600, and a 2 $\mu$m GaInP disk with InP quantum dots lased at room temperature with $Q\approx 4000$ and a threshold of about 1 $\mu$W.
• Microring: The same whispering-gallery mode guided in a narrow ring; electrically pumped InAs quantum-dash rings of 25 to 100 $\mu$m outer radius and 4 $\mu$m width lased continuously at 1.55 $\mu$m, which makes rings of this size microlasers and not nanolasers.
• Photonic-crystal cavity: A single defect in a two-dimensional hole array in a thin slab, with the photonic crystal confining the light laterally and total internal reflection in the slab confining it vertically; the 1999 defect-mode laser (InGaAsP quantum wells, 1.5 $\mu$m, 143 K, pulsed) held the light in 2.5 cubic half-wavelengths, about 0.03 $\mu {\mathrm{m}}^3$, and a zero-cell (H0) cavity with InAs quantum dots reaches about $0.23(\lambda /n{)}^3$.
• Gain material: InGaAsP quantum wells, InAs and InP quantum dots, InGaN quantum dots, InAs quantum dashes and monolayer $\mathrm{WS}{\mathrm{e}}_2$ have all been used.
• Pumping: Most nanolasers are pumped optically; electrical injection has been shown in a single-cell photonic-crystal laser (threshold about 260 $\mu$A at room temperature) and in the quantum-dash ring lasers.

2. Working Principle

• Purcell effect: The spontaneous emission rate into the resonant mode is multiplied by $F_P=(3/4{\pi }^2)(\lambda /n{)}^{3}Q/V$, so a small mode volume and a high quality factor increase it; the factor can also be below unity.
• Coupling factor $\beta$: The fraction of the spontaneous emission that enters the lasing mode is about $10^{-5}$ to $10^{-3}$ in macroscopic lasers, $10^{-2}$ to $10^{-3}$ in microdisk lasers and up to about 0.5 in semiconductor nanolasers, and it tends to 1 as the mode volume shrinks.
• Rate equations: The photon equation carries the source term $\Gamma {\beta }_{sp}{R}_{sp}$, so that a large $\beta$ feeds the mode with spontaneous emission from the start.
• Soft turn-on: As $\beta$ approaches 1 the kink of the output-input curve disappears and the laser is called thresholdless, so the onset of lasing is judged from the inflection point of the curve, from a mean photon number of one in the mode, or from $g^{(2)}(0)$ approaching 1.
• Whispering-gallery cavities: The mode is resonant when the circumference holds a whole number of wavelengths, $2\pi r{n}_{eff}=m\lambda$, with neighbouring orders spaced by ${\lambda }^2/(2\pi r{n}_g)$; the quality factor is limited by radiation loss, which rises steeply as the radius is reduced, by roughness of the side wall and by absorption.
• Microdisk lasers: The quality factor of microdisks lies between about $10^3$ and $10^6$, which lowers the threshold, while $\beta$ is only about $10^{-2}$ to $10^{-3}$, so the output-input curve keeps a clear kink.
• Photonic-crystal cavities: The hole pattern sets $Q$ and $V$; measured $Q$ was 25,300 for an L3 cavity and 33,000 for a zero-cell cavity in separate studies, and in one comparison the zero-cell cavities lased at about one third of the threshold of the L3 cavities (280 nW against 780 nW on average).
• Current injection: The contact must not spoil the quality factor; the single-cell photonic-crystal laser injected current through a sub-micrometre semiconductor wire at the centre of the mode with little loss of $Q$.

3. Applications

• Low-power on-chip light sources (research and development): Nanolasers are motivated by ultralow thresholds, integrability and reduced power consumption for photonic chips.
• Cavity quantum electrodynamics and single-photon sources (research and development): The wavelength-scale photonic-crystal laser was presented as a step toward thresholdless lasers and single-photon sources, and quantum-dot photonic-crystal cavities are used for cavity quantum electrodynamics.
• Nanospectroscopy and light–matter interaction (research and development): In a photonic-crystal nanocavity laser the lasing field is confined in the air-hole region.
• Studies of lasing at high $\beta$ (research): Photon statistics, $g^{(2)}(0)$ and the definition of threshold are tested on these devices.

Name	Formula	Related Formulae	Variables/Constants
Purcell factor and mode volume	$F_P=\frac{3}{4{\pi }^2}{\left(\frac{\lambda }{n}\right)}^3\frac{Q}{V}$	$V=\frac{\iiint \epsilon \left(\vec{r}\right){|\vec{E}\left(\vec{r}\right)|}^2{d}^{3}r}{\max \left\{\epsilon \left(\vec{r}\right){|\vec{E}\left(\vec{r}\right)|}^2\right\}}$ $F_{tot}=\frac{3}{4{\pi }^2}\frac{{\left(\lambda /n\right)}^3}{V}{\left(\frac{1}{Q_{res}}+\frac{1}{Q_{emit}}\right)}^{-1}$ $F_P\approx 0.076\frac{Q}{V/{\left(\lambda /n\right)}^3}$	$\begin{array}{r}{F}_P:Purcellfactor,thefactorbywhich\\ thespontaneousemissionrateintothe\\ cavitymodeexceedsthebulkrate,foran\\ emitteronresonance,atthefield\\ maximum,withanemissionlinenotwider\\ thanthecavityresonance\end{array}$ $\begin{array}{r}\lambda :free-spacewavelengthandn:refractive\\ indexattheemitter,sothat\lambda /nisthe\\ wavelengthinthematerial\end{array}$ $Q:qualityfactorofthecavitymode$ $\begin{array}{r}V:modevolume,definedastheenergyof\\ themodedividedbyitspeakenergy\\ density\end{array}$ $\begin{array}{r}\epsilon \left(\vec{r}\right),\vec{E}\left(\vec{r}\right):permittivityandelectric\\ fieldofthemodeattheposition\vec{r}\end{array}$ $\begin{array}{r}{F}_{tot}:Purcellfactorofthetotaldecay\\ whentheemissionlinehasitsownquality\\ factor{Q}_{emit}=\omega /2\gamma ,Q_{res}beingthatofthe\\ cavity\end{array}$ $\begin{array}{r}0.076=3/4{\pi }^2:numericalformwhenVis\\ measuredinunitsof{\left(\lambda /n\right)}^3\end{array}$
Quality factor, photon lifetime and linewidth	$\frac{1}{{\tau }_p}=\frac{\omega }{Q}$	$Q\equiv \omega \frac{\mathrm{energy}\mathrm{stored}}{\mathrm{power}\mathrm{lost}}$ $Q=\frac{\lambda }{\Delta \lambda }$ $\frac{1}{{\tau }_p}=v_g\left({\alpha }_i+{\alpha }_m\right)$	${\tau }_p:photonlifetimeofthecavitymode$ $\omega :angularfrequencyofthemode$ $\begin{array}{r}Q:qualityfactor,theratiooftheenergy\\ storedtotheenergylostperradian\end{array}$ $\begin{array}{r}\Delta \lambda :fullwidthathalfmaximumofthe\\ cavityresonance,fromwhichQismeasured\\ belowthreshold\end{array}$ $\begin{array}{r}{v}_g,{\alpha }_i,{\alpha }_m:groupvelocity,internalloss\\ andmirrorlossofaFabry-Perotcavity,\\ forwhichthegeneraldefinitionreduces\\ tothisform\end{array}$
Spontaneous-emission coupling factor	$\beta =\frac{R_{sp,cav}}{R_{sp,cav}+R_l}$	$\beta =\frac{F_m}{F_{tot}}$ ${\beta }_{sp}\approx \frac{q}{I_{th}{\tau }_p}\frac{n_{sp}}{{\eta }_i{\eta }_r}$ ${\beta }_{tot}={\eta }_r\beta$	$\begin{array}{r}\beta :fractionofthespontaneousemission\\ thatentersthelasingmode\end{array}$ $\begin{array}{r}{R}_{sp,cav}:spontaneousemissionrateintothe\\ lasingmode\end{array}$ $\begin{array}{r}{R}_l:spontaneousemissionrateintoall\\ othermodes\end{array}$ $\begin{array}{r}{F}_m:Purcellfactorofthelasingmode\\ aloneand{F}_{tot}:Purcellfactorofthetotal\\ decay\end{array}$ $\begin{array}{r}{\beta }_{sp}:valueatthresholdofthefactorused\\ intherateequationsofadiodelaser,\\ foundfromthethresholdcurrent{I}_{th},the\\ photonlifetime{\tau }_p,thepopulation\\ inversionfactor{n}_{sp},theinjection\\ efficiency{\eta }_{i}andtheradiativeefficiency\\ {\eta }_r\end{array}$ $\begin{array}{r}{\beta }_{tot}:thefactorwhennonradiative\\ recombinationisalsocountedamongthe\\ competingchannels\end{array}$ $q:electroncharge$
Rate equations with the coupling factor	$\frac{dN}{dt}=\frac{{\eta }_{i}I}{qV}-\left(R_{sp}+R_{nr}\right)-v_{g}g{N}_p$ $\frac{d{N}_p}{dt}=\left(\Gamma v_{g}g-\frac{1}{{\tau }_p}\right)N_p+\Gamma {\beta }_{sp}{R}_{sp}$	$N_p=\frac{\Gamma {\beta }_{sp}{R}_{sp}{\tau }_p}{1-\Gamma v_{g}g{\tau }_p}$ $\Gamma v_g{g}_{th}=\frac{1}{{\tau }_p}$	$\begin{array}{r}N,N_p:carrierdensityintheactive\\ regionandphotondensityinthemode\end{array}$ $I:injectedcurrent$ ${\eta }_i:injectionefficiency$ $V:volumeoftheactiveregion$ $q:electroncharge$ $\begin{array}{r}{R}_{sp},R_{nr}:totalspontaneousandnonradiative\\ recombinationratesperunitvolume\end{array}$ $g:gain,v_g:groupvelocity,{\tau }_p:photonlifetime$ $\begin{array}{r}\Gamma =V/V_p:confinementfactor,theratioof\\ theactivevolumetothemodevolume,\\ whichappearsinthephotonequationand\\ notinthecarrierequation\end{array}$ $\begin{array}{r}{\beta }_{sp}:spontaneous-emissioncouplingfactor,\\ whichmultipliesthespontaneousemission\\ thatfeedsthemode\end{array}$ $\begin{array}{r}{g}_{th}:gainatthreshold,wherethephoton\\ equationhasnonetlossintheabsenceof\\ thesourceterm\end{array}$
Threshold of a nanolaser and photon statistics	$n_{e,th}=\frac{n_0}{2}+\frac{{\gamma }_c}{2{\gamma }_r}$	${\gamma }_c=\frac{{\omega }_c}{Q_c}$ ${\gamma }_r\propto \frac{1}{V_p}$ $g^{(2)}\left(0\right)=\frac{⟨n_p\left(n_p-1\right)⟩}{{⟨n_p⟩}^2}$ $⟨n_p⟩\ge 1$	$\begin{array}{r}{n}_{e,th}:numberofexcitedemittersatwhich\\ thestimulatedemissionintothemode\\ equalsthelossofthemode,for{n}_0\\ two-levelemitters\end{array}$ $n_0/2:numberofexcitedemittersattransparency$ ${\gamma }_c:photondecayrateofthemode$ ${\omega }_c:itsfrequencyand{Q}_c:itsqualityfactor$ $\begin{array}{r}{\gamma }_r:spontaneousemissionrateperemitter\\ intothemode,whichvariesastheinverse\\ ofthemodevolume{V}_p\end{array}$ $\begin{array}{r}{g}^{(2)}\left(0\right):second-ordercoherenceatzero\\ delay,largerthan1belowthresholdand\\ equalto1foracoherentfield\end{array}$ $\begin{array}{r}⟨n_p⟩\ge 1:criterionofameanphoton\\ numberofatleastoneinthemode,called\\ thequantumthreshold\end{array}$
Resonance and mode spacing of a whispering-gallery cavity	$2\pi r{n}_{eff}=m\lambda$	$\Delta {\lambda }_{FSR}\approx \frac{{\lambda }^2}{2\pi r{n}_g}$ $n_g=n_{eff}-\lambda \frac{\partial n_{eff}}{\partial \lambda }$ $n_c\frac{J_{l+1}\left(k{n}_{c}R\right)}{J_l\left(k{n}_{c}R\right)}=n_{cl}\frac{H_{l+1}^{(1)}\left(k{n}_{cl}R\right)}{H_l^{(1)}\left(k{n}_{cl}R\right)}$	$\begin{array}{r}r:radiusofthedisk,ortheradiusof\\ thecentrelineofthering\end{array}$ $n_{eff}:effectiveindexoftheguidedmode$ $\begin{array}{r}m:azimuthalorder,thenumberof\\ wavelengthsalongthecircumference\end{array}$ $\begin{array}{r}\Delta {\lambda }_{FSR}:spacingbetweenneighbouring\\ azimuthalorders(freespectralrange)\end{array}$ $\begin{array}{r}{n}_g:groupeffectiveindex,whichincludes\\ thematerialandmodaldispersion\end{array}$ $\begin{array}{r}{J}_l,H_l^{(1)}:BesselandHankelfunctionsof\\ orderlofatwo-dimensionaldiskof\\ radiusR\end{array}$ $\begin{array}{r}{n}_c,n_{cl}:indicesofthediskandofits\\ surroundingsandk:thewavenumber,which\\ iscomplexbecausethemodeleaks\end{array}$
Quality factor of a whispering-gallery cavity	$\frac{1}{Q}=\frac{1}{Q_{rad}}+\frac{1}{Q_{ss}}+\frac{1}{Q_{abs}}$	$Q_{abs}=\frac{2\pi n_g}{\alpha \lambda }$ $Q=\frac{k_r}{2k_i}$ $\frac{1}{Q_{loaded}}=\frac{1}{Q}+\frac{1}{Q_{coup}}$	$\begin{array}{r}{Q}_{rad}:qualityfactorsetbytheradiation\\ (bending)lossofthecurvedboundary,\\ whichfallssteeplyastheradiusis\\ reducedortheindexcontrastislowered\end{array}$ $\begin{array}{r}{Q}_{ss}:qualityfactorsetbyscatteringat\\ theroughnessandwavinessoftheside\\ wall\end{array}$ $\begin{array}{r}{Q}_{abs}:qualityfactorsetbyabsorption,\\ with\alpha thepowerabsorptioncoefficient\\ and{n}_{g}thegroupindex\end{array}$ $\begin{array}{r}{k}_r,k_i:realandimaginarypartsofthe\\ complexwavenumberoftheleakymode\end{array}$ $\begin{array}{r}{Q}_{loaded},Q_{coup}:loadedqualityfactorandthe\\ qualityfactoroftheloadingbyacoupled\\ waveguide,presentonlywhenlightis\\ takenoutthroughone\end{array}$
Mode-volume units and effective mode volume	$V_n=\frac{V}{{\left(\lambda /n\right)}^3}=\frac{V_{2n}}{8}$	$V_{2n}=\frac{V}{{\left(\lambda /2n\right)}^3}$ $\frac{1}{V_{eff}}=\frac{\Gamma }{V_{act}}$	$\begin{array}{r}{V}_n:modevolumeinunitsofthecubeof\\ thewavelengthinthematerial\end{array}$ $\begin{array}{r}{V}_{2n}:modevolumeinunitsofthecubeof\\ halfthatwavelength,whichsomepapers\\ use(2.5cubichalf-wavelengthsis0.31\\ cubicwavelengthsinthematerial)\end{array}$ $\begin{array}{r}{V}_{eff}:effectivemodevolumeseenbythe\\ gainregion\end{array}$ $\begin{array}{r}{V}_{act}:itsvolumeand\Gamma :theconfinement\\ factorofthemodeinit\end{array}$

Important Questions

Ques: Why does a small cavity with a high quality factor make it easier to reach lasing?

In a cavity the spontaneous emission can enter only the modes that the cavity allows, and the rate into the resonant mode is multiplied by the Purcell factor $F_P=(3/4{\pi }^2)(\lambda /n{)}^{3}Q/V$, which grows with the quality factor and falls with the mode volume. With $V$ measured in units of $(\lambda /n{)}^3$ this is $F_P\approx 0.076Q/V$. For the zero-cell cavity ($Q$ = 33,000, $V$ = 0.23 $(\lambda /n{)}^3$) the ideal value is about $1.1\times 10^4$, and for the L3 cavity ($Q$ = 25,300, $V$ = 0.7 $(\lambda /n{)}^3$) about $2.7\times 10^3$ (computed). These are upper limits for an emitter exactly at the field maximum and on resonance, and the real enhancement of a quantum dot is lower. The enhanced rate puts more of the recombination into the lasing mode, and so less pumping is needed to reach lasing. Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 4.4.4; Quantum optics and cavity QED with quantum dots in photonic crystals, arXiv:1402.2541; Zero-cell photonic crystal nanocavity laser with quantum dot gain, arXiv:1008.3489.

Ques: Why does a very high quality factor stop raising the Purcell factor?

The general expression is $F_{tot}=\frac{3}{4{\pi }^2}\frac{(\lambda /n{)}^3}{V}{\left(\frac{1}{Q_{res}}+\frac{1}{Q_{emit}}\right)}^{-1}$, in which the emission line of the gain material has its own quality factor $Q_{emit}$. When the cavity is much broader than the line, $Q_{res}\ll Q_{emit}$, the enhancement is proportional to $Q_{res}/V$; when the cavity is narrower than the line, $Q_{emit}$ takes over and a higher cavity $Q$ gives nothing more. With an assumed $Q_{emit}$ = 100, a cavity $Q_{res}$ of $10^3$, $10^4$ and 33,000 gives factors proportional to 91, 99 and 99.7 (computed), so a tenfold higher $Q$ adds only 9%. The book states the same condition: the effect is noticeable when the linewidth of the transition is not wider than that of the cavity mode. A high $Q$ therefore pays only with a narrow emitter, and with a broad one the mode volume is the lever. Reference: Active Nanophotonics, arXiv:1912.03801; Coldren et al., 2nd ed., Section 4.4.4.

Ques: Why does shrinking the mode volume push β toward one, and is β the same thing as the Purcell factor?

The spontaneous emission rate per emitter into the mode varies as $1/V_p$, while the emission into leaky modes and the nonradiative recombination are not changed by the cavity in the model, so the fraction $\beta ={\gamma }_r/({\gamma }_r+{\gamma }_{bg})$ rises. A simulation of pillar lasers with a fixed $Q$ = 235 found $\beta$ approaching unity when $V_{eff}$ was reduced from 0.5 to 0.0025 $\mu$m${}^3$. With an assumed starting value $\beta$ = 0.01 and a loss rate that stays fixed, a 200-fold smaller volume gives $\beta \approx 0.67$ (computed). The two quantities are not the same, because $\beta =F_m/F_{tot}$ is a ratio: it can be raised by suppressing emission into other modes, for example by a photonic bandgap, with no Purcell enhancement at all, so that the sources call them in general independent. Reference: Purcell Effect in the Stimulated and Spontaneous Emission Rates of Nanoscale Semiconductor Lasers, arXiv:1801.08879; The threshold of semiconductor nanolasers, arXiv:2305.15888; Active Nanophotonics, arXiv:1912.03801.

Ques: Why do different papers quote different values of β for the same kind of laser?

Two definitions are in use. In the first, $\beta$ is the rate into the lasing mode divided by the total spontaneous emission rate, which is the form of the book, of arXiv:1801.08879 and, with the total radiative decay rate in the denominator, of arXiv:1912.03801. In the second, the denominator also contains the nonradiative recombination, $\beta ={\gamma }_r/({\gamma }_r+{\gamma }_{bg})$ with ${\gamma }_{bg}$ including nonradiative loss (arXiv:2305.15888). The second is smaller by the radiative efficiency ${\eta }_r$, which the book puts at 50% to 80% for diode lasers, so the two values differ by 20% to 50% (computed). The first is the more likely to be meant, because it is the one in the book and in two of the three papers; the second is correct if it is used with a rate equation that does not also carry the nonradiative term separately, as in Eq. (5.3) of the book. Reference: Coldren et al., 2nd ed., Sections 4.4.3 and 5.2; arXiv:1801.08879; arXiv:1912.03801; arXiv:2305.15888.

Ques: Why does a laser with β close to one have no sharp threshold, and how is its threshold then defined?

The photon equation is fed by spontaneous emission from the start, so the output rises smoothly with the pump and the kink of the output-input curve disappears at $\beta \to 1$. Several criteria are then used: the inflection point of the curve (about 300 nW in a zero-cell cavity), a mean photon number $⟨n_p⟩\ge 1$, and second-order coherence, with $g^{(2)}(0)>1$ below threshold and $\to 1$ above it. In micropillar lasers $g^{(2)}(0)$ approached 1 only for 10 to 100 photons in the mode. A modelling study finds that for a high-$\beta$ laser the intensity jump and a coherence criterion of $g^{(2)}(0)$ = 1.1 can be separated, and that one photon is not enough for an extended gain medium, where coherence appears near 1000 photons. A quoted threshold therefore needs its criterion. Reference: Unveiling the physics of microcavity lasers, PMC6062310; F. Lohof, R. Barzel, P. Gartner and C. Gies, Do Nanolasers Benefit From a High β-Factor?, arXiv:1809.08976; Zero-cell photonic crystal nanocavity laser with quantum dot gain, arXiv:1008.3489; Active Nanophotonics, arXiv:1912.03801.

Ques: Does a larger β always lower the threshold of a nanolaser?

The sources do not agree in emphasis. One review states that raising the mode’s share of the emission and lowering the other emission increases $\beta$ and lowers the threshold. A modelling study of high-$\beta$ lasers finds that this holds for quantum-dot gain, where about 10 photons suffice for coherence at $\beta \to 1$, but not for extended gain media such as quantum wells or two-dimensional materials, where the cavity $Q$ is the better lever. The second is the more likely for a large number $n_0$ of emitters, because the threshold $n_{e,th}=n_0/2+{\gamma }_c/(2{\gamma }_r)$ consists of the transparency number $n_0/2$, which is set by the number of emitters and not by the cavity, and an excess ${\gamma }_c/(2{\gamma }_r)$ that scales as $V/Q$ (derived). Both statements can then be true of different gain media. Reference: Active Nanophotonics, arXiv:1912.03801; arXiv:1809.08976; arXiv:2305.15888.

Ques: Should the mode spacing of a microdisk or microring use the effective index or the group index?

The group index, because neighbouring azimuthal orders differ in frequency by the change of the phase constant per round trip, and that depends on the dispersion. The formula is $\Delta \lambda \approx {\lambda }^2/(2\pi r{n}_g)$. For the 100 $\mu$m radius quantum-dash ring with $n_g$ = 3.6 at 1.55 $\mu$m it gives 1.06 nm, which matches the fine mode spacing of about 1.06 nm reported for that group index (computed). For the 1 $\mu$m InGaN disk, one source writes the spacing with $n_{eff}$ = 2.48 and states about 20 nm, but that formula gives 22.6 nm at an assumed 420 nm (computed), and 20 nm needs $n_g$ ≈ 2.8. The group index is the correct quantity, and a value above $n_{eff}$ is plausible because the index falls with wavelength, so the stated 20 nm is the more likely to be right. Reference: Electro-optically tunable microring resonators in lithium niobate, arXiv:0705.2392; Y. Wan et al., Low-Threshold Continuous-Wave Operation of Electrically Pumped 1.55 μm InAs Quantum Dash Microring Lasers, ACS Photonics (2018); Low threshold, room-temperature microdisk lasers in the blue spectral range, arXiv:1208.6452.

Ques: Why does the quality factor of a microdisk fall when the radius is reduced, and what limits the measured value?

The curved boundary leaks light, and the sources state that the radiation loss rises exponentially with the curvature, while it can be lowered by a higher index contrast. In a calculation for an index of 1.65, a ring with inner and outer radii of 2.5 and 3.2 $\mu$m and azimuthal order 22 had a radiation-limited $Q$ of about 15,000, and a disk of 1 $\mu$m radius with order 5 only 12.3. For high-index semiconductor disks the radiation limit is far higher (above $10^{10}$ for a TE mode of a 200 nm thick GaAs disk of 2.5 $\mu$m radius), so the measured value is set by other losses: the best GaAs disks saturate near half a million because of surface absorption, and the 1 $\mu$m InGaN disks measured 6,600 against 18,000 to 25,000 simulated. Reference: Large Q factor with very small Whispering Gallery Modes resonators, arXiv:1901.05740; Electro-optically tunable microring resonators in lithium niobate, arXiv:0705.2392; Origin of optical losses in gallium arsenide disk whispering gallery resonators, arXiv:1507.04906; arXiv:1208.6452.

Ques: Is the quality factor limited by absorption 2πn/(αλ) or πn/(αλ)?

The photon lifetime of a cavity is $1/{\tau }_p=v_g\alpha$ when $\alpha$ is the loss of the power, so $Q=\omega {\tau }_p=2\pi n_g/(\alpha \lambda )$ (derived from the book). This is the form of the microdisk paper, $Q_{mat}=2\pi n_c/(\alpha \lambda )$. The microring paper writes $Q\approx \pi N/(\alpha \lambda )$, a factor 2 smaller, which equals the same result only if its $\alpha$ is the loss of the field amplitude. Which loss its $\alpha$ means is not clear from the formula as given, so the factor-2 form is more likely a convention than an error. With the power form, an assumed $\alpha$ = 1 cm${}^{-1}$ at 1.55 $\mu$m and $n_g$ = 3.6 give $Q\approx 1.5\times 10^5$ (computed). Reference: Coldren et al., 2nd ed., Eq. (5.7); Single-atom detection using whispering gallery modes of microdisk resonators, arXiv:quant-ph/0407229; arXiv:0705.2392.

Ques: Which wavelength factor belongs in the Purcell factor, (λ/n)³ or (λ/2n)³?

The book and several papers give $F_P=(3/4{\pi }^2)(\lambda /n{)}^{3}Q/V$, and a fourth source, a review of nanoscale cavities, writes $(3/4{\pi }^2)(Q/V)(\lambda /2n{)}^3$. If $V$ is an absolute volume, the second form is 8 times smaller. The first is the more likely to be correct, being the form of the book and of several papers and the one that gives $F_P=3/4{\pi }^2$ for $Q$ = 1 and $V=(\lambda /n{)}^3$. The second form probably comes from the habit of quoting $V$ in cubic half-wavelengths: with $V_{2n}$ in those units the prefactor is $8\times 3/4{\pi }^2=0.61$ (computed). The 1999 defect-mode laser, with 2.5 cubic half-wavelengths, has 0.31 $(\lambda /n{)}^3$ (computed). A volume should always be given with its unit. Reference: Coldren et al., 2nd ed., Section 4.4.4; arXiv:1801.08879; arXiv:1402.2541; Emerging Optics from Structured Nanoscale Optical Cavities, arXiv:2307.12587; O. Painter et al., Two-Dimensional Photonic Band-Gap Defect Mode Laser, Science 284, 1819 (1999).

Ques: How can current be injected into a wavelength-scale cavity without destroying its quality factor?

A contact or doped region that overlaps the mode adds loss, so the injection path has to be small compared with the mode. The single-cell photonic-crystal laser fed the current through a sub-micrometre semiconductor wire placed at the centre of the mode with little loss of cavity quality, and it lased at room temperature with a threshold of about 260 $\mu$A, in a nondegenerate monopole mode whose measured behaviour agreed with simulations of the fabricated dimensions. A quantum-dot photonic-crystal laser pumped by a lateral p-i-n junction made by ion implantation is reported, in a review, to lase continuously up to 150 K with 287 nA. Most nanolasers are still pumped optically, because the 1999 defect-mode laser needed only a pulsed optical pump at 143 K. Reference: H.-G. Park et al., Electrically driven single-cell photonic crystal laser, Science 305, 1444 (2004); Emerging Optics from Structured Nanoscale Optical Cavities, arXiv:2307.12587; Painter et al., Science 284, 1819 (1999).

Ques: Is an electrically pumped microring laser a nanolaser?

Not at the sizes reported. The quantum-dash rings have 25 to 100 $\mu$m outer radius and are 4 $\mu$m wide, so a 100 $\mu$m ring has a footprint of about $1.4\times 10^4$ $(\lambda /n{)}^2$ at 1.55 $\mu$m and $n_g$ = 3.6 (computed), which is far from wavelength size, so the Purcell effect is expected to be small. The ring is included here because it has the same whispering-gallery resonance and mode-spacing relations as the microdisk. Its threshold was 13 mA (528 A/cm${}^2$) in a 100 $\mu$m ring and 3.5 mA in the smallest, with continuous operation up to 55 ${}^{\circ }$C. The quality factor was not reported. Reference: Y. Wan et al., ACS Photonics (2018); arXiv:0705.2392; arXiv:1901.05740.

Related pages in this vault: Silicon-Compatible Lasers (Hybrid III-V, Epitaxial Quantum-Dot and Ge-GeSn), Vertical-External-Cavity Surface-Emitting Laser (VECSEL), Double-Heterostructure Fabry-Perot Laser, Quantum-Dot Laser, Distributed Bragg Reflector Laser (DBR), Silicon Microring Modulator, Superluminescent Diode (SLD), Resonant Cavity LED (RCLED), Semiconductor Lasers (Edge-Emitting and VCSEL).