Vertical-Cavity Surface-Emitting Laser (VCSEL)

1. Theory

• Definition: A vertical-cavity surface-emitting laser (VCSEL) emits perpendicular to the wafer from a cavity about one wavelength long, formed between two distributed Bragg reflector (DBR) mirror stacks, with quantum wells as the gain region.
• Short gain length: The active layers are only a few tens of nanometres thick, so each mirror needs a power reflectance above about 99% to reach threshold.
• DBR mirrors: Quarter-wave stacks of high- and low-index layers (e.g. AlAs/GaAs) grown in the same epitaxial run as the cavity; AlGaAs/GaAs reaches $R>0.999$ with fewer than about two dozen periods, while lattice-matched InGaAsP/InP pairs need many more.
• Standing-wave gain placement: The wells sit at the peaks of the cavity ${\left|E\right|}^2$ pattern, which raises the confinement factor by up to a factor of 2.
• Effective cavity length: The field penetrates each mirror by $L_{eff}$, so $L_{DBR}=L_c+L_{eff1}+L_{eff2}$ is the length used for gain, loss and efficiency.
• Lateral confinement: Proton implantation (large, deliberately multimode devices) or an oxidised AlAs layer close to the active region (small, single-mode, low-threshold devices) confines current and light; InP-based devices use an etched tunnel-junction mesa.
• Direct modulation: A VCSEL is modulated like an edge-emitting laser, by adding a data current to the bias (Directly Modulated Laser page). The same rate equations give a relaxation resonance $f_r=D\sqrt{I-I_{th}}$ and a damping $\gamma =K{f}_r^2+{\gamma }_0$, but the active volume and the mode volume are very small and the photon lifetime is a few $\mathrm{ps}$, so D-factors and MCEF values of 14 to 17 $GHz/m{\mathrm{A}}^{1/2}$ are reported (Chang and Coldren; Michalzik and Ebeling).
• Bandwidth limits: The $3\mathrm{dB}$ bandwidth is set by the smallest of five limits: the relaxation resonance ($f_{3dB}\approx 1.55f_r$), the damping limit $2\sqrt{2}\pi /K$, carrier transport and capture into the wells, the parasitic resistance-capacitance pole of the mesa and pad, and self-heating, which makes the bandwidth stop growing with current (Chang and Coldren; Michalzik and Ebeling; Xie et al.).

2. Working Principle

• Threshold: The one-pass modal gain $\Gamma g_{th}{L}_{DBR}$ equals $A_i+T_m$, where $T_m=\ln (1/R)$ is the mirror loss and $A_i={\alpha }_i{L}_{DBR}$ the one-pass internal loss.
• Differential efficiency: ${\eta }_d=F{\eta }_i{T}_m/(A_i+T_m)$, where $F$ is the fraction of the power leaving through the output mirror, which is made less reflective than the other.
• Single axial mode: The short cavity gives a very large mode spacing and the mirror band is narrow, so only one axial mode can lase.
• Lateral modes: Apertures of about 20 $\mu$m lase in many lateral modes for multimode-fibre links; smaller oxide apertures support a single mode.
• Temperature tuning: The mode moves about 0.1 nm/${}^{\circ }$C and the gain peak about 0.5 nm/${}^{\circ }$C, so the peak sweeps past the mode at about 0.4 nm/${}^{\circ }$C.
• Gain–mode offset: Placing the gain peak at a shorter wavelength than the mode at room temperature makes the threshold nearly flat, or even falling, as the device heats.
• Output beam: A small, circular, low-divergence beam allows passive alignment to multimode fibre, and devices can be tested at wafer scale.
• Photon lifetime: The lifetime follows $1/{\tau }_p=v_g({\alpha }_i+{\alpha }_m)$, with the mirror loss ${\alpha }_m=\ln (1/\sqrt{R_t{R}_b})/L_{eff}$. For mirrors of 99.5 per cent and $L_{eff}$ = 1.3 $\mu \mathrm{m}$ this is 2.5 $\mathrm{ps}$ (computed from Michalzik and Ebeling), and the high mirror reflectance of the threshold condition above is what keeps it from being much shorter.
• Small-signal response: A small current step modulates the photon density through the relaxation resonance. The response is flat up to about $f_r$ and then falls; with $K$ = 0.15 $\mathrm{ns}$ the damping alone would allow 59 $\mathrm{GHz}$ (computed), so in good devices the damping is not the first limit (Chang and Coldren).
• Transport: The carriers must first be captured into the wells from the barriers. This adds a pole $1/(2\pi {\tau }_{cap})$ and a factor $\chi =1+{\tau }_{cap}/{\tau }_{esc}$ that lowers D; in a simulation of an 850 nm device a capture time of 15 $\mathrm{ps}$ instead of 1 $\mathrm{ps}$ cut the bandwidth from 40 to 15 $\mathrm{GHz}$ (Grabowski et al.).
• Parasitics: The current reaches the wells through the mirror resistance and the capacitance of the oxide layers and pad. A pad over thick BCB has 47 $\mathrm{fF}$ instead of 3.3 $\mathrm{pF}$ over thin nitride, and extra oxide layers in the mesa can raise the parasitic $3\mathrm{dB}$ frequency from 22 to above 30 $\mathrm{GHz}$ (Chang and Coldren; Ou et al.).
• Self-heating: The thermal resistance is about $1/(2{\lambda }_c{D}_a)$, 3.6 $K/mW$ for a 3 $\mu \mathrm{m}$ aperture (computed, assumed conductivity), and the temperature rise lowers the differential gain and shifts the gain peak away from the mode, so $f_r$ and the output saturate before the damping limit is reached (Michalzik and Ebeling; Xie et al.).
• Beyond the carrier limit: Modulating the loss instead of the current rolls off at about 20 $\mathrm{dB}$ per decade instead of 40, and electro-absorption and electro-optic modulators integrated in the VCSEL reached 17 and 35 $\mathrm{GHz}$; injection locking to a separate laser reached 66 $\mathrm{GHz}$ (Chang and Coldren).

3. Applications

• Short-reach datacom: 850 nm multimode VCSELs with multimode fibre for Gigabit Ethernet, Fibre Channel and parallel-fibre ribbon links, valued for low cost and wafer-scale testing.
• Long-wavelength links: 1300 and 1550 nm InP-based VCSELs, and GaInNAs on GaAs for 1300 nm, for single-mode fibre links at about 10 Gb/s over tens of kilometres (mostly research and development).
• High-speed VCSELs: Bottom-emitting 980 nm oxide-aperture devices with integrated microlenses, and quantum-dot VCSELs at 20 Gb/s up to 85 ${}^{\circ }$C (research and development).
• 3D sensing and LiDAR: Arrays of small-aperture VCSELs as illumination sources for depth cameras and ranging.
• Datacom at 25 to 57 $Gbit/s$: 850 nm VCSELs with 7 to 8 $\mu \mathrm{m}$ apertures carried 47 to 57 $Gbit/s$ back to back and 44 to 55 $Gbit/s$ over 50 m of OM4 fibre in a Chalmers work, and 40 $Gbit/s$ up to 85 ${}^{\circ }\mathrm{C}$ (Haglund et al.); a 7 $\mu \mathrm{m}$ device carried 56 $Gbit/s$ with on-off keying and four-level pulse-amplitude modulation (Grabowski et al.).
• Record history of direct modulation: 20 $Gbit/s$ at 850 nm in 2001, 25 to 40 $Gbit/s$ at 1.1 $\mu \mathrm{m}$ in 2006 and 2007, and 35 $Gbit/s$ at 980 nm in 2007 with 3.5 $(Gbit/s)/mW$ (Chang and Coldren).
• Energy of parallel links: A 500 $Gbit/s$ aggregate from 48 channels used 3.3 $\mathrm{W}$, 6.6 $mW/(Gbit/s)$, and 160 $Gbit/s$ from a 4 by 4 array used 2.5 $\mathrm{W}$, 15.6 $mW/(Gbit/s)$ (computed from Chang and Coldren).
• Longer wavelengths: 1.28 $\mu \mathrm{m}$ devices carried 10 $Gbit/s$ over 9 $\mathrm{km}$ of single-mode fibre, 1.3 $\mu \mathrm{m}$ GaInNAs devices over 20 $\mathrm{km}$, and 1.55 $\mu \mathrm{m}$ devices 12.5 $Gbit/s$ over 3 $\mathrm{km}$ (Chang and Coldren).

Name	Formula	Related Formulae	Variables/Constants
Threshold and differential efficiency of a VCSEL	$\Gamma g_{th}{L}_{DBR}=A_i+T_m$	${\eta }_{dj}=F_j{\eta }_i\frac{T_m}{A_i+T_m}$ $T_m=\ln \frac{1}{R},A_i=⟨{\alpha }_i⟩L_{DBR}$ $L_{DBR}=L_c+L_{eff1}+L_{eff2}$	$\begin{array}{r}\Gamma g_{th}{L}_{DBR}:one-passthresholdmodalgainof\\ thecavity\end{array}$ $\begin{array}{r}{A}_i:one-passinternallossincludingthe\\ lossesofthemirrors,⟨{\alpha }_i⟩beingthe\\ internallossaveragedovertheeffective\\ cavity\end{array}$ $\begin{array}{r}{T}_m:mirrorlossperpass,whichisabout\\ themeanpowertransmissionofthemirrors\\ whentheyarelossless\end{array}$ $\begin{array}{r}R:meanmirrorpowerreflectance,R=\sqrt{R_1{R}_2},\\ whichmustbehigh,typicallyabove99\%,\\ becausethegainlengthissmall\end{array}$ $\begin{array}{r}{\eta }_{dj}:differentialquantumefficiencyoutof\\ sidej,F_{j}beingthefractionofthe\\ outputpowerleavingthroughthatside\\ (seetheDBRpage)\end{array}$ ${\eta }_i:injectionefficiency$ $\begin{array}{r}{L}_{DBR}:effectivecavitylength,thespacing\\ L_c=L_a+L_{p}betweenthereferenceplanesof\\ themirrorsplustheeffectivepenetration\\ lengths{L}_{eff1}and{L}_{eff2}intothetwomirrors\end{array}$
Reflectance of a DBR mirror stack	$\left|r_g\right|=\frac{1-b}{1+b},b={\prod }_i{\left(\frac{n_{1i}}{n_{2i}}\right)}^{2m_i}$	$R_g={\left|r_g\right|}^2$ $n_{1i}={\bar{n}}_i-\frac{{\chi }_i\Delta n_i}{2},n_{2i}={\bar{n}}_i+\frac{{\chi }_i\Delta n_i}{2}$ $L_{eff}\approx \frac{{\lambda }_0}{4\Delta n}\left(\left|r_g\right|\to 1\right)$	$\begin{array}{r}\left|r_g\right|:amplitudereflectivityofthe\\ mirrorattheBraggwavelength(lossless)\end{array}$ $\begin{array}{r}b:productoverthelayerpairsiofthe\\ stack,with{n}_{1i}thelowand{n}_{2i}thehigh\\ indexofthepairand{m}_{i}thenumberof\\ periodsofthatpair(therearetwo\\ interfacesperperiod;ifthefirst\\ interfaceiswithaspacerofindex{n}_3,\\ theproductismultipliedby{n}_1/n_{3}for\\ thatinterface)\end{array}$ $\begin{array}{r}{\bar{n}}_i,\Delta n_i:averageanddifferenceofthe\\ maximumandminimumindexofthepair\end{array}$ $\begin{array}{r}{\chi }_i:shapefactoroftheinterface,equal\\ to1forabruptinterfaces,\pi /4fora\\ sinusoidalgrating,2/\pi foratriangular\\ sawtoothand\sin x/xforatrapezoid,withx\\ from0(abrupt)to\pi /2(sawtooth)(graded\\ interfaceslowertheseriesresistanceat\\ asmallcostofreflectance)\end{array}$ $\begin{array}{r}{L}_{eff}:penetrationdepthintothestackwhen\\ itreflectsstrongly,with\Delta n=n_2-n_{1}and\\ {\lambda }_{0}thevacuumwavelength(thisisthe\\ limit\Lambda /4roftheDBRpage)\end{array}$ $R_g:powerreflectanceofthestack$
Axial confinement factor and standing-wave enhancement	$\Gamma ={\Gamma }_{xy}\frac{L_a}{L_{DBR}}\left[1+\cos \left(2\beta z_s\right)\frac{\sin \left(\beta L_a\right)}{\beta L_a}\right]$	${\Gamma }_z={\Gamma }_{fill}{\Gamma }_{enh},{\Gamma }_{fill}=\frac{L_a}{L_{DBR}},{\Gamma }_{enh}=1+\frac{\sin \left(\beta L_a\right)}{\beta L_a}$ $\beta =\frac{2\pi \bar{n}}{\lambda }$	${\Gamma }_{xy}:lateral(transverse)confinementfactor$ $L_a:totalthicknessoftheactive(gain)layers$ $L_{DBR}:effectivecavitylength$ $\begin{array}{r}{z}_s:shiftbetweenthecentreoftheactive\\ layerandthepeakofthestandingwave\\ (withtheactivelayeratapeak,z_s=0)\end{array}$ $\beta :propagationconstantofthemode$ $\begin{array}{r}{\Gamma }_{enh}:enhancementfactoroftheaxial\\ confinement,whichapproaches2forathin\\ activelayeratapeakof{\left|E\right|}^{2}andis\\ below1ifthelayerisatanode(aratio\\ ofthegainlayerstothewholecavityis\\ thefillfactor)\end{array}$
Temperature tuning of the mode and the gain peak	$\frac{d\left({\lambda }_{gain}-{\lambda }_{mode}\right)}{dT}\approx 0.4nm{/}^{\circ }C$	$\frac{d{\lambda }_{mode}}{dT}\approx 0.1nm{/}^{\circ }C$ $\frac{d{\lambda }_{gain}}{dT}\approx 0.5nm{/}^{\circ }C$ $\delta \lambda =\frac{{\lambda }^2}{2{\bar{n}}_g{L}_{DBR}}$	$\begin{array}{r}{\lambda }_{mode}:wavelengthofthesingleaxialmode,\\ whichmoveswiththetemperaturethrough\\ thechangeoftherefractiveindex\end{array}$ $\begin{array}{r}{\lambda }_{gain}:wavelengthofthegainpeak(the\\ energyseparationofthelowest\\ quantum-welllevels),whichmovesfaster\\ becauseofthechangeofthebandgap\end{array}$ $\begin{array}{r}\delta \lambda :spacingoftheaxialmodes,whichis\\ verylargebecause{L}_{DBR}isaboutone\\ wavelength(thespacingistheDBRformof\\ theFabry-Perotone,withtheoptical\\ lengthincludingthepenetrationintothe\\ mirrors)\end{array}$ ${\bar{n}}_g:groupindex$ $\begin{array}{r}(theratesaretypicalvaluesfor\\ GaAs-baseddevices,andthegainpeakis\\ deliberatelyplacedatashorter\\ wavelengththanthemodeatroom\\ temperature)\end{array}$

Direct Modulation of the VCSEL

Name	Formula	Related Formulae	Variables/Constants
Photon lifetime and threshold gain of a VCSEL	$\frac{1}{{\tau }_p}=v_g\left({\alpha }_i+{\alpha }_m\right)$ ${\alpha }_m=\frac{1}{L_{eff}}\ln \frac{1}{\sqrt{R_t{R}_b}}$	$g_{th}\approx \frac{{\alpha }_i{L}_{eff}+T_m}{{\Gamma }_{enh}{d}_a}$ ${\alpha }_m{L}_{eff}=T_m$	${\tau }_p:photonlifetime$ $\begin{array}{r}{v}_g:groupvelocity(c/3.6inMichalzikand\\ Ebeling)\end{array}$ ${\alpha }_i:internalloss$ ${\alpha }_m:mirrorloss$ $L_{eff}:effectivecavitylength$ $\begin{array}{r}{R}_{t}and{R}_b:powerreflectancesofthetop\\ andbottommirrors(T_{m}andthemeanRof\\ thethresholdrowabove)\end{array}$ $g_{th}:thresholdgainofthewells$ $\begin{array}{r}{\Gamma }_{enh}:axialenhancementfactor(written{\Gamma }_r\\ byMichalzikandEbeling)\end{array}$ $\begin{array}{r}{d}_a:totalthicknessofthewells,theform\\ of{g}_{th}isEq.14ofMichalzikandEbeling\\ withthewelllosssettozeroandL-d_a\\ takenas{L}_{eff}(mysimplification),Chang\\ andColdren:addinginternallosslowers\\ thephotondensitybymorethanit\\ shortens{\tau }_p,soitlowers{f}_r,Xieetal.:\\ shorteningthephotonlifetimeisa\\ misconceptionforVCSELsbecauseitlowers\\ thedampingandraisesthethreshold\end{array}$ $\begin{array}{r}(computedfortheexampleofMichalzikand\\ Ebeling,R_t=R_b=99.5percent,{\alpha }_i=10\\ \mathrm{c}{\mathrm{m}}^{-1},L_{eff}=1.3\mu \mathrm{m},{\Gamma }_{enh}=1.8,d_a=24\mathrm{nm}:\\ {\alpha }_m=38.6\mathrm{c}{\mathrm{m}}^{-1},{\tau }_p=2.47\mathrm{ps}againstthe\\ quoted2.5\mathrm{ps},and{g}_{th}=1461\mathrm{c}{\mathrm{m}}^{-1}against\\ thequoted1460\mathrm{c}{\mathrm{m}}^{-1};Grabowskietal.give\\ {\tau }_p=3.9\mathrm{ps}at7.7\mathrm{mA}foran850\mathrm{nm}\\ device)\end{array}$
Relaxation frequency, D-factor and transport factor of a VCSEL	$f_r=D\sqrt{I-I_{th}}$ $D=\frac{1}{2\pi }\sqrt{\frac{{\eta }_i{v}_{g}a}{q{V}_p\chi }}$	$\chi =1+\frac{{\tau }_{cap}}{{\tau }_{esc}}$ $\mathrm{MCEF}=\frac{f_{3dB}}{\sqrt{I-I_{th}}}$ $f_{3dB}\approx 1.55f_r$	$f_r:relaxationresonancefrequency$ $Iand{I}_{th}:biasandthresholdcurrent$ $D:D-factorinGHz/m{\mathrm{A}}^{1/2}$ ${\eta }_i:injectionefficiency$ $\begin{array}{r}{v}_g:groupvelocity,a=\partial g/\partial N:\\ differentialgain\end{array}$ $\begin{array}{r}{V}_p:mode(photon)volume,takenasthe\\ activevolumedividedbytheconfinement\\ factor(myreadingofEq.7.5ofChangand\\ ColdrenagainstEq.6ofGrabowskietal.)\end{array}$ $\chi :transportfactor$ $\begin{array}{r}{\tau }_{cap}and{\tau }_{esc}:captureandescapetimesof\\ carriersintoandoutofthewells\end{array}$ $MCEF:modulationcurrentefficiencyfactor$ $\begin{array}{r}{f}_{3dB}:small-signal3dBfrequency,1.55f_r\\ holdsforsmalldampingandparasitics\\ (ChangandColdren),theformofDisthat\\ oftheDirectlyModulatedLaserpagewith\\ thefactor\chi added,values:970nmVCSEL\\ of3by3\mu \mathrm{m}MCEF16.8GHz/m{\mathrm{A}}^{1/2},the\\ highestforquantum-welldevices(Chang\\ andColdren),850nmVCSELof4by4\mu \mathrm{m}of\\ Learetal.14.2GHz/m{\mathrm{A}}^{1/2}(afigure\\ label),MCEFabove14GHz/m{\mathrm{A}}^{1/2}(Michalzik\\ andEbeling)\end{array}$ $\begin{array}{r}(computed:the980nmdeviceofChangand\\ Coldrenwith15\mathrm{GHz}at1\mathrm{mA}anda\\ thresholdof0.14\mathrm{mA}gives16.2GHz/m{\mathrm{A}}^{1/2},\\ 3percentbelowitsquoted16.7;Dvaries\\ as1/\sqrt{V_p},sohalvingthemodevolume\\ raisesitby1.41;with{\tau }_{cap}=7\mathrm{ps}and{\tau }_{esc}\\ =55\mathrm{ps}ofGrabowskietal.,\chi =1.13)\end{array}$
Damping, K-factor and the damping limit of a VCSEL	$\gamma =K{f}_r^2+{\gamma }_0$ $K=4{\pi }^2\left({\tau }_p+\frac{ϵ\chi }{v_{g}a}\right)$	$f_{3dB,max}=\frac{2\sqrt{2}\pi }{K}\approx \frac{8.89}{K}$ ${\gamma }_0=\frac{1}{\chi {\tau }_n}$ $K_{min}=4{\pi }^2{\tau }_p\left(1+\frac{\bar{g}}{g_{th}}\right)$	$\gamma :dampingrate$ $K:K-factorin\mathrm{ns}$ $ϵ:gaincompressionfactor$ $a:differentialgain$ $\chi :transportfactor$ ${\gamma }_0:dampingoffset$ ${\tau }_n:carrierlifetimeinthewells$ $\begin{array}{r}{f}_{3dB,max}:highest3dBfrequencythatthe\\ dampingallows,reachedwhen\gamma /{\omega }_r=\sqrt{2}\\ (ChangandColdren)\end{array}$ $\begin{array}{r}{K}_{min}:thesmallestKforanoptimum\\ thresholdgain,with\stackrel{ˉ}{}gthegainconstant\\ ofthelogarithmicgainlaw(Eq.56of\\ MichalzikandEbeling),Kand{\gamma }_{0}areEqs.\\ 7to9ofGrabowskietal.,thelimitis\\ printedas\sqrt{2}/(2\pi K)byMichalzikand\\ Ebelingandas(\sqrt{2}/2\pi )KbyChangand\\ Coldreninthesummaries;botharemisprints,\\ sincemaximising{f}_{3dB}over{f}_{r}numerically\\ givesexactly2\sqrt{2}\pi /K(seethequestions),\\ smallestreportedK:0.15to0.2\mathrm{ns}\\ (MichalzikandEbeling)\end{array}$ $\begin{array}{r}(computed:K=0.15,0.153and0.2\mathrm{ns}give\\ limitsof59.2,58.1and44.4\mathrm{GHz},andthe\\ damping-limited58\mathrm{GHz}thatChangand\\ Coldrenquoteforthe850nmdeviceof\\ Learetal.correspondstoK=0.153\mathrm{ns};\\ thelowerbound4{\pi }^2{\tau }_{p}is0.154\mathrm{ns}for3.9\\ \mathrm{ps}and0.099\mathrm{ns}for2.5\mathrm{ps})\end{array}$
Carrier transport and the response of a VCSEL	$M(f)=\frac{1}{1+j2\pi f{\tau }_s}\cdot \frac{f_r^2}{f_r^2-f^2+jf\gamma /2\pi }$	$f_s=\frac{1}{2\pi {\tau }_s}$ $\chi =1+\frac{{\tau }_s}{{\tau }_e}$	$M:small-signalresponsefromcurrenttolight$ $\begin{array}{r}{\tau }_s:transporttimeofthecarriersinto\\ thewells(thecapturetime{\tau }_{cap}of\\ Grabowskietal.playsthisrole,my\\ reading)\end{array}$ ${\tau }_e:escapetime$ $f_s:poleofthetransport$ $\begin{array}{r}{f}_{r}and\gamma :asintherowsabove,theform\\ isEq.48ofMichalzikandEbeling,the\\ barrierandwellcarriersareseparate\\ reservoirsinGrabowskietal.,who\\ simulatea7\mu \mathrm{m}850nmdeviceat7.7\mathrm{mA}\\ (20kA/c{\mathrm{m}}^2)with{\tau }_{cap}=7\mathrm{ps}(swept1to15\\ \mathrm{ps}),{\tau }_{esc}ofabout55\mathrm{ps},awell\\ recombinationtimeofabout720\mathrm{ps}anda\\ measuredbandwidthof28\mathrm{GHz}at25{}^{\circ }C\end{array}$ $\begin{array}{r}(computed:\chi =1.02,1.13and1.27for{\tau }_{cap}\\ =1,7and15\mathrm{ps},thelastmatchingthe\\ quotedmaximumofabout1.3;f_{r}fallsby\\ thefactor\sqrt{1.018/1.273}=0.894,10.6per\\ cent,matchingthequoted10percent;the\\ pole1/(2\pi {\tau }_{cap})is159,22.7and10.6\mathrm{GHz}\\ for1,7and15\mathrm{ps},andthesimulated\\ bandwidthfallsby15/40=0.375,adrop\\ muchlargerthanthatof{f}_r)\end{array}$
Parasitic response of a VCSEL	$H_{par}(f)=\frac{1}{1+jf/f_{rc}}$ $f_{rc}=\frac{1}{2\pi {\tau }_{RC}}$	$|H(f){|}^2\propto \frac{1}{1+(f/f_{rc}{)}^2}\cdot \frac{f_r^4}{(f_r^2-f^2{)}^2+(\gamma f/2\pi {)}^2}$ $C_m=C_{mesa}+C_j,C_{mesa}=\frac{C_{ox}{C}_{int}}{C_{ox}+C_{int}}$ $C_{pad}=\frac{{\epsilon }_0{\epsilon }_{r}A}{t}$	$f_{rc}:parasitic3dBfrequency$ ${\tau }_{RC}:timeconstantoftheequivalentcircuit$ $C_p:padcapacitance(tenstohundredsof\mathrm{fF})$ $\begin{array}{r}{R}_m:mirrorresistance(mirror,sheetand\\ contactparts,dominatedbythemirror)\end{array}$ $C_m:mesaplusjunctioncapacitance$ $C_{ox}:capacitanceoftheoxidelayers$ $C_{int}:capacitanceoftheintrinsicregion$ $C_j:junctioncapacitance$ $R_j:junctionresistance$ $\begin{array}{r}{\epsilon }_r,Aandt:relativepermittivity,area\\ andthicknessofthedielectricundera\\ pad(parallel-plateform,background\\ knowledge)\end{array}$ $\begin{array}{r}BCB:benzocyclobutene,thecircuitandthe\\ productformoftheresponsearefromEqs.\\ 7.10and7.11andFigs.7.3and7.4of\\ ChangandColdren,MichalzikandEbeling\\ write{M}_{par}=1/(1+i2\pi \nu {\tau }_{RC}),thetable\\ ofChangandColdrenlists(C_{p}in\mathrm{fF},R_m\\ in\Omega ,C_{m}in\mathrm{fF},R_{j}in\Omega ,f_{rc}in\mathrm{GHz},f_{3dB}in\\ \mathrm{GHz}):Learetal.at850\mathrm{nm}(4by4\mu \mathrm{m})\\ 41.7,28.3,44.3,288.7,36.4,21.5;\\ Suzakietal.at1.1\mu \mathrm{m}56,63.3,133,\\ 71.6,23.6,20;Yashikietal.at1.1\mu \mathrm{m}\\ 33,38,130,134,21.0,24,Ouetal.:the\\ parasitic3dBfrequencyofaChalmers850\\ nmVCSELcanrisefrom22toabove30\mathrm{GHz}\\ withafewmoreoxidelayers\end{array}$ $\begin{array}{r}(computed:a100by100\mu \mathrm{m}padover200\mathrm{nm}\\ ofnitride({\epsilon }_r=7.5,assumed)is3.32\mathrm{pF}\\ andover5\mu \mathrm{m}ofBCB({\epsilon }_r=2.65,assumed)\\ is46.9\mathrm{fF},aratioof70.8,againstthe\\ quoted3.3\mathrm{pF},47\mathrm{fF}andabout70;a\\ circuitwitha50\Omega source(assumed),C_p\\ acrossit,R_{m}inseriesand{C}_{m}inparallel\\ with{R}_j,withthevoltagetakenacross{R}_j,\\ gives3dBfrequenciesof23.6and21.1\\ \mathrm{GHz}fortheSuzakiandYashikisets,equal\\ tothetabulated23.6and21.0,and39.1\\ \mathrm{GHz}forLearetal.against36.4;\\ combining21.5and36.4\mathrm{GHz}inquadrature\\ (assumed)leaves26.6\mathrm{GHz}forthe\\ intrinsicresponse)\end{array}$
Thermal resistance and self-heating of a VCSEL	$R_{th}=\frac{\Delta T}{P_{diss}}\approx \frac{1}{2{\lambda }_c{D}_a}$	$P_{diss}=IV-P_{opt}=IV(1-{\eta }_c)$ $\Delta \lambda =\frac{\partial \lambda }{\partial T}\Delta T$	$R_{th}:thermalresistance$ $\Delta T:temperatureriseoftheactiveregion$ $P_{diss}:dissipatedelectricalpower$ $IandV:drivecurrentandvoltage$ $P_{opt}:opticaloutputpower$ ${\eta }_c:powerconversionefficiency$ $\begin{array}{r}{\lambda }_c:thermalconductivityofthe\\ surroundingsemiconductor\end{array}$ $\begin{array}{r}{D}_a:active(aperture)diameter,the\\ estimateisEq.36ofMichalzikand\\ Ebelingandequalsthespreading\\ resistanceofadischeatsource\\ (backgroundknowledge),\partial \lambda /\partial T:wavelength\\ shiftwithtemperature,0.07nm/Kfor\\ VCSELsat800to1000nm(Michalzikand\\ Ebeling)andforthe980nmcavitymode\\ (ChangandColdren,whosegainpeakmoves\\ 0.3nm/K),ChangandColdren:the\\ dissipationscalesaboutwiththearea,\\ thethermalimpedancevariesinversely\\ withthediameter,andcopperplatingor\\ substratesofhigherconductivitylower\\ therise(copperisabout9timesGaAs),\\ MichalzikandEbeling:rolloveratroom\\ temperatureoccursatatemperaturerise\\ ofabout90\mathrm{K},andlaserturn-offnear200\\ {}^{\circ }Calmostindependentoftheheatsink\end{array}$ $\begin{array}{r}(computedwith{\lambda }_c=46W/(\mathrm{m}\mathrm{K})forGaAs\\ (assumed,consistentwithcopperat400\\ W/(\mathrm{m}\mathrm{K})being9timeslarger):R_{th}=3.62,\\ 2.17and1.55K/mWfor{D}_a=3,5and7\mu \mathrm{m};\\ the980nmdeviceofChangandColdren,\\ 1.3\mathrm{mW}dissipatedat1\mathrm{mA}forariseof\\ 4.3\mathrm{K},has3.3K/mWat3\mu \mathrm{m},9percent\\ belowtheestimate;ariseof4.3\mathrm{K}moves\\ themodeby0.30\mathrm{nm})\end{array}$
Self-heating and the bandwidth ceiling of a VCSEL	$T=T_0+R_{th}{P}_{diss}$ $f_r(T)=D(T)\sqrt{I-I_{th}(T)}$	$\frac{d{P}_{opt}}{dI}=0(\mathrm{thermal}\mathrm{rollover})$	$T_0:heat-sinktemperature$ $T:temperatureoftheactiveregion$ $\begin{array}{r}{I}_{th}(T):thresholdcurrent,whichisleast\\ nearthetemperaturewherethegainpeak\\ linesupwiththemode(seethe\\ temperature-tuningrowabove),D(T):\\ D-factor,whichfallsasthedifferential\\ gainfallswithtemperature(seethe\\ DirectlyModulatedLaserpage),the\\ rolloveristhecurrentatwhichheating\\ makestheoutputpowerstoprising,Xieet\\ al.:thebandwidthofa10Gbit/sdesign\\ saturatesat10to11\mathrm{GHz}athigh\\ temperaturewhiletheorypredicts22to27\\ \mathrm{GHz},andthermalrolloverlimitsitto\\ lessthanhalf,ChangandColdren:a6\mu \mathrm{m}\\ quantum-dotVCSELlosesMCEFfrom5.6to\\ 4.6GHz/m{\mathrm{A}}^{1/2}andmaximumbandwidthfrom15\\ to13\mathrm{GHz}between25and85{}^{\circ }C,Michalzik\\ andEbeling:thepracticallimitsare\\ self-heatingandtheonsetofmultimode\\ operation\end{array}$ $\begin{array}{r}(computed:thequantum-dotMCEFfallsto\\ 0.82andthemaximumbandwidthto0.87of\\ the25{}^{\circ }Cvalues;10/27to11/22is0.37\\ to0.50,sotheceilingisatorjust\\ belowhalfofthetheoreticalvalue)\end{array}$
Bandwidth needed for a bit rate and the 850 nm record	$f_{3dB}\approx {\kappa }_{B}B$	${\kappa }_B\approx 0.64\mathrm{to}0.72$ $B\approx \frac{f_{3dB}}{{\kappa }_B}$	$f_{3dB}:small-signalbandwidth$ $B:bitrateofnon-return-to-zerosignalling$ $\begin{array}{r}{\kappa }_B:ratioofthebandwidthtothebit\\ rate,takenfromthestatementofXieet\\ al.thata25Gbit/sVCSELneedsatleast\\ 16to18\mathrm{GHz},reportedresultsfor850nm\\ VCSELs:asmall-signalbandwidthof28\mathrm{GHz}\\ foranapertureofabout4\mu \mathrm{m}(Haglundet\\ al.),error-freetransmissionat47Gbit/s\\ backtobackand44Gbit/sover50mof\\ OM4fibrefora7\mu \mathrm{m}aperturewitha30\\ \mathrm{GHz}receiver,and57Gbit/sbacktoback,\\ 55Gbit/sover50mand43Gbit/sover100\\ mforan8\mu \mathrm{m}aperturewitha22\mathrm{GHz}\\ receiver(Haglundetal.),40Gbit/supto\\ 85{}^{\circ }C(Haglundetal.),and56Gbit/swith\\ bothon-offkeyingandfour-level\\ pulse-amplitudemodulationfora7\mu \mathrm{m}\\ device(Grabowskietal.)\end{array}$ $\begin{array}{r}(computed:16/25and18/25give0.64and\\ 0.72;47Gbit/sovera28\mathrm{GHz}bandwidthis\\ 1.68bit/sper\mathrm{Hz},and56Gbit/sover28\\ \mathrm{GHz}is2.0,whichneedsalargerratio\\ than{\kappa }_B=0.64to0.72allows(1.39to\\ 1.56bit/sper\mathrm{Hz}),sotheseresultsrely\\ onequalisationoronareceiver-limited\\ bandwidth(myreading);the28\mathrm{GHz}may\\ alsobelongtoadifferentdevicefromthe\\ 47to57Gbit/sones)\end{array}$
Current density in the aperture	$J=\frac{4I}{\pi D_a^2}$	$D_a=2\sqrt{\frac{I}{\pi J}}$	$J:currentdensity$ $I:biascurrent$ $\begin{array}{r}{D}_a:oxide-aperture(active)diameter,\\ valuesfromthesources:7\mu \mathrm{m}and7.7\mathrm{mA},\\ 20kA/c{\mathrm{m}}^2(Grabowskietal.),about6\mu \mathrm{m}\\ and9\mathrm{mA},about10kA/c{\mathrm{m}}^2(Changand\\ Coldren,fora39Gbit/s850nmdevice),3\\ \mu \mathrm{m}and4.4\mathrm{mA}(35Gbit/sat980nm,Chang\\ andColdren),3.5\mu \mathrm{m}and4.1\mathrm{mA}\\ (single-modedeviceofMichalzikand\\ Ebeling)\end{array}$ $\begin{array}{r}(computed:7.7\mathrm{mA}in7\mu \mathrm{m}is20.0kA/c{\mathrm{m}}^2,\\ asquoted;9\mathrm{mA}in6\mu \mathrm{m}is31.8kA/c{\mathrm{m}}^2,\\ and10kA/c{\mathrm{m}}^{2}at9\mathrm{mA}needs10.7\mu \mathrm{m};4.4\\ \mathrm{mA}in3\mu \mathrm{m}is62kA/c{\mathrm{m}}^2;4.1\mathrm{mA}in3.5\mu \mathrm{m}\\ is42.6kA/c{\mathrm{m}}^2)\end{array}$

Important Questions

Ques: Why must the mirrors of a VCSEL have a reflectance above 99%, when the cleaved facets of an edge-emitting laser reflect only about 30%?

In a VCSEL the light passes through the gain layers only once per pass, over a distance of a few tens of nanometres, instead of through hundreds of micrometres. The one-pass gain is therefore small, and the one-pass threshold condition $\Gamma g_{th}{L}_{DBR}=A_i+T_m$ can be met only if the mirror loss $T_m=\ln (1/R)$ is small too, so that the mean reflectance $R$ must be above about 99%. The same condition sets the efficiency: ${\eta }_d=F{\eta }_i{T}_m/(A_i+T_m)$ stays close to ${\eta }_i$ only if the internal loss of the whole effective cavity, including the mirrors, $A_i$, is kept below $T_m$. Design curves of the reference show that an AlGaAs/GaAs stack reaches $R>0.999$ with fewer than about two dozen periods. Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 3.6.4 (Eqs. 3.70 and 3.71, Fig. 3.18).

Ques: Why are the quantum wells of a VCSEL placed at a peak of the standing wave?

The mirrors make the field in the cavity a standing wave, with ${\left|E\right|}^2\propto {\cos }^2\beta z$, and the gain layers interact only with the field where they are. A thin active layer at a peak sees about twice the average intensity, so the axial confinement factor is ${\Gamma }_{fill}{\Gamma }_{enh}$ with ${\Gamma }_{enh}=1+\sin (\beta L_a)/\beta L_a\to 2$, and it falls if the layer is shifted by $z_s$ from the peak, by the factor $\cos (2\beta z_s)$. The reference states that this enhancement, of up to 2, is what lets most commercial VCSELs reach threshold. For the same reason the cavity is made one wavelength long, with the wells at its centre, and the doping and the grading of the mirror layers are arranged to keep the lossy regions at the nodes of the field. Reference: Coldren et al., 2nd ed., Section 3.6.4 (Eq. 3.74, Figs. 3.19 to 3.21) and Appendix 5 (Eq. A5.14, Fig. A5.2).

Ques: Why does a VCSEL lase in a single axial mode but often in several lateral modes?

The axial mode spacing is ${\lambda }^2/(2{\bar{n}}_g{L}_{DBR})$, and with a cavity of about one wavelength it is much larger than in an in-plane laser, and the mirror band is narrow, so only one axial mode falls within the bandwidth of the gain and of the mirrors and the short cavity does most of the selection. The lateral modes are set by the aperture: proton-implanted devices of more than 10 $\mu$m diameter, and the oxide-aperture devices of about 20 $\mu$m used for multimode-fibre datacom links, have many lateral modes, which suits the large core of the fibre. A smaller oxide aperture, which also gives a very low threshold current, supports a single lateral mode. The aperture confines the light by a lensing action, since it is a low-index insulating layer close to the active region, and it also confines the current. Reference: Coldren et al., 2nd ed., Section 3.6.3 (Fig. 3.17), Section 1.7 (Fig. 1.18) and Section 1.8 (Fig. 1.20).

Ques: Why is the threshold current of a VCSEL designed to stay flat or fall as the temperature rises?

The mode wavelength follows the refractive index and tunes at about 0.1 nm/${}^{\circ }$C, while the gain peak follows the bandgap and tunes at about 0.5 nm/${}^{\circ }$C, so the gain peak moves past the mode at about 0.4 nm/${}^{\circ }$C. The wells are therefore designed so that the gain peak is at a shorter wavelength than the mode at room temperature, and as the device heats it moves into alignment with the mode and then past it. As the peak approaches the mode the gain at the mode wavelength rises, which offsets the usual loss of gain with temperature, so the device has a wide region of relative insensitivity to temperature and the threshold current can even decrease with temperature. Reference: Coldren et al., 2nd ed., Section 1.8 (Fig. 1.20b and 1.20d and the text of the discussion).

Ques: Why are long-wavelength InP-based VCSELs harder to make than GaAs-based ones?

The reference lists three reasons. First, there are no well-developed lattice-matched mirrors with a high index contrast and a high thermal conductivity: the InGaAsP/InP pairs have a small index step, so many more periods are needed for the same reflectance (Fig. 3.18), and the available mirrors do not combine a high index contrast with a high thermal conductivity. Second, there is no simple oxidation technology for the aperture, as AlAs gives on GaAs. Third, the active materials have a lower gain per unit current and are more sensitive to temperature. Solutions explored are an etched tunnel-junction mesa for the current and the optical confinement together with a dielectric top mirror, wafer bonding of GaAs-based mirrors to an InP active region, a metamorphic GaAs/AlAs top mirror and lattice-matched AlGaAsSb mirrors, but the reference says that after some development none of the last three alternatives has been manufactured in any volume. GaInNAs and quantum-dot wells on GaAs are the alternative route to 1300 nm. Reference: Coldren et al., 2nd ed., Section 1.8 (Fig. 1.20c to 1.20e and the text) and Section 3.6.4 (Fig. 3.18).

Ques: Is the damping limit of the VCSEL bandwidth $2\sqrt{2}\pi /K$, $\sqrt{2}/(2\pi K)$ or $(\sqrt{2}/2\pi )K$?

The summaries print the limit as $\sqrt{2}/(2\pi K)$ (Michalzik and Ebeling, Eq. 55) and as $(\sqrt{2}/2\pi )K$ (Chang and Coldren), and both are damaged. The second has the unit of time, not of frequency. The first gives 1.5 $\mathrm{GHz}$ for $K$ = 0.15 $\mathrm{ns}$ (computed), which contradicts their reported bandwidths above 20 $\mathrm{GHz}$. I derive the limit from the condition $\gamma /{\omega }_r=\sqrt{2}$ stated by Chang and Coldren: with $\gamma =K{f}_r^2$ it gives $f_r=2\sqrt{2}\pi /K$, so 8.89/$K$ (computed). This gives 59.2 $\mathrm{GHz}$ for 0.15 $\mathrm{ns}$, and 58 $\mathrm{GHz}$ for 0.153 $\mathrm{ns}$, the damping-limited value quoted for the device of Lear et al. (computed). It is also the form on the Directly Modulated Laser page. A numerical maximisation of the 3 dB frequency of the two-pole response over $f_r$, with $\gamma =K{f}_r^2$, gives the same 59.24 $\mathrm{GHz}$ for $K$ = 0.15 $\mathrm{ns}$, at the point where the 3 dB frequency equals $f_r$, so $2\sqrt{2}\pi /K$ is correct and the two printed forms are extraction errors. Reference: Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context); Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled).

Ques: Is a 6 micrometre VCSEL at 9 milliamps at 10 or at 32 kiloamps per square centimetre?

Chang and Coldren list a 39 $Gbit/s$ 850 nm device of about 6 $\mu \mathrm{m}$ at a bias of 9 $\mathrm{mA}$ and give ‘about 10’ $kA/c{\mathrm{m}}^2$. The current density is $4I/(\pi D_a^2)$: 9 $\mathrm{mA}$ in 6 $\mu \mathrm{m}$ is 31.8 $kA/c{\mathrm{m}}^2$, and 10 $kA/c{\mathrm{m}}^2$ would need an aperture of 10.7 $\mu \mathrm{m}$ (computed). The other devices in the sources are 20.0 $kA/c{\mathrm{m}}^2$ for 7.7 $\mathrm{mA}$ in 7 $\mu \mathrm{m}$ (matching the 20 quoted by Grabowski et al.), 62 $kA/c{\mathrm{m}}^2$ for 4.4 $\mathrm{mA}$ in 3 $\mu \mathrm{m}$ and 42.6 for 4.1 $\mathrm{mA}$ in 3.5 $\mu \mathrm{m}$ (computed), so 32 is within the range of small high-speed devices, while 10 is on the low side. The summary is garbled in places, and I cannot tell whether the size or the density is wrong; I regard the density as the less certain figure (my reading). Reference: Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context); Grabowski, Gustavsson and Larsson, ‘Impact of Carrier Transport and Capture on VCSEL Dynamics’, IEEE Journal of Quantum Electronics 59(1):1-6 (2023), DOI 10.1109/JQE.2023.3236396 (fetch-tool summary; the equations were extracted imperfectly); Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled).

Ques: Does the VCSEL cavity mode move 0.1 or 0.07 nanometres per kelvin?

The temperature-tuning row of this page, from Coldren et al., gives about 0.1 $nm/K$ for the mode and 0.5 for the gain peak. Michalzik and Ebeling give 0.07 $nm/K$ for VCSELs at 800 to 1000 nm, and Chang and Coldren give 0.07 for the mode and 0.3 for the gain peak of a 980 nm device. A check from the refractive index: $\Delta \lambda /\Delta T=\lambda (dn/dT)/n$, with $dn/dT$ = $2.7\times 10^{-4}$ per $\mathrm{K}$ and $n$ = 3.5 for AlGaAs (assumed, background knowledge), gives 0.066 $nm/K$ at 850 $\mathrm{nm}$ and 0.076 at 980 $\mathrm{nm}$ (computed). Two independent sources and this estimate favour 0.07, and I regard 0.1 as a rounded value; the offset between the gain peak and the mode (0.23 to 0.4 $nm/K$ in the two sources that give both rates, computed) is the quantity that matters for the threshold. I have not changed the earlier row. Reference: Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled); Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context); Coldren, Corzine and Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 1.8 (the rates of the temperature-tuning row of this page).

Ques: Does the estimate $R_{th}=1/(2{\lambda }_c{D}_a)$ reproduce the measured thermal resistance?

Michalzik and Ebeling give $R_{th}\approx 1/(2{\lambda }_c{D}_a)$, the spreading resistance of a disc heat source. With ${\lambda }_c$ = 46 $W/(\mathrm{m}\mathrm{K})$ for GaAs (assumed; the 9-to-1 ratio to copper in Chang and Coldren is consistent with this) it gives 3.62, 2.17 and 1.55 $K/mW$ for apertures of 3, 5 and 7 $\mu \mathrm{m}$ (computed). Chang and Coldren report that a 3 $\mu \mathrm{m}$ 980 nm device dissipating 1.3 $\mathrm{mW}$ at 1 $\mathrm{mA}$ rises 4.3 $\mathrm{K}$, so 3.3 $K/mW$ (computed), 9 per cent below the estimate. The agreement is as good as the assumed conductivity, since a mirror stack conducts less than bulk GaAs and a metal-plated mesa conducts more. The rise at rollover, about 90 $\mathrm{K}$ (Michalzik and Ebeling), then fixes the dissipation that a given aperture can take. I regard the estimate as good to about 10 per cent for apertures of a few micrometres (my reading). Reference: Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled); Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context).

Ques: Why does a longer capture time cut the VCSEL bandwidth by 62 per cent when it lowers the resonance frequency by only 10?

Grabowski et al. simulate an 850 nm device. A capture time ${\tau }_{cap}$ rising from 1 to 15 $\mathrm{ps}$ lowers $f_r$ by about 10 per cent, since $\chi$ goes from 1.02 to 1.27 and $f_r\propto 1/\sqrt{\chi }$ (0.894, computed), but it lowers the $3\mathrm{dB}$ bandwidth from 40 to 15 $\mathrm{GHz}$, a ratio of 0.375 (computed). The summary does not give the cause. A transport pole at $1/(2\pi {\tau }_{cap})$ is 159 $\mathrm{GHz}$ at 1 $\mathrm{ps}$ and 10.6 $\mathrm{GHz}$ at 15 $\mathrm{ps}$ (computed), and Michalzik and Ebeling include such a pole (Eq. 48), so the pole, which acts on the whole response and not only on $f_r$, is the likely cause (my reading). At the nominal 7 $\mathrm{ps}$ it is 22.7 $\mathrm{GHz}$, below the 28 $\mathrm{GHz}$ measured for the device (computed), so a single pole is not the whole model. Reference: Grabowski, Gustavsson and Larsson, ‘Impact of Carrier Transport and Capture on VCSEL Dynamics’, IEEE Journal of Quantum Electronics 59(1):1-6 (2023), DOI 10.1109/JQE.2023.3236396 (fetch-tool summary; the equations were extracted imperfectly); Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled).

Ques: How much faster is the relaxation resonance of a VCSEL than that of an edge-emitting directly modulated laser at the same current?

The Directly Modulated Laser page computes $D$ = 1.26 $GHz/m{\mathrm{A}}^{1/2}$ for its reference DFB laser (10.6 $\mathrm{GHz}$ at 75 $\mathrm{mA}$). The VCSELs in the sources reach 14.2 (Lear et al.), 16.2 and 16.8 $GHz/m{\mathrm{A}}^{1/2}$ (Chang and Coldren), a factor of 11 to 13 in $D$ (computed), though two of these values are MCEF and not $D$, which differ by the factor 1.55 (so the factor in $D$ is 7 to 9, computed). Since $D^2\propto {\eta }_i\Gamma a/V$ with the active volume $V$, the factor of 53 to 74 in $D^2$ is the ratio of confinement factor over volume, which I do not compute because the confinement of the DFB page was not at hand (my reading). At 3 $\mathrm{mA}$ above threshold the VCSEL would have a resonance of 16 to 19 $\mathrm{GHz}$ against 2.2 $\mathrm{GHz}$ (computed). Reference: Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context); Michalzik and Ebeling, ‘Operating Principles of VCSELs’, book chapter, University of Ulm (year not stated, the references run to about 2000) (fetch-tool summary; some equations were garbled); Directly Modulated Laser page of this vault.

Ques: Which limit sets the 21.5 gigahertz bandwidth of the 850 nanometre VCSEL of Lear et al.?

Chang and Coldren list for this device a damping-limited bandwidth of about 58 $\mathrm{GHz}$, a parasitic frequency $f_{rc}$ of 36.4 $\mathrm{GHz}$ and a measured $f_{3dB}$ of 21.5 $\mathrm{GHz}$. Neither number alone gives 21.5. Combining the measured value with the parasitic pole in quadrature (assumed) leaves 26.6 $\mathrm{GHz}$ for the rest, and with $f_{3dB}\approx 1.55f_r$ the resonance would be 17 $\mathrm{GHz}$ (computed). So the parasitics, the resonance and the heating each take a share, and heating is not given in the summary. The same quadrature rule is not exact, since the damping also reshapes the peak. A threshold-and-D estimate of $f_r$ would need the bias current, which the summary does not give, so I cannot attribute the remainder (my reading). Reference: Chang and Coldren, ‘Design and Performance of High-Speed VCSELs’, Chapter 7 in R. Michalzik (ed.), VCSELs, Springer Series in Optical Sciences 166, pp. 233-262 (2013), DOI 10.1007/978-3-642-24986-0_7 (fetch-tool summary; the page text was garbled in places and the tool reconstructed equations from context).

Ques: Is the 28 gigahertz 850 nanometre VCSEL a 4 or a 7 micrometre device?

The Chalmers poster credits a small-signal bandwidth of 28 $\mathrm{GHz}$ to a VCSEL with an oxide aperture of about 4 $\mu \mathrm{m}$, and reports error-free transmission for devices of about 7 and 8 $\mu \mathrm{m}$. Grabowski et al. quote a flat response with a $3\mathrm{dB}$ bandwidth of 28 $\mathrm{GHz}$ at 25 ${}^{\circ }\mathrm{C}$ for a 7 $\mu \mathrm{m}$ primary aperture, from their reference 10, at 7.7 $\mathrm{mA}$. Both are 28 $\mathrm{GHz}$, but the apertures differ by 3 $\mu \mathrm{m}$, which changes the current density from 61 to 20 $kA/c{\mathrm{m}}^2$ at the same current (computed, assuming the same 7.7 $\mathrm{mA}$). They may be different devices of one design family, as both come from Chalmers; the summaries do not show the original publications (my reading). I regard the poster’s 4 $\mu \mathrm{m}$ as belonging to the highest-speed small device and the 7 $\mu \mathrm{m}$ as the transmission device, but this is not checked. Reference: Haglund et al. (Chalmers), ‘High-Speed 850-nm VCSELs Operating Error-Free Beyond 50 Gbit/s’, conference poster, Optics and Photonics in Sweden (2013), Chalmers publication 186542 (abstract page only, the poster was not read); Grabowski, Gustavsson and Larsson, ‘Impact of Carrier Transport and Capture on VCSEL Dynamics’, IEEE Journal of Quantum Electronics 59(1):1-6 (2023), DOI 10.1109/JQE.2023.3236396 (fetch-tool summary; the equations were extracted imperfectly).

Related pages in this vault: Distributed Bragg Reflector Laser (DBR), Distributed Feedback Laser (DFB), Directly Modulated Laser, Vertical-External-Cavity Surface-Emitting Laser (VECSEL), Double-Heterostructure Fabry-Perot Laser, Semiconductor Lasers (Edge-Emitting and VCSEL), Silicon-Compatible Lasers (Hybrid III-V, Epitaxial Quantum-Dot and Ge-GeSn), Resonant-Cavity-Enhanced Photodetector (RCE), Resonant Cavity LED (RCLED).