Directly Modulated Laser

1. Theory

• Definition: A directly modulated laser (DML) carries the data by changing its injection current. The laser is biased above threshold, a data current is added, and the output power follows the current, so no separate modulator is needed. Distributed-feedback (DFB), distributed-reflector, coupled-cavity and photonic-crystal lasers are all used this way (Lumentum; ETRI; NUSOD abstract; NTT).
• Speed limit: Carriers and photons exchange energy in a resonance, the relaxation oscillation, at $f_R=D\sqrt{I-I_{th}}$. The response is flat up to about $f_R$ and falls above it, so $f_R$ and the damping set the bandwidth (KIT; Hantschmann et al.).
• Damping: The damping rate is $\gamma =K{f}_R^2+{\gamma }_0$, so the bandwidth cannot exceed $2\sqrt{2}\pi /K$ however hard the laser is driven (Hantschmann et al.). From the arXiv table, with the gain compression corrected to $2\times 10^{-23}$ ${\mathrm{m}}^3$, I compute $K=0.418$ $\mathrm{ns}$ and a limit of 21.3 $\mathrm{GHz}$.
• Beyond the limit: An optical resonance between the main mode and a side mode, the photon-photon resonance, can lift the response far above the carrier limit: 46 $\mathrm{GHz}$ for a DFB+R laser (ETRI) and 110.6 $\mathrm{GHz}$ for a coupled-cavity DFB laser (NUSOD abstract).
• Chirp: The same carrier change that modulates the power also shifts the index, so the frequency moves with the power: $\Delta \nu =(\alpha /4\pi )[(1/P)dP/dt+\kappa P]$. The chirp broadens the spectrum and limits the fibre reach through dispersion (arXiv; RF Essentials).
• Figures of merit: $f_R$, the D-factor, the K-factor, the $3\mathrm{dB}$ bandwidth, the linewidth enhancement factor $\alpha$, the extinction ratio and TDECQ, and the energy per bit.
• Scope: The threshold, gain and cavity physics of these lasers are on the DFB, quantum-well and quantum-dot laser pages; the vertical-cavity laser has its own page. This page covers what changes when the current is modulated.

2. Working Principle

• Bias and drive: The bias sets the photon density $S_0$ and so $f_R$. The data current moves the output between a low and a high level, with $ER=(I_1-I_{th})/(I_0-I_{th})$. PhotonForge uses 45 and 70 $\mathrm{mA}$ above a 35.9 $\mathrm{mA}$ threshold.
• Carrier-photon dynamics: A current step raises the carrier density, the gain overshoots, the photon density builds up and depletes the carriers again, so the output rings at $f_R$ before it settles. The small-signal version of this is the second-order response $H(f)$.
• Damping: Gain compression ($ϵ$) and the finite carrier lifetime damp the ringing. Heavy damping (large $K$) flattens the response but lowers the bandwidth reached at a given current.
• Parasitics: The contact resistance and pad capacitance add a pole $f_p=1/(2\pi R{C}_p)$, and Finisar quotes an RC cut-off of about 22 $\mathrm{GHz}$ for its device.
• Photon-photon resonance: A side mode (or a mode of a coupled cavity) at a frequency $c\Delta \lambda /{\lambda }^2$ from the main mode makes a second peak. If it falls where the carrier response is rolling off, the bandwidth is extended (NUSOD; ETRI; Lumentum).
• Chirp and fibre: The edges carry a transient chirp and the levels an adiabatic chirp; with a positive $\alpha$ and fibre dispersion the pulses broaden or compress depending on the sign of $D$.
• Equalisation: For PAM4 at 53 $\mathrm{GBd}$ the Nyquist frequency is 26.6 $\mathrm{GHz}$, above a 20.1 $\mathrm{GHz}$ laser bandwidth, so the signal is pre- or post-equalised (computed $4.4\mathrm{dB}$ for a single pole).

3. Applications

• Data-centre transmitters: Uncooled 50 $Gbit/s$ PAM4 DMLs on four CWDM wavelengths reached 10 $\mathrm{km}$ (AOI, 2017), and DFB lasers 150 $\mu \mathrm{m}$ long reached 106 $Gbit/s$ PAM4 (53 $\mathrm{GBd}$) over 500 m and 2 $\mathrm{km}$ (Lumentum, 2022). A DML is simpler and smaller than an electro-absorption modulated laser (qualitative statement of Lumentum).
• Access networks: A 10 $Gbit/s$ DML over 20 $\mathrm{km}$ and a chirp-managed 25 $Gbit/s$ laser over 40 $\mathrm{km}$ (Finisar, TWDM PON).
• Long-reach 10 Gbit/s: A chirp-managed 1550 nm laser with an optical filter reached beyond 100 $\mathrm{km}$ (Azna article).
• On-chip optical links: Photonic-crystal lasers need only 4.4 to 14 $\mathrm{fJ}$ per bit (NTT; Matsuo), for short links on silicon.
• Quantum-dot lasers on silicon: A 1.3 $\mu \mathrm{m}$ laser on silicon had a small-signal bandwidth of 1.6 $\mathrm{GHz}$ because of its large K-factor (Hantschmann et al.).

Rate Equations and Static Behaviour

Name	Formula	Related Formulae	Variables/Constants
Rate equations with gain compression	$\frac{dS}{dt}=\frac{\Gamma g_0\left(N-N_0\right)S}{1+ϵS}-\frac{S}{{\tau }_p}+\frac{\Gamma \beta N}{{\tau }_n}$ $\frac{dN}{dt}=\frac{I}{qV}-\frac{N}{{\tau }_n}-\frac{g_0\left(N-N_0\right)S}{1+ϵS}$	$\frac{d\phi }{dt}=\frac{\alpha }{2}\left[\Gamma g_0\left(N-N_0\right)-\frac{1}{{\tau }_p}\right]$	$S:photondensity$ $N:carrierdensity$ $I:injectioncurrent$ $q:elementarycharge$ $V:activevolume$ ${\tau }_n:carrierlifetime$ ${\tau }_p:photonlifetime$ $\Gamma :confinementfactor$ $\begin{array}{r}{g}_0:gaincoefficientperunitcarrier\\ density(in{\mathrm{m}}^3/s)\end{array}$ $N_0:transparencycarrierdensity$ $ϵ:gaincompressionfactor(in{\mathrm{m}}^3)$ $\begin{array}{r}\beta :fractionofspontaneousemissionthat\\ entersthemode\end{array}$ $\alpha :linewidthenhancementfactor$ $\begin{array}{r}\phi :opticalphase,thethreeequationsare\\ Eqs.1to3ofthearXivpaperandhave\\ thesamestructureinthePhotonForge\\ guide,whichalsowritesthegainas{c}_0\\ a_0/(n_g(1+ϵp))(thearXivpaperusesa\\ logarithmicgaininitsEq.5witha\\ saturationdensitywhosevalueitdoesnot\\ give,soIusethelinearformforthe\\ analyticresults),arXivTable1:\Gamma =\\ 0.24,{\tau }_p=2.60\mathrm{ps},{\tau }_n=3.17\mathrm{ns},N_0=\\ 2.00\times 10^{24}{\mathrm{m}}^{-3},activecrosssection{\sigma }_g=\\ 3.34\times 10^{-20}{\mathrm{m}}^2,n_g=4,V=3.60\times 10^{-17}{\mathrm{m}}^3(36\\ \mu {\mathrm{m}}^3),\beta =1\times 10^{-3},{\eta }_0=0.2,ϵprintedas\\ 2.00\times 10^{23}{\mathrm{m}}^3(seethequestions),I_{th}=4.06\\ \mathrm{mA}\end{array}$ $\begin{array}{r}(computedwith{g}_0=v_g{\sigma }_g=2.50\times 10^{-12}\\ {\mathrm{m}}^3/s,where{v}_g=c/n_g=7.49\times 10^{7}m/sismy\\ assumptionforthegaincoefficient:N_{th}=\\ N_0+1/(\Gamma g_0{\tau }_p)=2.64\times 10^{24}{\mathrm{m}}^{-3}and{I}_{th}=qV\\ N_{th}/{\tau }_n=4.80\mathrm{mA},18percentabovethe\\ 4.06\mathrm{mA}thatthepaperreports)\end{array}$
Output power, photon density and slope	$P=\frac{SV{\eta }_{0}h\nu }{2\Gamma {\tau }_p}$	$P={\eta }_d\frac{h\nu }{q}\left(I-I_{th}\right)$ $SE=\frac{dP}{dI}={\eta }_d\frac{h\nu }{q}$	$P:opticaloutputpower$ $\begin{array}{r}{\eta }_0,{\eta }_d:differentialquantumefficiency\\ (0.20inthearXivtable)\end{array}$ $h\nu :photonenergy$ $\begin{array}{r}SE:slopeefficiency,thefirstformis\\ Eq.4ofthearXivpaper(thefactor2is\\ asprintedandisnotexplainedinthe\\ summary)andthelinearL-Iformis\\ backgroundknowledge,PhotonForge:\\ threshold35.9\mathrm{mA}andslopeefficiency\\ 0.159W/Afromafitabove45\mathrm{mA}\end{array}$ $\begin{array}{r}(computedat75\mathrm{mA}for\lambda =1500\mathrm{nm}(h\nu =\\ 0.827\mathrm{eV},thewavelengthofthepape{r}^{\prime }s\\ illustrativeeye):P=11.7\mathrm{mW},S_0=\\ 1.53\times 10^{22}{\mathrm{m}}^{-3}andϵS_0=0.31ifϵ=2\times 10^{-23}\\ {\mathrm{m}}^3(mycorrectionoftheprintedvalue))\end{array}$
Turn-on delay and the bias point	$t_d={\tau }_n\ln \left[\frac{I_p}{I_p+I_b-I_{th}}\right]\left(I_b<I_{th}\right)$	$I_{th}=\frac{qV{N}_{th}}{{\tau }_n},N_{th}=N_0+\frac{1}{\Gamma g_0{\tau }_p}$	$\begin{array}{r}{t}_d:delaybetweenthecurrentstepandthe\\ firstlight\end{array}$ $I_p:heightofthecurrentstep$ $I_b:biascurrent$ $I_{th}:thresholdcurrent$ $\begin{array}{r}{N}_{th}:thresholdcarrierdensity(spontaneous\\ emissionneglected),thestep-response\\ formulaisstandard(backgroundknowledge)\\ andisnotinthesourcesIread,\\ PhotonForgesimulatesa2.5Gbit/sNRZ\\ streambetween45\mathrm{mA}(zeros)and70\mathrm{mA}\\ (ones),bothaboveits35.9\mathrm{mA}threshold\end{array}$ $\begin{array}{r}(computedwith{\tau }_n=3.17\mathrm{ns}:astepfrom\\ zerobiasto2I_{th}gives{t}_d={\tau }_n\ln 2=2.20\\ \mathrm{ns},whichis55symbolsat25\mathrm{GBd}(40\mathrm{ps}\\ each),soadirectlymodulatedlaseris\\ keptabovethreshold)\end{array}$
Output levels, extinction ratio and modulation amplitude	$ER=\frac{P_1}{P_0}=\frac{I_1-I_{th}}{I_0-I_{th}}$	$OMA=P_1-P_0=SE\left(I_1-I_0\right)$ $I_{avg}=\frac{I_0+I_1}{2}$	$P_1,P_0:outputpowersoftheoneandzerolevels$ $\begin{array}{r}{I}_1,I_0:thecorrespondingcurrents(both\\ abovethreshold,linearL-Icurveassumed,\\ backgroundknowledge)\end{array}$ $\begin{array}{r}OMA:opticalmodulationamplitude,\\ Lumentum:3.0dBextinctionratio,upto\\ 35\mathrm{mA}peak-to-peak,threshold13.7to16.3\\ \mathrm{mA}at80degreesCelsius,20.1\mathrm{GHz}\\ bandwidthat70\mathrm{mA}and85degreesCelsius\end{array}$ $\begin{array}{r}AOI:extinctionratioupto6dBat50\\ Gbit/sPAM4\end{array}$ $\begin{array}{r}(computedfromthePhotonForgenumbers:P_0\\ =1.45\mathrm{mW},P_1=5.42\mathrm{mW},ER=3.75(5.7\\ dB)andOMA=3.97\mathrm{mW};computedfor\\ Lumentum:aratioof2witha35\mathrm{mA}swing\\ and{I}_{th}=13.7to16.3\mathrm{mA}putsthezero\\ levelat48.7to51.3\mathrm{mA}andthemeanat\\ 66.2to68.8\mathrm{mA},closetothe70\mathrm{mA}at\\ whichthebandwidthisquoted,whichisa\\ consistencycheckandnotareportedfact)\end{array}$

Small-Signal Dynamics

Name	Formula	Related Formulae	Variables/Constants
Relaxation oscillation frequency and D-factor	$f_R=D\sqrt{I-I_{th}}$ $D=\frac{1}{2\pi }\sqrt{\frac{{\eta }_i\Gamma v_{g}a}{qV}}$	${\omega }_R^2=\frac{\Gamma g_0{S}_0}{{\tau }_p}$ $\mathrm{MCEF}=\frac{f_{3dB}}{\sqrt{I-I_{th}}}$	$f_R:relaxationoscillationfrequency$ $D:D-factor(inGHz/m{\mathrm{A}}^{1/2})$ $\begin{array}{r}{\eta }_i:injectionefficiency,a=\partial g/\partial N:\\ differentialgain\end{array}$ $v_g:groupvelocity$ $S_0:steady-statephotondensity$ $MCEF:modulationcurrentefficiencyfactor$ $\begin{array}{r}KIT:{\omega }_r^2=G_d{N}_{P0}/({\tau }_{P}V)inphoton-number\\ form,whichisthesamelaw,Hantschmann\\ etal.:Eq.1intheDformwiththe\\ quantum-dotvolume,measuredDabout0.3\\ GHz/m{\mathrm{A}}^{1/2}inoneversionofthepaper(the\\ otherversiondoesnotgiveD)andan\\ efficiencyof0.4GHz/m{\mathrm{A}}^{1/2}atlowcurrent\\ fora2.5mmquantum-dotlaseronsilicon\end{array}$ $\begin{array}{r}(computed:thearXivvalue{f}_R=10.6\mathrm{GHz}\\ at75\mathrm{mA}isD=1.26GHz/m{\mathrm{A}}^{1/2};myestimate\\ fromitstableis9.5\mathrm{GHz}withoutgain\\ compressionand8.3\mathrm{GHz}withϵ=2\times 10^{-23}\\ {\mathrm{m}}^3;thequantum-dotlaserhas{f}_R=1.3\mathrm{GHz}\\ at20\mathrm{mA}abovethreshold,aDfourtimes\\ smaller)\end{array}$
Damping, K-factor and the bandwidth limit	$\gamma =K{f}_R^2+{\gamma }_0$ $K=4{\pi }^2\left({\tau }_p+\frac{ϵ}{v_{g}a}\right)$	$f_{3dB,max}=\frac{2\sqrt{2}\pi }{K}$ ${\gamma }_0\approx \frac{1}{{\tau }_{eff}}$	$\gamma :dampingrate(in{\mathrm{s}}^{-1})$ $K:K-factor(in\mathrm{ns})$ $\begin{array}{r}{\gamma }_0:dampingoffsetsetbytheeffective\\ carrierlifetime{\tau }_{eff}\end{array}$ $\begin{array}{r}{f}_{3dB,max}:highest3dBfrequencythedamping\\ allows,Hantschmannetal.:\gamma =K{f}_R^2+{\gamma }_0\\ withK=3.7\mathrm{ns}and{\gamma }_0=2.0\mathrm{n}{\mathrm{s}}^{-1}(0.5\mathrm{ns}),\\ thebound2\pi \sqrt{2}/K,andthe{f}_{R,max}varyingas\\ 1/\sqrt{ϵ},theexpressionK=4{\pi }^2({\tau }_p+ϵ/(v_g\\ a))ismyownlinearisationoftherate\\ equationsabove(theirKisprintedas\\ 4{\pi }^2{\tau }_{ph}(1+\Gamma \partial g/\partial S/\partial g/\partial N)inagarbled\\ summary),LoweryandNovak:complex\\ couplinginaDFBcanreduceKandtriple\\ themaximumintrinsicbandwidth(abstract\\ only,nonumbers)\end{array}$ $\begin{array}{r}(computedfromthearXivtable:ϵ/(v_{g}a)=\\ 8.0\mathrm{ps}against{\tau }_p=2.6\mathrm{ps},soK=0.418\\ \mathrm{ns}and{f}_{3dB,max}=21.3\mathrm{GHz};at{f}_R=10.6\mathrm{GHz}\\ thedampingis4.73\times 10^{10}{\mathrm{s}}^{-1}and\gamma /{\omega }_R=\\ 0.71;thequantum-dotlaserwithK=3.7\\ \mathrm{ns}islimitedto2.40\mathrm{GHz}andmeasured1.6\\ \mathrm{GHz})\end{array}$
Modulation response and 3 dB bandwidth	$H\left(f\right)=\frac{f_R^2}{f_R^2-f^2+jf\gamma /2\pi }\cdot \frac{1}{1+jf/f_p}$	${\omega }_{3dB}^2={\omega }_R^2-\frac{{\gamma }^2}{2}+\sqrt{{\left({\omega }_R^2-\frac{{\gamma }^2}{2}\right)}^2+{\omega }_R^4}$ $f_p=\frac{1}{2\pi R{C}_p}$	$\begin{array}{r}H:small-signaltransferfunctionfrom\\ currenttolight\end{array}$ $f_p:parasiticpoleofthecontactandpad$ $\begin{array}{r}Rand{C}_p:resistanceandcapacitanceof\\ thechipandthedriver\end{array}$ $\begin{array}{r}KIT:thesecond-orderpartandthe3dB\\ expression,Hantschmannetal.:the\\ three-poleform|H{|}^2,thelimitsare1.55\\ f_{R}for\gamma \to 0and{f}_{R}for\gamma =\sqrt{2}{\omega }_R(computed\\ fromtheformula;theKITsummarystatesa\\ maximumof\sqrt{2}{\omega }_R,whichdoesnotfollow\\ fromitsownexpressionandIdidnotuse\\ it),Finisar:RCcut-offabout22\mathrm{GHz}\end{array}$ $\begin{array}{r}(computedforthearXivtable:f_{3dB}=15.0\\ \mathrm{GHz}=1.41f_R,againstthepape{r}^{\prime }s16.4\\ \mathrm{GHz}fromitsapproximation1.55f_{R}andits\\ reported25.5\mathrm{GHz};a22\mathrm{GHz}polewith50\Omega \\ needs{C}_p=145fF(assumedresistance))\end{array}$
Photon-photon resonance	$f_{PPR}=\frac{c\Delta \lambda }{{\lambda }^2}$	$f_{PPR}\approx \frac{c}{2n_g{L}_{eff}}$	$\begin{array}{r}{f}_{PPR}:beatfrequencybetweenthemainmode\\ andasidemode\end{array}$ $\Delta \lambda :wavelengthspacingofthetwomodes$ $\begin{array}{r}{L}_{eff}:effectivelengthofthecoupled\\ cavity(backgroundknowledgeforboth\\ forms),aresponsepeakat{f}_{PPR}extendsthe\\ bandwidthwhen{f}_{PPR}liesneartheroll-off\\ ofthecarrier-photonresponse,NUSOD\\ abstract:coupled-cavityDFBlaser,110.6\\ \mathrm{GHz}at3.5timesthreshold\end{array}$ $\begin{array}{r}ETRI:DFB+Rlaserwitha3percentfront\\ reflectivity,about46\mathrm{GHz}at25degrees\\ CelsiusandPAM4above100Gbit/s,\\ Lumentum:20.1\mathrm{GHz}with{f}_{r}of3.8to4.4\\ (25degreesCelsius)and2.8to3.0(85\\ degreesCelsius),Finisar:about60\mathrm{GHz}at\\ 1310\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:aspacingof0.3\mathrm{nm}at1310\mathrm{nm}\\ (assumed)gives{f}_{PPR}=52\mathrm{GHz};apurely\\ carrier-photonresponsewouldneedK=\\ 0.44,0.19and0.08\mathrm{ns}for20.1,46and\\ 110.6\mathrm{GHz},against0.42\mathrm{ns}inthearXiv\\ table)\end{array}$

Chirp and Fibre Transmission

Name	Formula	Related Formulae	Variables/Constants
Frequency chirp of a directly modulated laser	$\Delta \nu \left(t\right)=\frac{\alpha }{4\pi }\left[\frac{1}{P}\frac{dP}{dt}+\kappa P\right]$ $\kappa =\frac{2\Gamma ϵ}{{\eta }_{0}h\nu V}$	$\alpha =-\frac{4\pi }{\lambda }\frac{dn/dN}{dg/dN}$ $\Delta {\nu }_{tr,mean}=\frac{\alpha }{4\pi }\frac{\ln \left(P_1/P_0\right)}{t_r}$	$\begin{array}{r}\Delta \nu :instantaneousfrequencydeviation,the\\ firsttermisthetransientchirpandthe\\ secondtheadiabaticchirp\end{array}$ $\kappa :adiabaticchirpfactor(in{\mathrm{J}}^{-1})$ $t_r:risetimeoftheedge$ $n:refractiveindex$ $\begin{array}{r}g:gain,arXiv:Eqs.9and10(novalues\\ of\alpha or\kappa arereported),RFEssentials:\alpha \\ about2to5forDFBlasers,3to7for\\ VCSELsand0.5to1.5forquantum-dot\\ lasers,thepagegives\alpha =\\ -(4\pi /\lambda )(dn/dN)/(dg/dN),transient\\ excursionsof1to10\mathrm{GHz},andno\\ equationsforthetwochirptypes,the\\ meantransientformismyintegrationof\\ thefirsttermoveranedge(background\\ knowledge)\end{array}$ $\begin{array}{r}(computedfromthearXivtablewith\alpha =3\\ (assumed):\kappa =1.01\times 10^{13}{\mathrm{J}}^{-1},anadiabatic\\ slopeof2.40GHz/mW(1.60for\alpha =2and\\ 4.01for\alpha =5),28\mathrm{GHz}at11.7\mathrm{mW}and12\\ \mathrm{GHz}fora5\mathrm{mW}swing;themeantransient\\ chirpis22\mathrm{GHz}fora6dBratioanda15\\ \mathrm{ps}edge(assumed))\end{array}$
Fibre transfer function and the first null	$H_{fibre}\left(f\right)\propto \cos \left(\frac{\pi {\lambda }^{2}DL{f}^2}{c}+\arctan \alpha \right)$	$f_{null}=\sqrt{\frac{c\left(\pi /2-\arctan \alpha \right)}{\pi {\lambda }^{2}DL}}$ $f_{null}\left(\alpha =0\right)=\sqrt{\frac{c}{2{\lambda }^{2}DL}}$	$\begin{array}{r}{H}_{fibre}:small-signalresponseofan\\ intensity-modulated,direct-detectionlink\end{array}$ $D:fibredispersioninps/(\mathrm{nm}\mathrm{km})$ $L:fibrelength$ $\lambda :wavelength$ $\begin{array}{r}c:speedoflight,RFEssentials:the\\ cosineform(thelateroftwoformsonthe\\ pagethatdisagree)and{f}_{null}=\sqrt{c/(2{\lambda }^{2}DL)}\\ forzerochirp,withafirstnullatabout\\ 5\mathrm{GHz}after20\mathrm{km}foraDFBlaser,the\\ generalnullfollowsfromsettingthe\\ cosineargumentto\pi /2(mystep)\end{array}$ $\begin{array}{r}(computedfor1550\mathrm{nm}withD=17\\ ps/(\mathrm{nm}\mathrm{km})(assumedstandardfibre):20\mathrm{km}\\ gives13.5\mathrm{GHz}for\alpha =0and6.1\mathrm{GHz}for\alpha \\ =3;10\mathrm{km}gives19.2and8.7\mathrm{GHz};2\mathrm{km}\\ gives42.8and19.4\mathrm{GHz})\end{array}$
Dispersion near the zero-dispersion wavelength	$D\left(\lambda \right)\approx S_0\left(\lambda -{\lambda }_0\right)$	$\Delta \tau =DL\Delta \lambda$	$\begin{array}{r}{S}_0:dispersionslopeatthe\\ zero-dispersionwavelength{\lambda }_0(about0.09\\ ps/(\mathrm{n}{\mathrm{m}}^2\mathrm{km})and1310\mathrm{nm},myassumptions\\ frombackgroundknowledge)\end{array}$ $\begin{array}{r}\Delta \tau :pulsespreadforaspectralwidth\Delta \lambda ,\\ Lumentum:fourlanesat1271,1291,1311\\ and1331\mathrm{nm}over500\mathrm{m}and2\mathrm{km}offibre\\ withazero-dispersionwavelengthnear\\ 1310\mathrm{nm},lanes0and1improvedwith\\ distanceandlanes2and3degraded\\ slightly,whichtheauthorsattributeto\\ chromaticdispersion\end{array}$ $\begin{array}{r}(computedwiththeassumed{S}_{0}and{\lambda }_0:D=\\ -3.5,-1.7,+0.1and+1.9ps/(\mathrm{nm}\mathrm{km})at\\ 1271,1291,1311and1331\mathrm{nm})\end{array}$

Energy and Equalisation

Name	Formula	Related Formulae	Variables/Constants
Energy per bit of a directly modulated laser	$E_{bit}=\frac{P_{el}}{B}=\frac{I{V}_d}{B}$	$P_{el}\approx I_{avg}{V}_d$	$E_{bit}:electricalenergyperbit$ $B:bitrate$ $I:drivecurrent$ $\begin{array}{r}{V}_d:voltageacrossthediode,driverand\\ impedance-matchinglossesnotincluded\\ (backgroundknowledge),NTTLEAPlaser:\\ threshold32\mu \mathrm{A},80\mu \mathrm{A}at10Gbit/swith\\ 10dBextinctionratioanda{2}^{31}-1pattern,\\ lessthan10\mathrm{fJ}perbit,NTT\\ photonic-crystallaser(Satoetal.):\\ threshold7.8\mu \mathrm{A}(390\mu \mathrm{A}fortheearlier\\ design),activeregion0.12\mu {\mathrm{m}}^3,about14\\ \mathrm{fJ}perbitestimatedat12.5Gbit/s,\\ Matsuo:4.4\mathrm{fJ}perbit\end{array}$ $\begin{array}{r}(computed:10\mathrm{fJ}at10Gbit/sis100\mu \mathrm{W},\\ so80\mu \mathrm{A}impliesatmost1.25V;14\mathrm{fJ}at\\ 12.5Gbit/sis175\mu \mathrm{W};fora70\mathrm{mA}laser\\ atanassumed1.5Vand106.25Gbit/sthe\\ figureis0.99\mathrm{pJ}perbit,70to225times\\ higher;thearXivactivevolumeof36\mu {\mathrm{m}}^3\\ is300timesthe0.12\mu {\mathrm{m}}^3)\end{array}$
PAM4 Nyquist frequency and the equalisation boost	$f_{Nyq}=\frac{R_s}{2},R_s=\frac{B}{{\log }_{2}M}$	$G=\sqrt{1+{\left(\frac{f_{Nyq}}{f_{3dB}}\right)}^2}$	$R_s:symbolrate$ $M:numberoflevels(4forPAM4)$ $\begin{array}{r}G:boostneededtoflattenasingle-pole\\ responseattheNyquistfrequency\\ (backgroundknowledge,thereallaser\\ responseisnotsingle-pole),Lumentum:\\ 106Gbit/sPAM4at53\mathrm{GBd}perlaser,\\ bandwidth20.1\mathrm{GHz},biterrorratiobelow\\ theKP4-FEClimit2.2\times 10^{-4}upto85degrees\\ Celsius,TDECQ2.6to3.0dBat25degrees\\ Celsiusand3.1to3.4dBat50degrees\\ Celsius(limit3.4dB)\end{array}$ $AOI:25\mathrm{GBd}PAM4(50Gbit/s),TDECQbelow2.5dB$ $\begin{array}{r}(computed:f_{Nyq}=26.6\mathrm{GHz}for53.1\mathrm{GBd},so\\ G=1.66(4.4dB)withabandwidthof20.1\\ \mathrm{GHz})\end{array}$
Temperature dependence of the relaxation frequency	$\frac{f_R\left(T_2\right)}{f_R\left(T_1\right)}=\sqrt{\frac{a\left(T_2\right)S_0\left(T_2\right)}{a\left(T_1\right)S_0\left(T_1\right)}}$	$f_R^2\propto a{S}_0$	$a:differentialgain$ $\begin{array}{r}{S}_0:photondensityattheoperating\\ current(fromtherelaxationfrequencylaw\\ above,backgroundknowledge),Lumentum:f_r\\ =3.8to4.4at25degreesCelsiusand2.8\\ to3.0at85degreesCelsius(unitsnot\\ stated,myreadingis\mathrm{GHz}),threshold13.7\\ to16.3\mathrm{mA}at80degreesCelsius\end{array}$ $\begin{array}{r}(computedfromthemid-points:f_{R}fallsto\\ 0.71ofitsvalue,sotheproductof\\ differentialgainandphotondensityfalls\\ to0.50;thebandwidthof20.1\mathrm{GHz}is\\ quotedonlyat85degreesCelsius)\end{array}$

Important Questions

Ques: Is the $3\mathrm{dB}$ bandwidth of the generic DFB laser 15.0, 16.4 or 25.5 GHz?

The arXiv paper lists $f_R$ = 10.6 $\mathrm{GHz}$ and $f_{3dB}$ = 25.5 $\mathrm{GHz}$ at 75 $\mathrm{mA}$, but its own approximation $f_{3dB}\approx 1.55f_R$ gives 16.4 $\mathrm{GHz}$ (computed). I linearised the rate equations with its table: $K$ = 0.418 $\mathrm{ns}$, a damping rate of $4.73\times 10^{10}$ ${\mathrm{s}}^{-1}$ at 10.6 $\mathrm{GHz}$, and a second-order response with $3\mathrm{dB}$ at 15.0 $\mathrm{GHz}$, which is 1.41 $f_R$ (computed). The damping bound $2\sqrt{2}\pi /K$ is 21.3 $\mathrm{GHz}$ (computed), so 25.5 $\mathrm{GHz}$ cannot come from the carrier-photon response of this table alone, and the paper does not say what else contributes (my reading). For the stated parameters I regard 15 to 16 $\mathrm{GHz}$ as more likely, and 25.5 $\mathrm{GHz}$ as the value of the simulated device. Reference: arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary); Exercise 08 and its solution, Institute of Photonics and Quantum Electronics, Karlsruhe Institute of Technology (ipq.kit.edu PDF; garbled text extraction read through a fetch-tool summary, which flagged its own maximum-bandwidth statement as unreliable); Hantschmann, Vasil’ev, Chen, Liao, Seeds, Liu, Penty and White, small-signal modulation of a 1.3 um InAs/GaAs quantum-dot laser on silicon (two accepted-manuscript files at discovery-pp.ucl.ac.uk, ids 10072362 and 10067505; journal and year not stated in the text read; read through fetch-tool summaries in which several symbols were garbled).

Ques: Is the gain compression factor $2\times 10^{23}$ or $2\times 10^{-23}$ cubic metres?

Table 1 of the arXiv paper prints $ϵ$ = $2.00\times 10^{23}$ ${\mathrm{m}}^3$ (as printed), which looks suspicious. The photon density at 75 $\mathrm{mA}$ follows from its Eq. 4: $S_0$ = $1.53\times 10^{22}$ ${\mathrm{m}}^{-3}$ (computed, with ${\eta }_0$ = 0.2 and 1500 nm), so $ϵS_0$ would be $3\times 10^{45}$ and the laser would not lase. With $2\times 10^{-23}$ ${\mathrm{m}}^3$ it is 0.31, a normal value, so I treat the exponent as a sign error (my reading). The choice matters: $ϵ/(v_{g}a)$ is then 8.0 $\mathrm{ps}$, three times ${\tau }_p$ = 2.6 $\mathrm{ps}$, so gain compression sets K (0.418 $\mathrm{ns}$, computed) and lowers the estimated $f_R$ from 9.5 to 8.3 $\mathrm{GHz}$ (computed). Reference: arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary).

Ques: Do the tabulated parameters reproduce the reported threshold and relaxation frequency?

Only roughly. With $g_0=v_g{\sigma }_g$ = $2.50\times 10^{-12}$ ${\mathrm{m}}^3/s$ (my assumption) the threshold is $N_{th}$ = $2.64\times 10^{24}$ ${\mathrm{m}}^{-3}$ and $I_{th}$ = 4.80 $\mathrm{mA}$ (computed), 18 per cent above the reported 4.06 $\mathrm{mA}$. The relaxation frequency at 75 $\mathrm{mA}$ comes out at 9.5 $\mathrm{GHz}$ without and 8.3 $\mathrm{GHz}$ with gain compression (computed), against the reported 10.6 $\mathrm{GHz}$, that is 10 to 22 per cent low. Three of my assumptions can account for the gap: the gain coefficient, the 1500 nm photon energy, and the linear gain, whereas the paper’s Eq. 5 is logarithmic with an unreported saturation density. I regard the reported values as what its simulation produced and my numbers as a check that the table is self-consistent to about 20 per cent. Reference: arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary).

Ques: How can a 150 um DFB laser reach 20.1 GHz when its $f_r$ is only 3 to 4?

Lumentum quotes $f_r$ = 3.8 to 4.4 at 25 degrees Celsius and 2.8 to 3.0 at 85 degrees Celsius, without units (I read them as $\mathrm{GHz}$), and a bandwidth of 20.1 $\mathrm{GHz}$ at 70 $\mathrm{mA}$ and 85 degrees Celsius. A second-order response peaks at $f_R$ and has its $3\mathrm{dB}$ point at most 1.55 $f_R$ above it, so 2.9 $\mathrm{GHz}$ would give about 4.5 $\mathrm{GHz}$ and 4 $\mathrm{GHz}$ about 6.2 $\mathrm{GHz}$ (computed), three to four times lower than 20.1. The article does not explain the difference, but it describes an asymmetric corrugation pitch with a normalised coupling of about 2.0, which suggests detuned loading and an optical resonance that carries the response (my reading). I regard the 20.1 $\mathrm{GHz}$ as measured and the small $f_r$ as the carrier-photon resonance only. Reference: Onga, Nakajima, Sekino, Nakanishi, Sasada, Nakajima, Sakamoto and Naoe (Lumentum Japan), ‘Easing data-centre demands with directly modulated lasers’, Compound Semiconductor 28(3), 12 April 2022 (compoundsemiconductor.net; fetch-tool summary; the article gives the relaxation frequency without units); Kim, Han, Yun, Park and Lee (ETRI and JBNU), DFB+R laser with detuned loading and photon-photon resonance, ICTC 2025, pp. 1353-1356, DOI 10.1109/ICTC66702.2025.11387824 (abstract page at pure.jbnu.ac.kr, read through a fetch-tool summary; the title is partly my reading of the page address).

Ques: Are bandwidths of 46 and 110.6 GHz consistent with the K-factor limit?

Only if the carrier-photon response is not the limit. A purely damped laser needs $K=2\sqrt{2}\pi /f_{3dB}$, which is 0.19 $\mathrm{ns}$ for 46 $\mathrm{GHz}$ and 0.08 $\mathrm{ns}$ for 110.6 $\mathrm{GHz}$ (computed), against 0.42 $\mathrm{ns}$ in the arXiv table and 3.7 $\mathrm{ns}$ for the quantum-dot laser. The ETRI DFB+R laser and the NUSOD coupled-cavity laser both name photon-photon resonance, and a side-mode spacing of 0.3 $\mathrm{nm}$ at 1310 nm would put a resonance at 52 $\mathrm{GHz}$ (computed, assumed spacing). The NUSOD page is only a teaser without length or wavelength, so I cannot check its numbers (my reading). Such a resonance gives a peak, not a flat band, so the response between $f_R$ and the peak must still be acceptable for the signal. Reference: Kim, Han, Yun, Park and Lee (ETRI and JBNU), DFB+R laser with detuned loading and photon-photon resonance, ICTC 2025, pp. 1353-1356, DOI 10.1109/ICTC66702.2025.11387824 (abstract page at pure.jbnu.ac.kr, read through a fetch-tool summary; the title is partly my reading of the page address); NUSOD 2020 abstract LD09, ‘Over 100 GHz 3-dB down bandwidth by direct modulation of a coupled cavity DFB-LD due to photon-photon resonance’ (nusod2020.polito.it; teaser page only; the author index lists T. Numai, my reading); arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary).

Ques: Why does a photonic-crystal laser use femtojoules per bit and a data-centre DML about a picojoule?

NTT reports less than 10 $\mathrm{fJ}$ per bit at 10 $Gbit/s$ and 80 $\mu \mathrm{A}$ for the LEAP laser, 14 $\mathrm{fJ}$ per bit at 12.5 $Gbit/s$ for the 7.8 $\mu \mathrm{A}$ laser and 4.4 $\mathrm{fJ}$ per bit in a later talk. The first is at most 100 $\mu \mathrm{W}$, so the voltage is at most 1.25 V (computed). A 70 $\mathrm{mA}$ laser at an assumed 1.5 V and 106.25 $Gbit/s$ needs 0.99 $\mathrm{pJ}$ per bit (computed, driver not included), 70 to 225 times more. The reasons are volume (0.12 $\mu {\mathrm{m}}^3$ against 36 $\mu {\mathrm{m}}^3$ in the arXiv table, 300 times, computed), threshold and output power: the small lasers emit microwatts into a waveguide, the large ones more than 6.7 $\mathrm{mW}$ into a fibre. Both numbers are right for their purposes; the sources give no energy per delivered optical power. Reference: NTT Basic Research Laboratories summary of a lambda-scale embedded active-region photonic-crystal (LEAP) laser (rd.ntt report13/report30E; fetch-tool summary; authors and publication not stated on the page); NTT summary of the photonic-crystal nanocavity laser of Sato, Takeda, Shinya, Nozaki, Taniyama, Hasebe, Kakitsuka, Notomi and Matsuo (rd.ntt report12/report30e; fetch-tool summary; the page cites Optics Express 20, 3773 (2012)); S. Matsuo, seminar abstract ‘Advanced semiconductor lasers: ultra-low operating energy and heterogeneous integration with Si photonics devices’, IEEE Ottawa, 3 December 2019 (4.4 fJ/bit; abstract page only); Onga, Nakajima, Sekino, Nakanishi, Sasada, Nakajima, Sakamoto and Naoe (Lumentum Japan), ‘Easing data-centre demands with directly modulated lasers’, Compound Semiconductor 28(3), 12 April 2022 (compoundsemiconductor.net; fetch-tool summary; the article gives the relaxation frequency without units); arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary).

Ques: How large is the chirp of a directly modulated laser?

The sources do not agree because they describe different things. Finisar quotes an adiabatic chirp of about 30 $\mathrm{GHz}$ for a 10 $Gbit/s$ DML after a filter, and a chirp-managed design with about 5 $\mathrm{GHz}$. From the arXiv table I compute 2.40 $GHz/mW$ for $\alpha$ = 3 (assumed), that is 12 $\mathrm{GHz}$ for a 5 $\mathrm{mW}$ swing and 28 $\mathrm{GHz}$ at 11.7 $\mathrm{mW}$ (computed), and a mean transient chirp of 22 $\mathrm{GHz}$ for a $6\mathrm{dB}$ ratio and a 15 $\mathrm{ps}$ edge (assumed). The paper itself gives neither $\alpha$ nor $\kappa$. The values scale with $\alpha$, from 1.6 to 4.0 $GHz/mW$ for 2 to 5. Tens of gigahertz are more likely for an ordinary DFB DML, and the 5 $\mathrm{GHz}$ belongs to the chirp-managed case. Reference: Matsui, Lin, Chen and Sudo (Finisar), ‘Wavelength-tunable DBR laser for burst mode TWDM PON applications’, IEEE 802.3 Super-PON presentation, 11-12 September 2018 (grouper.ieee.org PDF; fetch-tool summary; some slide formulas garbled); arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary); RF Essentials, chirp parameter glossary page (rfessentials.com; fetch-tool summary; the page gives two forms of the fibre transfer function that disagree, and I use the later one).

Ques: Why did the shorter-wavelength lanes improve with distance and the longer ones degrade?

Lumentum finds that its 1271 and 1291 nm lanes improved with distance (500 m and 2 $\mathrm{km}$) while the 1311 and 1331 nm lanes slightly degraded, and attributes this to chromatic dispersion. With an assumed slope of 0.09 $ps/(\mathrm{n}{\mathrm{m}}^2\mathrm{km})$ and a zero-dispersion wavelength of 1310 nm, the dispersion is -3.5, -1.7, +0.1 and +1.9 $ps/(\mathrm{nm}\mathrm{km})$ on the four lanes (computed). A laser with positive $\alpha$ is blue-shifted on the rising edge. With negative dispersion that edge arrives later and the pulse is compressed; with positive dispersion it arrives earlier and the pulse broadens. This is my explanation, not the authors’, and it needs the real zero-dispersion wavelength of the fibre, which can lie between 1300 and 1324 nm (background knowledge). Reference: Onga, Nakajima, Sekino, Nakanishi, Sasada, Nakajima, Sakamoto and Naoe (Lumentum Japan), ‘Easing data-centre demands with directly modulated lasers’, Compound Semiconductor 28(3), 12 April 2022 (compoundsemiconductor.net; fetch-tool summary; the article gives the relaxation frequency without units); RF Essentials, chirp parameter glossary page (rfessentials.com; fetch-tool summary; the page gives two forms of the fibre transfer function that disagree, and I use the later one).

Ques: Why can a 1550 nm DML reach only 20 km at 10 Gbit/s while an O-band DML carries 50 Gbit/s over 10 km?

The dispersion-length product decides. At 1550 nm with 17 $ps/(\mathrm{nm}\mathrm{km})$ over 20 $\mathrm{km}$ it is 340 ps/nm, and the first fibre null is at 6.1 $\mathrm{GHz}$ for $\alpha$ = 3 (computed), close to the 5 $\mathrm{GHz}$ that RF Essentials gives for a DFB laser at 20 $\mathrm{km}$ and just above the 5 $\mathrm{GHz}$ Nyquist frequency of 10 $Gbit/s$ NRZ. Finisar’s DML reached 20 $\mathrm{km}$ at 10 $Gbit/s$. In the O-band, |D| stays below about 3.5 $ps/(\mathrm{nm}\mathrm{km})$ (computed with assumed values), so 10 $\mathrm{km}$ gives 35 ps/nm, ten times less, which is why AOI reached 10 $\mathrm{km}$ at 50 $Gbit/s$ PAM4 with an extinction ratio of up to $6\mathrm{dB}$. Chirp management raises 1550 nm reach to 40 $\mathrm{km}$ at 25 $Gbit/s$ (Finisar). Reference: RF Essentials, chirp parameter glossary page (rfessentials.com; fetch-tool summary; the page gives two forms of the fibre transfer function that disagree, and I use the later one); Matsui, Lin, Chen and Sudo (Finisar), ‘Wavelength-tunable DBR laser for burst mode TWDM PON applications’, IEEE 802.3 Super-PON presentation, 11-12 September 2018 (grouper.ieee.org PDF; fetch-tool summary; some slide formulas garbled); Applied Optoelectronics press release, ‘Applied Optoelectronics Announces 50 Gbps PAM-4 Directly Modulated Lasers’, 8 August 2017 (investors.ao-inc.com; fetch-tool summary).

Ques: Can a 20.1 GHz laser carry 106 Gbit/s PAM4?

Lumentum shows that it can, with little margin. At 53 $\mathrm{GBd}$ the Nyquist frequency is 26.6 $\mathrm{GHz}$, so a single-pole response needs a boost of 1.66, or $4.4\mathrm{dB}$, to flatten it (computed; the real response is not single-pole). The bit error ratio stayed below the KP4-FEC limit of $2.2\times 10^{-4}$ up to 85 degrees Celsius, with TDECQ of 2.6 to $3.0\mathrm{dB}$ at 25 degrees Celsius and 3.1 to $3.4\mathrm{dB}$ at 50 degrees Celsius, against a limit of $3.4\mathrm{dB}$, so at 50 degrees Celsius the margin is zero to $0.3\mathrm{dB}$. The extinction ratio was set to $3.0\mathrm{dB}$, half (in dB) of the $6\mathrm{dB}$ that AOI quotes at 25 $\mathrm{GBd}$. The article does not say where the equalisation sits, so I cannot tell how much of the boost is in the laser driver. Reference: Onga, Nakajima, Sekino, Nakanishi, Sasada, Nakajima, Sakamoto and Naoe (Lumentum Japan), ‘Easing data-centre demands with directly modulated lasers’, Compound Semiconductor 28(3), 12 April 2022 (compoundsemiconductor.net; fetch-tool summary; the article gives the relaxation frequency without units); Applied Optoelectronics press release, ‘Applied Optoelectronics Announces 50 Gbps PAM-4 Directly Modulated Lasers’, 8 August 2017 (investors.ao-inc.com; fetch-tool summary).

Ques: Why is a quantum-dot laser on silicon limited to 1.6 GHz?

Hantschmann et al. measure a K-factor of 3.7 $\mathrm{ns}$ and a damping offset of 2.0 $\mathrm{n}{\mathrm{s}}^{-1}$, equal to a lifetime of 0.5 $\mathrm{ns}$. The damping bound is 2.40 $\mathrm{GHz}$ (computed), and the measured 1.6 $\mathrm{GHz}$ at 40 $\mathrm{mA}$ is below it. The arXiv DFB table gives 0.418 $\mathrm{ns}$ (computed), 8.9 times smaller. Their gain compression is 0.5 to $1.5\times 10^{-15}$ $\mathrm{c}{\mathrm{m}}^3$, 25 to 75 times the $2\times 10^{-17}$ $\mathrm{c}{\mathrm{m}}^3$ of the table after unit conversion (computed), although the definitions may differ (my reading). The authors point to a long photon lifetime, a low differential gain or strong gain compression. The D-factor is about 0.3 $GHz/m{\mathrm{A}}^{1/2}$ in one version of the paper and absent from the other, so I use it only as an indication. Reference: Hantschmann, Vasil’ev, Chen, Liao, Seeds, Liu, Penty and White, small-signal modulation of a 1.3 um InAs/GaAs quantum-dot laser on silicon (two accepted-manuscript files at discovery-pp.ucl.ac.uk, ids 10072362 and 10067505; journal and year not stated in the text read; read through fetch-tool summaries in which several symbols were garbled); arXiv:2405.09907 (2024), end-to-end optimisation of optical communication systems based on directly modulated lasers (title from the search result; authors not checked; arxiv.org PDF read through a fetch-tool summary); Quantum-Dot Laser page of this vault.

Related pages in this vault: Indium Phosphide Quantum-Well Electroabsorption Modulator, Silicon MOS-Capacitor Modulator, Silicon Carrier-Depletion Mach-Zehnder Modulator, Distributed Feedback Laser (DFB), Silicon Microring Modulator, Silicon Carrier-Injection PIN Modulator, Vertical-Cavity Surface-Emitting Laser (VCSEL), Nanolaser (Microdisk, Microring and Photonic Crystal), Semiconductor Lasers (Edge-Emitting and VCSEL).