Distributed Bragg Reflector Laser (DBR)

1. Theory

• Definition: A distributed-Bragg-reflector (DBR) laser is an in-plane laser with separate gain, passive phase and passive grating-mirror sections, giving single-mode output with an electrically tunable wavelength.
• Grating outside the gain section: Unlike a DFB, the Bragg grating acts as a wavelength-selective mirror at one or both ends of the gain region.
• Effective mirror length: The grating reflects gradually, equivalent to a mirror at depth $L_{eff}=\tanh (\kappa L_g)/2\kappa$; weak gratings behave like a mirror at their centre, strong ones within $1/2\kappa$.
• Wavelength-selective loss: The grating gives a single mirror-loss minimum near the Bragg wavelength, so the cavity mode nearest it lases.
• Multi-section design: Separate electrodes control gain, cavity phase and grating index independently.
• Fabrication cost: Active and passive waveguides must be butt-joined and the grating regrown without discontinuities.

2. Working Principle

• Threshold: $\Gamma g_{th}=⟨{\alpha }_i⟩+\ln \left[1/\left(r_1\left|r_g\right|\right)\right]/(L_a+L_p)$, with grating losses either lumped into $r_g$ or treated as propagation loss.
• Unequal output: Power splits between the ends according to each mirror’s transmission; grating loss reduces both reflected and transmitted power.
• Grating tuning: Injecting current into the grating changes its index and moves the Bragg wavelength.
• Mode hops: Cavity modes move less than the Bragg peak, so tuning the grating alone makes the wavelength jump by one mode spacing at a time.
• Phase section: Shifts the whole comb of cavity modes continuously, filling in wavelengths between hops for quasi-continuous tuning.
• Mode spacing: Set by the combined optical length of gain, phase and effective grating sections.

3. Applications

• Widely tunable telecom sources: Sampled-grating and multi-section DBR lasers for DWDM and coherent transceivers that switch channels in software.
• Photonic integrated circuits: Tunable lasers integrated with modulators and amplifiers on InP chips.
• Spectroscopy and sensing: Single-frequency tunable sources for gas detection and frequency-doubled visible lasers.

Name	Formula	Related Formulae	Variables/Constants
Effective length of a grating mirror	$L_{eff}=\frac{1}{2\kappa }\tanh \left(\kappa L_g\right)$	$L_{eff}=-\frac{1}{2}\frac{\partial \varphi }{\partial \beta }$ $L_{eff}\to \frac{L_g}{2}\left(\kappa L_g\ll 1\right)$ $L_{eff}\to \frac{1}{2\kappa }=\frac{\Lambda }{4r}\left(\kappa L_g\gg 1\right)$ $L_{pen}\approx L_{eff}$	$\begin{array}{r}{L}_{eff}:effectivelengthofthegrating\\ mirror,thatisthedistancefromthe\\ startofthegratingtotheplaneofan\\ equivalentdiscretemirrorofreflection\\ \left|r_g\right|(theexpressionneglectsthird-and\\ higher-ordertermsinrandisfor\\ \left|\delta L_g\right|\ll \pi )\end{array}$ $\begin{array}{r}\kappa :couplingconstantofthegrating(see\\ theDFBpage)\end{array}$ $L_g:lengthofthegrating$ $\begin{array}{r}\varphi :phaseofthereflectioncoefficientof\\ thegrating\end{array}$ $\beta :propagationconstant$ $\Lambda :periodofthegrating$ $\begin{array}{r}r:amplitudereflectionateachindexstep\\ ofthegrating\end{array}$ $\begin{array}{r}{L}_{pen}:penetrationdepthoftheoptical\\ energyintothegrating,whichisabout\\ equalto{L}_{eff}andsotheenergystoredin\\ themirrorisabouttheenergydensityat\\ itsinputmultipliedby{L}_{eff}\end{array}$
Threshold gain of the DBR laser	$\Gamma g_{th}=⟨{\alpha }_i⟩+\frac{1}{L_a+L_p}\ln \frac{1}{r_1|r_g|}$	${\alpha }_m=\frac{1}{L_a+L_p}\ln \frac{1}{r_1|r_g|}$ $\Gamma g_{th}=⟨{\alpha }_i{⟩}^{\prime }+\frac{1}{L_{DBR}}\ln \frac{1}{r_1|r_g^{\prime }|},L_{DBR}=L_a+L_p+L_{eff}$	$\begin{array}{r}{g}_{th}:materialgainoftheactivesectionat\\ threshold(thethresholdcondition\Gamma \\ g_{th}={\alpha }_i+{\alpha }_{m}oftheotherlaserpages;the\\ factor\Gamma waslostinthetextextraction\\ ofthereferenceequationandisrestored\\ here)\end{array}$ $\Gamma :confinementfactoroftheactivesection$ $\begin{array}{r}⟨{\alpha }_i⟩:internallossaveragedoverthe\\ activeandthepassivesectiononly(the\\ lossesinthegratingarein\left|r_g\right|)\end{array}$ $\begin{array}{r}{L}_a,L_p:lengthsoftheactiveandthe\\ passive(phase)section\end{array}$ $\begin{array}{r}{r}_1:amplitudereflectioncoefficientof\\ thediscretemirrorattheotherend\end{array}$ $\begin{array}{r}\left|r_g\right|:magnitudeoftheamplitude\\ reflectionofthegratingmirroratthe\\ Braggwavelength,includingitsloss(Case\\ a,thegratingisaseparateelement)\end{array}$ ${\alpha }_m:mirrorloss$ $\begin{array}{r}{L}_{DBR}:effectivecavitylengthofthe\\ effective-mirrormodel(Caseb),inwhich\\ thelossesofthegratingarepropagation\\ lossesover{L}_{eff}andtheprimedquantities\\ areaveragedover{L}_{DBR}andusethelossless\\ reflection\left|r_g^{\prime }\right|\end{array}$ $\begin{array}{r}(thetwomodelsgivethesameresultnear\\ theBraggfrequencyifthelossesarenot\\ toolarge;withtwogratingmirrors{r}_{1}is\\ replacedby\left|r_{g1}\right|)\end{array}$
Power out of the ends of a DBR laser	${\eta }_{d1}=F_1{\eta }_i\frac{{\alpha }_m}{⟨{\alpha }_i⟩+{\alpha }_m},{\eta }_{d2}=F_2{\eta }_i\frac{{\alpha }_m}{⟨{\alpha }_i⟩+{\alpha }_m}$	$F_1=\frac{t_1^2}{\left(1-r_1^2\right)+\frac{r_1}{|r_g|}\left(1-|r_g|{}^2\right)}$ $F_2=\frac{|t_g|{}^2}{\left(1-|r_g|{}^2\right)+\frac{|r_g|}{r_1}\left(1-r_1^2\right)}$ $\left|S_{g21}\right|{}^2\approx \left(1-\left|r_g\right|{}^2\right)e^{-2{\alpha }_i{L}_{eff}}$ $\left|t_g^{\prime }\right|{}^2=\left(1-\left|r_g^{\prime }\right|{}^2\right)e^{-{\alpha }_i{L}_{eff}}$	$\begin{array}{r}{\eta }_{d1},{\eta }_{d2}:differentialquantumefficiencies\\ outofthediscretemirrorend(1)andthe\\ gratingend(2)\end{array}$ $\begin{array}{r}{F}_1,F_2:fractionsoftheoutputpower\\ leavingeachend(Casea,withthegrating\\ scatteringparameters{r}_g=S_{g11}and{t}_g=S_{g21}\\ replacing{r}_{2}and{t}_2;thesumisbelowone\\ ifthegratingislossy)\end{array}$ ${\eta }_i:injectionefficiency$ $\begin{array}{r}{t}_1:amplitudetransmissionofthediscrete\\ mirror,t_1^2=1-r_1^{2}ifitislossless\end{array}$ $\begin{array}{r}\left|S_{g21}\right|{}^2:powertransmissionofalossy\\ gratingattheBraggconditionforasmall\\ uniformloss{\alpha }_i,whichfallsas{e}^{-2{\alpha }_i{L}_{eff}}\\ likethereflectedpower\end{array}$ $\begin{array}{r}\left|t_g^{\prime }\right|{}^2:transmissionthroughtheeffective\\ mirrorofCaseb,whichisnotthe\\ losslesstransmissionthroughthegrating\end{array}$ $\begin{array}{r}(theratio{F}_1/F_{2}isthesameinCaseaand\\ bandsois{\eta }_{d}approximately;the\\ effectivemirrormodeloverestimates{\eta }_d\\ slightly,butnotmuchfor{\alpha }_i{L}_{eff}<0.1)\end{array}$
Axial mode spacing and tuning of a three-section DBR laser	$\delta \lambda =\frac{{\lambda }^2}{2\left({\bar{n}}_{ga}{L}_a+{\bar{n}}_{gp}{L}_p+{\bar{n}}_{gDBR}{L}_{eff}\right)}$	$\frac{\Delta {\lambda }_g}{{\lambda }_g}=\frac{\Delta {\bar{n}}_{DBR}}{{\bar{n}}_{DBR}}$ $\frac{\Delta {\lambda }_m}{{\lambda }_m}=\frac{\Delta {\bar{n}}_a{L}_a+\Delta {\bar{n}}_p{L}_p+\Delta {\bar{n}}_{DBR}{L}_{eff}}{{\bar{n}}_{ga}{L}_a+{\bar{n}}_{gp}{L}_p+{\bar{n}}_{gDBR}{L}_{eff}}$ $\Delta {\bar{n}}_j=\frac{\partial \bar{n}}{\partial N}\frac{{\eta }_i\tau I_j}{q{V}_j},\frac{\partial \bar{n}}{\partial N}\approx -{\Gamma }_{xy}10^{-20}c{m}^3$	$\delta \lambda :spacingoftheaxialmodes$ $\lambda :wavelength$ $\begin{array}{r}{\bar{n}}_{ga},{\bar{n}}_{gp},{\bar{n}}_{gDBR}:groupindicesofthe\\ active,phaseandgratingsections\end{array}$ $\begin{array}{r}{L}_a,L_p,L_{eff}:lengthsoftheactiveandthe\\ phasesectionandtheeffectivelengthof\\ thegratingmirror\end{array}$ $\begin{array}{r}\Delta {\lambda }_g:shiftofthecentrewavelengthofthe\\ gratingfilterwhentheindexofthe\\ gratingsectionchangesby\Delta {\bar{n}}_{DBR}\end{array}$ $\begin{array}{r}\Delta {\lambda }_m:shiftoftheaxialmodecombwhenthe\\ indicesofthesectionschangeby\Delta {\bar{n}}_j\\ (thegroupindicesmustbeusedinthe\\ denominator;changesofcurrentinthe\\ activesectiongiveonlyasmallchangeof\\ indexbecausethecarrierdensityis\\ clamped)\end{array}$ $\begin{array}{r}\partial \bar{n}/\partial N:changeoftheeffectiveindex\\ withcarrierdensityinthepassive\\ section(negative:injectingcarriers\\ lowerstheindexandsoshiftsthe\\ wavelengthtoshortervalues)\end{array}$ ${\Gamma }_{xy}:transverse-lateralconfinementfactor$ $\begin{array}{r}{I}_j,V_j,\tau :currentintothesectionj,its\\ volumeandthecarrierlifetime(the\\ injecteddensityisfoundwithg=0and\\ dN/dt=0)\end{array}$ $\begin{array}{r}(indexchangesoftheorderof0.1\%are\\ quotedinthereference;areversebiasalso\\ changestheindexthroughtheelectro-optic\\ effects,andrespondsfasterthan\\ injection,whosespeedislimitedbythe\\ carrierlifetimetoafewhundred\\ megahertz,whileheatinggivestime\\ constantsoftensofmicroseconds)\end{array}$

Important Questions

Ques: Why does a DBR laser select a single longitudinal mode, and what is the price of putting the grating outside the gain section?

The grating mirror reflects strongly only near the Bragg wavelength, so the mirror loss ${\alpha }_m(\lambda )$ has one minimum there, instead of the many equal minima of a Fabry-Perot cavity or the repeating minima of coupled cavities. Of all the axial modes of the cavity, the one closest to the minimum has the lowest threshold gain. For a long in-plane laser the mirror loss rises quickly on both sides of the minimum, so the neighbouring axial mode has a large loss margin, which is added to the roll-off of the net modal gain. In a very short vertical cavity the mode spacing is so large that the short cavity and the roll-off of the gain do most of the selection. The price is the fabrication: the active and the passive waveguides have to be joined without an unwanted discontinuity, and the grating has to be patterned and regrown, which is why in-plane DBR lasers are made only when single-frequency operation and wavelength tuning with separate electrodes are required. In a DFB laser the grating lies along the gain region and no such junction is needed. Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 3.6.1 (Fig. 3.16) and Section 3.6.3 (Fig. 3.17).

Ques: Why is a grating mirror described by an effective length, and what does it depend on?

A grating does not reflect at one plane: the field penetrates it and is reflected gradually along its length, so the reflection has a phase that changes with the wavelength. Near the Bragg wavelength this can be replaced by a discrete mirror of the same reflection magnitude placed at a distance $L_{eff}=-\frac{1}{2}\partial \varphi /\partial \beta$ inside the grating, which also gives the approximate penetration depth of the optical energy. For a weak grating ($\kappa L_g\ll 1$) the equivalent mirror sits at the centre of the grating, $L_{eff}\approx L_g/2$, and for a strong grating ($\kappa L_g\gg 1$) the field decays within $1/(2\kappa )$ and a longer grating adds nothing. Two models follow from this. In Case (a) the grating is a separate element with its reflection $|r_g|$ (losses included), and the cavity length is $L_a+L_p$. In Case (b) the grating losses are propagation losses over $L_{eff}$, and the cavity length is $L_{DBR}=L_a+L_p+L_{eff}$. Both give about the same threshold gain near the Bragg wavelength if the losses are not large, and Case (b) needs the least computation, so it is used for design. As an illustration, in Example 3.5 of the reference a 100 $\mu$m grating with $|r_g{|}^2=0.2$ gives $\kappa L_g=0.48$ and $L_{eff}=46\mu$m (computed, taking the reflection as lossless), close to the value of 50 $\mu$m used in the example. Reference: Coldren et al., 2nd ed., Section 3.5.2 (Eqs. 3.62 and 3.63, Fig. 3.15) and Section 3.6.2 (Eqs. 3.64 to 3.66).

Ques: Why does the power leave a DBR laser unequally from the two ends, and why does the loss of the grating also reduce the transmitted power?

The fraction of the power leaving through each end depends on how much each mirror transmits compared with how much it reflects, so the end with the weaker mirror gets the larger fraction, as in the Fabry-Perot laser. For a discrete mirror with reflection $r_1$ at one end and a grating with $r_g$ and $t_g$ at the other, the fractions are $F_1=t_1^2/[(1-r_1^2)+(r_1/|r_g|)(1-|r_g{|}^2)]$ and $F_2=|t_g{|}^2/[(1-|r_g{|}^2)+(|r_g|/r_1)(1-r_1^2)]$, and the efficiencies are ${\eta }_{dj}=F_j{\eta }_i{\alpha }_m/(⟨{\alpha }_i⟩+{\alpha }_m)$. If the grating has a small uniform loss, the power transmitted through it at the Bragg condition falls as $e^{-2{\alpha }_i{L}_{eff}}$, like the reflected power, which the reference notes is unintuitive. Then $F_1+F_2<1$, because part of the power is absorbed in the grating. For the tunable laser of Example 3.5 (${\alpha }_m=34.4\mathrm{c}{\mathrm{m}}^{-1}$, $⟨{\alpha }_i⟩=12.5\mathrm{c}{\mathrm{m}}^{-1}$, ${\eta }_i=0.8$, $|r_g{|}^2=0.2$, $|t_g{|}^2=0.7$, a cleaved facet at the other end) this gives $F_1=0.40$, $F_2=0.52$, ${\eta }_{d1}=0.24$ and ${\eta }_{d2}=0.31$ (computed). Reference: Coldren et al., 2nd ed., Section 3.6.2 (Eqs. 3.67 to 3.69) and Example 3.5.

Ques: Why does the laser hop between modes when only the grating section is tuned, and what is the phase section for?

Changing the index of the grating section moves the Bragg wavelength in proportion to the index, $\Delta {\lambda }_g/{\lambda }_g=\Delta {\bar{n}}_{DBR}/{\bar{n}}_{DBR}$, and so moves the loss minimum along the wavelength axis. The axial modes move much less, because only the length $L_{eff}$ of the cavity is in the grating, and the shift is weighted by the group index. The lasing mode is always the cavity mode closest to the loss minimum, so the wavelength stays nearly constant while the Bragg wavelength moves, then jumps by one mode spacing when the next mode becomes closer (mode-hop tuning). The phase section changes the index of a passive part of the cavity, so it moves the whole comb of modes continuously under the grating peak; the grating and the phase section together give access to wavelengths between the mode-hop steps. The active section cannot do this, because the carrier density there is clamped above threshold. In the passive sections the index change is produced by the injected carriers, $\Delta {\bar{n}}_j=(\partial \bar{n}/\partial N){\eta }_i\tau I_j/(q{V}_j)$ with $\partial \bar{n}/\partial N<0$, or by reverse bias through the electro-optic effects, which is faster. As an illustration, for the laser of Example 3.5 a change of 0.01 in the index of the grating section moves the Bragg wavelength by 4.56 nm and the mode comb by 0.45 nm, while the mode spacing is 0.70 nm, so the lasing mode moves by about 4.7 nm in 6 mode steps (computed from the equations). Reference: Coldren et al., 2nd ed., Section 3.6.5 (Eqs. 3.75 to 3.77, Figs. 3.22 and 3.23) and Example 3.5.

Related pages in this vault: Distributed Feedback Laser (DFB), Double-Heterostructure Fabry-Perot Laser, Vertical-Cavity Surface-Emitting Laser (VCSEL), Vertical-External-Cavity Surface-Emitting Laser (VECSEL), Nanolaser (Microdisk, Microring and Photonic Crystal), Resonant-Cavity-Enhanced Photodetector (RCE), Superluminescent Diode (SLD), Resonant Cavity LED (RCLED), Semiconductor Lasers (Edge-Emitting and VCSEL).