Distributed Feedback Laser (DFB)

1. Theory

• Definition: A distributed-feedback (DFB) laser is an edge-emitting laser in which a grating along the whole gain region provides the feedback and selects a single longitudinal mode.
• Bragg condition: The grating period $\Lambda =m{\lambda }_B/2\bar{n}$ (about 241 nm for first order at 1550 nm in InP) reflects only near the Bragg wavelength.
• Coupling constant $\kappa$: Sets grating strength; the reflection of a grating section is $\tanh (\kappa L_g)$, and $\kappa L_g\approx 1$ to $2$ is a common design range (a reading of the threshold-gain charts).
• Distributed reflection: Feedback is spread through the cavity rather than coming only from facets.
• Uniform-grating degeneracy: A uniform grating with AR-coated ends has two equal-threshold modes either side of the Bragg wavelength, with no mode at it.
• Quarter-wave shift: A phase shift in the middle of the grating creates a single mode exactly at the Bragg wavelength.

2. Working Principle

• Mode selection: Only the mode satisfying the grating threshold condition $r_{g1}{r}_{g2}=\pm 1$ with the lowest gain lases.
• Cleaved-facet DFB: An uncoated facet breaks the degeneracy, but its random phase relative to the grating hurts yield.
• Quarter-wave-shifted DFB: Both ends AR-coated, giving a reproducible single mode at ${\lambda }_B$ and symmetric output.
• Threshold and efficiency: $\Gamma g_{th}=A/L_g+{\alpha }_i$, with $A$ read from $\kappa L_g$ charts; e.g. $\approx 72\mathrm{c}{\mathrm{m}}^{-1}$ and ${\eta }_d\approx 0.65$ for a 500 $\mu$m quarter-wave-shifted device.
• Side-mode suppression: The gain margin between the main and next mode gives the side-mode suppression ratio (48.8 dB in the textbook example; commercial devices are usually specified above about 30 to 40 dB).
• Wavelength tuning: The Bragg wavelength shifts slowly with temperature and current (of the order of 0.1 nm/K for InP lasers at 1550 nm, a typical value not taken from the reference), allowing fine tuning without mode hops.

3. Applications

• Telecom and data-centre transmitters: Single-frequency lasers for long-reach and DWDM fibre links, including directly modulated and electro-absorption-modulated lasers.
• Analog and wireless fronthaul: Low-noise lasers for cable TV and 5G radio-over-fibre links.
• Gas sensing: Tunable diode laser absorption spectroscopy for methane, $\mathrm{C}{\mathrm{O}}_2$ and other gases.

Name	Formula	Related Formulae	Variables/Constants
Bragg condition of the grating	$\Lambda =\frac{m{\lambda }_B}{2\bar{n}}$	${\lambda }_B=\frac{2\bar{n}\Lambda }{m}$ ${\beta }_0\Lambda =m\pi$ $\delta =\beta -{\beta }_0$	$\Lambda :periodofthegrating$ $\begin{array}{r}m:orderofthegrating(m=1isthefirst\\ order,inwhichtheperiodishalfofthe\\ averageopticalwavelengthinthemedium;\\ significantreflectionsalsooccuratthe\\ harmonicsofthisfrequency)\end{array}$ ${\lambda }_B:Braggwavelengthinvacuum$ $\begin{array}{r}\bar{n}:averageeffectiveindexoftheguided\\ modeinthegrating\end{array}$ $\begin{array}{r}{\beta }_0:propagationconstantattheBragg\\ wavelength,{\beta }_0=2\pi \bar{n}/{\lambda }_B\end{array}$ $\begin{array}{r}\beta :averagepropagationconstantofthe\\ gratingatthewavelengthconsidered\end{array}$ $\begin{array}{r}\delta :detuningfromtheBraggcondition,\\ whichiszeroattheBraggwavelength(in\\ thereference\delta =\pi \left(\nu -{\nu }_0\right)/{\nu }_{0}foraperiodof\\ aquarterwaveineachlayer;thegeneral\\ ordermandtheformofthefirstequation\\ arethestandardresultandthereference\\ statesthefirstorder)\end{array}$
Reflection and coupling strength of the grating	$r_g=\tanh \left(\kappa L_g\right)$	$\kappa L_g=2mr=\frac{L_g}{\Lambda }\frac{\Delta \bar{n}}{\bar{n}}$ $\kappa \approx \frac{2\Delta \bar{n}}{{\lambda }_B}$ $\left|r_g\right|\approx \kappa L_g\frac{\sin \left(\delta L_g\right)}{\delta L_g}\left(mr<0.2\right)$ $\Delta {\lambda }_{FWHM}\approx \frac{{\lambda }^2}{2{\bar{n}}_g{L}_g}\frac{2.78}{\pi }\left(mr<0.2\right)$	$\begin{array}{r}{r}_g:amplitudereflectioncoefficientofa\\ losslessgratingattheBraggwavelength\\ (theapproximation\tanh \left(2mr\right)isgoodto\\ secondorderinr)\end{array}$ $\begin{array}{r}\kappa :couplingconstant,thereflectionper\\ unitlengthofthegrating\end{array}$ $L_g:lengthofthegrating,equaltom\Lambda$ $m:numberofgratingperiods$ $\begin{array}{r}r:amplitudereflectionateachindex\\ step,r\approx \Delta \bar{n}/\left(2\bar{n}\right),andtherearetwosteps\\ perperiod\end{array}$ $\begin{array}{r}\Delta \bar{n}:differenceoftheeffectiveindices\\ ofthetwosegmentsoftheperiod\end{array}$ $\delta :detuningfromtheBraggcondition$ ${\bar{n}}_g:groupindex$ $\begin{array}{r}\Delta {\lambda }_{FWHM}:fullwidthathalfmaximumofthe\\ powerreflectionpeakofaweakgrating,\\ from{\left(\sin x/x\right)}^2=1/2atx=\delta L_g=1.392(thevalue\\ 3.76/\pi thatisalsoquotedisthewidthathalf\\ oftheamplitude\left|r_g\right|,wherethepowerisone\\ quarterofthepeak;fora\\ stronggratingthetopsaturatesatone\\ andthestopbandwidens;thecoupling\\ constantaboveisforasquare-wave\\ grating,andforasinusoidalgratingthe\\ referencestatesthatitisreducedby\\ \pi /4)\end{array}$
Threshold condition of the DFB laser	$r_{g1}\left({\tilde{\beta }}_{th}\right)r_{g2}\left({\tilde{\beta }}_{th}\right)=\pm 1$	$+1:\lambda /4shifted,-1:unshifted$ $\Gamma g_{th}={\alpha }_i+{\alpha }_m,{\alpha }_m\equiv \frac{A}{L_g}$ $A=\left(\Gamma g_{th}-{\alpha }_i\right)L_g$	$\begin{array}{r}{r}_{g1},r_{g2}:complexreflectioncoefficientsof\\ thetwohalvesofthegratingasseenfrom\\ thecavityatthecentre,withthegain\\ includedinthepropagationconstant\end{array}$ $\begin{array}{r}{\tilde{\beta }}_{th}:complexpropagationconstantofthe\\ gratingatthreshold(itsrealpartgives\\ thewavelengthanditsimaginarypartthe\\ gain)\end{array}$ $\begin{array}{r}\pm 1:theunshiftedDFBhasanactivecavity\\ ofaquarterwaveandnosolutionatthe\\ Braggwavelength,sotwomodes,equally\\ spacedonthetwosidesofit,reach\\ thresholdtogether;the\\ quarter-wave-shiftedDFB(ahalfwave\\ betweenthetwogratings)hasamodeat\\ theBraggwavelength\end{array}$ $\begin{array}{r}\Gamma ,g_{th}:confinementfactorandmaterial\\ gainatthreshold\end{array}$ ${\alpha }_i:internalloss$ $\begin{array}{r}{\alpha }_m:equivalentmirrorlossoftheDFB,\\ definedbythethresholdgain\end{array}$ $\begin{array}{r}A:normalizedthresholdmodalgain,which\\ dependson\kappa L_{g}andonthemodeandis\\ foundnumerically(thereferenceplotsit\\ forstandardandquarter-wave-shifted\\ laserswithantireflectioncoatings,and\\ for\kappa L_g=1itis3.1fortheshifted\\ laser)\end{array}$ $\begin{array}{r}(again-coupledDFB,inwhichthegrating\\ modulatesthegainandnottheindex,has\\ amodeattheBraggwavelengthwithoutthe\\ shift)\end{array}$
Differential efficiency and side-mode suppression	${\eta }_d={\eta }_i\frac{A}{\Gamma g_{th}{L}_g}={\eta }_i\frac{{\alpha }_m}{{\alpha }_i+{\alpha }_m}$	$SMSR=10\log \left[\frac{F_1\left({\lambda }_0\right){\alpha }_m\left({\lambda }_0\right)}{F_1\left({\lambda }_1\right){\alpha }_m\left({\lambda }_1\right)}\frac{{\alpha }_m\left({\lambda }_1\right)-{\alpha }_m\left({\lambda }_0\right)}{\delta G}+1\right]$ $\delta G=\left({\alpha }_i+{\alpha }_m\right){\beta }_{sp}{\eta }_r\frac{I_{th}}{I-I_{th}}$	$\begin{array}{r}{\eta }_d:differentialquantumefficiencyoutof\\ bothends(forthequarter-wave-shifted\\ DFBhalfofthepowerleaveseachend)\end{array}$ ${\eta }_i:injectionefficiency$ $\begin{array}{r}SMSR:side-modesuppressionratioindBof\\ themainmodeat{\lambda }_{0}overthenextmodeat\\ {\lambda }_1\end{array}$ $\begin{array}{r}{F}_1:fractionoftheoutputpowerthat\\ leavesthroughfacet1atthewavelength\\ indicated\end{array}$ $\begin{array}{r}{\alpha }_m\left({\lambda }_0\right),{\alpha }_m\left({\lambda }_1\right):equivalentmirrorlosses\\ ofthemainandthenextmode\end{array}$ $\begin{array}{r}\delta G:gainmarginthatthespontaneous\\ emissionofthemainmodegivestothe\\ nextmodeabovethreshold\end{array}$ ${\beta }_{sp}:spontaneousemissionfactor$ ${\eta }_r:radiativeefficiencyoftherecombination$ $\begin{array}{r}I,I_{th}:currentandthresholdcurrent(the\\ expressionneglectsthechangeofthegain\\ withthewavelengthbetweenthetwomodes,\\ andthesecondfactoristhedifferenceof\\ themirrorlossesdividedbythegain\\ margin)\end{array}$

Important Questions

Ques: Why does a plain DFB laser lase in two modes, and what does the quarter-wave shift do?

With a uniform grating and antireflection-coated ends the threshold condition is $r_{g1}{r}_{g2}=-1$, which has no solution at the Bragg wavelength, so the first modes are a pair at equal distances on the two sides of it, and they reach threshold together. In practice an uncoated cleaved facet breaks the degeneracy if the grating is not too strong ($\kappa L_g\le 1$), but the phase of the cleave relative to the grating is random, which is a problem for the yield; the best case is when the end reflection is in quadrature with the grating reflection. A phase shift in the middle of the grating, a quarter wave in the grating or half a wave of cavity between the two halves, gives $r_{g1}{r}_{g2}=1$ and a mode at the Bragg wavelength. The reference finds that a physical shift of somewhat less than a tenth of a quarter wave is already enough to break the degeneracy and give a good side-mode suppression, so the exact length is not critical, but the shift has to be close to the centre of the device. For $\kappa L_g=1$ the normalized threshold gain $A$ of the shifted laser is 3.1 and that of the next mode is 4.4, and for the operating point of Example 3.7 ($I=4I_{th}$, ${\beta }_{sp}=1.25\times 10^{-5}$, ${\eta }_r=0.8$) these give a side-mode suppression of 48.8 dB (the printed value, which I reproduced). Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 3.7.1 (Eqs. 3.86 and 3.87, Figs. 3.25 and 3.26), Section 3.7.2 (Figs. 3.27 to 3.29) and Section 3.7.3 with Example 3.7 (the 48.8 dB).

Ques: What are the threshold gain, the efficiency and the wavelength of a typical DFB laser?

The reference gives a worked example of a 1550 nm quarter-wave-shifted InGaAsP/InP multiple-quantum-well DFB laser with $\Gamma =0.06$, ${\eta }_i=75\%$, ${\alpha }_i=10$ cm${}^{-1}$, $\kappa =20$ cm${}^{-1}$, $\bar{n}=3.21$, $L=500$ $\mu$m and antireflection-coated facets. Here $\kappa L_g=1.0$ and the chart gives $A=3.1$, so $\Gamma g_{th}=3.1/L_g+{\alpha }_i=72$ cm${}^{-1}$ and ${\eta }_d=0.75\times 3.1/(72\times 0.05)=0.646$ for both ends (0.323 for one end), at the Bragg wavelength of 1550 nm. For the standard (unshifted) DFB with one antireflection-coated end and a high-reflectivity end of the best phase, $A=1.2$, so $\Gamma g_{th}=34$ cm${}^{-1}$ and ${\eta }_d=0.529$, nearly all from the antireflection-coated end, and the lasing mode has a detuning of $\delta L_g=0.35$. I reproduce all the gains and the efficiencies (computed). I do not reproduce the lasing wavelength printed for the last case, 1551.087 nm: with $\beta ={\beta }_0+\delta$, $\delta =0.35/L_g=7$ cm${}^{-1}$ and $2\pi \bar{n}/{\lambda }_0=1.3012\times 10^5$ cm${}^{-1}$ the wavelength is 1549.92 nm (or 1550.08 nm for the mode on the other side of the Bragg wavelength), a shift of 0.08 nm and not +1.09 nm, so the printed value is most likely a misprint for 1550.08 nm. The normalized values $A$ are read from the book’s charts and not from a formula. Reference: Coldren et al., 2nd ed., Section 3.7, Example 3.6 (Figs. 3.26b and 3.28b); the check of the wavelength is my own calculation.

Ques: What grating period and coupling strength does a 1550 nm DFB laser need?

In the first order the period is half the wavelength in the medium: for $\bar{n}=3.21$ and ${\lambda }_B=1550$ nm, $\Lambda ={\lambda }_B/(2\bar{n})=241$ nm (computed). For the coupling constant of the example above, $\kappa =20$ cm${}^{-1}$, the square-wave relation $\kappa =2\Delta \bar{n}/{\lambda }_B$ requires $\Delta \bar{n}=\kappa {\lambda }_B/2=1.6\times 10^{-3}$ (computed). Since $\kappa L_g=1$ was the case treated, a longer or stronger grating changes the mode gains: the value of $\kappa L_g$ between about 1 and 2 is the usual compromise in a design, which is my statement from the charts of the reference and not a statement it makes. For a sinusoidal grating the reference says that $\kappa$ is reduced by $\pi /4$ compared with the square wave, so a larger index modulation is needed. Reference: Coldren et al., 2nd ed., Section 3.5.1 (the Bragg condition, Eq. 3.43) and Section 3.5.2 (Eq. 3.59 and the following text for $\kappa$ and the factor of $\pi /4$, Eq. 3.55 for $\tanh (\kappa L_g)$); the numbers are my own calculation.

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