Double-Heterostructure Fabry-Perot Laser

1. Theory

• Definition: A double-heterostructure Fabry-Perot laser is an edge-emitting diode in which a thin low-gap active layer between two high-gap claddings confines carriers and light, and the cleaved facets form the optical cavity.
• Foundation page: Quantum-well, quantum-dot, DFB, DBR and VCSEL lasers reuse the threshold, power and mode relations of this page.
• Carrier confinement: The band-edge steps trap electrons and holes together in the 0.1 to 0.2 $\mu$m active layer, raising recombination.
• Optical confinement: The active layer’s higher refractive index forms a waveguide; the confinement factor $\Gamma$ is the fraction of the mode overlapping the active layer.
• Stimulated emission and gain: With population inversion, photons stimulate identical photons, giving optical gain $g$ that rises with carrier density.
• Fabry-Perot cavity: Cleaved facets ($R\approx 0.3$ for $n\approx 3.5$) provide feedback; mirror loss ${\alpha }_m=\ln (1/R)/L$.

2. Working Principle

• Below threshold: The device emits weak spontaneous light like an LED.
• Threshold: Lasing starts when modal gain equals total loss, $\Gamma g_{th}={\alpha }_i+{\alpha }_m$.
• Gain clamping: Above threshold the carrier density stays at $N_{th}$, so extra current turns into stimulated photons.
• Output power: $P={\eta }_i\left[{\alpha }_m/({\alpha }_i+{\alpha }_m)\right](h\nu /q)(I-I_{th})$; the differential efficiency falls for longer cavities.
• Parameter extraction: Plotting $1/{\eta }_d$ against cavity length gives $1/{\eta }_i$ from the intercept; the slope is ${\alpha }_i/[{\eta }_i\ln (1/R)]$, so ${\alpha }_i$ needs both.
• Longitudinal modes: Modes are spaced by ${\lambda }^2/2n_{g}L$ (about 0.27 nm for a 300 $\mu$m cavity at 850 nm), much less than the gain width, so several modes can lase and hop with current and temperature.

3. Applications

• Current status: Bulk double-heterostructure active layers have been largely replaced by quantum wells, but the Fabry-Perot cavity is still widely used.
• Multimode Fabry-Perot diode lasers: Optical disc readers, laser pointers and levels, and high-power multimode pump and materials-processing diode bars, where single-frequency output is not needed.

Name	Formula	Related Formulae	Variables/Constants
Threshold gain condition and mirror loss	$\Gamma g_{th}={\alpha }_i+{\alpha }_m$	${\alpha }_m=\frac{1}{2L}\ln \left(\frac{1}{R_1{R}_2}\right)$ $R={\left(\frac{n_{eff}-1}{n_{eff}+1}\right)}^2$ $\Gamma =\frac{{\int }_{-d/2}^{d/2}{E}^{2}dz}{{\int }_{-\infty }^{\infty }{E}^{2}dz}$	$\begin{array}{r}\Gamma :opticalconfinementfactor,the\\ fractionofthemodepowerintheactive\\ layer(Eisthefieldoftheguidedmode\\ acrossthelayersanddthethicknessof\\ theactivelayer)\end{array}$ $\begin{array}{r}{g}_{th}:material(power)gainoftheactive\\ layeratthreshold\end{array}$ $\begin{array}{r}{\alpha }_i:internallossperunitlength(the\\ absorptionandscatteringinthecladding\\ andtheactivelayerandthewaveguide\\ roughness)\end{array}$ $\begin{array}{r}{\alpha }_m:mirrorlossperunitlength,theloss\\ bythelightthatleavesthroughthetwo\\ facets\end{array}$ $L:lengthofthecavity$ $\begin{array}{r}{R}_1,R_2:powerreflectivitiesofthetwo\\ facets(theproductoftheamplitude\\ reflectioncoefficients,r_1{r}_2=\sqrt{R_1{R}_2},is\\ themeanmirrorreflectioncoefficientR\\ ofthereference,sothat{\alpha }_m=\ln \left(1/R\right)/Lfor\\ equalfacetstoo)\end{array}$ $\begin{array}{r}{n}_{eff}:effectiveindexoftheguidedmode\\ (theFresnelformulaisthereflectivity\\ ofacleaved,uncoatedfacetfornormal\\ incidence,about0.31for{n}_{eff}=3.5,and\\ about0.32isthevalueusedforGaAsand\\ InPlasers)\end{array}$ $\begin{array}{r}(theconditionistherequirementthatthe\\ fieldrepeatsitselfafteraroundtrip,\\ R_1{R}_2\exp \left[2\left(\Gamma g-{\alpha }_i\right)L\right]=1)\end{array}$
Gain, carrier density and threshold current density	$J_{th}=\frac{qd}{{\eta }_i}\left[A{N}_{th}+B{N}_{th}^2+C{N}_{th}^3\right]$	$g=g_{0N}\ln \left(\frac{N}{N_{tr}}\right)$ $N_{th}=N_{tr}\exp \left(\frac{{\alpha }_i+{\alpha }_m}{\Gamma g_{0N}}\right)$ $a=\frac{\partial g}{\partial N}=\frac{g_{0N}}{N}$ $J_{th}\approx \frac{qdB{N}_{th}^2}{{\eta }_i}$ $J_{th}\left(T\right)=J_0\exp \left(\frac{T}{T_0}\right)$	$\begin{array}{r}{J}_{th}:thresholdcurrentdensity(currentper\\ unitareaofthejunction;theequationis\\ thesteady-statebalance\\ {\eta }_i{I}_{th}/\left(qV\right)=AN+B{N}^2+C{N}^{3}atthreshold,with\\ thevolumeVequaltotheareatimesd)\end{array}$ $q:elementarycharge$ $\begin{array}{r}d:thicknessoftheactivelayer(the\\ carriersareconfinedinitbythe\\ heterojunctions,sothecurrentneededfor\\ agivencarrierdensityfallswiththe\\ thickness)\end{array}$ $\begin{array}{r}{\eta }_i:injectionefficiency,thefractionof\\ thecurrentthatisinjectedintothe\\ activelayerandrecombinesthere\end{array}$ $\begin{array}{r}A,B,C:coefficientsofthedefect\\ (trap),radiative(bimolecular)andAuger\\ recombination,asintheABCmodelofthe\\ LightEmittingDiodepage(avalueofBof\\ about10^{-10}c{m}^3/sisgivenfortheIII-V\\ lasers)\end{array}$ $\begin{array}{r}{N}_{th}:carrierdensityintheactivelayerat\\ threshold,whichstaysclampedatthis\\ valueabovethresholdbecausethegainis\\ pinnedatitsthresholdvalue\end{array}$ $\begin{array}{r}{g}_{0N}:empiricalgaincoefficientofthe\\ two-parameterlogarithmicfitofthegain\\ againstthecarrierdensity,validfor\\ positivegain\end{array}$ $\begin{array}{r}{N}_{tr}:transparencycarrierdensity,atwhich\\ thegainiszero\end{array}$ $\begin{array}{r}a:differentialgain,theslopeofthe\\ gainagainstthecarrierdensity,which\\ fallsasthedensityrisesinthismodel\\ (theapproximateexpressionfor{J}_{th}holds\\ forthebestmaterials,inwhichthe\\ recombinationatthresholdisdominatedby\\ theradiativeterm)\end{array}$ $\begin{array}{r}{T}_0:characteristictemperatureofthe\\ thresholdcurrent,asmallervaluemeaning\\ astrongerdependenceonthetemperature\\ (theexponentialformisanapproximate\\ modeloveralimitedrangeoftemperature;\\ valuesabove120Kareobservedforgood\\ GaAs/AlGaAsDHlasersnearroom\\ temperatureand50to70KforInGaAsP/InP\\ lasers)\end{array}$
Output power, differential efficiency and cavity length	$P={\eta }_i\frac{{\alpha }_m}{{\alpha }_i+{\alpha }_m}\frac{h\nu }{q}\left(I-I_{th}\right)$	${\eta }_d={\eta }_i\frac{{\alpha }_m}{{\alpha }_i+{\alpha }_m}$ $\frac{1}{{\eta }_d}=\frac{1}{{\eta }_i}\left[1+\frac{{\alpha }_i}{{\alpha }_m}\right]$ $S_e=\frac{dP}{dI}={\eta }_d\frac{h\nu }{q}$ $F_1=\frac{\left(1-R_1\right)\sqrt{R_2}}{\left(1-R_1\right)\sqrt{R_2}+\left(1-R_2\right)\sqrt{R_1}}$	$\begin{array}{r}P:totalopticalpowerleavingthrough\\ bothfacets,forI>I_{th}\end{array}$ $\begin{array}{r}{\eta }_i:injectionefficiency(abovethreshold\\ alltheextracarriersrecombineby\\ stimulatedemission)\end{array}$ $\begin{array}{r}\frac{{\alpha }_m}{{\alpha }_i+{\alpha }_m}:fractionofthestimulated\\ photonsthatleavethroughthemirrorsand\\ arenotlostinside\end{array}$ $h\nu :photonenergy$ $I,I_{th}:currentandthresholdcurrent$ $\begin{array}{r}{\eta }_d:externaldifferentialquantum\\ efficiency,thephotonsoutperelectron\\ injectedabovethreshold\end{array}$ $\begin{array}{r}{S}_e:slopeefficiencyofthelight-current\\ curve(inW/A,forbothfacetstogether)\end{array}$ $\begin{array}{r}{F}_1:fractionofthepowercoupledoutby\\ themirrorsthatleavesthroughfacet1\\ forlosslessfacets(itisonehalffor\\ equalfacetsandoneiffacet2isa\\ perfectreflector,andthedifferential\\ efficiencyoffacet1is{F}_1{\eta }_d)\end{array}$ $\begin{array}{r}(plotting1/{\eta }_{d}againstthecavitylengthL\\ givesastraightlinewhoseinterceptis\\ 1/{\eta }_{i}andwhoseslopeis\\ {\alpha }_i/\left[{\eta }_i\ln \left(1/R\right)\right],\\ since{\alpha }_i/{\alpha }_{m}risesinproportiontoL;{\alpha }_i\\ followsfromtheslopetimes{\eta }_i\ln \left(1/R\right))\end{array}$
Longitudinal modes of the cavity	$\Delta \lambda =\frac{{\lambda }^2}{2n_{g}L}$	$2n_{eff}L=m\lambda$ $n_g=n_{eff}-\lambda \frac{d{n}_{eff}}{d\lambda }$	$\begin{array}{r}\Delta \lambda :wavelengthspacingofneighbouring\\ longitudinalmodesoftheFabry-Perot\\ cavity(thefreespectralrange)\end{array}$ $\lambda :wavelengthinvacuum$ $\begin{array}{r}{n}_g:groupindexoftheguidedmode,which\\ istypically20to30percentlargerthan\\ theindex(valuesofabout4.5at850to\\ 980nm,4at1300nmand3.8at1550nm\\ areusedinthereference)\end{array}$ $L:cavitylength$ $m:integerorderofthemode$ $\begin{array}{r}{n}_{eff}:effectiveindex(theresonance\\ conditionisthattheround-tripphaseis\\ amultipleof2\pi ,andthelasingmodesare\\ thoseofthissetthatlieunderthegain\\ spectrumwiththehighestnetgain)\end{array}$
Carrier and optical confinement of the double heterostructure	$\Gamma \approx \frac{D^2}{2+D^2}$	$D=\frac{2\pi d}{\lambda }\sqrt{n_a^2-n_c^2}$ $n\left(A{l}_{x}G{a}_{1-x}As\right)\approx 3.590-0.710x+0.091x^2$ $\Delta E_c\approx 0.65\Delta E_g,\Delta E_v\approx 0.35\Delta E_g$	$\begin{array}{r}\Gamma :opticalconfinementfactorofa\\ symmetricslabwaveguideofanactive\\ layerbetweentwocladdings(an\\ approximateexpressionforthefundamental\\ modewhentheindexdifferenceissmall)\end{array}$ $\begin{array}{r}D:normalizedfrequency(normalized\\ thickness)oftheactivelayer\end{array}$ $d:thicknessoftheactivelayer$ $\begin{array}{r}{n}_a,n_c:refractiveindicesoftheactive\\ layerandofthecladding\end{array}$ $\begin{array}{r}x:aluminiumfractionofthecladding(the\\ indexfitisthatofH.C.CaseyandM.B.\\ Panish,HeterostructureLasers,Academic\\ Press,1978,foraphotonenergyof1.38eV,\\ near900nm;itschordtox=1istheoften\\ quotedlineardifferenceofabout0.62x)\end{array}$ $\begin{array}{r}\Delta E_c,\Delta E_v:conduction-bandand\\ valence-bandoffsetsatthe\\ heterojunction,and\Delta E_{g}thedifferenceof\\ thebandgaps(theratiosarethoseofthe\\ AlGaAs/GaAssystemforthedirect-gap\\ rangex\lesssim 0.45,inwhichabouttwothirdsof\\ theoffsetisintheconductionband(values\\ from0.6to0.67arequoted),andfortheInGaAsP/InP\\ systemonly\\ about40percentis;theoffsetsconfine\\ theelectronsandholesintheactive\\ layer,andthelowerindexofthecladding\\ guidesthelight)\end{array}$

Important Questions

Ques: What does the threshold condition mean, and how much of the threshold gain is due to the mirrors?

At threshold the gain of the active layer must replace the losses of the cavity: the internal loss ${\alpha }_i$ and the mirror loss ${\alpha }_m$, where the light leaving through the facets is counted as a loss because it is not available for the amplification. Above threshold the gain is clamped at this value and the carrier density stays at $N_{th}$, so the extra current produces only stimulated photons. As an illustration with assumed values, take cleaved facets with $n_{eff}=3.5$, so $R=0.31$, a length of 500 $\mu$m and ${\alpha }_i=10\mathrm{c}{\mathrm{m}}^{-1}$. The mirror loss is ${\alpha }_m=\ln (1/R)/L=23.5\mathrm{c}{\mathrm{m}}^{-1}$ and the modal gain needed is 33.5 cm${}^{-1}$, of which 70% is the mirror loss. For a confinement factor of 0.05 (also assumed) the material gain needed is $g_{th}=670\mathrm{c}{\mathrm{m}}^{-1}$ (computed). A coating of high reflectivity on one facet or a longer cavity lowers ${\alpha }_m$ and so the threshold gain, at the cost of a lower differential efficiency. Reference: L.A. Coldren, S.W. Corzine and M.L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 2.5, Eqs. (2.21) to (2.28) (the round-trip condition, $\Gamma g_{th}=⟨{\alpha }_i⟩+{\alpha }_m$ with ${\alpha }_m=(1/L)\ln (1/R)$ and $R=r_1{r}_2\approx 0.32$ for cleaved GaAs or InP); the carrier clamping is Eqs. (2.33) and (2.34). The numbers are mine.

Ques: Why did the double heterostructure replace the homojunction laser?

In the double heterostructure a thin slab of undoped low-gap active material (typically 0.1 to 0.2 $\mu$m) is placed between p-type and n-type claddings of higher gap, so that a potential well is formed in the band edges for the electrons and the holes injected from the two sides, and they are captured and confined together, which raises their probability of recombining. The active layer also has a higher refractive index than the claddings, so that a dielectric waveguide is formed that confines the light to the same layer. A homojunction has neither step, so neither the carriers nor the light are held in a thin layer. The simple p-n junction theory, which assumes that the carriers crossing the depletion region do not recombine in it, is not appropriate for a laser, in which all the injected carriers should recombine in the active layer; the carriers that escape over the heterobarrier are the leakage current, which is part of ${\eta }_i$. The thickness of the active layer is a compromise: a thinner layer lowers the current needed for a given carrier density, but for a small normalized thickness the confinement factor falls as $V^2$ (the slab expression of the page), which raises the carrier density needed. A historical account gives about 30 kA/cm${}^2$ for the earlier GaAs lasers, which broke quickly from the heat, against about 1 kA/cm${}^2$ for the double heterostructure, with continuous operation at room temperature reported in 1970 by Alferov and co-workers and, at about the same time, by Hayashi and Panish at Bell Telephone Laboratories (the first confirmation at 24 ${}^{\circ }$C on 1 June 1970 in that account). Reference: Coldren et al., Diode Lasers and Photonic Integrated Circuits, 2nd ed., Section 1.4 (the confinement of the carriers and the light, the thickness of 0.1 to 0.2 $\mu$m, the statement that the double heterostructure made the diode laser practical and led to two Nobel prizes in physics in 2000, the leakage current) and Eq. (A3.16) of Appendix 3 ($\Gamma \approx V^2/(2+V^2)$ for the fundamental mode of the symmetric slab at a small index difference, with the exact Eq. (A3.15)); Shimura, “Betting on the Room Temperature Continuous Oscillation Laser,” shmj.or.jp/shimura/shimura_E/ssis_shimura3_11E.html (the current densities and the dates; a secondary source whose own references were not checked); I. Hayashi, M.B. Panish, P.W. Foy and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17, 109 (1970).

Ques: How does the slope of the light-current curve depend on the cavity length?

The differential efficiency is ${\eta }_d={\eta }_i{\alpha }_m/({\alpha }_i+{\alpha }_m)$, and ${\alpha }_m$ falls as $1/L$, so a longer cavity loses a larger part of its photons inside. For $R=0.31$ and an assumed ${\alpha }_i=10\mathrm{c}{\mathrm{m}}^{-1}$ and ${\eta }_i=1$ the differential efficiency is 0.82 for $L=250\mu$m, 0.70 for 500 $\mu$m and 0.54 for 1000 $\mu$m (computed), and a lower ${\eta }_i$ scales all of them. The plot of $1/{\eta }_d$ against $L$ is a straight line, and this is the standard way to extract ${\eta }_i$ (from the intercept) and ${\alpha }_i$ (from the slope) of a wafer, from lasers of several lengths cut from it. The slope efficiency in W/A is ${\eta }_{d}h\nu /q$, for example about 1.0 W/A for ${\eta }_d=0.70$ at 850 nm (computed, $h\nu =1.46$ eV), counting both facets. Reference: Coldren et al., Diode Lasers and Photonic Integrated Circuits, 2nd ed., Eqs. (2.39) to (2.42) for the power and the differential efficiency, and Section 2.8.1, Eq. (2.59), $1/{\eta }_d=\left[{\alpha }_i/({\eta }_i\ln (1/R))\right]L+1/{\eta }_i$, for the extraction of ${\alpha }_i$ and ${\eta }_i$ from lasers of several lengths; the fraction emitted from each facet for unequal facets is Eqs. (3.31) and (3.32) of Section 3.3. The numbers are mine.

Ques: How many longitudinal modes does a Fabry-Perot laser have?

The cavity supports a comb of modes with the spacing $\Delta \lambda ={\lambda }^2/(2n_{g}L)$. For $\lambda =850$ nm, $L=300\mu$m and $n_g=4.5$ (the value suggested in the reference for 850 to 980 nm) the spacing is 0.27 nm (computed), and the gain spectrum is tens of nanometres wide, so many modes are possible. Above threshold the modes compete for the same carriers, and in a well-built laser the mode of the highest gain takes most of the power, but a Fabry-Perot laser has no element that selects one mode, so several modes often lase together and the mode that dominates can jump with the current and the temperature. This is the reason for the distributed-feedback and distributed-Bragg-reflector lasers, which are the next pages. Reference: Coldren et al., Diode Lasers and Photonic Integrated Circuits, 2nd ed., Section 2.5, Eqs. (2.29) and (2.32) for the mode condition and the spacing, Table 2.1 for the group indices, and Example 2.1 (0.395 nm for 1.55 $\mu$m, 800 $\mu$m and $n_g=3.8$, which the formula reproduces); the statement on mode competition is a general description, not traced to a specific passage of the book.

Related pages in this vault: Distributed Bragg Reflector Laser (DBR), Distributed Feedback Laser (DFB), Vertical-Cavity Surface-Emitting Laser (VCSEL), Quantum-Well Laser (Single and Multiple Quantum Well), Nanolaser (Microdisk, Microring and Photonic Crystal), Superluminescent Diode (SLD), Semiconductor Lasers (Edge-Emitting and VCSEL), Light Emitting Diode (Homojunction and Heterojunction LED), Resonant-Cavity-Enhanced Photodetector (RCE).