Traveling Wave Model for Frequency Comb Generation in Single Section Quantum Well Diode Lasers
Mark Dong, Niall M. Mangan, J. Nathan Kutz, Steven T. Cundiff, Herbert G. Winful
aa r X i v : . [ phy s i c s . op ti c s ] J u l Traveling Wave Model for Frequency Comb Generation in SingleSection Quantum Well Diode Lasers
Mark Dong ∗ Department of Electrical Engineering and Computer Science, University of Michigan,1301 Beal Avenue, Ann Arbor, 48109-2122 andDepartment of Physics, University of Michigan,450 Church Street, Ann Arbor, MI 48109-1040
Niall M. Mangan † and J. Nathan Kutz ‡ Department of Applied Mathematics, University of WashingtonLewis Hall 202, Box 353925, Seattle, WA 98195-3925
Steven T. Cundiff § Department of Physics, University of Michigan,450 Church Street, Ann Arbor, MI 48109-1040 andDepartment of Electrical Engineering and Computer Science, University of Michigan,1301 Beal Avenue, Ann Arbor, 48109-2122
Herbert G. Winful ¶ Department of Electrical Engineering and Computer Science, University of Michigan,1301 Beal Avenue, Ann Arbor, 48109-2122 (Dated: September 12, 2018) bstract We present a traveling wave model for a semiconductor diode laser based on quantum wells.The gain model is carefully derived from first principles and implemented with as few phenomeno-logical constants as possible. The transverse energies of the quantum well confined electrons arediscretized to automatically capture the effects of spectral and spatial hole burning, gain asym-metry, and the linewidth enhancement factor. We apply this model to semiconductor opticalamplifiers and single-section phase-locked lasers. We are able to reproduce the experimental re-sults. The calculated frequency modulated comb shows potential to be a compact, chip-scale combsource without additional external components. ∗ [email protected] † [email protected] ‡ [email protected] § cundiff@umich.edu ¶ [email protected] . INTRODUCTION Optical frequency combs have had a great impact on the fields of ultrafast and nonlinearoptics, frequency metrology, and optical spectroscopy in the past few decades [1]. Frequencycombs are useful in many applications, including absolute frequency measurement [2], multi-heterodyne spectroscopy [3], optical atomic clocks [4], and arbitrary waveform synthesis [5].Current methods for comb generation include the mode-locking of Ti:Sapphire laser [6] andfiber lasers [7], as well as parametric frequency conversion due to the Kerr nonlinearity inpassive microresonators [8]. These approaches, however, require many discrete optical orfiber components, careful alignment, and bulky pump lasers and amplifiers, thus limitingtheir general utility outside of laboratories. There is thus a need for portable, efficient,robust, and chip-scale comb sources that can be deployed in the field and greatly extend theusefulness of frequency combs.Mode locked diode lasers offer the possibility of direct generation of frequency combsfrom a chip-scale device [9, 10]. Typically, passively mode-locked diode lasers comprise twosections: a gain section and a reverse-biased saturable absorber section that leads to theformation of a periodic train of short pulses and hence a comb in the frequency domain.The major obstacle in generating short pulses in diode lasers stems from the nonlinear phaseshifts that occur due to fast carrier dynamics [11], essentially limiting the pulse width insidethe cavity. However, single-section diode lasers without saturable absorbers can also operatein a multimode phase-synchronized state known as frequency-modulated (FM) mode locking[12]. In the ideal FM mode locked state, the output is a continuous wave in time but thefrequency modulation results in a set of comb lines with a fixed, non-zero phase difference.Such FM modelocked operation has been studied most intensively in quantum dot (QD)[13, 14] and quantum dash [15] (QDash) lasers, but has also been observed in quantum well(QW) [16, 17] and bulk semiconductor lasers [12]. While some theoretical work has beendone for how these combs emerge in a QD single-section laser [13], a detailed model for FMcomb generation in QW diode lasers is still lacking.There have been many models published for semiconductor quantum well lasers withvarying degrees of complexity. The simplest models include only a single rate equationand photon density variable [18, 19], while more complex models may use multiple rateequations and more complex forms of the material polarization [20–25] with varying degrees3f phenomenological expressions and constants inserted. However, the existing models areusually insufficiently detailed to explain why FM combs arise in some QW lasers and notothers, nor do they indicate which parameters need to be optimized for comb generation.The difficulty in modeling these types of diode lasers stems from the many nonlinear effectsin the semiconductor laser cavity that must be properly accounted for.In this paper we present a detailed traveling wave model of Fabry-Perot QW diode lasersthat elucidates the origin of FM self-mode locking and the formation of frequency combs inthese lasers. The model takes into account the multiple cavity modes as a modulation ofthe electric field envelope, spectral and spatial hole burning, carrier induced refractive indexshift, some intraband carrier dynamics, and cavity dispersion. The gain is derived from firstprinciples, starting from the modified Semiconductor Bloch equations with carrier-carrierinteractions described through rate equations. Our approach follows that of previous works[13, 26] but tailored to quantum well nanostructures.
II. THEORETICAL MODEL
We start by giving an overview of our model from a physical perspective and write downonly the essential equations to be solved while the detailed mathematical derivation is rele-gated to the appendices. The basic schematic for the model is shown in Figure 1. Electronsinjected from the n side (holes from the p side) drop down to the separate confinement het-erostructure (SCH) layer, and become trapped in the quantum well. The most importantdifference between our quantum well model and previous models is that, for the carrierstrapped in the quantum well, we have discretized the carrier equations in energy space andcombined them with a truly multimode wave equation. While this approach does increasethe number of carrier equations to solve, it captures all the important dynamics of the mul-tiple Fabry-Perot cavity modes and their interactions with carriers at different transverseenergies.In a semiconductor, the carriers are typically confined in some type of nanostructure,such as a 2-D quantum well, a 1-D quantum wire or a 0-D quantum dot or dash, with anenergy distribution determined by the N -dimensional density of states D N − Dr and occupationprobability for electrons ( e ) or holes ( h ) ρ e,h . We assume that the microscopic coherencedecays sufficiently quickly such that each individual carrier emits light in a characteristic4orentzian spectral lineshape with a homogenous linewidth 2Γ as determined by intrabandrelaxation effects. However, each group of carriers will emit at a different central frequency.In quantum wells in particular, the carriers have momenta in the unconfined directions thatwe quantify as the transverse energy E t , and it is these energies that modify the transitionfrequency for all carriers with energy E t [27]. By integrating all carrier Lorentzians inenergy space for each quantum well confined state, we have a gain term that accountsfor homogenous and inhomogenous broadening, the asymmetric nature of the gain due tooccupation levels and density of states, and the carrier-induced refractive index change.These complex Lorentzians also offer a simple way to calculate the real and imaginary partsof the gain without resorting to the Kramers-Kronig relations.The electric field of the light wave in the cavity is taken as a sum of forward and backwardcomponents E ( z, t ) = E + ( z, t ) e ik z + E − ( z, t ) e − ik z (1)whose amplitudes satisfy the slowly-varying envelope equation ± ∂∂z E ± ( z, t ) + 1 v g ∂∂t E ± ( z, t ) = Γ xy ω ik c ǫ h P tot ( t ) e ∓ ik z i (2)where the angular brackets signify averaging over a few wavelengths. Here, v g = c/n is thegroup velocity, n is the group refractive index, Γ xy is the transverse confinement factor, ω is the central photon frequency (the choice of ω can be arbitrary but is generally chosen tobe the transition frequency at the band edge), and k = n ω /c . The material polarization P tot is obtained from the Bloch Equations as tailored to semiconductors [28]: i ~ ∂p ( k , t ) ∂t = ( ~ ω − ∆ E cv ( k )) p ( k , t ) − d cv E ( k , z, t )( ρ e ( k , t ) + ρ h ( k , t ) − − i ~ p ( k , t ) T (3a) ∂ρ e ( k , t ) ∂t = − ~ Im [ d ∗ cv E ∗ ( z, t ) p ( k , t )] + ∂ρ e ( k , t ) ∂t | relax (3b) ∂ρ h ( k , t ) ∂t = − ~ Im [ d ∗ cv E ∗ ( z, t ) p ( k , t )] + ∂ρ h ( k , t ) ∂t | relax (3c)5here p ( k , t ) is the microscopic polarization, ρ e,h ( k , t ) is the occupation probability of elec-trons and holes, d cv is the dipole matrix element, ∆ E cv ( k ) is the transition energy betweenthe conduction and valence bands, and T = 1 / Γ is the intraband relaxation time whichgives rise to homogenous broadening. It is important to note that these equations are inthe time domain but are parameterized by the wavevector k and hence represent the timeevolution of the subset of carriers with momentum k .A key simplification in our model is to assume that the intraband scattering is sufficientlyfast to warrant the microscopic polarization adiabatically following the changes in carrierpopulation. For modeling ultra-short pulses, this assumption may no longer hold and a fullset of polarization equations will need to be solved dynamically. Integrating Eq. 3a, weobtain a time domain expression for the microscopic polarization in terms of the occupationprobabilities and the electric field: p ( k , t ) = id cv E ( k , z, t )2 ~ Z t −∞ dt ′ E ( z, t ′ ) e − ( i ∆ Ecv ( k ) ~ − ω ) ( t − t ′ ) − Γ( t − t ′ ) ( ρ e ( k , t ′ ) + ρ h ( k , t ′ ) − / Γ and can be takenout of the integral, with t ′ replaced by t . The remaining convolution integral is then definedas the filtered field [13] F ( k , z, t ) = Γ Z t −∞ dt ′ e i ( ∆ Ecv ( k ) ~ − ω )( t − t ′ ) − Γ( t − t ′ ) E ( z, t ′ ) (5)The filtered field consists of all the components that interact with the population ρ e,h ( k , t ). Here the transition frequency is defined such that ~ ω is the transition energy fora confined electron-hole pair with zero transverse energy and satisfies∆ E cv ( k ) ~ − ω = E t ( k ) ~ Thus each discretized carrier group will have a different filtering frequency defined by thetransverse energy E t . The time-dependent microscopic polarization reduces to a simpleexpression: 6 ( k , t ) = id cv ~ Γ F ( k , z, t ) (6)Here we note that physically, the k dependence of the confined carriers in the quantumwell is due to a momentum k in the two transverse directions, and we therefore define atransverse energy with a simple parabolic band structure: E t = ~ | k | m ∗ r (7)where m ∗ r is the reduced effective mass. Hence to save space, we interchangeably write ρ e,h ( k , t ) ↔ ρ e,hE t . We can also rewrite the filtered field by interchanging F ( k , z, t ) ↔ F ( E t , z, t ).The total polarization per volume is a summation over all carrier groups with momentum k . Therefore, the total polarization for a 2-D quantum well can be written: P tot ( t ) = 2 V X k d ∗ ev p ( k , t ) = i | d cv | ~ Γ 2 V X k ( ρ eE t + ρ hE t − F ( E t , z, t ) (8)The k -summation can be converted to a transverse energy integral. We use a simpleparabolic dispersion relation for the conduction and valence bands: E c = E g + E e + ~ | k | m ∗ e (9a) E v = E h − ~ | k | m ∗ h (9b) ~ ω = E g + E e − E h (9c)where E g is the band gap energy, E e is the confined electron energy, E h is the confined holeenergy, m ∗ e,h is the electron and hole effective mass (we have assumed only a single confinedelectron state). Rewriting Eq. 8 with an energy integral, we obtain: P tot ( t ) = i | d cv | ~ Γ Z dE t D Dr ( ρ eE t + ρ hE t − F ( E t , z, t ) (10)7he dipole matrix element can be rewritten as the momentum matrix element via | d cv | = q m ω | ˆe · p | where q is the electron charge and m the electron mass. The macroscopicpolarization calculated in Eq. 10 serves as a source term for the forward and backwardpropagating electric fields in the laser. The constants on the RHS of 2 can be combined toyield a gain coefficient g = Γ xy q D Dr | ˆe j · p cv | n cǫ m ΓTo complete the derivation of the propagation equations, we include the effects of carriergratings resulting from the interference between forward and backward waves. Our approachto modeling this spatial hole burning (SHB) is to follow the techniques of [18], [29] and [30]and expand the QW population into its second harmonic in space. In this formulation, thepopulation becomes ρ e,hE t = ρ e,hqw,E t + ρ g,E t e i k z + ρ ∗ g,E t e − i k z + ... (11)For simplicity, we have used a single variable for the carrier gratings for both electronsand holes. The filtered field in the polarization also consists of forward and backwardcomponents: F = F + e − ik z + F − e ik z (12)Inserting Eqs. 10 11 12 in Eq. 2 and keeping only the phase-matched terms we obtainthe electric field equations: ± ∂E ± ∂z + 1 v g ∂E ± ∂t = g Z dE t ~ ω ( ρ eqw,E t + ρ hqw,E t − F ± ( E t , z, t )+ g Z dE t ~ ω ρ ( ∗ ) g,E t F ∓ ( E t , z, t ) (13)We note that the grating term ρ ( ∗ ) g,E t is associated with the forward wave equation andits conjugate with the backward wave. Finally, we simply add the additional terms in Eq.13 that describe standard linear and nonlinear effects, and scale via n qw , the number ofquantum wells to obtain 8 ∂E ± ∂z + 1 v g ∂E ± ∂t + i k ′′ ∂ E ± ∂t = − α E ± − (cid:16) α S iβ S (cid:17) ( | E ± | + 2 | E ∓ | ) E ± + S sp + n qw g Z dE t ~ ω ( ρ eqw,E t + ρ hqw,E t − F ± ( E t , z, t )+ n qw g Z dE t ~ ω ρ ( ∗ ) g,E t F ∓ ( E t , z, t ) (14)where k ′′ is the dispersion coefficient, α is the linear waveguide loss, and α S , β S are respec-tively the two-photon absorption and Kerr nonlinear coefficients, and S sp is the spontaneousemission term derived in the Appendix.These field equations are coupled with the carrier rate equations for the SCH and QWsections. The QW equations are labeled with the transverse variable for each discretized binyielding ∂ρ e,hsch ∂t = ηJ in qN c,v,sch h sch (1 − ρ e,hsch ) − ρ e,hsch τ sp + n qw X E t " ρ e,hqw,E t (1 − ρ e,hsch ) τ e,he − ρ e,hsch (1 − ρ e,hqw,E t ) τ e,hc (15) ∂ρ e,hqw,E t ∂t = h sch N c,v,sch n qw h qw N r,qw ρ e,hsch (1 − ρ e,hqw,E t ) τ e,hc − ρ e,hqw,E t (1 − ρ e,hsch ) τ e,he ! − ρ e,hqw,E t τ sp − R st − R g (16) ∂ρ g,E t ∂t = − ρ g,E t τ sp − k Dρ g,E t − g ∆ E t ( ~ ω ) h qw W N r,qw × (cid:20)
12 ( E ∗ + F − + F ∗ + E − )( ρ eqw + ρ hqw −
1) + 2Re( E ∗ + F + + E ∗− F − ) ρ g,E t (cid:21) (17) R st = 2 g ∆ E t ( ~ ω ) h qw W N r,qw ( ρ eqw,E t + ρ hqw,E t − E ∗ F ) (18) R g = 2 g ∆ E t ( ~ ω ) h qw W N r,qw (cid:0) ( E + F ∗− + F + E ∗− ) ρ g,E t + ( E ∗ + F − + F ∗ + E − ) ρ ∗ g,E t (cid:1) (19)where N c,v,sch = 2 (cid:16) m ∗ e,h k B T ~ π (cid:17) / , N r = m ∗ r ∆ E t ~ πh qw are the effective 3-D and 2-D density of states, D is the ambipolar diffusion coefficient, τ sp is the spontaneous emission lifetime, τ e,hc isthe capture lifetime, and τ e,he is the escape lifetime. The recombination rates R st and R g govern population decay due to stimulated emission and the carrier grating respectively.The escape times τ e,he are particularly important in our model as they phenomenologicallyrepresent intraband interactions. As shown in the Appendix, they are given by9 ee = τ ec exp(( δE c − m ∗ r m ∗ e E t ) /k B T ) (20) τ he = τ hc exp(( δE v − m ∗ r m ∗ h E t ) /k B T ) (21)The value of these escape times is tailored specifically to allow the rate equations 15, 16 torelax to the Fermi-Dirac distribution. III. NUMERICAL RESULTS FOR PULSE AMPLIFICATION
We solve the forward and backward wave equations (Eq. 14), coupled with the carrierrate equations (Eqs. 15,16,17) numerically using a first order Euler scheme very similar toreference [31]. We have chosen a time step of ∆ t = 30 fs with the full simulation parameterslisted in Table I.In order to solve the full set of equations, one must specify the limits to E t as well asthe number of different energy bins. The maximum transverse energy can be set to thequantum well barrier height, as there will not be any confined carriers with a total energythat surpasses this value. However, the maximum can be lower if the pump current is nottoo large, as then the high energy carriers will not significantly contribute to the total gain.For our simulations, we have chosen the values max( E t ) = 50 meV with 25 energy bins fora total of 75 quantum well carrier equations (25 for both electron and hole equations, 25for the grating term). The energy step ∆ E t = 2 meV is small relative to the homogenousFWHM (2Γ) to ensure reasonable accuracy in the gain integral.We first solve the equations for a pulse passing once through the laser cavity without facetreflections, acting as a semiconductor optical amplifier (SOA) in order to test that the phaseshifts are accurately modeled. It is important to model these phase distortions accurately,as they typically work against mode locking. The results of our simulation are shown inFigure 2. The pulse phase varies as expected, which for the long pulse (5 ps) resembles amore linear shape while the shorter pulse (0.5 ps) retains a cubic shape due to the carrierinduced refractive index change. The population depletion and recovery, shown in Figure3 are consistent in behavior with results from simpler impulse response models [11]. Forthe long pulse (5 ps), the population depletion is mostly monotonic and follows a smoothcurve. However, a short pulse (0.5 ps) will deplete the population quickly but the gain will10artially recover due to carrier cooling. These fast carrier dynamics are the primary cause ofthe cubic phase shifts in the amplified pulse and are detrimental to the generation of ultrashort pulses. In our simulations, carrier cooling occurs as additional carriers drop downfrom the SCH layer to fill the vacant QW states depleted by the short pulse. This capturetime is on the order of a picosecond, thus only pulses much shorter than this time will seethe effects of carrier cooling. These results verify the accuracy of our gain calculations withprevious pulse amplification experiments [11]. IV. NUMERICAL RESULTS FOR A DIODE LASER
While we have successfully modeled the phase dynamics in single-pass pulse amplification,the primary application of our model is to investigate FM frequency comb generation in asingle-section diode laser. We simulate 200 ns of a cleaved facet laser starting from noise andmonitor the output. The results are plotted in Figure 4. The temporal output and spectrummatch well with the experimental results for a single-section laser found in references [16]and [17], with a significant number of strong comb lines spanning about 30 nm in bandwidthwith a mode spacing of ν fsr = 85 . t > ρ g,E t , consistent with previous work [13], [29]. Thisterm allows several modes to lase at once and acts as a conduit for four-wave mixing. Weverify this by turning off the grating term and we only obtain a single lasing mode after theinitial relaxations as shown in Figure 5.To show that the modes are indeed locked, we plot the spectrum and spectral phasein linear scale in Figure 6a. This quadratic phase can be compensated by propagationthrough anomalous dispersion fibers [16], transforming the output into a series of shortpulses. We simulate this compensation by multiplying our spectrum by the transfer function H ( ω ) = e − i GDD ω [13] where GDD is the group delay dispersion, calculated to be 0 .
41 ps .11fter applying the inverse Fourier transform, we see a series of short pulses ( ≈
390 fsFWHM) emerge, which is indicative of mode locking[15]. The original field and dispersioncompensated field are plotted in Figure 6b for comparison. We note that the compensationis not perfect, as there is a small side pulse in front of the main pulse that indicates that theoutput pulses have higher order chirp that is not compensated by the simple application ofquadratic phase [13]. However, the fact that the phase compensation can result in a seriesof short pulses suggests the field inside the cavity is actually a train of highly chirped pulses.In order for this comb to be practical, the linewidth of each mode must be very narrowfor many of the high resolution comb spectroscopy techniques to be used. Unfortunately wecould not obtain an exact value for the linewidth of our comb as an accurate measurementrequires a very lengthy sample of data in the time domain, which is difficult to obtainfrom a computational standpoint. We have run simulations up to 1.5 µ s and attempted tomeasure the linewidth but even at such time scales, the linewidth was still limited by thetime window. Despite this, we calculate an upper bound of 1 MHz for the linewidth, whilethe real RF linewidth may be much smaller in the tens of kHz range [17]. V. DISCUSSION
The results shown in Figures 4, 6 show that single-section QW diode lasers have thepotential to produce useful frequency combs. The FM nature of the comb and the abilityto convert FM into a series of pulses via external dispersion compensation may prove usefulfor probing either fragile samples that require low intensity or samples that benefit fromhigh pulse power. Moreover, the planar processes used in manufacturing such diodes arewell developed and allow many lasers to be made at once. While the bandwidth is alreadysufficiently large, a wider bandwidth may be achieved by combining several lasers together,each with an offset to the central lasing frequency by adjusting the bandgap of the semicon-ductor material. The mode spacing can also be adjusted by changing the length of theselasers anywhere from a few hundred microns to several millimeters, and perhaps even on afiner scale by adjusting the pump current [16] for multiheterodyne measurements. Becausethe entire comb is generated on the chip itself without any external mirrors or components,the single-section QW diode laser represents a highly portable source of frequency combs.We have found that several material parameters are vital to the generation of FM combs.12irst, the homogenous linewidth 2Γ should be reasonably large compared to the mode spac-ing, primarily to facilitate strong four-wave mixing (FWM) interactions to lock the modestogether. In addition, too small of a homogenous linewidth may allow additional modes tolase independently from decreased gain competition, with additional gain coming from theinhomogenously distributed carriers. Once this occurs, there is no mechanism for lockingthese disparate modes, as FWM is no longer effective due to these modes falling outside ofthe homogenous linewidth. Second, for effective multimode lasing, we need a SHB effect, ora low enough diffusion constant, in order to see comb generation. The InGaAsP QW systemis well suited to satisfy this requirement, as the laser operates in the near-IR so that thehalf-wavelength grating spacing exceeds the diffusion length. Compared to other materialssuch as GaAs, the quaternary alloy InGaAsP has low measured values of diffusion [32]. Itis the persistence of the spatially burnt holes that leads to gain suppression [33] as well asmultimode operation.Moreover, we have found, perhaps surprisingly, that other effects have very little impacton the generation of combs. Second order material and waveguide dispersion, as modeled bythe parameter k ′′ , has only a very minor effect on mode locking, as the laser produces an FMmode locked state regardless of the inclusion of dispersion. The third-order Kerr nonlinearityand two- photon absorption also do not significantly alter the FM output, consistent withprevious findings in QD systems [13].We have used typical values for many of the physical parameters appropriate to an In-GaAsP system and we see these combs emerge naturally through spatial hole burning andfour-wave mixing. However, the interaction of the various physical phenomena is rathercomplex and we will present a more thorough study on the physics behind these combs in afuture work. VI. CONCLUSION AND ACKNOWLEDGMENTS
In conclusion, we have presented a comprehensive traveling wave model for a quantum-well based semiconductor laser. We have validated the accuracy of the calculations byreplicating a few experimental results, particularly generating frequency combs from single-section diode lasers. This model should serve as a suitable platform for additional studiesinto the physics that enables these combs to be generated and possibly discover new ways to13chieve stable mode-locking in these diode lasers. Long-wavelength QW lasers show muchpromise as practical, chip-scale sources of FM combs with the necessary bandwidth andlinewidth for the many applications of frequency combs.This research was developed with funding from the Defense Advanced Research ProjectsAgency (DARPA) through the SCOUT program. The views, opinions and/or findings ex-pressed are those of the author and should not be interpreted as representing the officialviews or policies of the Department of Defense or the U.S. Government. This research wasalso supported in part through computational resources and services provided by AdvancedResearch Computing at the University of Michigan, Ann Arbor.
Appendix A: Derivation of Rate Equations
Use of the microscopic polarization Eq. 6 in Eqs. 3b, 3c yields the rate equations ∂ρ e,h ( k , t ) ∂t = − | d cv | ~ Γ ( ρ e ( k , t ) + ρ h ( k , t ) − E ∗ F ) + ∂ρ e,h ( k , t ) ∂t | relax (A1)We rewrite the electric fields in a convenient form: the electric field is scaled to be in unitsof √ Watts via the expression | E ( z, t ) | → xy h qw W cn ǫ | E ( z, t ) | , where Γ xy is the confinementfactor, h qw is the height of the quantum well and W is the width. ∂ρ e,h ( k , t ) ∂t = − Γ xy | ˆe j · p cv | q n cǫ m Γ 1( ~ ω ) h qw W ( ρ e ( k , t ) + ρ h ( k , t ) − E ∗ F ) + ∂ρ e,h ( k , t ) ∂t | relax (A2)We define the gain coefficient g and the energy-discretized, reduced density of states, N r,qw =∆ E t D Dr and rewrite the rate equations: ∂ρ e,h ( k , t ) ∂t = − g ∆ E t ( ~ ω ) h qw W N r,qw ( ρ e ( k , t ) + ρ h ( k , t ) − E ∗ F ) + ∂ρ e,h ( k , t ) ∂t | relax (A3)So far, we have only applied a two-level approach to the rate equations even though asemiconductor is actually a four-level system [28]. However, because we solve the electronand holes separately based on the input current and charge conservation, we allow for the14ases in which an electron may exist but no hole, and vice versa. In this case, the occupationprobabilities ρ e,h no longer obey two-level relations ( ρ e − ρ h = 0) but can take on any valuebetween 0 and 1 according on the relaxation and pump terms. Appendix B: Evaluation of Carrier Relaxation Terms
In order to progress further, we need to evaluate the electron and hole relaxation terms ∂ρ e,h ( k ,t ) ∂t | relax . We follow the capture and escape approach presented in [13]. First, we startwith simple rate equations (without the presence of photons) for carrier number in the SCHand QW layers that satisfy charge conservation. dN sch dt = − N sch τ c + N qw τ e (B1a) dN qw dt = N sch τ c − N qw τ e (B1b) ddt ( N sch + N qw ) = 0 (B1c)We can convert this to occupation probability equations using the relations N sch = N c,v,sch W h sch ∆ zρ e,hsch (B2a) N qw = N r,qw W h qw ∆ zρ e,hqw,E t (B2b)We also add Pauli blocking terms, which results in the following differential equations forthe occupation probabilities: ∂ρ e,hsch ∂t = − ρ e,hsch τ e,hc (1 − ρ e,hqw,E t ) + N r,qw h qw N c,v,sch h sch ρ e,hqw,E t τ e,he (1 − ρ e,hsch ) (B3a) ∂ρ e,hqw,E t ∂t = N c,v,sch h sch N r,qw h qw ρ e,hsch τ e,hc (1 − ρ e,hqw,E t ) − ρ e,hqw,E t τ e,he (1 − ρ e,hsch ) (B3b)The steady state solutions to Eqns B3 should relax into a Fermi-Dirac distribution. Weassume the solutions for the electrons (holes follow a similar expression) are of the form:15 esch = 11 + exp (cid:16) E sch − E f k B T (cid:17) (B4a) ρ eqw,E t = 11 + exp (cid:18) E qw + m ∗ rm ∗ e E t − E f k B T (cid:19) (B4b)where E f is the electron Fermi level. We can use these solutions in Eqs. B3 and solve forthe proper escape time in terms of the capture time such that the occupation probabilitiessettle into a Fermi-Dirac distribution. We find the resulting expressions for the escape timesand the relaxation to be: τ ee = τ ec (cid:18) N r,qw h qw N c,sch h sch (cid:19) exp(( δE c − m ∗ r m ∗ e E t ) /k B T ) (B5a) τ he = τ hc (cid:18) N r,qw h qw N v,sch h sch (cid:19) exp(( δE v − m ∗ r m ∗ h E t ) /k B T ) (B5b)Here, δE c = E sch − E qw (and analogously, δE v ) is the energy difference between the SCH layerand the confined carrier with zero transverse energy, visually labeled in Figure 1. Lastly, weremove the bracketed fraction and write it explicitly in the rate equations, allowing us todefine the escape lifetimes more simply as: τ ee = τ ec exp(( δE c − m ∗ r m ∗ e E t ) /k B T ) (B6a) τ he = τ hc exp(( δE v − m ∗ r m ∗ h E t ) /k B T ) (B6b)While we have shown the derivation for only a single quantum well carrier group, there areactually multiple quantum well rate equations. Thus the SCH equation must sum up thecapture and escape contributions from every group of quantum well carriers. Appendix C: Carrier Grating Terms
The stimulated emission term in the rate equations contains the product:Re( E ∗ F ) = Re( E ∗ + F + + E ∗− F − ) + 12 ( E ∗ + F − + F ∗ + E − ) e i k z + 12 ( E + F ∗− + F + E ∗− ) e − i k z − D ∂ ∂z on the RHS, where D is the diffusion coefficient. The resultingequations are ∂ρ e,hqw,E t ∂t = ... − g ∆ E t ( ~ ω ) h qw W N r,qw × h Re( E ∗ + F + + E ∗− F − )( ρ eqw,E t + ρ hqw,E t − E + F ∗− + F + E ∗− ) ρ g,E t + ( E ∗ + F − + F ∗ + E − ) ρ ∗ g,E t i (C1) ∂ρ g,E t ∂t = − k Dρ g,E t − g ∆ E t ( ~ ω ) h qw W N r,qw × (cid:20)
12 ( E ∗ + F − + F ∗ + E − )( ρ eqw,E t + ρ hqw,E t −
1) + 2Re( E ∗ + F + + E ∗− F − ) ρ g,E t (cid:21) (C2)The stimulated emission rate and the photon-grating interaction as are now clearly identifiedas: R st = 2 g ∆ E t ( ~ ω ) h qw W N r,qw ( ρ eqw,E t + ρ hqw,E t − E ∗ F ) R g = 2 g ∆ E t ( ~ ω ) h qw W N r,qw (cid:0) ( E + F ∗− + F + E ∗− ) ρ g,E t + ( E ∗ + F − + F ∗ + E − ) ρ ∗ g,E t (cid:1) Combining all the elements together and adding in the pump J in = I in /W L and spontaneousemission terms, we have the final form of the rate equations: ∂ρ e,hsch ∂t = ηJ in qN c,v,sch h sch (1 − ρ e,hsch ) − ρ e,hsch τ sp + X E t " ρ e,hqw,E t (1 − ρ e,hsch ) τ e,he − ρ e,hsch (1 − ρ e,hqw,E t ) τ e,hc (C3) ∂ρ e,hqw,E t ∂t = h sch N c,v,sch h qw N r,qw ρ e,hsch (1 − ρ e,hqw,E t ) τ e,hc − ρ e,hqw,E t (1 − ρ e,hsch ) τ e,he ! − ρ e,hqw,E t τ sp − R st − R g (C4) ∂ρ g,E t ∂t = − ρ g,E t τ sp − k Dρ g,E t − g ∆ E t ( ~ ω ) h qw W N r,qw × (cid:20)
12 ( E ∗ + F − + F ∗ + E − )( ρ eqw + ρ hqw −
1) + 2Re( E ∗ + F + + E ∗− F − ) ρ g,E t (cid:21) (C5)17 ppendix D: Derivation of the Gain Spectrum We can take a Fourier transform of the gain term in the traveling wave equation in orderto visualize the gain spectrum. We assume the carriers are in steady state so that thepopulations obey Fermi-Dirac statistics. In this case, the Fourier transform evaluates to F (cid:20) g Z dE t ~ ω ( ρ eqw,E t + ρ hqw,E t − F ± ( E t , z, t ) (cid:21) = g Z dE t ~ ω ( ρ eqw,E t + ρ hqw,E t − E ± ( z, ω ) − i ( ω + E t / ~ ) − Γ (D1)and hence the field gain is g ( ω ) = g Z dE t ~ ω ( ρ eqw,E t + ρ hqw,E t −
1) 1 − i ( ω + E t / ~ ) − Γ (D2)In this form, we see that the gain spectrum consists of a series of Lorentzians centered atdifferent transition energies. A plot of Eq. D2 is shown in Figure 7 for varying levels ofcarrier population.
Appendix E: Derivation of Spontaneous Emission
Lastly, the spontaneous emission term S sp is derived more phenomenologically. Thespontaneous emission term is found by following the approach in [31] in which the powerspectrum follows the quantum well gain spectrum. | S sp ∆ z | = X modes τ sp × photon energy × coupling factor= X modes n qw N r,qw h qw W ∆ z πτ sp ρ eqw,E t ρ hqw,E t ~ ωβ sp S sp ≈ X E t s n qw N r,qw h qw W β sp ~ ωρ eqw,E t ρ hqw,E t πτ sp ∆ z F sp ( E t ) (E1) F sp ( E t ) = Γ Z t −∞ dt ′ e i ( ∆ Ecv ~ − ω )( t − t ′ ) − Γ( t − t ′ ) e iφ ( z,t ′ ,E t ) (E2)18ere, φ ( z, t, E t ) is a random phase value between 0 and 2 π , and ∆ z = c ∆ t/n is the spatialdiscretization size. [1] S. T. Cundiff and J. Ye, Rev. of Mod. Phys. , 325 (2003).[2] T. Udem, J. Reichert, R. Holzwarth, and T. W. H¨ansch, Phys. Rev. Lett. , 3568 (1999).[3] I. Coddington, W. C. Swann, and N. R. Newbury, Phys. Rev. Lett. , 13902 (2008).[4] S. A. Diddams, T. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M.Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, Science , 825 (2001).[5] S. T. Cundiff and A. M. Weiner, Nat. Phot. , 760 (2010).[6] D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller,V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. , 631 (1999).[7] M. E. Fermann and I. Hartl, Nat. Phot. , 868 (2013).[8] T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L.Gorodetsky, and T. J. Kippenberg, Nat. Phot. , 480 (2012).[9] V. Moskalenko, J. Koelemeij, K. Williams, and E. Bente, Opt. Letters , 1428 (2017).[10] R. Rosales, K. Merghem, A. Martinez, A. Akrout, J.-P. Tourrenc, A. Accard, F. Lelarge, andA. Ramdane, IEEE J. Sel. Top. Quantum Electron. , 1292 (2011).[11] P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, andJ. P. Heritage, IEEE J. Quantum Electron. , 2203 (1992).[12] L. F. Tiemeijer, P. I. Kuindersma, P. J. A. Thijs, and G. L. J. Rikken, IEEE J. QuantumElectron. , 1385 (1989).[13] M. Gioannini, P. Bardella, and I. Montrosset, IEEE Sel. Topics Quantum Electron. ,1900811 (2015).[14] R. Rosales, K. Merghem, C. Calo, G. Bouwmans, I. Krestnikov, A. Martinez, and A. Ram-dane, App. Phys. Lett. , 221113 (2012).[15] R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P.Barry, and A. Ramdane, Optics Express , 8649 (2012).[16] K. Sato, IEEE J. Sel. Top. Quantum Electron. , 1288 (2003).[17] C. Cal`o, V. Vujicic, R. Watts, C. Browning, K. Merghem, V. Panapakkam, F. Lelarge, A. Mar-tinez, B.-E. Benkelfat, A. Ramdane, and L. P. Barry, Opt. Express , 26442 (2015).
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Semiconductor-Laser Physics (Springer-Verlag,1994).[29] J. Javaloyes and S. Balle, IEEE J. Quantum Electron. , 431 (2009).[30] M. Homar, J. V. Moloney, and M. S. Miguel, IEEE J. Quantum Electron. , 553 (1996).[31] M. Rossetti, P. Bardella, and I. Montrosset, IEEE J. Quantum Electron. , 139 (2011).[32] D. Marshall, A. Miller, and C. C. Button, IEEE J. Quantum Electron. , 1013 (2000).[33] C. B. Su, IEEE Electron. Lett. , 370 (1988). arameter Description Value L Length of device 500 µ m W Width of waveguide 4 µ m h sch Height of SCH layer 50 nm h qw Height of quantum well 5 nm n Group refractive index 3.5 n qw Number of quantum wells 2 α Intrinsic waveguide loss 5 cm − Γ xy Optical confinement factor 0.01 α S Two-photon absorption 2750 W − m − β S Kerr coefficient 430 W − m − k ′′ Dispersion coefficient 1.25 ps /m ~ ω Central transition energy 800 meV | ˆe · p | Momentum matrix element 21 meV × m / ~ m ∗ e,h,sch Effective mass of electrons, holes in the SCH layer 0 . m , 0 . m m ∗ e,h,qw Effective mass of electrons, holes, in the InGaAsP QW 0 . m , 0 . m τ e,h,qwc electron, hole capture time 1, 10 ps δE c Conduction band quantum well barrier 50 meV δE v Valence band quantum well barrier 75 meV β sp Spontaneous emission coupling factor 1 × − τ sp Spontaneous emission lifetime 1 ns D Ambipolar diffusion coefficient 7.2 cm /s [32]TABLE I. Simulation parameters for QW traveling wave model for the InGaAsP system. CHInGaAsP QWn-type p-type δ E c E qw E sch I in τ c τ e(k) τ sp R st τ sp, FIG. 1. A schematic of the quantum well laser diode with current injection into the SCH layerwhich is captured via τ c into the quantum well. The captured electrons have a distribution oftransverse energies that can escape the quantum well via τ e ( k ). b Time (ps) P o w e r ( m W ) -0.01-0.00500.0050.010.0150.020.0250.03 P h a s e / P h a s e / Time (ps)
Pulse IntensityPulse Phase P o w e r ( m W ) FIG. 2. The results of sending an unchirped gaussian pulse through a single pass of the laser cavity.The amplified pulse shape and phase are plotted for a) a 0.5 ps pulse b) a 5 ps pulse. Time (ps) P o p u l a t i o n I n v e r s i o n FIG. 3. The population depletion and recovery as the pulses pass through. b
100 150 200
Time (ns) O u t p u t P o w e r ( m W ) Wavelength (nm) P o w e r S p e c t r a l D e n s i t y ( d B m / H z ) c
150 150.01 150.02 150.03 150.04
Time (ns) O u t p u t P o w e r ( m W )
150 150.01 150.02 150.03 150.04194195196197198199200201 I n s t a n t a n e o u s F r e q u e n c y ( T H z ) Time (ns)
FIG. 4. a) The temporal output of the single-section quantum well device at ηI in = 25mA with azoomed inset to show the detailed dynamics. The output is quasi-CW except for a short burst thatrepeats every round trip. A steady state is reached for t >
110 ns b) The power spectral densityof the last 100 ns of the temporal output in log scale showing a broad comb c) the instantaneousfrequency of the laser output, which is also sweeping periodically, showing the FM nature of thecomb. b
40 60 80
Time (ns) O u t p u t P o w e r ( m W ) Wavelength (nm) P o w e r S p e c t r a l D e n s i t y ( d B m / H z ) FIG. 5. a) the temporal output of the laser with the population grating term set to zero. Theoutput relaxes to a single mode after some time b) the spectrum of the above output which showsa single mode dominating, in stark contrast to the case when the grating is on (Figure 4b). b
194 195 196 197 198 199
Frequency (THz) P o w e r S p e c t r a l D e n s i t y ( m W / H z ) -20246810 S p e c t r a l P h a s e ( r a d / ) Power SpectrumSpectral Phase
20 20.01 20.02 20.03
Time (ns) P o w e r ( m W ) Original FieldDispersion Compensated Field
FIG. 6. a) the power spectral density in linear scale along with the spectral phase b) The spectrumis compensated for dispersion and inverse Fourier transformed to produce a series of short pulsesseparated by the cavity round trip time. The group delay dispersion is calculated to be 0.41 ps .
400 1500 1600 1700Wavelength (nm)-50-40-30-20-10010203040 G a i n c m - G a i n c m - G a i n c m - -50-40-30-20-10010203040-50-40-30-20-10010203040 Real(g( ))Imag(g( ))Real(g( ))Imag(g( ))Real(g( ))Imag(g( )) abc