Design optimization of passively mode-locked semiconductor lasers with intracavitygrating spectral filters

Finbarr O’Callaghan, David Bitauld,∗ and Stephen O’BrienTyndall National Institute, Lee Maltings, University College Cork, Cork, Ireland

We consider design optimization of passively mode-locked two-section semiconductor lasers thatincorporate intracavity grating spectral filters. Our goal is to develop a method for finding theoptimal wavelength location for the filter in order to maximize the region of stable mode-locking asa function of drive current and reverse bias in the absorber section. In order to account for materialdispersion in the two sections of the laser, we use analytic approximations for the gain and absorptionas a function of carrier density and frequency. Fits to measured gain and absorption curves thenprovide inputs for numerical simulations based on a large signal accurate delay-differential modelof the mode-locked laser. We show how a unique set of model parameters for each value of thedrive current and reverse bias voltage can be selected based on the variation of the net gain alongbranches of steady-state solutions of the model. We demonstrate the validity of this approachby demonstrating qualitative agreement between numerical simulations and the measured current-voltage phase-space of a two-section Fabry-Perot laser. We then show how to adapt this method todetermine an optimum location for the spectral filter in a notional device with the same materialcomposition, based on the targeted locking range, and accounting for the modal selectivity of thefilter.

I. INTRODUCTION

Semiconductor mode-locked lasers have the potentialto address a great number of applications in advancedtelecommunications and signal processing [1–4]. Becausemany of these applications place stringent requirementson the laser source, a series of design innovations havebeen suggested that can enhance the timing and phase-noise performance of semiconductor mode-locked [ML]lasers. In particular, devices that incorporate intracav-ity spectral filters and pulse shapers have seen signifi-cant progress. Recent examples include integration of aMach-Zehnder interferometer for flattening of the gainspectrum [5], integration of arrayed waveguide gratingsand phase modulators for pulse shaping [6], and harmonicmode-locking of extended cavity devices with integratedring resonator filters for phase-noise reduction [7]. Inparallel with these developments, a further series of in-teresting experimental techniques and devices have beendemonstrated that exploit the very large quality factorsof whispering gallery mode microresonators in order togenerate low phase-noise optical frequency combs andstable sources of ultrafast pulses. Many of these inno-vations are based on principles of multiwavelength ex-citation of parametric processes [8, 9] or on coupling ofconventional lasing and nonlinear resonators [10, 11].

In recent work we have demonstrated a number of two-section Fabry-Perot lasers with engineered spectra de-fined by an intracavity grating spectral filter [12, 13].The grating filter in these devices is designed to select afinite number of predetermined lasing modes so that pre-cise tailoring of the comb line spectrum is possible. These

∗Current address: Nokia Research Center, 21 J J Thomson Avenue,Cambridge CB3 0FA, UK

filter designs can also be adapted for integrated lightwavecircuits based on open grating resonators to provide on-chip sources of tailored and phase-locked lasing modes[14].

In conventional two-section semiconductor lasers, thepeak emission wavelength can vary strongly as the cur-rent and the reverse bias applied to the short section arevaried [15]. Although we have succeeded in mode-lockinga variety of devices that incorporated grating spectralfilters [16], we have found that the extent of the sta-ble mode-locking region was very limited compared toFabry-Perot [FP] lasers with the same material compo-sition. The ability to adjust the laser drive parameterswhile remaining in a mode-locked state is an importantrequirement for applications, as this will enable tuningof the spectral profile and comb line frequencies. Thelatter property is of great practical importance, as it willfacilitate locking of a comb to an external pump beam,or to an extended underlying or external cavity. Herewe develop a design strategy for optimising the mode-locking range of two-section semiconductor lasers withintracavity grating spectral filters.

In order to optimize the mode-locking range of thesedevices for applications, efficient dynamical models arecrucial for understanding the structure of the mode-locking region in the phase-space of drive parameters.In general, frequency domain models extend the rateequation description of a two-section semiconductor laser[17, 18] to include phase-sensitive modal interactions[19, 20]. In this case, the mode-locked state can be de-scribed as a mutually injection locked steady-state withzero net group velocity dispersion. Although these mod-els are attractive and convenient given the natural modalpicture that follows from our design approach [18], cur-rent frequency domain models are valid for small gainand loss. In addition, because these models describe theML state as a steady-state rather than a steady peri-

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odic state, these models are small signal models of themode-locked laser [21]. They therefore may not providea complete and accurate picture of the various dynamicalprocesses that can destabilize the ML state [22, 23]. Fullydistributed time domain simulations that describe thespatio-temporal dynamics of the carriers and the prop-agating fields provide a quantitative picture of the dy-namics in semiconductor ML lasers [24–26]. A lumpedelement time domain model has also been developed thatcan describe the large gain and losses and strong satura-tion of the absorption that are typical in a semiconductorlaser [27]. We will employ this delay differential modelfor our numerical simulations, which has advantages ofcomputational efficiency and potential for analytic anal-ysis [22].

Here we first present experimental results that illus-trate the complex dynamical phase-space structure thatis typical of a two-section FP laser. We map regionsof self-pulsations, stable mode-locking, and Q-switchedmode-locking as a function of injected current and re-verse bias applied to the short section of the laser. Wealso present optical spectra showing how the peak emis-sion can vary strongly and exhibit large shifts over rel-atively small changes in the drive parameters. We thenprovide measurements from a device with a tailored spec-trum, which show an unexpected variation of the net gainat the location of the spectral filter.

Our design method requires us to describe the fre-quency dispersion of gain and saturable absorption andto capture their effects on the dynamics and stability ofmode-locked states of these devices. To calibrate ourmodel parameters, we fit an approximate analytic func-tion to the measured modal gain and to the modal ab-sorption as a function of reverse bias. To reproduce themeasured phase-space data, we propose a direct methodto choose a wavelength reference based on the variationof the net gain along steady-state solutions of the delaydifferential equation [DDE] model. This approach leadsto good qualitative agreement between the DDE modeland the results of our experiments.

We next consider the problem of finding the optimalwavelength location for an intracavity spectral filter. Keyproperties of interest are the stability of the ML spec-trum as defined by the grating filter and tunability ofthe ML state. Because the ML states of our devices arecharacterised by relatively narrow spectral bandwidths,we consider a quasi-static limit, and we again use steady-state solutions to estimate the variation of the net gain atdetuned wavelengths over a defined region in the phase-space of drive parameters. Based on measured resultsfor the selectivity of typical grating filters and the struc-ture of the simulated mode-locking region, our resultsindicate that we can expect stable mode-locking in opti-mized devices over a large range of voltages and currents.We conclude by discussing possible improvements to theproposed method and prospects for future work.

1540 1550 1560 1570wavelength (nm)

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nsity

(dB

)

1540 1550 1560 1570 1540 1550 1560 1570

delay (25 ps/div) delay (25 ps/div) delay (25 ps/div)

-0.5 V70 mA

-0.9 V70 mA 100 mA

-1.6 V

FIG. 1: Upper panel: Measured phase space data (left)and power output (right) of a two-section plain Fabry-Perotlaser. Labeled phase-space regions are SP (self-pulsation),ML (mode-locking), QSML (Q-switched mode-locking), andHML (harmonic mode-locking). Lower panel: Optical spec-tra and intensity autocorrelation measurements for a series ofdrive parameters as indicated.

II. EXPERIMENTAL MEASUREMENTS OFTHE IMPACT OF MATERIAL DISPERSION ONPASSIVE MODE-LOCKING OF TWO-SECTION

SEMICONDUCTOR LASERS

In this section we present experimental measurementstaken from two-section indium phosphide-based ridge-waveguide FP lasers. The devices are high-reflection[HR] coated, with quantum-well active regions and theabsorber section placed adjacent to the HR mirror. Thefirst device is a plain FP laser of length 545 µm, witha saturable absorber section of length 30 µm, while thesecond is a device of length 875 µm, with a saturableabsorber section of length 60 µm. The second deviceincludes an intracavity grating spectral filter defined byetched features in the laser ridge-waveguide.

For the plain FP device the measured threshold currentwith a single current density over the whole device length[single section FP] was 12.5 mA. A measured phase-spacemap of the dynamical states of the device as the devicecurrent and reverse bias voltage applied to the short sec-tion are varied is shown in the upper panel of Fig. 1.One can see that a large region of self-pulsations [SP] isfound near to threshold, and that these tend to evolveinto mode-locked [ML] operation through a region of Q-switched mode-locking [QSML]. The current range whereQSML is found becomes wider as the reverse bias is in-creased, and harmonic mode-locking [HML] is also found

3

FIG. 2: Optical spectrum of the two-section plain Fabry-Perot laser as the reverse bias voltage is varied. The currentin the gain section is 100 mA.

in a small region for large negative values of the absorberbias. A map of the power output of the device is alsoshown in the upper panel of Fig. 1. This shows an in-creasing threshold current until approximately -1.5 V,where an anomalous decrease is found for increasing neg-ative bias.

In the lower panel of Fig. 1 we have included opticalspectra and the corresponding intensity autocorrelationmeasurements at three points in the phase space of Fig.1. We observe incomplete mode-locking at small valuesof the reverse bias around -0.5 V. Well developed mode-locking of the FP device is found near a device currentof 70 mA, and a reverse bias voltage of -1.0 V. In thisregion the device generates pulses of approximately 1.7ps duration at a repetition rate of 80 GHz. These pulses,which are not transform limited, become shorter as thereverse bias is increased, reaching a minimum durationof approximately 1.5 ps near -1.5 V. Note that the shiftof the peak emission wavelength with increasing reversebias is initially towards longer wavelengths and that thistrend is reversed for larger reverse bias voltages. Thisreversal also occurs in the region of reverse bias voltagevalues where the threshold variation with increasing biasbecomes anomalous.

Fig. 2 illustrates the dramatic shifts in the peak emis-sion wavelength that occur in the region of anomalousthreshold variation. Here we have shown the optical spec-trum of the device as a function of saturable absorbervoltage at a fixed device current in the long section of100 mA. At reverse bias voltages up to approximately-1.5 V, one can see that the peak emission wavelength isrelatively constant, with a small drift towards long wave-length. Beyond this region, with increasing reverse bias,we observe a much more rapid shift of the peak emissiontowards shorter wavelengths.

Experimental data from the second device of length875 µm are presented in Fig. 3 [13]. In this examplethe filter was designed to select a comb of six primarymodes with 100 GHz spacing at a wavelength of 1545nm. Net gain measurements near threshold made using

1540 1550wavelength (nm)

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delay (20 ps/div)

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1540 1550 1560wavelength (nm)

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gain

(cm

-1)

1540 1544 1548wavelength (nm)

inte

nsity

(20

dB

/div

)

-2.2 V

(a)

(b)

120 mA-2.0 V, 52 mA

-1.0 V, 60 mA-1.5 V, 55 mA ∆ν=100 GHz

HR

V-

Lc= 875 µm

FIG. 3: (a) Net gain measured below threshold for the driveparameters indicated. The inset shows a calculation of themodal threshold gain. The lower panel is a device schematicshowing the two-section geometry and the construction of thegrating spectral filter. The device length is 875 µm. (b)Mode-locked spectrum of the device. The inset shows an in-tensity autocorrelation measurement. The dashed line is theimplied spectral bandwidth derived from the autocorrelationmeasurement.

the Hakki-Paoli technique are shown in Fig. 3 (a). Thesedata show that the net gain is an increasing function ofreverse bias near the location of the spectral filter for re-verse bias voltages larger than -1.0 V. Note also that Fig.3 (a) provides us with a measure of the filter selectivitythat can be achieved with etched features in devices suchas these. In the device shown, the selectivity is somewhatnon-uniform but it reaches a maximum of approximately2 cm−1. We found that the device of Fig. 3 mode-lockednear a drive current of 120 mA and reverse bias voltageof -2.2 V to obtain a transform limited 100 GHz pulsetrain with 2 ps pulse duration [Fig. 3 (b)]. The timingjitter of the pulse train and the phase noise of the indi-vidual comb lines also showed significant improvement ascompared to an equivalent plain FP laser [16], which weattribute to the beneficial effects of thinning of the op-tical spectrum. The optical spectrum also shows strongsuppression of cavity modes between the primary comblines, which could otherwise be a source of so-called su-permode noise [3, 7]. However, although we obtained ahigh quality mode-locking spectrum, the stable ML re-gion was far too small to allow significant tunability ofthe comb spectrum.

The measurements of Fig. 2 and Fig. 3 (a) suggestthat the limited mode-locking range of these devices wasdue to the complex dispersion of the net modal gain atthe location of the spectral filter. In order to address thisissue, we have developed a general method for determin-ing the optimal wavelength location for a spectral filterthat can lead to a large ML region and associated tun-ability in the parameter region where the variation of thepeak wavelength of the net gain with drive parameters is

4

less pronounced. We have noted that a spectral filter willconstrain the active modes of a device so that the carrierwave frequency in the mode-locked state cannot adjust asin the FP. Although a quantitative optimization methodwould explicitly account for the action of the spectral fil-ter, we will illustrate the application of the method byapplying it to a notional device with the same materialcomposition as the two-section FP laser of Fig. 1. How-ever, the method is general, and it could be adapted tomodels of the two-section device that account for the fil-ter explicitly. We also argue that our results here can beregarded a qualitatively accurate, as we expect model-locking of a narrow-band comb defined by an intracavityspectral filter will not be more difficult than a plain FPlaser, provided that the additional modal dispersion dueto the filter is limited.

III. MODEL PARAMETER EXTRACTIONBASED ON AN ANALYTICAL MODEL OF THE

SEMICONDUCTOR MODAL GAIN ANDABSORPTION

In this section we describe the analytic model of themodal gain and absorption that we use to fit measureddata. Before we proceed, it is instructive to first intro-duce the well-known rate equation description of a laserwith a saturable absorber [LSA model]. By consideringthese equations, which are also referred to as the Yamadamodel [28], we can introduce the various model parame-ters that we must obtain in their physical context, and wemay also motivate the scaling of the dynamical variablesthat we will employ in our simulations.

Although the LSA model cannot describe mode-lockedstates of the device, it is the simplest model that candescribe phenomena such as self-pulsations and the ap-pearance of a bistable region at threshold in devices withsaturable absorbers [17]. These equations make a totalfield approximation to describe the intensity of the FPlaser, and the saturable absorber section of the laser isdescribed as an unpumped region, with an unsaturatedloss determined by the applied voltage. In physical unitsthe LSA model reads

S = [(1− ρ)Gm(Ng) + ρQm(Nq)− γ]S

Ng = j − Ng

τs−Gm(Ng)S (1)

Nq = −Nq

τq−Qm(Nq)S

Here S is the average photon density, Gm is the modalgain and Qm is the modal absorption. Ng and Nq arethe carrier densities in the gain and absorber sections,respectively. The total field losses are γ = αm + αint,where αm are the mirror losses, and αint are the inter-nal losses of the device. The current density in the gainsection is j, while the carrier lifetimes in the gain andabsorber sections are τs and τq respectively.

To proceed we must establish the dependence of thegain and absorption in the device as a function of car-

rier density and frequency. Reference [29] provides aconvenient model that can be fitted to measured gainand absorption obtained using the Hakki-Paoli method.This work developed an analytical expression for the sus-ceptibility of a quantum-well semiconductor material atlow-temperature and has been used extensively to modelthe gain and saturable absorption in simulations of free-running and mode-locked semiconductor lasers [15, 26].While the model results cannot be regarded as quantita-tive, they nevertheless provide a qualitative descriptionof the material susceptibility that can be used to obtainphysically appropriate parameters for dynamical simula-tions.

The result for the modal gain is [29]

Gm(λ,Dg) = G0(tan−1[u]− 2 tan−1[u−Dg]− π/2),

where λ is the wavelength, Dg is the carrier density nor-malised to the transparency value, G0 is the material gaincoefficient, and

u =2πc

γGp

(1

λ− 1

λGbg

)+ σD1/3

g .

Here, λGbg is the nominal transition wavelength, σ de-scribes the bandgap shrinkage with increasing carrierdensity, and γGp is the linewidth of the optical transi-tion. We use a different carrier density variable in theexpressions for the modal gain and absorption in Eqns 1as we will choose to normalise the model variables withrespect to the differential gain, rather than the trans-parency value of the carrier density.

We now fit this expression to the modal gain of the ac-tive material, Gm, which we determine using the Hakki-Paoli technique. For these measurements a single currentdensity over the full length of the device was maintained.The negative offset at long-wavelength gives an estimateof the internal losses, which are assumed to be wave-length independent and uniformly distributed over thedevice length. The results of the fit that we obtainedfor two values of the device current are shown in Fig. 4(a). The mirror losses of the device are αm = 12.2 cm−1.We find that the internal losses, αint = 18.0 cm−1, asindicated in Fig. 4 (a).

From the resulting fit to the data, we determine thevalue of the carrier density at threshold for the singlesection FP laser. Fixing the wavelength at the location ofthe gain peak at threshold, we determine the differentialgain at threshold for the FP laser. We then define thethreshold gain and differential gain at the gain peak tobe equal to 1 in normalized units. This scale then definesour normalized carrier density variable, Ng. We will alsodefine the scaled current density, js, where the thresholdcurrent density in the single section FP laser is definedto be equal to 1 in normalised units. The variation ofthe modal gain at the wavelength of the gain peak atthreshold as a function of carrier density is shown in Fig.4 (b). Here have plotted the data in physical units andin normalized units. The value of the differential gain

5

1.5 1.53 1.56 1.59 1.62wavelength (µm)

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0.0 1.5 3.0

carrier density, Dg

-1

-0.5

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(un

its o

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γ = αm

+ αint

αm

-αint

(a) (b)

[9.5, 10.5] mA

FIG. 4: (a) Measured modal gain of the Fabry-Perot laser[dashed lines] for two values of the device current as indicated.Solid lines are fits to these data and a plot of the gain curveat threshold defined by the fitted gain function. The mirrorlosses are αm and the background losses are αint. (b) Varia-tion of the fitted modal gain as a function of carrier density ata fixed wavelength λ0. This wavelength [1554 nm] is at the lo-cation of the peak gain at threshold for the Fabry-Perot laserindicated by the vertical dashed line in (a). The thresholdmodal gain, γ, is equal to the sum of the mirror and inter-nal losses in the device. The differential gain at threshold isdefined to be equal to one in normalised units.

λGbg G0 γG

p σ λ0 gg

[nm] [cm−1] [1013 s−1] [nm] [cm−1]

1550.0 27.0 2.3 0.73 1554.0 16.0

TABLE I: Fitting and derived parameters for the gain func-tion

at threshold is gg = 16.0 cm−1[≡ 1]. A table of fittingparameters for the susceptibility function is presented intable I. The values of Dg and Ng at threshold are 2.38and 1.26 respectively.

To determine the modal absorption in the reverse biassection, Qm, we employ a differential Hakki-Paoli tech-nique. The modal absorption is found from the differencebetween the modal gain measured with a uniform currentdensity over the whole device length and the modal gain,G

′

m, measured with the same current density applied tothe long section and a given voltage applied to the shortsection. The relationship is [15, 30]

ρQm(V, λ) = G′

m(V, j, λ) + αint − (1− ρ)Gm(j, λ).

To fit the derived absorption curve we use the expres-sion

Qm(λ,Dq) + αabs =

Q0[V](tan−1[u]− 2 tan−1[u−Dq]− π/2),

where Dq is the carrier density normalised to the trans-

parency value as before, the material absorption coeffi-cient, Q0[V], is now a function of voltage, and

u =2πc

γQp

(1

λ− 1

λQbg[V]

).

Here we have neglected the σ parameter as the carrierdensity is taken to be much smaller in the reversed bi-ased section. However, in order to obtain a reasonablefit to the data we have had to introduce a voltage depen-

dent bandgap wavelength λQbg[V]. We have also includedan additional carrier density and voltage independentcontribution to the absorption in the absorber section,αabs. This was necessary because, while the above ex-pression can fit the absorption function accurately nearthe bandedge, we found that the derived carrier den-sity was close to the transparency value, which led toan unphysically large value for the differential absorp-tion of the material. We therefore included a small off-set, αabs = 2.5 cm−1, in order to obtain accurate fits ofthe function with small values of the carrier density ap-propriate for a reverse biased section. Note also that thetransition linewidth in the absorber section was taken tobe γQp = 0.85 × 1013 s−1. This change in the value ofthe transition linewidth and the voltage dependence of

λQbg are consistent with the quantum-confined Stark ef-

fect [15].Experimental and fitted absorption curves for four val-

ues of the reverse bias voltage are shown in Fig. 5 (a). Ineach case, at short wavelengths, the experimental curvesshow a region of decreasing absorption with decreasingwavelength. This non-monotonic dependence is consis-tent with the measurements of Fig. 3 and it is not de-scribed by our model absorption function. The result isthat fits to these absorption curves are only accurate forwavelengths above a characteristic cutoff that dependson the applied voltage. In all cases however, one cansee that the fits are very close to the measured data forwavelengths larger than approximately 1555 nm. Fromthe optical spectra of the lower panel of Fig. 1 and fromFig. 2, we can see that the peak emission wavelengthbegins to shift towards short wavelengths and approach1555 nm only for reverse bias voltage values of -1.5 V orlarger. These values also define the boundary of the re-gion of rapid variation of the peak emission with voltage,which will result in poor tunability of the ML spectrum.For this reason we will illustrate the application of ouroptimization method targeting reverse bias voltages be-tween -1.0 V and -1.5 V, and assume that our fits to theabsorption are accurate at all wavelengths of interest overthis voltage range. A table of fitting parameters for themodal absorption function obtained for voltages rangingfrom -0.5 to -1.5 V are shown in table II. Here D0

q isthe value of the carrier density used to fit the measured[unsaturated] absorption.

In order to define our model parameters, we mustchoose a reference wavelength and a carrier density valuefor each device section where the modal gain and ab-

6

bias λQbg ρQ0 D0

q λ0 ρQ0m N0

g gg ρgq

[V] [nm] [cm−1] [nm] [γ]

-0.5 1558.0 22.75 0.33 1560.0 0.38 1.36 0.79 1.39

-1.0 1564.0 20.75 0.25 1565.0 0.43 1.43 0.68 1.40

-1.5 1572.0 18.25 0.18 1571.5 0.49 1.60 0.54 1.41

TABLE II: Fitting and derived parameters obtained for theabsorber section as a function of reverse bias voltage.

sorption functions are linearized. For the gain vari-able, we expand around the carrier density value thatcorresponds to the threshold gain assuming a transpar-ent absorber section: Gm = G0

m + gg(Ng − N 0g), where

G0m = Gm(N 0

g) = (1 − ρ)−1γ. For the absorber section,we linearize the absorption function at the saturationvalue of the carrier density, and define the carrier densityvariable so that the unsaturated modal losses, Q0

m, cor-respond to zero carrier density in the absorber section.The modal absorption is then given by Qm = gqNq−Q0

m.We will see that our choice of linearization point for theshort section is appropriate given the location of the sta-ble mode-locking region far from threshold, where theabsorber is strongly saturated.

1.53 1.56 1.59wavelength (µm)

-45

-30

-15

0

15

30

45

gain

(cm

-1)

0 0.5 1 1.5 2

carrier density, Dq

-15

0

15

0 1 2scaled carrier density, Ng

-0.5

0

0.5

1

1.5

gain

(un

its o

f γ

)

-0.25 0 0.25 0.5 0.75scaled carrier density, Nq

-0.5

0

0.5

gain

(un

its o

f γ)

0 2 4carrier density, Dg

-15

0

15

30

45γ = α

m+ α

int

(1- ρ)Gm

ρQm[-0.5, -2.0]V

1565 nm

(1- ρ)Gm

ρQm

(1- ρ)Gm

(a)

(b)

(c)

+ ρQm [-1.0 V]

1565 nm

1565 nm-1.0 V

Dq0

ρQm0

FIG. 5: (a) Fits to the modal absorption at a series of valuesof the reverse bias as shown. The modal gain corrected forthe finite absorber length, and the net gain in the two-sectiondevice for a reverse bias of -1.0 V are also shown. (b) Varia-tion of the modal gain in the two-section device with carrierdensity at the reference wavelength indicated in (a). Verticallines indicate the threshold carrier density in the FP and thethreshold carrier density in the two-section device assuming atransparent absorber section. (c) Variation of the modal ab-sorption in the two-section device with carrier density. Thewavelength is the same as in (b) The scaled carrier density iszero in the unsaturated device, and the differential absorptionis defined at the saturated value.

Results for a reverse bias voltage of -1.0 V are illus-trated in Fig. 5 (b) and (c), where we have plotted themodal gain and absorption as a function of carrier den-sity at the reference wavelength indicated by the verticalline in Fig. 5 (a). We will explain how this wavelengthwas chosen in the next section. The differential gain atthe threshold carrier density assuming a transparent ab-sorber section is gg = 0.68 or 10.8 cm−1 in physical units,and the differential absorption at the saturation value isρgq = 1.4 or 22.4 cm−1 (here we have quoted values forthe differentials that account for the difference in the de-vice section lengths). The calculated differential gain ggand differential absorption ρgq for voltages ranging from-0.5 to -1.5 V are shown in table II. Here we have alsoincluded the reference wavelength λ0, to be derived inthe next section, the unsaturated losses, ρQ0

m, and thethreshold carrier density N 0

g as defined above. The dif-ferentials are given in dimensionless units defined by thetotal losses γ.

IV. DELAY-DIFFERENTIAL MODE-LOCKEDLASER MODEL AND DEVICE OPTIMIZATION

METHOD

The results of the previous section have provided uswith a qualitatively accurate picture of the variation ofthe modal gain and absorption in the FP device as func-tions of carrier density, wavelength and reverse bias volt-age. In this section we introduce the delay-differentialmodel of the two-section device, and we describe ourmethod for finding the optimum wavelength location fora spectral filter based on the measured material disper-sion. In order to test the accuracy of the delay-differentialmodel, our first objective will be to reproduce a portionof the phase-space structure shown in Fig. 1, concentrat-ing on the region where the variation of the thresholdcurrent with reverse bias voltage is normal.

The delay-differential lumped element model that weemploy eliminates the spatial dependence of distributedtime-domain models in favour of a delay-differential equa-tion for the field variable. As we have discussed, thismodel accounts for the large gain and loss and the strongsaturation of the absorption that are typical of mode-locked semiconductor lasers. It is also suitable for an-alytic analysis, and it requires minimal computing re-sources to implement. Although the model is derivedassuming a ring-cavity geometry, previous results haveshown qualitative agreement with distributed time do-main simulations of mode-locking in linear two-sectiongeometries with both quantum well and quantum dotactive regions [31, 32].

The system of equations is [22, 27, 33]

γGE(t) = −E(t) +√κR(t− τ)E(t− τ)

G = g0 − γrG− exp−Q(expG− 1)|E|2 (2)

Q = q0 −Q− s(1− exp−Q)|E|2

7

where

R(t) = exp

[1

2(1− iαg)G−

1

2(1− iαq)Q− iφ

]and

κ = exp [−(αm + αint)Lc] .

In the above, τ is the round-trip time in the cav-ity, the dynamical variable for the gain is defined asG =

∫(1−ρ)Lc

dzGm(z, t), and the saturable absorption

is described by the variable Q =∫ρLc

dzQm(z, t). Here

Gm(z, t) and Qm(z, t) are respectively the modal gainand absorption, which are now assumed to be spatiallyvarying. γr = γg/γq = τq/τg is the ratio of the recoverytimes in the gain and absorber sections, and s = gq/ggis the ratio of the material differential gain and absorp-tion. αg,q are the linewidth enhancement factors in thegain and absorber sections of the device, and φ is the de-tuning of the gain peak from the nearest cavity mode.With the linearized gain and absorption variables de-fined as in the previous section, the scaled pump cur-rent is g0 =

∫(1−ρ)Lc

dz[γ−1q ggj − (ggN

0g − G0

m)] and

the parameter describing the unsaturated absorption isq0 =

∫ρLc

dzQ0m. The parameter γG determines the band-

width of the gain medium, which is included in theseequations through the action of a linear Lorentzian filter[22].

We have seen that the peak emission wavelength of theFP laser will in general evolve continuously as the driveparameters are varied. On the other hand, for each biasvoltage value, each wavelength defines a unique set ofmodel parameters through the modal gain and absorp-tion functions. In order to reproduce the structure of thephase-space of Fig. 1, our approach therefore is to asso-ciate a unique wavelength with each point in the phase-space of drive parameters. To do this, we calculate thevariation of the net gain along branches of steady-statesolutions of Eqns 2. The voltage value and material pa-rameters at each wavelength define a unique branch ofsteady-state solutions in the current density variable. Tocompare the model and experiment, we choose the branchwhose wavelength coincides with the peak of the net gainat the current density point of interest. This method pro-vides a consistent means to compare the measured phase-space data and the simulation results over the entire pa-rameter space, provided the variation of the net gain iswell described by our fitted modal gain and absorptionfunctions. To avoid unnecessary details, we will take -1.5V as defining a boundary beyond which fitting errors canlead to disagreement between the calculated and mea-sured location of the peak emission wavelength. In prac-tice, one should locate this boundary precisely as a func-tion of device current and define the target ML region forvoltages inside this boundary. Note that we should alsoin principle linearize our modal absorption function atthe carrier density value where the net gain peak and the

wavelength of the steady-state solution branch in ques-tion coincide. However, finding the correct steady-statebranch in this case requires a much larger numerical ef-fort, as the dispersion of the net gain at any particularwavelength will vary as the linearized absorption functionis changed. Instead, we linearized the modal absorptionaround the transparency point, which is only approachedasymptotically. However, we will see that the absorptionis already well saturated inside the large stable mode-locking region of Fig. 1.

To find the correct steady-state solution branches wemust first calculate the steady-state solutions of themodel. Assuming slowly varying fields with respect toτ , we neglect the time derivative term in Eqn 2 and mul-tiply the resulting equation by its complex conjugate toobtain P (t+τ) = κ exp[G(t)−Q(t)]P (t), where P = |E|2is the optical power. The assumption of slowly varyingfields implies that P (t + τ) = P (t) + τP (t) so that weobtain an ordinary differential equation for the opticalpower [23]

τP = −[1− exp(G−Q+ lnκ)]P. (3)

This equation, together with the equations for the gainand absorption variables in Eqn 2, can be regarded asdefining an improved LSA model that is accurate for largegain and loss. In order to find the steady-state solutionbranches we must specify the values of the carrier re-covery times in the gain and absorbing sections of thedevice. We can estimate the carrier recovery time in thegain section from a measurement of the relaxation oscil-lation frequency of the single section FP device. If wemake the approximation that ggN

0g /γ = 1 for the plain

FP laser, this quantity is expressed in physical units as

νRO =1

2π

√γphtn(js − 1)

τg, (4)

where γphtn = γvg is the photon lifetime in the cavityand vg is the group velocity. The relaxation oscillationfrequency can in turn be determined from the intensitynoise power spectrum [13]. At a device current of 40mA [js ' 3], we found that the relaxation oscillation fre-quency is approximately 4 GHz, from which we determinethat γτg ' 240 or τg = 0.9 ns. To complete the improvedLSA model we must fix the value of the carrier lifetime inthe absorber section. This quantity is in general smallerthan the recovery time in the gain section, and it is knownto have a strong dependence on the applied voltage. Todetermine τq for our simulations, we assumed that therecovery time varies as τq = τ0q exp[V/V0] [25, 34] and

took values of τ0q = 85 ps and V0 = 3.0 V that providedthe best agreement with experiment. These values thendetermine τq[-0.5, -1.0, -1.5] V = [72, 61, 52] ps.

Results showing the variation of the carrier density inboth sections of the device along a steady-state solutionbranch located at a wavelength of 1565 nm for a reversebias voltage of -1.0 V are shown in the left panel of Fig.

8

0 5 10scaled current density, js

0

1

2

3

4

carr

ier

dens

ity

1.562 1.564 1.566 1.568 1.57

wavelength (µm)

29

30

31

net g

ain

(cm

-1)

Dg

Dq

js TC

js TC3.0 [js ]

9.0

Dqtrns

Dq0

Dg0

Dgthr

γ

1565.0 nm

-1.0 V 1565.0 nm

5.0

7.0

FIG. 6: Left: Carrier densities Dg and Dq in both sectionsof the device calculated along the steady-state branch at awavelength of 1565 nm for a reverse bias voltage of -1.0 V. Thesolid vertical line is the threshold current density, jTC

s , for thisbranch of solutions. The dashed vertical line is the currentdensity value [j∗s = 7] where the peak of the net gain along thesteady-state branch coincides with the wavelength reference.Solid horizontal lines are the threshold carrier density in thegain section, Dthr

g , the threshold carrier density in the gainsection, assuming a transparent absorber section, Dthr

q , andthe saturated, Dtrns

q , and unsaturated, D0q , carrier densities in

the absorber section. Right: Variation of the net gain alongthe steady-state branch of the figure on the left for a series ofcurrent density values as indicated.

6. In the figure the threshold current density, jTCs , for

this branch of solutions is indicated. Horizontal lines in-dicate the threshold carrier density in the gain section,Dthrg , the threshold carrier density in the gain section,

assuming a transparent absorber section, D0g , and the

saturated, Dtrnsq , and unsaturated, D0

q , carrier densitiesin the absorber section. The wavelength reference wasdetermined by requiring that the peak of the net gainalong the steady-state branch coincided with the wave-length reference at a scaled current density j∗s = 7 [83mA]. This value was chosen as it lies near the center of theregion where we found the highest quality mode-lockingin the FP device. The variation of the net gain along thebranch of steady-state solutions is shown in right panel ofthe figure. One can see that the peak of the net gain is lo-cated relatively far [5 nm] from the wavelength referenceat threshold, with an excess of the net gain of approx-imately 1.5 cm−1. As the current density is increased,the peak of the net gain shifts rapidly towards the wave-length reference, where the net gain is clamped at thevalue of the total losses γ. Because the current densityregion we are targeting here is located far above thresh-old, we find that the variation of the net gain is minimalover a large range of current density values near the tar-get. This is due to the fact that the absorption is alreadywell saturated in this region, and it justifies our expand-ing the absorption function around the saturation value

10-6

10-4

10-2

100

10-8

10-6

10-4

10-2

100

inte

nsity

(ar

b. u

nits

)

inte

nsity

(20

dB

/div

)

0 3 6 9 12 15scaled current density, js

10-8

10-6

10-4

10-2

100

time (25 ps/div) detuning(100 GHz/div)

0 30 60 90 120 150device current (mA)

90 mA

150 mA

76 mA

-0.5 V

-1.0 V

-1.5 V

SP

SP

SP

MLQSML

ML

ML*

1560 nm

1565 nm

1571.5 nm

-0.5 V

-1.0 V

-1.5 V

QSML

FIG. 7: Left panels: Numerical bifurcation diagrams cal-culated at three reverse bias values with model parametersdefined at the reference wavelengths as indicated. Dashedvertical lines indicate mode-locking stability boundaries cal-culated using the generalized New stability criterion of Ref.[22]. Right panels: Intensity time traces, and correspondingoptical spectra at values of the current density located in themode-locking regions as indicated.

of the carrier density. These results also suggest that wecan be confident that our choice of model parameters atthe wavelength reference will be appropriate for currentdensity values in excess of js ' 5 [59 mA].

We now present numerical bifurcation diagrams cal-culated using the delay-differential model and comparethe results with the experimental measurements from theplain FP laser of Fig. 1. For these simulations thevalue of γ−1

G , which determines the bandwidth of theLorentzian filter in Eqns 2, was set as 0.15 ps. Thisvalue corresponds to a bandwidth of approximately 80cavity modes, and it was chosen so that the curvature ofthe modal gain function at the gain peak was accuratelyreproduced by the Lorentzian filter. Further parameterswere αg = αq = 2.6, and φ = 0. The value of α chosenis consistent with our previous work modeling optical in-jection and feedback experiments in devices with similarmaterial composition [35].

Numerical bifurcation diagrams calculated for reversebias values ranging from -0.5 V to -1.5 V are shown inFig. 7. In each case the value of the wavelength refer-ence is indicated. One can see that the variation of thereference wavelength with bias voltage from 1560 nm at-0.5 V [j∗s = 5] to 1565 nm at -1.0 V [j∗s = 7] is in broadagreement with the measured variation of the peak emis-sion wavelength visible in the optical spectra shown inFig. 1. The further red-shift in the reference wavelengthto 1571.5 nm that we find for a bias voltage of -1.5 V[j∗s = 9] is not in agreement with the measured datafor -1.6 V, as this voltage is inside the region where theanomalous dispersion of the material absorption comes

9

into play. However, we will see that the dynamical statesof the laser are qualitatively reproduced in this voltageregion. If we first compare the simulated device dynamicswith measured data at -0.5 V, we find SPs at threshold,followed by a transition to a dynamical state where theintensity is weakly modulated at a frequency close to theround trip time in the cavity. Although SPs at thresh-old were not resolved in our measurements at this volt-age, for larger currents we recall that the correspondingmeasured behavior was incomplete mode-locking, wherea large number of FP modes were above threshold but theintensity modulation at the round trip frequency was alsoweak. For comparison, we have included a simulated in-tensity time trace and corresponding optical spectrum ata scaled current density of js = 6.5 [76 mA]. One can seethat the optical spectrum is almost single mode for theseparameters. We have therefore referred to these statesas “starred” mode-locked states of the laser, [ML∗], asthe intensity modulation at the round-trip frequency isvery weak in this region. The tentative correspondencebetween these simulations and our measurements is an in-dication that the delay-differential model with calibratedparameters can capture subtle physical variations thatwere found across the ML region in our experiment.

At a reverse bias of -1.0 V, we see that the SP regionat threshold increases in size, and we find a narrow re-gion of QSML before entering a region of well developedmode-locking near a device current of 90 mA. This nar-row region of QSML is also evident in our experimentalmeasurements for larger reverse bias values. The MLregion extends as far as 140 mA, where mode-lockingbecomes unstable. When compared to our experimentalmeasurements, we can see that the ML region here opensat a larger value of the device current, and that it extendsover a larger current range in the simulation. However,one should note that the incomplete ML region near -0.5V was recorded as a CW region in our measurements,which indicates that there may be a larger region of in-complete ML surrounding the labeled ML region in Fig.1. If we examine the simulations for -1.5 V in the lowerleft panel of Fig. 7, one can see that the general trendsthat were observed in our measurements are well repro-duced by the simulations. These include the increasingsize of the QSML region, and the rapid expansion of theML region with increasing reverse bias. One can see alsothat the pulse duration becomes progressively shorter asthe reverse bias is increased. Although the numericalpulse duration at -1.0 V [2.7 ps] is considerably longerthan the experimental value, the pulse duration is 1.6 psat -1.5 V, which is in broad agreement with measureddata.

Dashed vertical lines in the bifurcation diagrams ofFig. 7 indicate regions of stable mode-locking calculatedusing the generalized New stability criterion [GNSC] forthe DDE model formulated in Ref. [22]. The New sta-bility criterion for mode-locked states requires that thenet gain must be negative at the beginning and end ofthe slow-stage of the dynamics between pulses when the

intensity is small [36]. Although the agreement with theGNSC is poor for smaller reverse bias voltages, at largerreverse bias, where the pulses are well developed, we findthis criterion gives a very good estimate for both sta-bility boundaries [the upper ML stability boundary at-1.5 V near js = 16 is not shown in this figure]. Theutility of these semi-analytic results for the ML stabilityboundaries obtained from Eqns 2 highlight an attractivefeature of the DDE model, but they also demonstratethat the DDE model has a much richer dynamics and amuch larger stable ML region than the GNSC would ingeneral suggest.

The results presented in Fig. 7 demonstrate thatthe delay-differential model with experimentally cali-brated parameters can provide reasonably good quali-tative agreement with experimental data. Overall, themodel seems to overestimate the drive current required toreach the ML region at a given voltage, and it also over-estimates the size of the various regions of dynamics inboth the voltage and device current variables. It is clearhowever that the region of simulated high quality mode-locking evolves from a region of incomplete mode-lockingas the reverse bias voltage is increased, which means thatwe can judge if a simulated ML region is likely to vanishfor a small change in the model parameters.

With these qualifications in mind, we now consider theproblem of finding the optimal wavelength location fora spectral filter in order to maximize the stable mode-locking region and tunability of the device. If we imaginea relatively narrow-band filter, and a quasi-static pictureof the dynamics, then the current and voltage range overwhich we can expect stable mode-locking at the desiredwavelength will be determined by the variation of thenet gain along the corresponding steady-state branchesat the wavelength location of the spectral filter. To avoidtransfer of the optical power to adjacent cavity modes,we require that the excess of the net gain along a steady-state branch at the filter location cannot exceed the filterselectivity at detuned wavelengths. This condition mustbe then be satisfied over a continuous region in the cur-rent and voltage parameters, accounting for the fact thateach voltage determines a different steady-state branchat the wavelength of the spectral filter.

An experimentally calibrated example illustrating theapplication of this method is shown in Figs 8 and 9, wherewe have found that a filter located at 1566.5 nm will leadto a large and continuous region of high quality mode-locking near 140 mA for voltages between -1.0 and -1.5V. Left panels in Fig. 8 show the carrier density alongthe steady-state solution branches for the two voltages,while the right panels show the variation of the net gainalong these branches for js between 7 [83 mA] and 13 [154mA]. Notice how the wavelength reference here is locatedcloser to the peak of the net gain at -1.0 V, which leadsto a larger excess of the net gain along the branch at -1.5V. This was necessary in order to ensure overlapping MLregions in the phase-space as shown in the left panels ofFig. 9. Time traces in the right panels of Fig. 9 indicate

10

0

2

4

29.5

30.0

30.5

31.0

0 3 6 9scaled current density, js

0

2

4

carr

ier

dens

ity

1.564 1.568 1.572wavelength (µm)

30.0

31.0

32.0

net

gai

n (c

m-1

) γ

γ

Dg

Dg

Dq

Dq

-1.0 V 1566.5 nm

-1.5 V 1566.5 nm

7.0 [ js ]9.0

11.0

13.0 [ js ]

FIG. 8: Carrier densities calculated along the steady-statebranches at 1566.5 nm (left panels). Variation of the net gainfor a series of current density values on the same branch asindicated (right panels).

50 75 100 125 150device current (mA)

10-8

10-6

10-4

10-2

100

3 6 9 12scaled current density, js

10-8

10-6

10-4

10-2

100

inte

nsity

time (25 ps/div)

inte

nsity

(ar

b. u

nits

)

-1.0 V 143 mA

-1.5 V 143 mA

SP

SP

ML

ML

QSML

QSML

-1.0 V 1566.5 nm

-1.5 V 1566.5 nm

FIG. 9: Bifurcation diagrams, time traces, and optical spec-tra. Top panels: -1.0 V, and lower panels -1.5 V. The wave-length that determined the model parameters is 1566.5 nm,and the pump parameter value for each of the time traces wasjs = 12.

well developed pulses over the ML region. Results in Fig.8 indicate that a spectral filter will require a minimumselectivity of ca. 3 cm−1 to ensure stable emission atthe desired wavelength over this parameter region. Al-though this value of the filter selectivity exceeds the mea-sured value from the FP device of Fig. 3, we note thatthis device was based on a low-density grating structurethat was defined by etched features in the laser ridgewaveguide. Numerical simulations of distributed feed-back structures that define comb laser spectra indicatethat more advanced grating designs can readily achievethis level of selectivity [14, 37].

The results of Fig. 9 indicate that a grating spectralfilter centered at a wavelength of 1566.5 nm will allow fortuning of the ML state of the device over a wide rangein the drive current and voltage parameters. Althoughthe overall device length and absorber section length ofthe device of Fig. 3 were not identical to the FP laserwe considered in order to illustrate the application of themethod, the material composition was identical. It isnotable therefore that the result for the optimal locationof the grating filter is some 20 nm from the designedwavelength location of the device of Fig. 3. We note alsothat the optimized wavelength location is more than 12nm from the position of the gain peak at threshold in theFP device.

Because the agreement between the simulated andmeasured phase-space data for the FP device was onlyqualitative, our results for the optimized wavelength lo-cation and the predicted dynamics are approximate. Inparticular, we did not account explicitly for any addi-tional dispersion or change in modal thresholds that agrating filter might introduce in the cavity. Numericalresults have shown that the grating induced dispersioncan become much larger than the underlying materialdispersion in devices with strongly scattering filters [37].However, it is also possible in general to correct for thisin narrow-band examples, which are the focus of our in-terest here. We also note that the comparison betweenthe simulations and experiment seemed to indicate thepresence of some systematic errors, where the onset ofmode-locking and the width of the ML region in the cur-rent variable was overestimated. It would be interestingto establish if these errors could be eliminated with amore accurate model of the semiconductor susceptibility.

We note also that although the DDE model is a largesignal model of the mode-locked laser, our method wasbased on steady-state solutions of the model and is there-fore small signal in nature. We justified this limitationon the grounds that we are interested primarily in mode-locked devices with relatively narrow-band spectra. Itwould however be interesting to attempt to extend themethod in order that it could be applied to devices de-signed to support wide bandwidth combs, with the poten-tial to generate short pulses. Under these circumstances,the spectrum could be strongly influenced by dynamicaleffects such as self-phase modulation and the efficiency ofthe absorber saturation in a mode-locked state [24, 26].

Although we believe that these results demonstratethe utility of the DDE model, our work has also high-lighted a number of its limitations. The DDE model isderived assuming a linear Lorentzian spectral filter thatcorresponds to the gain bandwidth of the semiconductor.However, we obtained unphysical results if we attemptedto adjust the filter bandwidth parameter to values thatmight describe a device such as in Fig. 3. In addition, wefound that adjusting the nominal value of the detuningof the gain peak from the nearest cavity mode throughthe φ parameter had a significant effect on the simulatedphase-space. Given that the gain bandwidth was of order

11

80 cavity modes, this is also an unphysical dependenceof the model. Similarly, we found a strong dependenceon the alpha factor values chosen for both sections of thedevice. Although this dependence is in some respects tobe expected, we were unable to interpret our results ina consistent way. We therefore set the alpha parametersto be equal for both sections, and we leave a detailed ex-ploration of the combined roles of the alpha factors andφ parameter for future work. Overall, it seems that fur-ther development of the DDE or a similar model is bothnecessary and desirable given its obvious strengths. Wenote that some progress towards generalizing the actionof the spectral filter to describe waveguide dispersion inreal devices has been already been made [38].

V. CONCLUSIONS

In conclusion, we have presented a method for de-termining the optimal wavelength location for a grat-

ing spectral filter in a two-section semiconductor mode-locked laser. The goal of this method was to maximisethe region of stable mode-locking in the phase-space ofdrive parameters of the device. We accounted for ma-terial dispersion in the two sections of the laser using asimple analytic model for the semiconductor susceptibil-ity, which provided good agreement with measured dataover the parameter region of interest. Our dynamicalsimulations were based on a delay-differential model ofthe device, which we found could qualitatively describethe structure of a measured dynamical phase-space whencalibrated with experimental parameters. By consideringthe variation of the net gain along steady-state solutionsof the model, we were able to optimize the location ofthe spectral filter with respect to regions of stable mode-locking in the device. Our results indicate that it will bepossible to obtain a large region of stable mode-lockingin devices with practical values of the grating selectivity.

The authors acknowledge financial support from Sci-ence Foundation Ireland under grant SFI13/IF/I2785.

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