## Abstract

We present a detailed analysis of a semiconductor hybrid laser exploiting spectral control from an external photonic waveguide circuit that provides frequency-selective feedback. Based on a spatially resolved transmission line model (TLM), we have investigated the output power, emission frequency, and the laser spectral linewidth. We find that, if the feedback becomes weaker, the spectral linewidth is larger than predicted by previous models that are based on a modified mean-field approximation, even if these take a strong spatial variation of the gain into account. The observed excess linewidth is caused by additional index fluctuations that are associated with strong spatial gain variations.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Widely tunable, narrow linewidth diode lasers are of significant relevance, ranging from terrestrial applications such as fiber-optic communications [1] or optical sensing [2], to space-based applications such as laser cooling [3] and atomic clocks [4] in global positioning systems (GPS). In the evolution of such applications, monolithic single-frequency diode laser sources, specifically distributed feedback (DFB) lasers and distributed Bragg reflector (DBR) lasers, approach their limits due to a relatively small tuning range [5] and large linewidths at the MHz level [6], the latter caused by the short cavity photon lifetime in these lasers. In comparison, so-called hybrid diode waveguide lasers, which have been subject of recent, extensive research, can offer much longer photon lifetimes, and therefore narrower linewidths, in addition to wider tunability. The concept of such hybrid lasers is based on optically coupling a semiconductor gain medium to a low-loss passive waveguide circuit that provides a significantly extended resonator length and a highly frequency selective feedback. The benefits to be gained with low-loss waveguide feedback circuits are that: a) flexible filtering schemes can be applied which allow for wide wavelength tunability, b) narrow linewidths can be achieved due to increased photon lifetime that is associated with an extended cavity length [7], and that c) such lasers are ideal for integrating into subsequent waveguide circuitry fabricated on the same waveguide chip.

Making use of high-quality intra-cavity microring resonators (MRRs) in the feedback circuit has introduced a new paradigm on how to pursue chip-based narrow linewidth semiconductor laser sources. As opposed to Bragg gratings, the fabrication of MRRs does not demand complicated lithography processes and can provide lower losses. A related important merit of using low-loss MRRs is that they effectively extend the optical length of the laser resonator by an appreciable factor, due to the multiple roundtrips in the MRR [8]. This length extension contributes again to a narrowing of the laser linewidth.

The tuning properties of such hybrid lasers are generally well-understood. However, the laser linewidth remains challenging to predict due to the increased complexity of the cavity design which is ruled by a larger set of experimental parameters. So far, for a coarse estimation of the linewidths of such lasers [9], it is common to employ Henry’s theory which is using the mean-field approximation (MFA) corrected with the so-called linewidth enhancement factor, *α* [7]. However, this approach is rather incomplete since the effect of linewidth narrowing or broadening induced by the spectral selectivity and dispersion of the external feedback [10] is not taken into account.

To overcome these shortcomings, refined models were developed [10] and used for comparison with experimental linewidth data, and qualitative agreement was found [11]. However, still these theories are based upon the mean-field approximation (MFA) in the gain section. Specifically for hybrid lasers this approximation is questionable and can be insufficiently accurate. The reason is that hybrid lasers, depending on the strength of the feedback, may show strong spatial variations of the intensity in the semiconductor gain element which introduces strong variations of the carrier density as well. We note that in monolithic diode lasers with a single cavity of short length such spatial effects [12] and nonlinear gain effects [13] cause an increase of the laser linewidths as compared to mean-field models.

In this work, we present the first modeling of external cavity diode lasers (ECDLs), also called extended cavity diode lasers, that takes into account the complex feedback obtained with waveguide resonator circuits such as in [11] and [14] and that simultaneously takes into account the detailed spatial variation of the diode-internal intensity and carrier density, such as in [12] and [13]. The goal of our detailed modeling is to enable a comparison with the existing simplified theories and, thereby, check their validity under different conditions. To this end, we make use of a commercially available, advanced transmission line model (TLM) [15] which involves spatially resolved rate equations for the complex-valued electric field and spatially resolved rate equations for the carrier density [16,17] in the active section of the laser cavity that contains the semiconductor optical amplifier. The remaining, external part of the cavity that comprises the linear optical waveguide feedback and spectral filtering circuit is modeled as a complex-valued frequency dependent scattering matrix. A first test of this model has been described in our previous conference publication [18], showing satisfying agreement in terms of linewidth within a factor of about 1.5 in the 20–50 kHz range, with experimental data obtained with a relatively short InP-Si_{3}N_{4} waveguide hybrid laser with feedback from two MRRs, and with a fixed optical coupling strength between the InP gain section and the Si_{3}N_{4} waveguide chip. In order to cover different regimes of interest, here we systematically vary this power coupling strength, *β*, between the InP gain section and the Si_{3}N_{4} waveguide chip. This variation brings the diode internal field and carrier distribution from spatially high uniformity (*β* ≈ 1 and strong feedback from the waveguide resonators, where the MFA remains justified) towards a strong spatial variation (*β* ≈ 0 or weak resonator feedback) where a detailed modeling has not been performed so far. The drive current of the diode amplifier is used as an adjustable input parameter because also in a typical experimental investigation the electric drive current can be varied most easily. For each setting of the chip-to-chip coupling efficiency we calculate the power spectral density of the frequency noise from which we derive a value for the laser linewidth, to compare with values obtained with a modified MF theory.

## 2. Operation principle

The hybrid laser that we model comprises a semiconductor active gain section coupled to a passive external cavity as shown in Fig. 1. For definiteness, as required in a numerical model, and to guide our experimental efforts, we chose a typical InP diode as amplifier while the feedback is provided via a Si_{3}N_{4} waveguide chip.

The laser diode gain section has one high-reflectivity facet with power-reflectivity *R*_{1} and one facet with low power-reflectivity *R*_{2}. The overall feedback provided by the Si_{3}N_{4} waveguide chip circuitry is lumped into a frequency dependent power reflectivity, *R _{eff}*(

*ω*), which comprises both propagation through the passive straight waveguides and reflection from two microring resonators (MRRs) of slightly different free spectral range (FSR). The two MRRs provide a so-called Vernier mirror, and act as wavelength-selective reflective filters. Wavelength tuning can be achieved via the thermo-optic effect, i.e., by heating of the MRRs, which increases the effective waveguide index and thereby the optical length of the MRRs. Further details about MRR Vernier filters can be found in [19–21].

Radiation generated by spontaneous and stimulated emission in the amplifier chip enters the Si_{3}N_{4} waveguide chip with a power coupling coefficient, *β*, that depends on the overlap (matching) of the mode profiles in the gain waveguide and Si_{3}N_{4} waveguide [22].

In the Si_{3}N_{4} chip, light is first divided by a symmetric 50/50 splitter (Y-junction), and then guided sequentially through the two cascaded MRRs before being combined and fed back to the gain section. For the hybrid laser to oscillate at the frequency of a specific longitudinal mode of the entire cavity, that frequency must also be supported by the gain spectrum of the specific semiconductor material used, and coincide with the transmission spectra of both MRRs. A second symmetric 50/50 splitter (Y-junction) combines the light that passes by the throughputs of the MRRs, which can be used as laser output.

## 3. Revisiting laser linewidth theories

The currently most complete analytical theory [10, 23] for external cavity diode lasers with spectrally filtered feedback suggests that the linewidth of the hybrid laser, Δ*ν*, can be calculated along the following lines: first, the hybrid laser is modeled in the same manner as a solitary Fabry-Perot laser with a back facet field amplitude reflectivity of ${r}_{1}=\sqrt{{R}_{1}}$, which is frequency-independent and real-valued. The front facet is given a complex-valued amplitude reflectivity, *r _{eff}*(

*ω*) =

*r*(

*ω*)

*exp*(−

*j**

*φ*(

_{eff}*ω*)). The real-valued amplitude factor in this expression,

*r*(

*ω*), represents the overall field feedback from the external passive circuit, and is given by

*β*, and any additional loss that occurs in the external feedback path. The frequency dependent phase,

*φ*(

_{eff}*ω*), takes into account the optical delay that the feedback circuit provides. Specifically, this phase varies with the light frequency because it depends on the detuning from the resonance frequency of the Vernier filter and on other optical parameters of the external feedback path, such as the length of the waveguide to and from the MRRs. After lumping the external feedback into

*r*(

_{eff}*ω*) for the front facet, Δ

*ν*is given by: In this equation, Δ

*ν*is the Schawlow-Townes linewidth of the equivalent solitary Fabry-Perot diode laser [23]:

_{ST}*L*

_{1}and having real-valued field reflectivities

*r*

_{1}and

*r*(

*ω*).

In Eq. (3), *v _{g}* =

*c/n*

_{g1}is the group velocity in the diode,

*h*is the Planck constant,

*ν*is the laser emission frequency,

*n*is the spontaneous emission factor,

_{sp}*γ*is the total loss (mirror loss plus diode internal loss),

_{tot}*γ*= 1/(2

_{m}*L*

_{1}) ln(1/(

*R*

_{1}

*r*

^{2}(

*ω*))) is the distributed mirror loss,

*α*is Henry’s linewidth enhancement factor, and

*P*

_{0}is the output power from the back diode facet. This facet was chosen due to its frequency independent transmission such that the output is simply proportional to the laser internal power.

The denominator in Eq. (2) is the so-called linewidth reduction factor which describes laser line narrowing due to the frequency dependency of the external feedback,

whereAfter substituting Eq. (1) into Eq. (6), the expression of B is then written as

In Eq. (5) and Eq. (7), *τ _{LD}* = 2

*n*

_{g1}

*L*

_{1}/

*c*is the roundtrip time in the solitary diode. The term A can be physically interpreted as the ratio between the external cavity optical length and the solitary diode optical length, which describes a linewidth reduction resulting solely from the extension of the cavity length. As described above,

*ϕ*(

_{eff}*ω*) contains the effect of propagation in the waveguides to and from the MRRs. It also contains (via a maximum of $\frac{d}{d\omega}{\phi}_{\mathit{eff}}(\omega )$ at the peak of the filter resonance) the cavity length enhancement resulting from multiple roundtrip passes through the MRRs. The term B describes an additional linewidth reduction that occurs only at the rising edge of the Vernier filter’s reflection peak, where $\frac{d}{d\omega}\text{ln}r(\omega )$ is positive.

The problem with the Schawlow-Townes expression in Eq. (3) is that it is based on the standard mean-field approach, which treats any lumped mirror loss as uniformly distributed along the cavity. However, such treatment is only valid for diode lasers that have symmetric and relatively high reflectivities (*r*_{1} ≈ *r*_{2} ≈ 1) such that the diode internal intensity is approximately uniform (spatially homogeneous). It should be noted that only with such uniform intensity, the carrier density is uniform as well, which is required to justify a uniform gain-index coupling in the form of the simple space-independent *α*-factor in Eq. (3). Hybrid lasers, specifically, do not generally fulfill these assumptions because the chip-to-chip coupling efficiency, *β*, might be rather low (at the 10% level without adiabatic tapering [20]), because the circuit reflectivity might be low and because the back facet reflectivity might also be very low, for instance, for the purpose of extracting more usable output power [24]. In these situations, the reflectivities of the equivalent solitary FP laser can become highly asymmetric, and the *α*-factor may strongly vary throughout the amplifier, which can increase the laser linewidth to beyond what is predicted in Eq. (2).

The limitations of the mean-field approximation had been realized by several independent researchers [25–27]. Specifically, lumping strongly asymmetric and low facet reflectivities in the Schawlow-Townes expression in Eq. (3) underestimates the coupling of amplified spontaneous emission into the main laser mode because low facet reflectivities result in a large single pass gain. To take these effects into account to some extent, it was suggested [25–27] to modify Eq. (2) with another multiplicative factor, *F _{R}*, called spontaneous emission enhancement factor, given by

*r*

_{2}is substituted by

*r*(

*ω*) (modified mean-field model).

However, even though modifying Eq. (2) with *F _{R}* takes into account additional noise by locally less depleted gain, this still does not address the associated index-effect that would lead to an increased

*α*-factor. What is required to quantify this effect, for the first time in hybrid lasers, is to conduct detailed calculations which also include the spatially varying index effect in the laser linewidth.

## 4. Spatially resolved modeling of a hybrid laser

#### 4.1. Implementation of the transmission line model

In order to conduct a more detailed and accurate linewidth analysis, which includes the spatial variations of the gain and index, we have chosen to apply an advanced transmission line model (TLM) for the gain section. Summarized, the implementation of the TLM [16,17] is a time-domain model that iterates optical fields travelling back and forth along the semiconductor amplifier to generate optical waveforms at the facets. Both the time and space coordinates are taken as discrete variables. Figure 2 shows the structure of the implemented TLM. The active gain section is divided into multiple short longitudinal TLM sections, each of which is short enough to be treated as homogeneous. For an efficient implementation, the TLM sections were chosen to have a length that is equal to the distance the light propagates within a model time step, Δ*z* = *c*Δ*t/n*_{g1}. For each section there is a so-called *scattering node* that represents, e.g., the electric field, carrier density, gain, loss and noise being present at that local section. Within each section, uncorrelated spontaneous emission noise is added as Langevin noise sources [7] for both forward and backward fields. The spectral shape of the spontaneous emission is determined by the gain spectrum, and the power spectral density is proportional to the local carrier density. The passive external resonator section is modeled in the frequency domain based on the scattering-matrix method [28] which makes use of well-known analytical solutions of the involved basic optical components, such as straight waveguide sections, Y-junctions, microring resonators [29–31]. A Fourier transform is performed in order to switch between the time-domain calculation and frequency-domain calculation. The configuration parameters for numerical modeling are given in Table 1. The laser parameters introduced into the TLM are summarized in Table 2 (diode parameters) and Table 3 (feedback parameters). The reason for our particular choice of parameters is that it enables a comparison of the numerically calculated linewidth with previously measured, experimental values as obtained with an according hybrid laser [32]. The first part of parameters is taken from the geometric and feedback waveguide parameters of that experimental setup. The remaining parameters are typical (standard) values for InP semiconductor amplifiers, which is why they are implemented also in the library of the used, commercially available TLM code [15].

The goal is to model the propagation and amplification of waves at frequencies of hundreds of THz, however, this would require numerically unfeasible sampling rates of about a petahertz if trying to resolve each optical cycle to full detail. To avoid this problem, we follow an alternative solution where the waves are modeled as a carrier wave (with a fixed, known frequency) multiplied with a somewhat slower varying envelope. This approximation is justified because the laser gain bandwidth is only a small fraction, about 5%, of the laser center frequency. In all of our modeling the center frequency was set to *f*_{0} = 193.1 THz, which is equivalent to an output wavelength around 1550 nm. Each calculation run includes a large number of iterations, called scattering steps, taking place depending on the sample rate and the time window settings. Multiple runs can be carried out subsequently, each one taking as the initial values the last results of the former run. For the continuous-wave, steady-state laser modeled here, results from initial runs containing laser startup and transients can be discarded, and results from remaining runs, after the laser reaches steady-state, can be used for averaging of the frequency noise spectra.

In order to illustrate what feedback (*r _{eff}*(

*ω*)) we use in the numerical model, we plot in Fig. 3(a) the magnitude of

*r*,

_{eff}*r*(

*ω*) = |

*r*|, as obtained with the parameters in Table 3 for three different values of

_{eff}*β*(

*β*= 0.1, 0.5 and 1). In Fig. 3(b) we plot the values for A, B and F, assuming a typical value for the linewidth enhancement factor,

*α*= 3. As the laser will operate at a frequency around the Vernier peak frequency (

*f*

_{0}), the horizontal axis is normalized to show the light frequency as a detuning with respect to

*f*

_{0}. It can be seen in Fig. 3(a) that the MRRs generate a spectral feedback maximum at zero detuning, where the height of the maximum decreases in proportion with

*β*. The linewidth reduction terms A and B, which are associated with a spectrally varying phase (

*ϕ*(

_{eff}*ω*)) and amplitude (

*r*(

*ω*)) of the feedback, and the resulting total linewidth reduction, F, are plotted in Fig. 3(b). It can be seen in Fig. 3(b) that the MRRs’ spectrally filtered feedback causes a) a linewidth reduction brought by an enhanced cavity length due to multiple roundtrips (see peak of A at zero detuning); b) a linewidth reduction in the lower-frequency wing of the filter resonance (see asymmetric peak in B).

In the following sections, we present typical results obtained from the numerical transmission line model and compare them with what is predicted by the modified mean-field model as given by Eqs. (2)–(8). In brief, we show that the two models agree well with each other, however, that the mean-field model underestimates the linewidth for the case of weak coupling between the two chips.

#### 4.2. Output power and laser emission frequency

With the TL model, the output power (*P*_{0}) from the diode back facet and laser emission frequency (Δ*f*, relative to *f*_{0}) was calculated for a range of coupling coefficients, from *β* = 0.1 to 1.0 in steps of 0.1, and for various different drive currents (*I*) up to 100 mA. Examples of typical results are presented in Fig. 4 (*P vs I*) and in Fig. 5 (Δ*f vs I*), for three different coupling efficiencies, *β* = 1, 0.7, 0.3. It can be seen in Fig. 4 that the output power follows an approximately linear growth with the current interrupted by discontinuous hops, and that the overall power level drops as *β* decreases. Discontinuities are also seen in the output frequency (Fig. 5) and these occur at the same current values as in Fig. 4. Variations of the output frequency of lasers as in Fig. 5 are actually well known. They represent so-called mode hopping of the laser frequency to that of a neighboring longitudinal mode of the overall hybrid laser cavity. The frequency change of such a hop is given by the free spectral range (longitudinal mode spacing) of the hybrid laser cavity which is here approximately 14 GHz.

It can also be seen in Fig. 5 that between mode hops the frequency increases linearly with the drive current (such as the output power in Fig. 4). This might be understood follows: at higher drive currents, higher output powers and photon densities are reached. This results in a lower differential gain parameter and thus a higher carrier density for balancing the same threshold gain. A higher carrier density results in a lower refractive index [33], which leads to an increased laser frequency.

#### 4.3. Frequency noise spectra and laser spectral linewidth

In order to calculate the laser spectral linewidth in a computationally feasible way, we calculate the power spectral density (PSD) of the frequency noise instead of directly resolving the full width at half maximum of the laser field power spectrum, because the latter requires kHz-level resolution and consequently a prohibitively long calculation time. The PSD is obtained via squaring the Fourier transform of the temporally varying (instantaneous) laser frequency. The fundamental (quantum) noise limited spectral linewidth can be retrieved from the limit value of the PSD as the noise frequency approaches zero, i.e., from the PSD in the flat, low-frequency part of the noise spectrum [34]. Two examples of such frequency noise spectra are shown in Fig. 6, calculated for the same value of the chip-to-chip coupling (*β* = 0.7) but for two different drive currents (20 mA and 60 mA). In both spectra, two peaks can be seen. The first, shallower and broader peak corresponds to damped relaxation oscillations (between 2 and 3 GHz). The second, higher and narrower peak corresponds to beating between adjacent longitudinal modes (at about 14 GHz). It can be seen that the average PSD value at low frequencies (below 1 GHz) decreases from about 10.25 × 10^{3}*Hz*^{2}/*Hz* at 20 mA to about 2.36 × 10^{3}*Hz*^{2}/*Hz* at 60 mA, indicating (via multiplication by *π* [34]) a linewidth reduction from about 32.2 kHz to about 7.4 kHz.

In order to conduct a first, qualitative comparison between the linewidth values from the transmission line model and the values obtained with the modified mean-field model, the linewidth for *β* = 0.7 is plotted as a function of the inverse output power, 1/*P*, in Fig. 7. The motivation for plotting the linewidth *vs 1/P* is that Eq. (3) predicts a 1/*P* dependence. The symbols in Fig. 7 represent values calculated with the transmission line model and the error bars represent the standard deviation of the PSD at low frequencies from the average PSD in that range. The dashed line is a least mean square fit with a straight line as expected from Eq. (2). The numerical data clearly confirm the expected overall trend of linewidth narrowing with increasing output power, i.e., towards smaller values of 1/*P* on the horizontal axis. All other linewidth data calculated for other values of *β* (not shown here) displayed the same power dependence as long as no mode hop was present within the inspected range of power variation.

#### 4.4. Comparison between numerical results and analytical results

To carry out a quantitative comparison between the numerically calculated linewidths and linewidths predicted by the mean-field theory, it is crucial to determine from the transmission line model an effective value of the linewidth enhancement factor, *α*, because, different from the analytical theory, such linewidth enhancement factor cannot be inserted a priori as parameter in the numerical model. Instead, the numerical model calculates explicitly the time dependent local carrier density, *N*(*z*, *t*), from which it explicitly calculates the local refractive index, *n*(*z*, *t*), and the according enhancement in laser frequency excursions. This means that, in the numerical model being much closer to a complete description, there is no generally valid *α*-factor. Instead the gain-index coupling is a local, space-dependent parameter of which some overall, effective (space averaged) *α*-factor may be derived that depends on the specific carrier and intensity distribution in the gain section. However, if the steady-state carrier density along the diode section is computed as a function of current at different values of *β*, this enables a calibration of the effective value of *α* comparable to the value of *α*-factor in the mean-field model, making use of the fact that *α* is linearly proportional to the carrier density.

We have carried out the named calibration as follows: at first, an operation point is identified in terms of drive current that gives rise to steady-state operation and that satisfies the mean-field approximation. Here we chose *β* = 1 because then *r*(*ω*) is close to unity as well (see Fig. 3(a)), as is *r*_{1} (*r*_{1} = 0.92). As a result, the carrier density, *N*_{0}, shows a negligible spatial variation. Secondly, Eq. (2) is fitted to the linewidth obtained from the transmission line model, where *α* is the only fit parameter. For a drive current of 20 mA, this yields an *α*-value of 3.4, which we denote as *α*_{0}. In a third step, we obtain *α*-values for lower values of *β* via multiplying *α*_{0} by *N*/*N*_{0} where *N* is the spatially averaged carrier density, regardless of how strong the spatial variation is. As the last step, the theoretical linewidth value (Δ*ν*) obtained with Eqs. (2–8) is calculated, where the *α*-value from the previous step is inserted.

In Fig. 8, the ratio between the linewidth from the transmission line model and from the modified mean-field model, denoted as Δ*ν _{TL}*/Δ

*ν*, is plotted as a function of

*r*(

*ω*) across the inspected range of

*β*(0.1 to 1) and across the inspected range of drive currents (20 mA to 60 mA). It can be seen in Fig. 8 that at

*r*(

*ω*) ≈ 1 this ratio is approximately unity (at the calibration point as expected) but that it deviates increasingly from unity as the effective reflectivity decreases. The vertical spread of data points can be associated with uncertainties in linewidth retrieval from frequency noise spectra. Nevertheless, clearly an overall increasing trend of the ratio towards small values of

*r*(

*ω*) can be seen. For convenient practical use in predicting the influence of

*r*(

*ω*) on the laser linewidth also for other hybrid lasers, without the need to perform an extensive numerical modeling again, we have summarized the numerical data in a closed-form expression which we call a linewidth correction factor,

*F*. The closed form is obtained by fitting a heuristic function with matching shape to the numerically calculated data in Fig. 8, under the assumption that

_{c}*Fc*is equal to 1 when

*r*(

*ω*) ≈ 1 for physical correctness. The fitted function in the form of a simple exponential is given by

## 5. Conclusion

Our theoretical modeling provides for the first time a comprehensive comparison between the existing, analytical linewidth theory (a modified-field model) and numerical calculations based on an advanced, spatially resolved transmission line model for hybrid diode lasers with frequency-selective feedback. It is found that with a weak optical coupling between the diode amplifier and the feedback (here a waveguide circuit on a chip), the modified mean-field model tends to underestimate the laser spectral linewidth. This underestimation becomes more apparent with decreasing coupling. We ascribe the discrepancy between the transmission line and mean-field model mainly to an inadequacy of the mean-field model in scaling the effect of a locally varying gain-index coupling into a single number for the *α*-linewidth enhancement factor.

While the direct implications of strong spatial gain variations may be well-included in the mean-field model via a spontaneous emission enhancement factor, this does not account for the associated variation in index. Our numerical modeling reveals that these index effects can be seen as an effective *α*-factor that increases the laser linewidth with decreasing feedback strength, and that the increase can be approximately accounted for with a heuristic linewidth correction factor, *F _{c}* = 0.94 + 1.06exp(−3.06

*r*(

*ω*)), where

*r*(

*ω*) is the coefficient that describes the effective amplitude reflectivity of the feedback circuit.

Our findings have the following implications for predicting the spectral linewidth of hybrid lasers in order to find a proper design of the waveguide feedback circuit parameters before fabrication and hybrid integration:

- For a proper prediction of the laser linewidth in the case of strong feedback (
*β*close to unity), it is fully sufficient to use the current, modified mean-field theory as guidance. The laser linewidth will decrease as expected with increasing output power, with increasing mode coupling efficiency, and by implementing feedback circuits with low loss and a large linewidth reduction term (*F*^{2}in Eq. (4)); - For hybrid lasers with weak coupling (
*β*≤ 0.1) where a strong spatial variation in the carrier density is present, the analytical theory should be applied with more caution although the laser linewidth still decreases with increasing output power. Namely one should be aware that the presence of a strong spatial variation in carrier density will broaden the linewidth due to an additional index related effect. For instance, at a weak coupling of*β*= 0.1 (which is still fully sufficient for laser operation due to the huge gain provided by diode amplifiers) the linewidth will be larger by a factor of about 2.

Examples of the second situation are hybrid lasers where the feedback mode is not well matched to the diode waveguide mode, or when the feedback waveguide circuit itself is providing high losses, such as would typically occur with feedback from waveguide circuits fabricated from Si. Also other types of diode lasers show a strong variation of the carrier density and should thus exhibit linewidths larger than expected. These are diode lasers that are deliberately equipped with a low reflectivity of the back facet (at the few percent level or below) [24] as is often chosen to increase the usable output power, or heterogeneous integrated hybrid DFB lasers [35] that might show strong spatial carrier density variations such as through spatial hole burning [12].

## 6. Appendix

## 6.1. Model descriptions

In this section we briefly recall [7,36–40] the basic equations for the TLM model that is used in the diode gain chip. The electric field inside each TLM section, *E*(*z*, *t*), can be represented as

*F*(

*z*,

*t*) and

*B*(

*z*,

*t*) are the complex envelopes for the forward and backward propagating field respectively [36],

*q*is the propagation constant, and

*ω*

_{0}is the operating center angular frequency. The field propagation is described by the following two coupled equations [37]:

*v*is the group velocity;

_{g}*α*is the internal loss;

_{i}*Q*and

_{spf}*Q*are stochastic Langevin noise terms [7] caused by spontaneous emission that is coupled into the propagation fields;

_{spb}*g*is the modal gain and its dependency on carrier density and photon density (

*S*) can be found in [38], and is given by where Γ is the modal confinement factor;

*g*

_{0}is the gain coefficient;

*N*is the transparency carrier density; and

_{tr}*∊*is the gain compression factor. The carrier dynamics is governed by the following two coupled rate equations:

*N*, and the other in the multiple quantum wells,

_{SCH}*N*. The coefficients

_{MQW}*A*,

_{d}*B*, and

_{d}*C*describe the single-electron recombination, bimolecular and Auger recombination, respectively [37]. The two cross sectional areas exchange carriers with each other at time scales that are determined by two time constants (capture time

_{d}*τ*and thermionic emission time

_{cap}*τ*) [39,40].

_{em}## 6.2. Numerical parameters

## Funding

Rijksdienst voor Ondernemend Nederland (RVO) (IPD12009); Dutch Technology Foundation STW (10442).

## Acknowledgment

This work is supported by funding within the framework of the Promis2Day project, for which the authors gratefully acknowledge the support of the IOP Photonic Devices program of Rijksdienst voor Ondernemend Nederland (RVO), part of the Netherlands Ministry of Economic Affairs and the Netherlands Ministry of Education, Culture and Science. This work is also partly supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organization for Scientific Research (NWO). We thank Julien Javaloyes (Universitat de les Illes Balears, Spain) for fruitful discussions.

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