The integrated tunable external cavity diode laser (ECDL) plays an important role in many emerging applications, such as LiDAR, integrated free-space optical communication, integrated nonlinear optics, quantum information processing, and data center optical interconnects. Based on the III-V/silicon or III-V/silicon nitride hybrid integration, chipscale tunable ECDL could achieve a broad wavelength tuning range (>100nm) and very narrow linewidth (<10kHz) with the help of micro-ring resonators. However, most hybridly integrated ECDLs exhibit a low slope efficiency (<0.1W/A), which severely limits the laser performance. We have developed a general numerical model for the hybrid laser and investigated several methods to improve its slope efficiency. We show that the slope efficiency of the hybrid laser can be efficiently increased by controlling the external cavity phase, improving the coupling between the laser chip and passive chip, using optimized anti-reflection and high reflection coatings, and reducing the laser intrinsic loss. We have also experimentally built an arrayed waveguide grating (AWG) based hybrid laser and shown that the numerical simulation results match well with the experimental results.

## I. INTRODUCTION

Fully integrated high power, narrow linewidth, tunable diode lasers are crucial for many emerging applications, such as integrated free-space optical communication, integrated nonlinear optics, quantum information processing, and data center optical interconnects.^{1–5} Traditional narrow linewidth, tunable diode laser sources are based on the external cavity diode laser (ECDL), where an external diffraction grating is used to select the single longitudinal mode and realize the wavelength tuning.^{6} To obtain the narrow linewidth, the external cavity is about half meter long so that the cavity photon lifetime is greatly increased. The wavelength tuning is obtained by mechanically moving the diffraction grating. Thus, the whole system is bulky and expensive. Integrated photonic approach is needed to reduce the system cost, size, weight and power consumption (CSWaP) for new applications.

Integrated photonic approach is to replace the diffraction grating external cavity with a photonic integrated circuit (PIC) and create the whole laser system through hybrid photonic integration,^{7–9} resulting in so-called external cavity hybrid lasers (ECHLs). Current approaches are based upon either III-V/silicon integration or III-V/silicon nitride integration.^{10} The PIC in ECHL usually has a double-ring resonator based wavelength-selective partial reflector. The tunability is achieved by tuning the ring resonance and the double ring structure is used to increase the tuning range.^{11} The ring resonators also introduce a large amount of time delay at resonance, effectively increasing the cavity length and reducing the laser linewidth. With this approach, many groups have demonstrated various hybrid lasers with impressive performance, including very broad wavelength tuning range (>100nm) and very narrow linewidth (<10KHz).^{12,13} However, most hybrid lasers demonstrated so far exhibit low slope efficiency (<0.1W/A),^{14–16} much smaller than conventional diode lasers in the same wavelength range. For many emerging applications, it is critical to increase the laser output power and efficiency.

In this manuscript, we aim to understand the limitations of the hybrid laser slope efficiency. We have developed a general numerical model for the hybrid laser and investigated several methods for efficiency improvement. It is shown that the ECHL slope efficiency can be efficiently increased by controlling the external cavity phase, improving the coupling between the laser chip and passive chip, using optimized anti-reflection and high reflection coatings, and reducing the laser intrinsic loss. We have also experimentally set up a hybrid laser and shown that the numerical simulation results based on the model developed here match well with the experimental results.

## II. THEORY

### A. Effective cavity model

Figure 1 shows the simplified configuration of an ECHL. The wavelength selective components, such as ring resonators or gratings, and output couplers are represented by a frequency selective partial reflector. The laser diode (LD) output is directly coupled to a waveguide on a separate external chip, where the partial reflector is integrated with the same waveguide. Optical feedback for the hybrid laser is provided by the high reflection (HR) coating on the LD back facet and the partial reflector. Using the effective cavity model (ECM), this ECHL system can be analyzed without loss of generality. We simply model the system as a single cavity with an effective reflectivity *R*_{eff} for the front facet and high reflectivity *R*_{1} for the back facet.^{17}

Referring to the left side of Fig. 1, *R*_{1} and *R*_{2} are the reflectivities of the back facet and front facet of the LD, respectively. *P*_{1} and *P*_{2} correspond to the output power extracted from the back facet and front facet of the LD, respectively. The partial reflector shown in Fig. 1 has a reflectivity *R*_{p}, and the output power of the ECHL is *P*_{3}. As described previously, this ECHL can be simplified to the ECM as shown on the right of Fig. 1. This ECM has the reflectivity *R*_{1} at the effective back facet and *R*_{eff} at the effective front facet.

In our analysis of the ECHL output characteristics, we will first look at the output characteristics of a LD by use of the scattering matrix (SM) of each component. The SM of the back facet, the internal medium and the front facet of the LD are illustrated in Fig. 2.

It can be shown that:

*r*_{1} and *r*_{2} are the reflection coefficient of back facet and front facet. *t*_{1} and *t*_{2} are the transmission coefficient of the back facet and front facet. Γ is the optical confinement and *g* is the gain coefficient. *α*_{i} is the intrinsic loss coefficient. *L* is the length of the LD. *β* is the propagation constant. Additionally, the total scattering matrix of the LD can be found by *S*_{t} = *S*_{1} ⊗ *S*_{2} ⊗ *S*_{3}, where ⊗ is a Redheffer Star Product.^{19}

According to the ECM, the effective reflectivity *R*_{eff} and effective transmissivity *T*_{eff} are not only dependent on simple single reflections but complicated by the multiple internal reflections of the system. Given that the reflections are ideal, *R*_{eff} and *T*_{eff} are described by the equations:^{20}

*r*_{eff} is effective reflection coefficient and *t*_{eff} is effective transmission coefficient. *η*_{c} is the coupling efficiency between the LD and external cavity. *ϕ*_{1} is the phase induced by the external cavity. *r*_{ext} is the reflection coefficient of the external reflector. The total output extracted from the LD cavity should be $b12+b22$ and our interested output is the optical output extracted from the front facet $b22$. We define a distribution ratio, *Q*_{LD}, to analyze the power coupled into the external waveguide.

Replacing the front facet reflectivity *R*_{2} with *R*_{eff} yields the following distribution ratio for the ECHL:

The benefit of the distribution ratio *Q*_{ECHL} is that we can easily obtain the output power from the hybrid laser front facet by multiplying *Q*_{ECHL} with the total output power. The *T*_{eff} should be revised as $Teff\u2032$ if the partial reflector is not lossless ($rext2+text2\u22601$).

The above discussion is for the basic configuration of one external cavity, but it can be easily extended to *N* cavities. Consider the configuration as illustrated in Fig. 3.

In Fig. 3, $\eta cii=2,3,\u2026,N$ is the coupling efficiency between the (*i* − 1)^{th} cavity and the *i*^{th} cavity. *η*_{c1} is the coupling efficiency between the first external cavity and the LD. $Rpii=1,2,3,\u2026,N$ is the reflectivity of the *i*^{th} cavity. If there is reflection loss in the front facet of the *i*^{th} cavity, it should be incorporated into that cavity’s distribution ratio as mentioned in the previous paragraphs.

### B. Steady state solutions of ECHL

Since the LD is the only active component in an ECHL, the output of the LD is the key to investigate the ECHL performance. The rate equation^{21} of an LD is described as:

*N* is the carrier density. *I* is the injection current. *V* = *WLd*_{a} is the active region volume. *W* is the active region width. *L* is the internal cavity length and *d*_{a} is the active region thickness. *q* is the electron charge. *c* is the speed of light. *n*_{a} is active region refractive index. *g*_{m} and *S*_{m} are the gain coefficient and photon density of the *m*^{th} mode. *τ*_{s} is the carrier lifetime and *τ*_{r} is the radiative recombination time. Γ is the optical confinement. *α*_{i} is the intrinsic loss. *R*_{1} and *R*_{2} are the reflectivity of the back and front facet. *γ* is the spontaneous emission factor. The values of some parameters used in the following calculation are shown: *W* = 1um, *L* = 0.5mm, *d*_{a} = 10nm, *n*_{a} = 3.5, *τ*_{s} = 2ns.

This rate equation is commonly solved using the Runge-Kutta (RK) method.^{22} A steady state solution can be used to find the threshold condition of the LD, in which the gain equals the total loss. The total loss *α*_{t} = *α*_{i} + *α*_{m} includes the mirror loss $\alpha m\u2009=\u200912Lln1R1R2$ and the intrinsic loss *α*_{i}. The mirror loss directly corresponds to the total output of the LD, so the front facet output power of the LD is:

Where *Q*_{LD} is the distribution ratio explained before. *η*_{LD} is the slope efficiency, $\eta EE=\alpha m\alpha t$ is the extraction efficiency and *η*_{IDQE} is the internal differential quantum efficiency. The typical value of *η*_{IDQE} is almost 1 which indicates that the injected electrons are efficiently converted to photons. The threshold current *I*_{th} under steady-state condition is

From the *m*^{th} mode gain in equation 8, the carrier density can be determined. The threshold current can be obtained by use of equation 7 and 8.

*B* is the differential gain coefficient. *λ*_{p} and *λ*_{m} are the peak gain wavelength and *m*^{th} mode lasing wavelength. Δ*λ* is the 3dB linewidth of the gain. *N*_{0} is the transparency carrier density. Some values related to these parameters are shown: *λ*_{p} = 1.55um, *N*_{0} = 1.02 × 10^{−20}m^{−3}, Δ*λ* = 20nm, *N*_{0} = 1 × 10^{−6}m^{−3}.

The output of the ECHL is shown in equation 9 based on the ECM. Δ*I* = *I* − *I*_{th} is the difference between the injected current and the threshold current.

### C. Slope efficiency of ECHL

From Sec. II B, the slope efficiency of an ECHL is $\eta ECHL=QECHL\eta EE\u2032\eta IDQEh\nu e$. In contrast, the slope efficiency of an LD is $\eta LD=QLD\eta EE\eta IDQEh\nu e$. The ECHL equation replaces *Q*_{LD} as *Q*_{ECHL} and corrects *η*_{EE} for external cavity. The impacts of the external cavity are manifested in both *Q*_{ECHL} and $\eta EE\u2032$.

We first consider the influence of the external cavity phase by changing the cavity length, as shown in Fig. 4. As the external cavity length varies, the ECHL slope efficiency changes periodically. The period is about 0.5um. The pink, blue, and red line correspond to 10%, 5% and 1% LD front facet AR coating, respectively. As clearly shown in the graph, the reflectivity of the AR coating has a marked effect on the fluctuation of the slope efficiency. A better–less reflective–AR coating lowers the slope efficiency fluctuation induced by the changes in cavity length. The slope efficiency fluctuation is caused by the change of effective front facet reflectivity *R*_{eff} that is strongly dependent on the external cavity phase. To obtain the highest slope efficiency, the external cavity phase needs to be well controlled. The period of *η*_{ECHL} is *λ*/2*n*_{ex}, where *λ* is the lasing wavelength and *n*_{ex} is the effective refractive index of the external cavity. Supposing a lasing wavelength of 1.55um and an *n*_{ex} of about 1.43 (silica waveguide), the period of ECHL is about 0.54um. This matches well with the period of 0.5um shown in Fig. 4.

The next important factor related to the ECHL slope efficiency is the partial reflector reflectivity *R*_{p}. Here we assume that the external cavity is exactly in phase for simplicity. As shown in Fig. 5, it is clear that a small *R*_{p} leads to a high slope efficiency. This is as expected since a small *R*_{p} results in a high extraction efficiency for the hybrid laser. However, the threshold current increases with the decrease of the *R*_{p}. The high threshold current will introduce nonlinear effects such as two-photon absorption (TPA) and free-carrier absorption (FCA), which may broaden the linewidth of ECHL.^{23} Detailed discussions are beyond the scope of this paper and will be addressed elsewhere.

The slope efficiency can be further improved by the optimized implementation of HR and AR coatings. In Fig. 5a, the AR coating reflectivity is fixed. In Fig. 5b, the HR coating reflectivity is fixed. Figure 5a indicates that a good HR coating could improve the slope efficiency. Figure 5b means that a better AR coating leads to a larger output when *R*_{p} is small (<0.15). However, for a high partial reflector reflectivity, the better AR coating leads to a smaller slope efficiency. The reason is that both of the AR coating of front facet and partial reflector reflectivity have an impact on the *R*_{eff} which ultimately determines the slope efficiency.

The influence of the coupling efficiency between the LD and passive element is illustrated in Fig. 6. It is obvious that a good coupling efficiency between the LD and external cavity improves the slope efficiency of the ECHL in Fig. 6. The slope efficiency can be divided into two characteristic parts, *Q*_{ECHL} and $\eta EE\u2032$. Both of these values are illustrated in Fig. 7.

As shown in Fig. 7, the average value of *Q*_{ECHL} is about 0.11 ∼0.16, and the output thus has an additional loss of 8 ∼10dB. This means that changing from the LD to the ECHL lowers power substantially. $\eta EE\u2032$ has a less substantial effect on the output power, adding a loss of only 2 ∼3dB to the output power. It can be concluded from Fig. 7, when parameters are optimally chosen, *Q*_{ECHL} is the major factor in lowering slope efficiency when compared with $\eta EE\u2032$. To improve *Q*_{ECHL}, the most direct path is to increase the coupling efficiency between the LD and the external cavity, as illustrated in Fig. 6. $\eta EE\u2032$, on the other hand, can be improved by lowering the intrinsic loss. Figure 8 shows how the ECHL slope efficiency changes as a function of *α*_{i} and *R*_{p}. It is important to note that a large intrinsic loss results in not only poor output power but also a high threshold current. Lower intrinsic loss can be achieved through epitaxial optimization and specializing geometry design for the LD.

### D. Output comparison with the LD

The output comparison of the LD and the ECHL is shown in Fig. 9. The red line indicates the slope efficiency of LD as a function of the front facet reflectivity with a HR coating reflectivity about 95% on the back facet. In the inset, the blue line shows the slope efficiency of ECHL as a function of the phase of external cavity with a fixed partial reflector reflectivity *R*_{p}. Here we assume that the front facet reflectivity of LD is the same as the partial reflector reflectivity of ECHL. The slope efficiency of ECHL is generally smaller than the LD due to additional losses. As illustrated in Fig. 9, lines are drawn where *R*_{p} equals 6% and 48% because those are the values used in the experiment.

In conclusion, the *Q*_{ECHL} is the major parameter in lowering the slope efficiency of an ECHL and is a main target for future improvement. To increase the slope efficiency of ECHL, possible methods include improving coupling efficiency between the LD and external cavity, applying an HR coating on the back facet of the LD, an AR coating on the front facet of the LD, optimizing the output coupler reflectivity, and reducing the LD intrinsic loss. A better coupling between the active component and passive part can greatly improve the slope efficiency of the ECHL and can be achieved by proper taper design and geometry optimization.

## III. EXPERIMENT

The experimental setup used in this paper is depicted in Fig. 10. We use two different types of semiconductor optical amplifiers (SOA) as the active components. The first type SOA (SOA1) has a HR coating on its back facet and the second type SOA (SOA2) requires a HR retroreflector. An Array Waveguide Grating (AWG) is inserted to provide the wavelength selection mechanism and measure multiple configurations at the same time. The loss in each channel of the AWG is measured before the experiment. The average loss of each channel is about 5.1dB. The external cavity is formed by the AWG, optical fiber and a partial reflector. The output is collected by a power meter, and the experimental data and simulated result are compared in Fig. 11. It should be noted that the partial reflector is not lossless in experiment. Thus the corrected $Teff\u2032$ described in theory part IIB must be considered in our calculations.

The output power of SOA1 and SOA2 are measured in experiment as a function of inject current, shown as the dotted lines in Fig. 11. The theoretical calculation results show good agreement with the experimental ones, which means that the output power of the ECHL can be predicted well with the model developed earlier in this paper.

## IV. CONCLUSION

In this paper, we have used the effective cavity model to perform a detailed analysis of the ECHL slope efficiency. The influences of different parameters are discussed in details in the theory section, where we demonstrate the importance of efficient coupling between the LD and external cavity. Additionally, we show that the ECHL slope efficiency can be improved by reducing the laser intrinsic loss and optimizing the laser facet coatings and output coupler reflectivity. The experiment results agree well with our theoretical analysis, indicating that the ECHL slope efficiency can be well predicted using the ECM. In all, this work helps pave the way for further improvement of the slope efficiency for ECHL.

## ACKNOWLEDGMENTS

The authors would like to thank the funding support from the Army Research Office (W911NF-18-1-0176) and Office of Naval Research (N00014-17-1-2556).

## REFERENCES

_{3}N

_{4}hybrid platform