Linewidth narrowing in self-injection locked lasers: Effects of quantum confinement

Quantum well (QW) laser diodes are widely used in photonic products,^1,2 offering low power consumption, extended lifespan, light weight, and compactness. Meanwhile, quantum-dot (QD) laser diodes are increasingly recognized for their unique features arising from the zero-dimensional (0D) carrier density of states, which resemble the discrete electronic structure of atoms.^3–5 These diodes promise reduced threshold currents, enhanced thermal stability, and reduced sensitivity to external optical feedback and material defects.⁶ Notably, the successful epitaxy growth on silicon-based platforms underscores their potential for broad application.⁷ 

However, both QW and QD lasers currently face challenges. When it comes to achieving narrow linewidth, QW lasers suffer from high linewidth enhancement factors (LEFs), while QD lasers are limited by inhomogeneous broadening.^8–10 An established technique for narrowing the linewidth of semiconductor lasers is self-injection locking (SIL), in which the laser is coupled to a high-Q cavity providing frequency-selective optical feedback. Since the first demonstration in 1998,¹¹ high-Q crystalline microring resonators have been integrated with laser diodes to produce narrow linewidth lasers,^12–17 alongside low-noise photonic microwave oscillators¹⁸ and soliton comb generators.^19,20 Over the recent years, considerable effort has been concentrated on developing novel resonator geometries to enhance the laser–resonator coupling and increase the strength of the reflected signal based on drop-port reflection,^21,22 hole defects,²³ photonic crystal ring resonators,²⁴ and intracavity Sagnac loops.²⁵ 

For effective device engineering, a laser model that can predict the properties of the active medium using only epitaxial growth characteristics (such as layer structure and thickness) as inputs is beneficial. Such a model should be able to accurately predict the effects of transitioning from one-dimensional QW to three-dimensional QD carrier confinement without relying on phenomenological or fitting parameters. Distinguishing between intrinsic and extrinsic effects is crucial; the former helps guide engineering design, while the latter indicates the quality of growth and fabrication. This approach surpasses traditional models based on rate equations for class B lasers,^26–28 which combine intrinsic and extrinsic factors to reduce input parameters to the gain coefficient and LEF. While the simplification facilitates fitting to experimental data, a model that offers predictive capabilities and ties directly to the band structure is more valuable for understanding the physical principles and enhancing the device performance.

Section II presents the theoretical framework for laser operation. The approach starts with describing laser fields using cavity modes, establishing a link to quantum-mechanical theory via electron–hole polarization. These modes are explored for both single and composite cavity lasers. Following this, this paper outlines the microscopic gain theory, derived from quantum mechanics, to compute parameters in the gain region. Subsequently, the multimode laser equations are introduced, which comprise a system of coupled differential equations for the intensities and relative phases of each cavity mode. In Sec. III, this multimode laser theory is applied to analyze single-cavity QW and QD lasers, reproducing experimental results, such as the L–I curve and the linewidth–current curve. In Sec. IV, the model is extended to QW and QD lasers coupled to a high-Q resonator, examining linewidth narrowing in self-injection locked configurations.^29,30

We treat the integrated laser and free space as a combined optical system,^31–33 providing a framework applicable for arbitrary optical coupling between the laser and the resonator, as well as between the laser and free space. In addition, it allows a more rigorous derivation of the laser equations by providing an orthonormal basis for an open laser cavity and also accommodating modal projections leading to laser equations.

Both the linear and nonlinear mode confinement factors have roles in calculating the laser parameters below.

The solution is repeated for the coupled cavities, with the effective transmission T₁ determining the coupling between the laser and resonator cavities. The resonance spectrum is shown in Fig. 1(d), and the confinement factor is plotted vs the relative passive-mode frequency Ω_n − Ω₀. The customary mode confinement factor describing the overlap of the gain region with the optical mode within the laser cavity is $Γnn,l(1)/(Γnn,l(1)+Γnn,r(1))$⁠, where $Γnn,r(1)=Nc−1∫resonatordzn2(z)un(z)un(z)$⁠. Each resonance is composed of multiple composite-cavity modes, yielding a finite linewidth Δ_cav due to the free-space coupling. The coupled-cavity plot shows three types of resonances. At points where the uncoupled cavity resonances align, there are two closely spaced composite-cavity resonances (rn and rw), with their splitting determined by the coupling T₁. The other resonances (such as nr) originate from the eigenmodes of the longer resonator cavity and are observable in the laser cavity due to optical coupling. We note that the optical phase between the laser and resonator cavity has an important effect on laser linewidth²⁷ and determines the operating regime, leading to the generation of frequency combs at certain values of the phase.^37,38 In this work, the phase is fixed at the optimal value for SIL operation.

Figure 2(a) shows the dependence of linewidth of a QD DFB laser on the injection current for different passive-cavity linewidths ΔΩ_cav, and Fig. 3(a) shows the same for a QW DFB laser. Two mechanisms contribute to the linewidth narrowing. The first is gain clamping, as in the Schawlow–Townes laser linewidth treatment, where the saturated gain $gnsat$ remains pegged to the cavity loss rate $γncav$ after threshold is reached.⁴⁵ This appreciably reduces the number of composite-cavity modes contributing to the emission, as shown in Fig. 2(b), where the orange curve corresponds to the cavity resonance shape and the blue curve corresponds to mode intensity at twice the lasing threshold. There is a second mechanism described by the term in Eq. (18) containing the relative phase ψ_nm. It tends to lock the composite-cavity modes to a common frequency, i.e., dψ_nm/dt = 0 for all lasing modes. However, all simulations for single-cavity lasers show only partial locking, which produces a $S̃(f,t)$ with time-varying width and spectral position. The resulting time-averaged lasing spectrum is shown in Fig. 2(c).

There are important quantitative differences between the QW and QD cases. Linewidth narrowing in QW single-cavity lasers saturates around the MHz level [Fig. 3(a)], while linewidth narrowing in QD single-cavity lasers saturates considerably lower, around tens of kHz. The difference, as indicated by comparing Figs. 3(d) and 3(e), is due to the significantly smaller frequency pulling coefficient σ_n in QD lasers, resulting in a considerably reduced LEF, given by α_H = 2σ_n/g_n, with the unsaturated gain and saturation intensity shown in Figs. 3(b) and 3(c).

This section discusses the laser linewidth when a high-Q resonator is optically coupled to the QD or QW DFB lasers, as modeled in the previous section. Figures 5(a) and 5(b) show the computed spontaneous emission limited lasing linewidth vs injection current for the QW and QD lasers described by the parameters in Table II. The passive-cavity linewidths ΔΩ_cav = 1 GHz, 100 MHz, and 10 MHz are obtained by lasing at the rw, rn, and nr resonances, respectively, with L₁ = 3 mm, T₁ = 0.040, T₂ = 0.035. For ΔΩ_cav = 100 and 10 MHz, both QD and QW lasers exhibit significant linewidth narrowing to less than 10⁻⁶ × ΔΩ_cav, which is a significant improvement over the single-cavity results. The similarity in linewidths for QW and QD lasers suggests that the SiN cavity can mitigate the carrier-induced refractive index, facilitating complete frequency locking of the composite-cavity modes and, thus, defining the linewidth solely by $Snϕ$⁠. Conversely, the ΔΩ_cav = 1 GHz curve in Fig. 5(b) does not show the same significant linewidth narrowing as in the other scenarios. Examination of the input–output (L–I) dependence indicates that intracavity intensity must be sufficiently high to achieve complete frequency locking.

Phase dynamics, as governed by Eq. (18), plays an equally important role in linewidth narrowing as gain clamping. Within the range of injection current where significant linewidth narrowing occurs, partial frequency locking leads to both QD and QW lasers exhibiting complex dynamical behaviors that are related to those observed and predicted for self-injection locking.^19,51 To illustrate this, we track the time dependencies of the lasing frequencies, dψ_n/dt, in composite-cavity modes n = 0, ±2, ±4, ±6 in the QD laser with ΔΩ_cav = 100 MHz. At 24 mA, the time traces show significant fluctuation, with dψ_n/dt spiking randomly [Fig. 6(a), top]. The resulting time-averaged spectrum has a linewidth of ΔΩ_L = 5 MHz [Fig. 6(a), bottom]. A noticeable change in dynamical behavior occurs at the slightly higher injection current of 26 mA. The time traces in Fig. 6(b) indicate a focusing of dψ_n/dt traces and synchronization in the spiking. The resulting time-averaged spectrum has a linewidth of 1 MHz. For injection current values over 32 mA, complete frequency locking takes place. Figure 6(c) shows that at 100 mA, all dψ_n/dt time traces merge, yielding a remarkably narrow laser linewidth of 10 Hz.

We also compare our model to the results of external-cavity locking (ECL) experiments, in which a 16 m long optical fiber reflector is coupled to a QD or QW laser.⁴⁴ This coupled-cavity configuration is highly suitable for our composite cavity model. Figure 7 shows the comparison. The extrinsic laser properties (γ_nr = 2 × 10⁹ s⁻¹, η = 0.7, Δ_inh = 10 meV for the QD laser, and γ_nr = 10⁸ s⁻¹, η = 0.7 for the QW laser) are determined by fitting our model to the experimentally measured L–I curves [Figs. 7(a) and 7(b)]. With the choices of ΔΩ_cav as indicated in Figs. 7(c) and 7(d), we obtain relatively good agreement with the experiment. The hollow markers and dashed lines are for the free-running lasers, while the solid markers and solid lines are for the lasers coupled to the optical fiber.

For complex systems, such as integrated self-injection locked lasers, the vast parameter space presents numerous opportunities and challenges for design engineering.⁵² Often, enhancing one performance parameter might compromise another. In this section, we discuss an example where our model can facilitate parametric studies to yield informed design decisions, especially in applications requiring a balance between output power and narrow linewidth.

The modeling performed so far is based on gain structures from our own fabricated lasers.^43,44 However, the L–I curves suggest that they may not be optimal for III–V/SiN lasers. For example, fewer QW layers may lower the threshold current, as shown in Fig. 8(a). Reducing from seven to four or fewer QW layers can significantly reduce the threshold current by almost an order of magnitude. However, the downside to this reduction is a decrease in output power and the onset of L–I rollover. Furthermore, the single-QW result indicates a limit in decreasing the number of QW layers, with the linewidth plateauing at 1 MHz regardless of the injection current. Similar to the QD ΔΩ_cav = 1 GHz case in Fig. 5(b), a minimum intracavity intensity is necessary to achieve the full benefits of the SiN resonator. Otherwise, the laser’s behavior is akin to that with unfiltered feedback.

For the QD III–V/SiN laser, appreciable L–I rollover suggests the potential benefit of incorporating additional QD layers. Figure 8(b) indicates that increasing the number of QD layers from five enhances the output power without a significant penalty in threshold current or linewidth. Notably, the linewidth reaches Hz-level before any L–I rollover, which is beneficial to modulation speed. However, surpassing nine QD layers may introduce engineering challenges, such as maintaining current uniformity over the entire active region. Similar benefits are observed when increasing the QD density within each layer.

As is clear from Figs. 8(a) and 8(b), L–I rollover poses a constraint on output power, attributable to two primary factors. First, gain compression from spatial and spectral hole burning leads to saturation. This is accounted for in our model by performing the electron–hole polarization derivation to the third order in the light-carrier interaction. With the onset of gain compression, the carrier population is not clamped to the threshold value but continues to increase with the increasing injection current. Second, gain saturation occurs at high carrier densities in both QW and QD gain structures. For QWs, the saturation stems from the step function carrier density of states, while in QDs, it results from both the finite density of QD and the inhomogeneous broadening, presently limited by the current epitaxial growth capability. Working together, these two factors result in the L–I rollover. Class B rate-equation treatment [see, e.g., Eq. (5.17) in Ref. 1] predicts a linear L–I behavior in the absence of ad hoc gain compression factors.

This paper describes a study of the effects of the gain medium on linewidth narrowing in integrated self-injection locked III–V QW and QD lasers. The theoretical approach accounts for (a) multimode laser interaction resulting in mode competition and wave mixing, (b) the quantum-optical contributions from spontaneous emission, and (c) the coupling dynamics among optical components via composite laser/free-space eigenmodes. This extension beyond traditional rate equation models allows predictions directly connected to band structure while emphasizing the distinction between intrinsic and extrinsic effects. The former guides engineering design, while the latter indicates fabrication quality.

Applying this theory to single-cavity lasers, such as DFB, DBR, or coated-facet lasers, reveals two primary physical mechanisms for linewidth narrowing: gain clamping, as in the Schawlow–Townes model, and frequency locking of composite laser/free-space modes. For single-cavity lasers, there is only partial frequency locking, resulting in linewidths limited by frequency drifts of the lasing spectra. For integrated III–V/SiN lasers, complete frequency locking can be achieved due to the mitigation of carrier-induced refractive index with a passive high-Q cavity, thereby reaching Hz-level spontaneous emission limited linewidths for both QW and QD lasers. Parametric studies suggest that the 2D QW carrier density of states helps achieve high output power, while the 0D QD carrier density of states benefits low threshold currents. Studies over a broad parameter space⁵² show that integrated III–V/SiN QW and QD lasers offer complementary solutions to meet the diverse requirements of photonic applications with highly stable semiconductor lasers.

The theoretical framework is also validated with experimental data from InAs QD and InGaAsP QW DFB and DBR lasers. Recent experiments with QD and QW lasers coupled to an optical fiber in an external cavity configuration reinforce the composite mode approach for modeling coupled laser and resonator cavities. The application of cavity-QED physics has been partially validated by accurately capturing the deviation of the lasing spectrum from a Lorentzian function in nanolaser experiments.⁵³ Furthermore, the model’s precision in describing III–V optical nonlinearities has been confirmed through four-wave mixing and pump–probe experiments,⁴⁶ as well as by reproducing the measured pulse train and RF spectrum of self-mode-locked InAs QD lasers.⁴⁷ 

Despite these advances, our theory is still evolving. The current free-carrier gain theory, while effective in describing the differences between 3D QD confinement and 1D QW confinement, does not fully account for many-body effects. This omission may lead to inaccurate predictions, especially regarding inhomogeneous broadening and the impact of doping on QD carrier-induced refractive index. Moreover, integrating the effects of cavity-QED into semiclassical laser theory poses a challenge. Work is under way to develop a comprehensive quantum optical treatment for integrated cavity-locked lasers and to refine the strong-signal multimode description—a challenge persisting in atomic, molecular, and optical (AMO) laser physics.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-NA0003525. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. We acknowledge King Abdullah University of Science and Technology (KAUST) under Award Nos. RFS-OFP2023-5558, ORA-2022-5314, ORA-2022-5313, and SDAIA-KAUST Center of Excellence in Data Science and Artificial Intelligence (SDAIA-KAUST AI). Frédéric Grillot acknowledges the support of the Institut Mines-Télécom.

The authors have no conflicts to disclose.

A.P. and W.W.C. contributed equally to this paper.

Artem Prokoshin: Conceptualization (equal); Investigation (equal); Writing – original draft (equal). Weng W. Chow: Conceptualization (lead); Investigation (lead); Writing – original draft (equal). Bozhang Dong: Investigation (equal). Frederic Grillot: Supervision (equal). John Bowers: Supervision (equal); Writing – review & editing (equal). Yating Wan: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.