1 Introduction

SOAs find application in various optoelectronic devices utilized for communications, biomedical imaging, ranging and detection, and holography (Mørk and Berg 2003; Sobhanan et al. 2022). To facilitate spatial beam coherence, SOAs contain waveguides featuring a core region where population inversion and optical confinement occur under electrical pumping, which is known as the active region. Given that a semiconductor medium’s refractive index depends on its carrier density, refractive index fluctuations arise during the population inversion process. These fluctuations are particularly pronounced near the transition frequency (Qasaimeh 2005; Melnik et al. 2006). Consequently, when an input optical beam is amplified through the active region, the occurrence of refractive index fluctuations induces phase distortion and compromises the spatial and temporal coherence of the amplified beam (Zhang et al. 2019; Qiu, et al. 2021). Since maintaining the coherence of an amplified laser beam holds great importance in almost all applications, it becomes crucial to minimize refractive index fluctuations. The parameter indicative of such fluctuations within the SOA active region is the linewidth enhancement factor (LWEF, also denoted by α) (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Henry 1982; Haug and Haken 1967). Numerous studies have underscored the adverse effects of a high LWEF on the performance of optoelectronic devices, particularly in terms of stability and coherence (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Henry 1982; Haug and Haken 1967; Schmidt et al. 2020). Consequently, previous research endeavors have concentrated on devising SOAs with low LWEFs that are compatible with practical optoelectronic devices (Xiao and Huang 2008; Chuang et al. 2014; Ruan et al. 2018). In doing so, it was observed that measuring or computing the LWEF of an SOA poses challenges (Chuang et al. 2014; Ruan et al. 2018; Harder et al. 1983; Fan et al. 2015; Miloszewski et al. 2009; Ohtoshi and Chinone 1989; Holzinger et al. 2018; Aşırım and Jirauschek 2022; Zubov et al. 2015). Experimental determination involves intricate procedures due to the LWEF’s sensitivity to fundamental SOA parameters like steady-state carrier density, photon density, and operational frequency (as in frequency-swept lasers) (Qasaimeh 2005; Chuang et al. 2014; Ruan et al. 2018; Harder et al. 1983; Fan et al. 2015; Miloszewski et al. 2009; Ohtoshi and Chinone 1989; Holzinger et al. 2018). Moreover, computational modeling is challenging due to the strong interdependence of each SOA parameter (Qasaimeh 2005; Schmidt et al. 2020; Xiao and Huang 2008; Aşırım and Jirauschek 2022). Therefore, the development of a simple and effective algorithm for accurately evaluating the LWEF of an SOA holds significant importance, necessitating the systematic formulation of a corresponding design strategy (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Schmidt et al. 2020; Aşırım and Jirauschek 2022). Previous experimental investigations on the LWEF have revealed LWEF values ranging from 1 to 9 for the majority of SOAs (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Chuang et al. 2014; Ruan et al. 2018; Harder et al. 1983; Fan et al. 2015; Miloszewski et al. 2009; Ohtoshi and Chinone 1989; Holzinger et al. 2018; Aşırım and Jirauschek 2022; Zubov et al. 2015; Ri et al. 2020), with only a few studies delving into design strategies aimed at achieving minimal LWEF (Qiu, et al. 2021; Aşırım and Jirauschek 2022; Zubov et al. 2015). Certain recent studies propose that quantum confinement could potentially reduce LWEF to near-zero levels (Zhang et al. 2019; Qiu, et al. 2021; Aşırım and Jirauschek 2022; Zubov et al. 2015). However, conflicting findings from other studies cast doubt on such assertions (Ding et al. 2023; Miloszewski et al. 2009; Dagens et al. 2005). Most importantly, the vast majority of the studies on the LWEF of SOAs have treated the LWEF as constant throughout the duration of the SOA operation, which may not always be adequate. For example, significant fluctuations in optical power can occur in optical cavities due to nonlinearity and dispersion that heavily affect the carrier density in the active region. Another example in which case the LWEF cannot be treated as a constant are frequency-swept lasers, where the frequency is often swept over the entire operation bandwidth of an SOA at each roundtrip, which would greatly alter the LWEF within a roundtrip as the LWEF has been shown to be strongly dependent on the operation frequency (Qasaimeh 2005; Harder et al. 1983; Miloszewski et al. 2009; Aşırım and Jirauschek 2022; Green et al. 1987). Therefore, a more realistic model that accounts for both the time and frequency dependence of the LWEF is needed, particularly for QD SOAs where the carrier lifetime is usually smaller than SOAs that are not based on 3D quantum confinement, leading to a stronger time dependence of the fundamental QD SOA parameters.

QD SOAs allow for adjustability of the LWEF at the design stage via quantum-confinement-enabled tunability of the emission frequency and carrier density of the employed Quantum-Dots (QDs) (Grillot et al. 2020; Kano et al. 1991; Dagens et al. 2005). Utilizing this tunability, one can greatly decrease the LWEF of QD SOAs and employ them in optoelectronic devices requiring high coherence. In this study, it is our goal to uncover how this decrease may occur. Therefore, we focus on formulating the LWEF of QD SOAs using a straightforward numerical procedure in light of empirical findings and examine how low the LWEF can get in QD SOAs. Then, we aim to find optimal QD SOA configurations to minimize the LWEF.

To effectively manage the LWEF in QD SOAs, a detailed yet simple algorithm is employed. Central to the algorithm is the characterization of the LWEF, denoted by α, which is mainly determined based on the refractive index of the active medium \({n}_{AC}\). The refractive index of the active medium, in turn, depends on several parameters, including the concentration of quantum dots (QDs) in the active region, the carrier density within the QDs, intraband collision-time of the carriers, and the emission frequency of the QDs (Dagens et al. 2005; Tan et al. 2009; Green et al. 1987; Markel 2016; Vahala et al. 1983). Hence, these parameters are of main interest in this study, and the LWEF variation is observed by tuning each of these parameters separately.

The concentration of QDs within the active region plays a critical role in shaping the LWEF and the refractive index based on the Maxwell–Garnett effective medium approximation theorem (Markel 2016). Different QD concentrations lead to different QD carrier densities, each leading to a different effective refractive index for the active region. Similarly, variations in carrier density within the QDs influence the refractive index of the QDs which comprise the active region. Higher QD carrier densities result in stronger gain, but often also in a higher LWEF due to increased carrier-induced phase distortion (Melnik et al. 2006; Zhang et al. 2019). Moreover, the emission frequency of the QDs has an impact on the refractive index and the LWEF. By detuning the frequency of the laser beam away from the QD emission frequency, the gain factor is reduced, though the LWEF can be lowered depending on the carrier collision rate, as described later in this paper.

To account for the carrier dynamics within the QDs and in the entire active region, the algorithm presented in this paper also encompasses population-rate equations describing the time evolution of the carrier density, N, and the photon density, S, within the active region (Vahala et al. 1983; Huang 1995; Saleh and Teich 2019; Silfvast 2004). These equations capture the profile of the carrier and photon interactions, variation of the carrier and photon lifetimes, variation of the carrier collision-rate, and the associated gain dynamics that determine the LWEF variation over time (Wang et al. 2007). To keep things straightforward, here, we will assume that the QDs have one dominant transition, which occurs between the highest occupied state in the valence band and the lowest unoccupied state in the conduction band, often referred to as the “first excitonic transition.” This is a reasonable assumption as it is generally observed that the first excitonic transition dominates the optical properties compared to higher energy transitions, since it is often an order of magnitude larger in terms of oscillator strength and transition probability (Saleh and Teich 2019; Silfvast 2004).

In summary, our proposed algorithm provides a comprehensive framework for understanding and controlling the LWEF in QD SOAs. By manipulating parameters such as QD concentration, QD carrier density, emission frequency, and carrier collision rate, one can tailor the behavior of QD SOAs to meet specific LWEF requirements in optoelectronic devices.

2 Methods

2.1 Maxwell–Garnett effective medium approximation

Despite being usually similar to the background active region in terms of electrical properties, owing to their super-small size, embedded QDs can display different electrical properties than the background active region in a QD SOA. This means that the entire active region of a QD SOA is essentially composed by two different media. Hence, the active region of a QD SOA should be considered as a composite medium that features the background active region as host, and the QDs as inclusions (Markel 2016). Since the carrier and photon densities, and the photon lifetime depend on the refractive index of the composite active region in a QD SOA, one needs to evaluate the equivalent refractive index of the composite active region. For composite media with a spatially homogenous host medium that contains much smaller inclusions in terms of volume just like the one in Fig. 1, the Maxwell–Garnett effective medium approximation theorem is often used to compute the equivalent refractive index of the composite medium where the cumulative volume of the tiny inclusions corresponds to a fraction of the entire composite medium. Assuming all inclusions are identical in terms of electrical properties, volume, and geometry, the effective refractive index of a composite medium that is formed by adding an inclusion fraction of Δ to the host medium can be evaluated as (Markel 2016)

$$n_{c} = \sqrt {\upvarepsilon_{i} \frac{{2\left( {1 - \Delta } \right)\upvarepsilon_{i} + \left( {1 + 2\Delta } \right)\upvarepsilon_{h} }}{{\left( {2 + \Delta } \right)\upvarepsilon_{i} + \left( {1 - \Delta } \right)\upvarepsilon_{h} }}}$$
(1)
$$\upvarepsilon_{i} :Permittivity\, of\, the\, inclusions,\, \upvarepsilon_{h} :Permittivity\, of\, the\, host\, medium$$
$$n_{c} :Effective\, refractive\, index\, of\, the\, composite\, medium,\, \Delta :Fraction\, of\, the\, cumulative\, volume\, of\, inclusions$$
Fig. 1
figure 1

Top view of a composite medium formed by adding inclusions inside a host medium

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It should be noted that unless all inclusions are identical, Eq. 1 cannot be used to evaluate the effective refractive index with high accuracy. In that case, a slightly modified formulation of the Maxwell–Garnett theorem is applied. To keep things simple, in this paper we will assume that all QDs are essentially identical.

2.2 Computation of the LWEF of a QD SOA

As explained in the introduction section, the LWEF is a measure of the change in the refractive index of the active region, which can be mathematically stated as the ratio of the change in the real part of the refractive index with respect to the carrier density, and the change in the imaginary part of the refractive index with respect to the carrier density. Considering the active region of a QD SOA, the LWEF is mathematically represented as (Nagarajan et al. 1999)

$$\alpha \approx - \frac{{\frac{{\partial Re\{ n_{AC} \} }}{\partial N}}}{{\frac{{\partial Im\{ n_{AC} \} }}{\partial N}}}, N:Carrier\, density,\, n_{AC} :Refractive\, index\, of\, the\, active\, region$$
(2)

To determine the refractive index of the active region, we need the dielectric permittivity of the QDs. In QDs, the traditional concept of dielectric permittivity, which is a macroscopic parameter associated with space-averaged dipole moments, must be generalized to account for the quantum mechanical nature of electronic transitions within nanoscale structures. Unlike bulk materials, where permittivity reflects the averaged response of a large number of dipoles, QDs exhibit unique quantum confinement effects, leading to discrete energy levels and modified electronic transitions. Hence, QDs require a specialized permittivity expression as their interaction with electromagnetic fields is fundamentally different due to size quantization and the finite number of electronic states. A generalized permittivity model allows for the effective description of polarization and dielectric response in QDs, which is essential for accurately predicting optical and electronic behavior in nanoscale devices. Concerning inhomogeneous broadening, temperature effects, and electron–phonon interactions, for a single QD, the permittivity \({\varepsilon }_{QD}(\omega )\) is expressed as (Holmström et al. 2010; Haug and Koch 2009; Fox 2010; Conwell 1967; Natelson 2015)

$$\varepsilon_{QD} \left( \omega \right) = \varepsilon_{\infty } + \frac{{e^{2} }}{{\hbar \varepsilon_{0} V}}\mathop \sum \limits_{i,j} \mathop \smallint \limits_{ - \infty }^{\infty } \frac{{f\left( {E_{ij}{\prime} ,T} \right)dE_{ij}{\prime} }}{{(E_{ij}{\prime} - \hbar \omega - \Delta E_{ph} \left( T \right))^{2} + \left( {\hbar \gamma_{ij} \left( T \right) + \Gamma_{ph} \left( T \right)} \right)^{2} }}\frac{{\left| {\mu_{ij} } \right|^{2} N_{QD} }}{{E_{ij} }}$$
$$where, f\left( {E_{ij}{\prime} ,T} \right) = \frac{1}{{\sqrt {2\pi } \sigma \left( T \right)}}exp\left( { - \frac{{\left( {E_{ij}{\prime} - E_{ij} } \right)^{2} }}{{2\sigma^{2} \left( T \right)}}} \right),\, \sigma : Standard\, deviation\, \left( {temperature\, dependent} \right)$$

e: Elementary charge, \(\hbar\): Reduced Planck’s constant, V: Volume of the quantum dot, \(\left|{\upmu }_{\text{ij}}\right|\): Dipole matrix element between states i and j, \({\text{E}}_{\text{ij}}={\text{E}}_{\text{j}}-{\text{E}}_{\text{i}}\): Central energy difference between quantum states i and j, \(\upomega\): Angular frequency of the incident electromagnetic wave, \({\upgamma }_{\text{ij}}\left(\text{T}\right)\): Homogeneous broadening factor due to non-radiative processes (temperature-dependent), \(\Delta {\text{E}}_{\text{ph}}(\text{T})\): Energy shift due to electron–phonon interactions (temperature-dependent), \({\Gamma }_{\text{ph}}\left(\text{T}\right):\) Broadening due to electron–phonon interactions (temperature-dependent), \({\text{f}}\left( {{\text{E}}_{{{\text{ij}}}}{\prime} ,{\text{T}}} \right)\): Distribution function representing inhomogeneous broadening (often modeled as a Gaussian), \({\upvarepsilon }_{\infty }\): High-frequency permittivity of the surrounding material, \({\text{E}}_{{{\text{ij}}}}{\prime}\): Variable of integration representing the energy spread due to inhomogeneous broadening.

Here, the sum captures all possible electronic transitions between discrete energy levels \({\text{E}}_{\text{i}}\) and \({\text{E}}_{\text{j}}\). The dipole matrix element \({\upmu }_{\text{ij}}\) quantifies the transition strength, and the denominator reflects resonance and damping effects. The term \({\upgamma }_{\text{ij}}\) accounts for energy dissipation, such as non-radiative recombination. This formulation provides a detailed quantum mechanical description of the QD’s dielectric response, accommodating inhomogeneous broadening, electron–phonon interactions, temperature effects, resonant absorption, and dispersion. However, to simplify calculations, we can approximate this detailed model to a more practical form that still captures the essential physics of the QD’s optical response under the presence of a single dominant transition, dominance of homogeneous broadening over inhomogeneous broadening, assumption of near-perfect carrier confinement, and exclusion of temperature effects. Then the dielectric permittivity can approximate that of a bulk material as given in Eq. 3 (detailed reasoning behind this approximation is given in the discussion section). Therefore, in the case of a QD SOA, the refractive index \({n}_{AC}\) of the effective active medium can be expressed as (Markel 2016; Holmström et al. 2010),

$$n_{AC} = \sqrt {\upvarepsilon_{QD} \frac{{2\left( {1 - \Delta } \right)\upvarepsilon_{QD} + \left( {1 + 2\Delta } \right)\upvarepsilon_{h} }}{{\left( {2 + \Delta } \right)\upvarepsilon_{QD} + \left( {1 - \Delta } \right)\upvarepsilon_{h} }}} , \upvarepsilon_{QD} \left( \omega \right) = \upvarepsilon_{h,0} + \left\{ {\frac{{\frac{{N_{QD} e^{2} }}{{m_{0} \upvarepsilon_{0} }}}}{{\omega_{QD} \left( {\varvec{d}} \right)^{2} - \omega^{2} - i2\Omega \left( {N_{QD} } \right)\omega }}} \right\},$$
$$\upvarepsilon_{h} \left( \omega \right) = \upvarepsilon_{h,0} + \left\{ {\frac{{\frac{{N_{h} e^{2} }}{{m\upvarepsilon_{0} }}}}{{\omega_{h}^{2} - \omega^{2} - i2\gamma \omega }}} \right\}$$
(3)
$$\begin{gathered} {\text{N}}_{{{\text{QD}}}} {\text{:QD carrier density, d:QD dimension vector, e:Unit charge, m}}_{{0}} {\text{:Electron mass, }}\upvarepsilon_{{0}} {\text{:Free space permittivity,}} \hfill \\ \omega_{{\text{h}}} {\text{:Angular transition frequency of the host, N}}_{{\text{h}}} {\text{:Host carrier density}} \hfill \\ \end{gathered}$$
$$\upvarepsilon_{{{\text{QD}}}} :{\text{Permittivity of QDs}},{ }\upvarepsilon_{{\text{h}}} :{\text{Permittivity of the semiconductor host medium}}$$
$$\omega_{{{\text{QD}}}} :{\text{QD angular transition frequency}},{ }\omega :{\text{Beam frequency}},{ }\Omega :{\text{Transition linewidth of the QDs}}$$
$$\upvarepsilon_{{\text{h}}} :{\text{Permittivity of the host material}},{ }\gamma :{\text{ Transition linewidth of the semiconductor host medium}}$$

In Eq. 3, the QD transition frequency \({\omega }_{QD}\) is determined based on the dimensions of the employed QDs

$$\upomega_{QD} \left( {\varvec{d}} \right) = \upomega_{0} + \frac{{h\left( {k_{1}^{2} + k_{2}^{2} + k_{3}^{2} } \right)}}{4\pi m}, k_{1} = \frac{{q_{1} \pi }}{{d_{1} }}, k_{2} = \frac{{q_{2} \pi }}{{d_{2} }}, k_{3} = \frac{{q_{3} \pi }}{{d_{3} }} , {\varvec{d}} = \left( {d_{1} ,d_{2} ,d_{3} } \right)$$
$$q_{1} = 1,2,3, \ldots , q_{2} = 1,2,3, \ldots , q_{3} = 1,2,3, \ldots$$
(4)
$$\omega_{0} :{\text{Angular frequency corresponding to the bandgap of the QDs}}$$
$${\text{d}}_{{\text{i}}} :{\text{Length of the QDs along dimension i}},{\text{ k}}_{{\text{i}}} :{\text{Wavenumber in dimension i}},{\text{ h}}:{\text{Planck}}{\prime} {\text{s constant}}$$

In such composite media, a large portion of the carriers would be captured by the QDs. Therefore, the QD carrier density \({N}_{QD}\) is related to the equivalent carrier density \(N\) of the composite active region through the following relation (Aşırım and Jirauschek 2022; Grillot et al. 2020; Tan et al. 2009; Markel 2016)

$$N = \left( {N_{QD} \times \Delta } \right) + \left[ {N_{h} \times \left( {1 - \Delta } \right)} \right], \Delta = \frac{{M \times V_{QD} }}{V}, 0 \le \Delta < 1$$
(5)
$$N_{QD} = \frac{N \times \zeta }{\Delta }, N_{h} = \frac{{\left[ {N \times \left( {1 - \zeta } \right)} \right]}}{1 - \Delta }$$
$${\text{M}}:{\text{Number of QDs in the active region}},{\text{ V}}_{{{\text{QD}}}} :{\text{Volume of the QDs}},{\text{ V}}:{\text{Volume of the active region}}$$
$$\upzeta :{\text{QD carrier confinement ratio}},{ }0 < { }\upzeta < 1$$

The carrier and photon dynamics of the overall active region is governed by the population-rate equations, in which the carrier and photon lifetimes depend on the QD carrier density. Hence, the QD carrier density \({N}_{QD}\) influences the overall active region carrier density \(N\) and vice versa. The gain factor \({G}_{AC}\) of the QD SOA is computed based on \(N\). However, since the electron dynamics are strongly based on the electronic properties of the QDs under effective quantum confinement, the gain linewidth mainly depends on \({N}_{QD}\). The beam confinement factor \({\Gamma }_{AC}\) is determined based on the refractive index of the active and cladding regions, the spot size of the beam, and the ratio of the active region thickness to the overall QD SOA waveguide thickness, which corresponds to the fraction of the beam intensity in the active region. In this study, we assume \(\upzeta \approx 1\) and \({N}_{h}\approx 0\) for simplicity. The population-rate equations in this case are stated as (Xiao and Huang 2008; Aşırım and Jirauschek 2022; Grillot et al. 2020; Huang 1995; Saleh and Teich 2019; Silfvast 2004; Wang et al. 2007),

$$\frac{dN}{{dt}} = \frac{{\xi_{in} I}}{eV} - \frac{N}{{\tau_{c} }} - G_{AC} \left( \omega \right)\frac{c}{{Re\{ n_{AC} \} }}S$$
(6)
$$\frac{dS}{{dt}} = \Gamma_{AC} G_{AC} \left( \omega \right)\frac{c}{{Re\left\{ {n_{AC} } \right\}}}S - \frac{S}{{\tau_{p} }} + \Gamma_{AC} \frac{N}{{\tau_{r} }}, \tau_{p} = \frac{{Re\left\{ {n_{AC} } \right\}L}}{c}, \tau_{r} = \frac{1}{{BN_{QD} }}$$
(7)
$$G_{AC} \left( \omega \right) = - 2\frac{\omega }{c}Im\left\{ {n_{AC} } \right\}$$
(8)

The carrier lifetime, intraband collision time, and the gain linewidth are computed as (Huang 1995; Saleh and Teich 2019; Silfvast 2004; Wang et al. 2007)

$$\tau_{c} \approx 1/\left( {A + BN_{QD} + CN_{QD}^{2} } \right), T_{c} \approx K/N_{QD} , \Omega \approx 2\pi \left[ {\left( {1/\tau_{c} } \right) + \left( {1/T_{c} } \right)} \right]$$
(9)

which are coupled with the carrier and photon densities through Eqs. 6–7. Note that while the transition linewidth of the QDs is considered to be a function of \({N}_{QD}\), the transition linewidth of the host medium (\(\gamma\)) is considered to be a constant as most of the carriers are assumed to be captured by the QDs, such that \({N}_{h}\ll {N}_{QD}\).

$${\text{Waveguide region}}:\left( {{\text{x}}_{0} < {\text{x}} < {\text{x}}_{1} ,{\text{ z}}_{0} < {\text{z}} < {\text{z}}_{1} } \right),{\text{ Active region}}:{ }\left( {{\text{x}}_{2} < {\text{x}} < {\text{x}}_{3} ,{\text{ z}}_{2} < {\text{z}} < {\text{z}}_{3} } \right)$$
$${\text{P}}_{{{\text{AC}}}} :{\text{Refractive index distribution profile, W}}:{\text{ Beam spot size}},{\text{ N}}_{{{\text{th}}}} :{\text{Threshold carrier density for lasing}}$$
$${\text{S}}:{\text{Photon density}},{\text{ N}}:{\text{Electron density}},{ }\uptau_{{\text{p}}} :{\text{Photon lifetime}},{ }\uptau_{{\text{c}}} :{\text{Carrier lifetime}},{ }\Gamma_{{{\text{AC}}}} :{\text{Optical confinement factor}}$$
$${\text{G}}_{{{\text{AC}}}} :{\text{Spectral gain function}},{ }\upomega_{{{\text{QD}}}} :{\text{Angular QD transition frequency}},{ }\Omega :{\text{Gain linewidth}}$$
$${\text{n}}_{{{\text{AC}}}} :{\text{ Effective refractive index of the active region}},{\text{ I}}:{\text{Pump current}},{\text{ K}}:{\text{Collision}} - {\text{time constant}}$$
$$\upomega { }:{\text{Angular laser beam frequency}},{ }\uplambda :{\text{ Wavelength}},{\text{ L }}:{\text{Active region length}}$$
$$\upxi_{{{\text{in}}}} :{\text{ Injection efficiency}},{ }\uptau_{{\text{r}}} :{\text{Radiative recombination time, T}}_{{\text{c}}} :{\text{ Intraband collision time}}$$
$${\text{A}}:{\text{ Trap}},{\text{ impurity}},{\text{ or defect based nonradiative recombination coefficient}}$$
$${\text{B}}:{\text{ Radiative recombination coefficient, C}}:{\text{ Auger recombination coefficient}}$$

Figure 2 shows the cross section of a QD SOA waveguide involving the cladding region and the active region. The cladding region is necessary for optical confinement within the waveguide. The red color indicates the density of carriers in the active region, which are mostly concentrated within the QDs as a result of quantum confinement. The photon density follows a similar distribution pattern as the carrier density. However, usually some of the photons pass through the cladding region as a result of imperfect optical confinement, which is quantified by the optical confinement factor \({\Gamma }_{\text{AC}}\). Many experimental studies on QDSOAs have reported a value for \({\Gamma }_{\text{AC}}\) between 0.2 and 0.6 corresponding to 5 to 15 QD layers. In Fig. 2, it is implied that although most carriers are effectively confined within the QDs, the carrier confinement ratio is not precisely equal to 1 as a small portion of the carriers remain within the host medium of the active region. Note that the Maxwell–Garnett theorem in Eq. 1 is applied within the active region and does not include the cladding.

Fig. 2
figure 2

QD SOA waveguide (left) and its cross section (right) involving the active and the cladding regions. Under strong beam confinement within the active region, the carrier and photon densities are greater within the QDs

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Figure 3 illustrates the flowchart of LWEF computation as it is discussed here. At each time step, the process involves solving the QD SOA population-rate equations and the corresponding update of the QD carrier density, followed by the associated modification of the carrier and photon lifetimes and the resulting application of the Maxwell–Garnett theorem for updating the effective refractive index of the active region to be used in the population-rate equations at the next time step.

Fig. 3
figure 3

Flowchart representing the computation of the LWEF of a QD SOA at a given time step

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3 Results

3.1 Simulation parameters

In our simulations, we aimed for using the typical values for QD SOAs (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Xiao and Huang 2008; Zubov et al. 2015; Dagens et al. 2005; Tan et al. 2009; Vahala et al. 1983; Saleh and Teich 2019; Silfvast 2004; Wang et al. 2007), which are given in Table 1. As we want to perform sweep-analyses for the key QD SOA parameters mentioned earlier, which considerably influence the LWEF, we included the higher and lower ends of the practical value range for each influential parameter (the first 5 parameters in Table 1). Some of the parameter values, or parameter-ranges mentioned in Table 1, were not directly provided in the literature (such as the collision-time constant), but derived in accordance with their mathematical relation to other parameters whose practical value range is well-known.

Table 1 Main parameters and their values for our QD SOA LWEF simulations
Full size table

The QD SOA operation bandwidth stated here is based on the configuration in Qasaimeh (2005). The QD transition frequency is often slightly detuned from the operation bandwidth to reduce gain saturation effects or improve the stability against optical feedback. Here we also intend to investigate the change in the LWEF when the QD transition frequency is highly detuned from the operation bandwidth, therefore we set the range of investigation for the QD transition frequency to reside between 220 and 240 THz. The composition ratio is varied within the practical value range [0.01,0.12] to examine the effect of including more QDs to the active region. Detailed information on the operation parameters of QD SOAs can be found in Melnik et al. (2006); Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Xiao and Huang 2008; Zubov et al. 2015; Liu and "High efficiency, high gain and high saturation output power quantum dot SOAs grown on Si and applications," in 2020; Sugawara 2003).

3.2 Variation of the LWEF based on pump current and photon density

Based on the practical QD SOA parameters given in Table 1, Fig. 4 illustrates the variation of the LWEF against the pump current for near-perfect \(\left(\zeta \approx 1\right)\) QD carrier confinement at \(\text{f}=\frac{\upomega }{2\uppi }=245\text{THz}\) based on various QD inclusion fractions (\(\Delta \approx \frac{\text{N}}{{\text{N}}_{\text{QD}}}\), the composition ratio), assuming a collision-time constant of \(\text{K}={1\times 10}^{11}{\text{m}}^{-3}\text{s}\) and given the QD transition frequencies \({\text{f}}_{\text{QD}}=\){\(240\text{THz},220\text{THz}\)}. Because of the unique properties of QDs, modulating the pump current of QD SOAs is critical for many reasons including gain control, high-speed optical switching, wavelength conversion, and noise reduction. This analysis enables one to determine the corresponding change in the LWEF when the pump current is modulated at a certain strength when performing such operations.

Fig. 4
figure 4

LWEF versus pump current for different values of Δ, for \({\text{f}}_{{{\text{QD}}}} = 240{\text{THz }}\) (a) and \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz }}\) (b) assuming a collision-time constant of \(\text{K}{=1\times 10}^{11}{\text{m}}^{-3}\text{s}\)

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Based on Fig. 4a, one can see that the increase of the pump current leads to an increase in the LWEF. The rate of increase in LWEF strongly depends on the QD composition ratio (Δ). Notably, higher Δ values yield a lower LWEF, suggesting the use of high cumulative QD volume in the active region. Figure 4b shows that when the QD transition frequency is decreased from 240 to 220THz, a sharp decrease in LWEF occurs for all Δ values. However, Fig. 5, for which the collision-time constant K is increased to \({10}^{12}{\text{m}}^{-3}\text{s}\), illustrates a contradiction concerning the absolute value of the LWEF regarding the increase or decrease of the value of Δ for minimizing LWEF magnitude. The magnitude of the LWEF is often the main concern regarding refractive index fluctuations (Green et al. 1987; Vahala et al. 1983). In Fig. 5b, an increase in the value of Δ leads to an increase in the absolute value of the LWEF (though the increase is small). Nevertheless, if one disregards the absolute value, based on Figs. 4and 5, one can conclude that the LWEF decreases with increasing Δ. Importantly, the QD transition frequency and the collision-time constant K seem to play a big role in altering the value of the LWEF for all values of Δ, which will be investigated in our upcoming sweep-analyses.

Fig. 5
figure 5

LWEF versus pump current for different values of Δ, for \({\text{f}}_{{{\text{QD}}}} = 240{\text{THz }}\) (a) and \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz}}\) (b) assuming a collision-time constant of \(\text{K}{=1\times 10}^{12}{\text{m}}^{-3}\text{s}\)

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Figures 6a and 7b illustrate the associated variation of the LWEF with respect to the photon density as a result of the respective sweeps of the pump current in Figs. 4a and 5b. Figures 6a and 7b also indicate the maximum value that the photon density can reach via sweeping the pump current from 10 to 1000 mA, based on each value of Δ. This analysis is essential for determining the temporal variation of the LWEF in nonlinear and/or dispersive optoelectronic devices where a significant change in photon density occurs over many roundtrips. As expected, the variation of the LWEF against the photon density is similar to its variation against the pump current. Importantly, we observe that the change in the LWEF is more steep for lower values of Δ. In addition, the strongest dependance of the LWEF to the photon density occurs when the collision-time constant is low and the detuning in \({\text{f}}_{\text{QD}}\) is small. Therefore, one can infer that in optical cavities employing QD SOAs, where variation in power remains insignificant over the entire duration of operation, the LWEF can be assumed as constant. In contrast, if power variation is significant, especially when the collision-time constant of the employed QD SOA is low and/or the QD emission frequency is close to the operation bandwidth, one should assume that the LWEF is time-dependent. Our presented model in Sect. 2 is quite useful to handle strong variations in carrier and photon densities for modeling the LWEF due its instant modification of each SOA parameter that depends on the population of photons and carriers.

Fig. 6
figure 6

LWEF versus photon density for different values of Δ, for \({\text{f}}_{{{\text{QD}}}} = 240{\text{THz }}\) (a) and \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz }}\) (b) assuming a collision-time constant of \(\text{K}{=1\times 10}^{11}{\text{m}}^{-3}\text{s}\)

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Fig. 7
figure 7

LWEF versus photon density for different values of Δ, for \({\text{f}}_{{{\text{QD}}}} = 240{\text{THz }}\) (a) and \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz }}\) (b) assuming a collision-time constant of \(\text{K}{=1\times 10}^{12}{\text{m}}^{-3}\text{s}\)

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3.3 Frequency dependence of the QD SOA LWEF in the steady-state based on different QD carrier densities

Using the given parameters in Table 1 and based on an unmodulated (constant) electrical pump current, the variation of the LWEF (steady-state value unless stated otherwise) against the operation frequency is shown in Fig. 8 for two different QD transition frequencies using a practical value for the collision-time constant (\(\text{K}{=10}^{11}{\text{m}}^{-3}\text{s}\)). Here, the analysis focuses on the QD carrier density instead of Δ as the LWEF is usually expressed as an explicit function of the QD carrier density in the literature (Zhang et al. 2019; Qiu, et al. 2021; Ding et al. 2023; Xiao and Huang 2008; Zubov et al. 2015; Holmström et al. 2010). Figure 8a shows the case when the QD transition frequency is equal to\({\text{f}}_{\text{QD}}=240\text{THz}\), which is closer to the operation bandwidth (244THz < f < 259 THz). In this case the LWEF assumes higher values, which decreases either via a reduction of the QD carrier density or through an increase of the beam frequency within the bandwidth such that the beam frequency moves away from the QD transition frequency.

Fig. 8
figure 8

LWEF versus operation frequency based on various QD carrier densities for \({\text{f}}_{{{\text{QD}}}} = 240{\text{THz}}\) (a) and for \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz }}\) (b), assuming a collision-time constant of \(\text{K}{=10}^{11}{\text{m}}^{-3}\text{s}\)

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Figure 8b shows the variation of the LWEF versus frequency for a QD transition frequency of \({\text{f}}_{\text{QD}}=220\text{THz}\), which is detuned from the operation bandwidth. Here the maximum attainable value of the LWEF is greatly decreased (compared to the case in Fig. 8a). However, for low values of the QD carrier density such as \({\text{N}}_{\text{QD}}=6\times {10}^{24}{\text{m}}^{-3}\) and \({\text{N}}_{\text{QD}}=4\times {10}^{24}{\text{m}}^{-3}\), the increase of the beam frequency does not necessarily yield a reduction of the absolute value of the LWEF, which is what truly matters as the sign of the LWEF is irrelevant due to the fact that the LWEF associated phase-shift (carrier fluctuation) is the actual concern rather than its direction. A striking observation here is that when the LWEF is increasing in negative values, the curve is always linear (as opposed to its exponential profile when the LWEF is usually positive in value). We will make use of these LWEF curve profiles indicating sign when we observe the variation of the maximum absolute value of the LWEF against key QD parameters in our sweep-analyses.

Figure 9 provides elaboration on the scenario in Fig. 8. In this case the collision-time constant is increased to \(\text{K}{=1\times 10}^{12}{\text{m}}^{-3}\text{s}\), which is still in the practical range (Qasaimeh 2005; Melnik et al. 2006; Zhang et al. 2019; Qiu, et al. 2021; Westbrook and Adams 1988; Ding et al. 2023; Xiao and Huang 2008; Zubov et al. 2015; Dagens et al. 2005; Tan et al. 2009; Saleh and Teich 2019; Silfvast 2004; Wang et al. 2007). Here, an increase of the operation frequency leads to a sharp increase in the absolute value of the LWEF for low QD carrier densities, causing more refractive index fluctuations. This also indicates that a decrease of the QD transition frequency does not always yield LWEF improvement (see Fig. 9b), particularly when the collision-time constant is high and the QD carrier density is low. A concurrent examination of Figs. 8 and 9 suggests that there is an optimal set of values for \({\text{f}}_{\text{QD}}\) and \(\text{K}\) that minimizes the maximum value of the LWEF magnitude at a given QD carrier density. For this reason, in Sect. 3.4, we analyze the variation of the maximum value (within the operation bandwidth) of the LWEF magnitude with respect to \({\text{f}}_{\text{QD}}\), \(\text{K}\), and \({\text{N}}_{\text{QD}}\) to assess the degree of refractive index fluctuations under different operation conditions, and come up with a minimization strategy.

Fig. 9
figure 9

LWEF versus operation frequency based on various QD carrier densities for \({\text{f}}_{\text{QD}}=240\text{THz}\) (a) and for \({\text{f}}_{{{\text{QD}}}} = 220{\text{THz }}\) (b), assuming a collision-time constant of \(\text{K}{=10}^{12}{\text{m}}^{-3}\text{s}\)

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3.4 Sweep analysis for the maximum absolute value of the LWEF against key QD parameters

As observed in Figs. 8and 9, the maximum absolute value of the LWEF \(\left( {\max \left( {\left| \alpha \right|} \right)} \right)\) within the operation bandwidth is greatly influenced by the QD carrier density in a given active region. Importantly, the active region carrier density and the composition ratio may not always be indicative of the QD carrier density, as in the case of poor or imperfect quantum confinement. Hence a sweep analysis for \(\text{max}\left(\left|\alpha \right|\right)\) regarding QD carrier dynamics should be made explicitly based on the QD carrier density, rather than the active region carrier density or the composition ratio.

Figure 10 shows two distinct relations between \(\text{max}\left(\left|\alpha \right|\right)\) and QD carrier density, corresponding to the lower and higher ends of the practical range of the collision-time constant. In Fig. 10a, for a collision-time constant of \(\text{K}{=10}^{11}{\text{m}}^{-3}\text{s}\), \(\text{max}\left(\left|\alpha \right|\right)\) assumes its minimum values among lower QD carrier densities, and beyond a small threshold value of the QD carrier density, the value of \(\text{max}\left(\left|\alpha \right|\right)\) starts to increase linearly, which is the characteristic for negative values of \(\alpha\). By contrast, in Fig. 10b, an increase in the QD carrier density allows for an exponential decrease in \(\text{max}\left(\left|\alpha \right|\right)\) for a much greater range of QD carrier densities, and the threshold value of the QD carrier density beyond which \(\text{max}\left(\left|\alpha \right|\right)\) increases linearly, is comparably high, particularly for a high detuning of \({\text{f}}_{\text{QD}}\).

Fig. 10
figure 10

Maximum absolute value of the LWEF versus \({\text{N}}_{\text{QD}}\) for \(\text{K}{=10}^{11}{\text{m}}^{-3}\text{s}\) (a) and \(\text{K}{=10}^{12}{\text{m}}^{-3}\text{s}\) (b)

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Therefore, a clear observation here is that the increase of the QD carrier density inside a QD with a high collision-time constant minimizes the value of \(\text{max}\left(\left|\alpha \right|\right)\) for a high detuning of \({\text{f}}_{\text{QD}}\). For QDs with low collision-time constant, the QD carrier density should be carefully adjusted for minimizing the value of \(\text{max}\left(\left|\alpha \right|\right)\) in accordance with the degree of detuning in \({\text{f}}_{\text{QD}}\). Based on these observations, a good strategy is to employ QDs with high collision-time constant, maximize the QD carrier density, and detune the transition frequency to reduce the value of \(\text{max}\left(\left|\alpha \right|\right)\) to the desired range within the operation bandwidth.

Importantly, one can see that the value of the collision-time constant has a great influence on the value of \(\text{max}\left(\left|\alpha \right|\right)\). Figure 10 implies that a gradual increase in the collision-time constant allows for a greater minimization of \(\text{max}\left(\left|\alpha \right|\right)\) under increasing QD carrier density, which concurrently allows for a greater optical gain. To see the direct relation between the collision-time constant and \(\text{max}\left(\left|\alpha \right|\right)\), a sweep-analysis for K is more informative. Additionally, based on Fig. 10, the QD transition frequency also appears to greatly influence \(\text{max}\left(\left|\alpha \right|\right)\). Hence, a sweep-analysis is also needed for \({\text{f}}_{\text{QD}}\) which is provided below.

Figure 11 shows the relation between the value of \(\text{max}\left(\left|\alpha \right|\right)\) and the QD transition frequency for a collision-time constant of K=\({10}^{11}{\text{m}}^{-3}\text{s}\) based on different QD carrier densities. For such low values of the collision-time constant, the value of \(\text{max}\left(\left|\alpha \right|\right)\) can be sharply decreased via a slight detuning of the QD transition frequency. For higher values of the collision-time constant, the detuning should be done more precisely as the value of \(\text{max}\left(\left|\alpha \right|\right)\) starts to increase beyond a certain range of detuning, which gets narrower with decreasing QD carrier density (see Fig. 12). Hence, for active media with a low collision-time constant, detuning the QD transition frequency is a good strategy to minimize \(\text{max}\left(\left|\alpha \right|\right)\), especially for an active region with a high QD carrier density. By contrast, detuning of the QD transition frequency should be done more carefully for active media with a high collision-time contant as the practical (concerning QD carrier density) optimal value of \(\text{max}\left(\left|\alpha \right|\right)\) resides somewhere in between low (\({\text{f}}_{\text{QD}}=240\text{THz})\) and high (\({\text{f}}_{\text{QD}}=220\text{THz}\)) detuning in accordance with the QD carrier density, as shown in Fig. 12. Fortunately, for active media with a low collision-time constant, a large detuning of \({\text{f}}_{\text{QD}}\) yields an almost 20-fold decrease in \(\text{max}\left(\left|\alpha \right|\right)\), while leading to a reduction of only 30 to 40% in the maximum optical gain (see Fig. 11). Likewise, for active media with a high collision-time constant, the required detuning of \({\text{f}}_{\text{QD}}\) yields a significant decrease in \(\text{max}\left(\left|\alpha \right|\right)\) for large QD carrier densities while causing a relatively low reduction in the maximum optical gain (see Fig. 12).

Fig. 11
figure 11

Maximum absolute value of LWEF versus \({\text{f}}_{\text{QD}}\) for K = \({10}^{11}{\text{m}}^{-3}\text{s}\), Δ = 0.1 (a) and the corresponding maximum optical gain (b)

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Fig. 12
figure 12

Maximum absolute value of LWEF versus \({\text{f}}_{\text{QD}}\) for K = \({10}^{12}{\text{m}}^{-3}\text{s}\), Δ = 0.1 (a) and the corresponding maximum optical gain (b)

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We conclude our analysis with Fig. 13, illustrating the variation of \(\text{max}\left(\left|\alpha \right|\right)\) against the collision-time constant K for different values of the QD carrier density. Figure 13 shows that for a low detuning of the QD transition frequency (\({\text{f}}_{\text{QD}}=240\text{THz}\)), \(\text{max}\left(\left|\alpha \right|\right)\) assumes larger values for small values of K. A sharp decrease in the value of \(\text{max}\left(\left|\alpha \right|\right)\) is evident via an increase in the value of the collision-time constant up to its optimal value which minimizes \(\text{max}\left(\left|\alpha \right|\right)\). Beyond this value, \(\text{max}\left(\left|\alpha \right|\right)\) displays a slight increase with increasing K (due to the LWEF assuming negative values beyond the optimal value of K). By contrast, for a large detuning of the QD transition frequency, the value of \(\text{max}\left(\left|\alpha \right|\right)\) can be minimized down to 0.23 via a careful selection of K (among smaller values) in accordance with the QD carrier density, as observed in Fig. 13b. Once again, beyond the optimal value of K, \(\text{max}\left(\left|\alpha \right|\right)\) increases almost linearly, but this time rapidly with increasing K. Therefore, based on Fig. 13, one should keep the QD transition frequency detuning low for a medium with a high collision-time constant, and vice versa. The rate of increase or decrease in the value of \(\text{max}\left(\left|\alpha \right|\right)\) against the collision-time constant is determined by the QD carrier density, as shown in Fig. 13. Based on the entire analysis, it appears that the relation between the LWEF and key QD parameters such as QD carrier density, QD transition frequency, and collision-time constant is not straightforward and requires careful modeling. In the next section, we will state a simple strategy for suppressing the LWEF in QD SOAs based on the given results in this section.

Fig. 13
figure 13

Maximum absolute value of LWEF versus collision-time constant for \({\text{f}}_{\text{QD}}=240\text{THz}\), Δ = 0.1 (a), \({\text{f}}_{\text{QD}}=220\text{THz}\), Δ = 0.1 (b)

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4 Discussion and comparison with empirical results

We have seen that contrary to what is sometimes claimed in literature, QD SOAs do not always have a low LWEF. In fact, their LWEF can be exceedingly high for certain design scenarios, as we have seen in previous figures. However, a thorough analysis of our results reveals two straightforward scenarios for designing a QD SOA with a very small LWEF. The first one can be easily inferred from Fig. 11a, where the detuning of the QD transition frequency (\({\text{f}}_{\text{QD}}=220\text{THz}\)) is high, but the collision-time constant and the QD carrier density are relatively low when the maximum absolute value of the LWEF is small. In this case, it is observed that the maximum absolute value of the LWEF can get as small as 0.2, a 20-fold decrease (see the green curve) with respect to the case when \({\text{f}}_{\text{QD}}=240\text{THz}\) while only sacrificing 35% in the maximum optical gain (see Fig. 11b). Fortunately, for high QD carrier densities, this compromise in the maximum optical gain via a detuning of \({\text{f}}_{\text{QD}}\) is much lower (less than 10%). However, in those cases, although the LWEF can still be decreased sharply, it does not converge to near-zero values. Hence, unless the intended application requires an ultra-small QD SOA LWEF, a significant compromise in optical gain is not crucial.

The second scenario can be inferred via Fig. 12, in which case the collision-time constant is high, and the attainment of very low LWEF values is not as straightforward as it is for the first scenario. Here, the detuning of the QD transition frequency must be adjusted based on the QD carrier density. One can see that the QD transition frequency should be set somewhere between 230 and 240THz for our parameters, which should be set closer to \({\text{f}}_{\text{QD}}=240\text{THz}\) for lower QD carrier densities, and closer to \({\text{f}}_{\text{QD}}=230\text{THz}\) for higher QD carier densities (see Fig. 12). Notably, as the QD carrier density gets higher, \(\text{max}\left(\left|\alpha \right|\right)\) gets slightly lower. In this second scenario, we observed that the maximum absolute value of the LWEF gets as small as 0.48. Based on our evaluation of all figures, there are clearly other scenarios in which the maximum absolute value of the LWEF can get very small. These scenarios can be uncovered through an optimization procedure which we do not go through in this study.

Since the measurement of the LWEF is experimentally complicated (Ruan et al. 2018; Holzinger et al. 2018), validated numerical models are essential for obtaining an estimate for the LWEF of QD SOAs. Due to this fact, there are only a few studies that report a conclusive empirical formulation for QD SOAs in accordance with experimental observations. A comprehensive study on the LWEF of QD SOAs was carried out in Qasaimeh (2005), along with a generalized empirical formulation. To verify the accuracy of our numerical results, we use the empirical formula given in Qasaimeh (2005) within the same operation bandwidth. Based on this formula, when the QD transition frequency is slightly detuned from the operation bandwidth, the LWEF decreases exponentially with increasing frequency in the operation bandwidth, which is in agreement with our observations.

To quantify the difference between our numerical results and the ones in Qasaimeh (2005), Fig. 14 shows the comparison of our numerical results for practical values (\({\text{N}}_{\text{QD}}=1\times {10}^{25}{\text{m}}^{-3}\), K \(={2\times 10}^{11}{\text{m}}^{-3}\text{s}\), \({\text{f}}_{\text{QD}}=240\text{THz}\)) with the empirical formula. Based on Fig. 14a, we observe good agreement between the numerical and empirical results, although some deviation between the results is clearly present, corresponding to a relative error of around 7%. At the highest operation frequency (259THz), this deviation is slightly more evident. This is because the carrier density, collision-time constant, and the composition ratio are not directly specified in Qasaimeh (2005). Instead, the joint effect of these parameters on the LWEF is modeled via the standard deviation in the gain linewidth, which is set by the QD carrier density in our formulation. We also observe that this deviation is gradually suppressed for higher QD carrier densities and higher collision-time constants. Concerning an alternative scenario to point this out, Fig. 14b shows the comparison of the results for \({\text{N}}_{\text{QD}}=3\times {10}^{25}{\text{m}}^{-3}\), K \(={3\times 10}^{11}{\text{m}}^{-3}\text{s}\), \({\text{f}}_{\text{QD}}=240\text{THz}\). Here we observe better agreement between the numerical and empirical results for all frequencies in the operation bandwidth, corresponding to a relative error of less than 4%. These relative errors are acceptable concerning the practical significance of LWEF values (Aşırım et al. 2022; Mahmoud et al. 2023).

Fig. 14
figure 14

Comparison of numerical and empirical LWEF values versus frequency for \({\text{N}}_{\text{QD}}=1\times {10}^{25}{\text{m}}^{-3}\), K \(={2\times 10}^{11}{\text{m}}^{-3}\text{s}\), \({\text{f}}_{\text{QD}}=240\text{THz}\), Δ = 0.1 (a) and \({\text{N}}_{\text{QD}}=3\times {10}^{25}{\text{m}}^{-3}\), K \(={3\times 10}^{11}{\text{m}}^{-3}\text{s}\), \({\text{f}}_{\text{QD}}=240\text{THz}\), Δ = 0.1 (b)

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Throughout this study, certain formulations and parameters are taken as approximations to simplify the analysis. For example, Eq. 3, which is based on Maxwell’s equations and the Lorentz dispersion model, is an approximation. Since Maxwell’s Equations and the Lorentz model remain fundamentally valid within the spatial scale of quantum dots (QDs), the use of the given permittivity expression for QDs is justified, provided that one incorporates the quantization of energy levels and the influence of QD size in the dielectric function. Studies such as (Holmström et al. 2010; Grinolds et al. 2015; Bayrak et al. 2023; Krevchik et al. 2023; Aspnes 1982) support the practical utility of this permittivity expression for QDs. Furthermore, a rigorous description of an effective medium is necessary for our case, as the permittivity of QDs can increase dramatically due to resonant enhancement which can cause the permittivity of QDs to become much larger than that of the host medium. Importantly, our model abstracts away the spectral complexity of QDs and the host material, but it does so within the context of the operational conditions and the physical phenomena that is of importance for our application. The approximation of a single dominant resonance can be reasonable in QDs, since, at typical operating conditions such as room temperature and specific excitation energies, the electronic transitions are predominantly between the ground state and the first excited state. Other transitions contribute less significantly due to larger energy gaps and lower transition probabilities. In this study, the QDs are assumed to have minimal size distribution, reducing inhomogeneous broadening and reinforcing the dominance of a single resonance. For the host material, the permittivity expression in Eq. 3 is justified by the operational frequency regime, where higher energy transitions do not substantially affect the optical properties. While this is a simplification, it is based on the understanding that the dominant transition in the wavelength range of interest has the greatest oscillator strength, allowing the host to be approximated as having a narrow linewidth response similar to a two-level system (Holmström et al. 2010; Kotb and Zoiros 2013).

Inhomogeneous broadening in QDs, typically arising from size, shape, and compositional variations, leads to a distribution of transition energies. In our study, the QDs are assumed to be identical in size, shape, and composition, minimizing such broadening. The focus is on the average response of the QD ensemble within a specific operational regime dominated by the ground to the first excited state transition, allowing us to average out the broadening effects and model the QDs effectively with a dominant transition energy. The model’s treatment using effective medium theory (EMT) simplifies the response of the quantum dot ensemble by averaging the optical properties over all QDs. This approach is justified when the operational conditions, such as excitation wavelength and intensity, interact primarily with the dominant transition across the ensemble. Even with inhomogeneous broadening, the collective effect can be approximated by a dominant transition model. Of course, having all QDs being identical in size, shape, and composition is unattainable in practice. However, if the fluctuations in size, shape, and composition are small, meaning that most quantum dots are still close to the average parameters (e.g., average size or shape), the overall optical response can still be reasonably approximated by using the average parameters in the formula. In such a scenario, the inhomogeneous broadening does not drastically change the central frequency or the shape of the permittivity function. The formula’s accuracy relies on the ensemble average, which remains centered around the mean values. Therefore, the model effectively represents the average dielectric response of the quantum dots if the deviations from the average do not lead to substantial shifts in the resonance frequency or significant changes in the damping behavior. As a result, Eq. 3 can still be reliably used if the fluctuations in size, shape, and composition are small, the average parameters of the quantum dots align with those used in Eq. 3, or if the distribution of quantum dot properties is narrow and centered around these average values, making our model a reasonable approximation despite some inhomogeneous broadening.

Here, we considered a constant value for the carrier confinement factor (\(\zeta \approx 1\)) to simplify the analysis. Using a constant value allows us to isolate and understand the primary effects of other variables without the added complexity of varying the carrier confinement factor. Previous research supports the use of constant carrier confinement in QDSOAs (Xiao and Huang May 2008; Ma et al. 2009; Ju et al. 2006), particularly under low temperatures and/or large bandgap differences. Since we assume near-perfect carrier confinement in this study, our model is only valid for quantum dots (QDs) with large potential depths. This is because the QD carrier confinement ratio increases with the potential depth and approaches 1 for QDs with very deep potentials. In such cases, the number of carriers remaining in the reservoir is negligible, indicating effective quantum confinement. For shallower QDs, where the potential depth is lower, a significant fraction of carriers could escape, affecting the carrier density and, consequently, the linewidth enhancement factor.

Likewise, the chosen constant value of 0.5 for the optical confinement factor is within the typical range observed in experimental studies of QDSOAs and represents a practical value. Many experimental studies have reported optical confinement factors in the range of 0.2 to 0.6, supporting our choice as a reasonable approximation. Similar simplifications have been adopted in studies such as (Kotb and Zoiros 2013; Hakimian et al. 2020). It is worth noting that increasing the number of QD layers can enhance the optical confinement factor because a greater portion of the optical mode overlaps with the active region, boosting gain. However, if there are too many layers, the active region may become too thick, reducing Γ as the mode spreads, leading to increased scattering losses and gain saturation. Conversely, too few layers decrease Γ, as there is less overlap of the optical mode with the quantum dots, resulting in insufficient gain. Thus, an optimal number of QD layers, typically between 10 and 15, is necessary to maximize Γ and achieve effective amplification.

Regarding the use of parameters A, B, and C to describe the recombination processes in QDs, the recalibration of coefficients A, B, and C to account for quantum confinement effects is supported by experimental research. Studies such as (Frost et al. 2014; Li et al. 2000) detail how this model is used under quantum confinement in QDs. Similarly, the inverse relation between intraband collision-time and the QD carrier density is demonstrated in Xiao and Huang May (2008); Harbold et al. Nov. 2005), supporting the accuracy of the intraband collision-time model employed in this study.

This work not only models the LWEF against four key QDSOA parameters, but it also proposes a strategy for strict LWEF minimization as we intend to minimize the maximum absolute value of the LWEF (max(|α|)) within the entire operation band. In optoelectronic devices employing SOAs, which usually operate in the form of a closed-loop (feedback) structure, the LWEF must be kept small for all frequencies within the operation bandwidth due to the fact that even a very brief jump in LWEF value over a given cavity roundtrip can trigger instabilities, which degrade the performance of frequency-swept lasers, such as Fourier Domain Mode-Locked (FDML) lasers (Aşırım et al. 2023).

5 Conclusion

Upon careful adjustment of the QD SOA parameters at the design stage, a great reduction in the maximum absolute value of the LWEF (\(\text{max}\left(\left|\alpha \right|\right)\) is possible. Based on our findings, the key parameters that influence the LWEF of a QD SOA are the QD carrier density, QD transition frequency, and the collision-time constant. Here we have observed that although the minimization of \(\text{max}\left(\left|\alpha \right|\right)\) requires a careful selection of these parameters, there are two basic scenarios that yield a low value for \(\text{max}\left(\left|\alpha \right|\right)\) without requiring any explicit minimization technique. One scenario is to use an active region with a low collision-time constant, low QD carrier density, and high QD transition frequency detuning. The other scenario involves using an active region with a high collision-time constant, careful adjustment of the QD transition frequency detuning (depending on the QD carrier density), and a high QD carrier density for minimization. Concerning gain performance, the second scenario is often a better but less straightforward one. Based on these simple design settings that require no optimization technique, we have observed that the value of \(\text{max}\left(\left|\alpha \right|\right)\) can be decreased to below 0.5 within the operation band, which improves the coherence and stability of optical devices employing QD SOAs.