## Abstract

Coherent coupling between two laterally coupled quantum dot semiconductor lasers based on longitudinal modes in the weak coupling regime has been investigated. Considering the effect of homogeneous and inhomogeneous broadenings, the spectral behavior of lasing modes in the coupled lasers depends on the coupling coefficient and the frequency detuning between the lasers. For cavity lengths with a minute difference and specific homogeneous broadening, full phase-locking along with single-mode operation is guaranteed provided that the coupling coefficient reaches a critical value. This result can be generalized for various cavity lengths and homogeneous broadenings covering almost all practical temperatures in the intermediate homogeneous broadening regime. The effect of coherent coupling on the emission spectrum, especially at sufficiently low temperatures at which the lasing spectrum shows broadband emission in the absence of coupling, has also been discussed. For constant cavity lengths and relatively high temperatures, onset of full phase-locking occurs at lower values of the coupling strengths, while longer cavity lengths lead to higher critical coupling strengths at a constant temperature. From a practical view, coherent coupling based on longitudinal modes has been examined in the case of a differential bias condition, which proves the electronically controllable coherent coupling of optical output signals.

© 2021 Optical Society of America

## 1. INTRODUCTION

Coherent beam combination of coupled semiconductor laser arrays has evolved recently as an alternative approach to scaling up emitted beam radiance [1–4]. Radiance improvement using spatial supermodes has also been studied over the years [5–8]. Although these structures have been studied in other applications such as bandwidth modulation [9,10] and beam steering in the far field [11–13], radiance improvement using longitudinal modes specifically in lasers with many longitudinal modes is as promising as an earlier method [14].

It has been theoretically and experimentally proven that the phase-locking of the laser elements is the key step in achieving coherent beam combination [15]. The phase-locking process in coupled lasers that are longitudinally single-mode has been investigated by many groups [16–19]. On the other hand, in two coupled lasers, each operating with many longitudinal modes, phase-locking remains subjected to mode competition, and the longitudinal modes that are ultimately dominant over the other modes determine the system’s final phase state. It is well known that the full phase-locking of two multilongitudinal mode lasers occurs for a sufficient coupling strength in which only common longitudinal modes survive and lead the coupled system to the full phase-locked state or coherent coupling [20–23].

Among semiconductor lasers, structures with quantum dots (QDs) in their active gain media are expected to exhibit promising features, including ultralow threshold current, temperature-insensitive operation, and high bandwidth [24–32]. However, broadband multimode lasing emission, i.e., inhomogeneous broadening of the spectrum in QD lasers due to the size fluctuations of dots, is considered a controversial aspect of the QD growth process. Methods like bandpass filtering and optical injection locking have already been recommended, thus paving the way toward generating a narrow linewidth emission spectrum [33–35]. Although multipopulation coupled QD lasers with a common wetting layer have been studied in [36], and the potential application of such structures in generating tunable THz radiation is discussed in [37], the possibility of producing one single longitudinal supermode out of the coupling between two laterally coupled yet separately functioning QD lasers merits further investigation.

In this paper, we present our investigations for two laterally coupled quantum dot lasers considering the homogeneous and inhomogeneous broadening effects, while the frequency detuning between spectral modes is introduced by a minute difference in cavity lengths. In particular, we show that, for each specific value of homogeneous broadening, only common modes, i.e., central lasing modes that experience zero detunings with respect to each other survive and lase, provided that the coupling coefficient reaches a critical value, leading the coupled system to a full phase-locked state along with single-mode operation. According to our results, this can be generalized for typical values of homogeneous broadening and various cavity lengths. Our results provide further evidence that coupled QD lasers are viable candidates for optical integration to achieve robust communications and practical signal processing mechanisms.

The paper is organized as follows: Section 2 includes details on our theoretical model and numerical approach; achieved results are presented in Section 3; Section 4 concludes the paper.

## 2. MODEL DESCRIPTION

The multimode rate equation model introduced in [38] has been applied to describe the carriers and photons dynamics in two laterally coupled QD lasers. Figure 1 exhibits the schematic view of the two laterally coupled QD lasers. Exciton approximation guarantees the charge neutrality in QDs. Also, the separate confinement heterostructure (SCH) level, wetting layer (WL), and a single energy state inside QDs are considered as three energy levels to form each InGaAs/GaAs QD active region. The energy diagram of the QD laser active region and the relaxation evolution of the carriers injected to the SCH along with the recombination and re-excitation processes are also shown in the Fig. 1 inset [38].

The associated time constants in this model are as follows: ${\tau _s}$ and ${\tau _d}$ represent the relaxation time constant to the lower state from the SCH layer and the WL, respectively. Also, ${\tau _e}$ and ${\tau _{{qe}}}$ are indicators for carrier re-excitation time constant to the upper state from the QD ground-state and the WL, respectively. Finally, carrier recombination time constants in the SCH region, the WL, and the QD ground-state are represented as ${\tau _{{sr}}}$, ${\tau _{{qr}}}$, and ${\tau _r}$, respectively. Considering the size fluctuation of QDs, the rate equation model takes into account the inhomogeneous broadening of the quantized energies, along with the homogeneous broadening for the optical gain of every single dot due to the polarization dephasing [38]. In two laterally coupled QD lasers, each QD ensemble has been divided into $n = {{1}},\;{{2}}, \ldots ,\;{{2}}M + {{1}}$ groups, quantized depending on their interband transition energies, where the $n = M$ corresponds to the central group at the resonant energy of ${E_{{cv}}}$ [38]. The resonant energy of the $n$th group of each QD laser in the coupled structure is expressed by ${E_{{na}}} = {E_{{cva}}} + (n - M)\Delta {E_a}$ and ${E_{{nb}}} = {E_{{cvb}}} + (n - M)\Delta {E_b}$, where ${E_{{cva}}}$ and ${E_{{cvb}}}$ are assumed to be equal in our model and represent the resonant energy of ${E_{{cv}}} = 1\;{\rm{eV}}$ for both central groups of lasers A and B. Likewise, the QD groups, the longitudinal modes of each laser cavity, have been divided into $n = {{1}},\;{{2}}, \ldots ,\;{{2}}M + {{1}}$ modes, where the resonant energy of each mode coincides with the same energy of the corresponding QD group. Assuming $\Delta {E_a} = hc/2{n_r}{L_a}$ and $\Delta {E_b} = hc/2{n_r}{L_b}$ as longitudinal mode energy separation for each cavity, with ${L_a}$ and ${L_b}$ being the cavity lengths of lasers A and B, one can introduce frequency detuning between two corresponding modes by a minute difference. Here, the parameters $h$, $c$, and ${n_r}$ are indicators for Planck’s constant, speed of light in the vacuum, and the refractive index of the active region. The impact of the inhomogeneous broadening due to the QD size fluctuations has been considered in laser cavity $i$ ($i = a$, $b$) using Gaussian distribution functions with standard deviations of ${\xi _0}$ as follows:

Here, the condition $\sum_{n = 1}^{2M + 1} {{G_{{ni}}}} = 1$ is satisfied for laser element $i$ and ${\Gamma _0} = 2\sqrt {2\ln 2} {\xi _0}$, i.e., ${\Gamma _0} = 2.35{\xi _0}$ [36,38] gives the full width at half maximum (FWHM) of the inhomogeneous broadening. The linear optical gain impact factor between the $n$th QD active region and the $m$th photon group in each cavity $i$ are expressed as [38]

In the above equations, ${| {P_{{cv}}^\sigma} |^2}$ is the transition matrix element, and the parameters $e$, $\hbar$, ${N_D}$, ${\varepsilon _0}$, and ${m_0}$ stand for the electron charge, the reduced Planck’s constant, volume density of dots, the vacuum permittivity, and the electron rest mass, respectively. ${P_{{ni}}}$ is the occupation probabilities of the $n$th QD group in laser element $i$, which can be calculated through ${P_{{ni}}} = {N_{{ni}}}/{D_g}{N_D}{V_i}{G_{{ni}}}$. ${N_{{ni}}}$ is the number of carriers in $n$th QD group in laser element $i$. Considering QDs with a height of $H$, the parameter ${V_i}$ implies the total active region volume of each QD laser defined as ${V_i} = HW{L_i}{N_w}$, where $W$ is the strip cavity width of each laser and ${N_w}$ indicates the assumed number of dot layers in the active region of each laser constituent. ${B_{{cvi}}}$ is Lorentzian-shaped function implying the homogeneous broadening or implicit indicator of the effect of temperature on the gain spectrum in laser element $i$ and can be described as

${\Gamma _{{cvi}}}$ designates the FWHM of the homogeneous broadening in each laser. The coupled equations subsuming the weak interaction between the electric fields of two QD lasers based on the rate equations model evolves as follows. First, we begin with the rate equations for QD laser element $i$, in the absence of coupling:

In the above equations, ${N_{{si}}}$, ${N_{{qi}}}$, and ${N_{{ni}}}$ are the carrier numbers in the SCH layer, the wetting layer, and the $n$th QD group in laser $i$. ${S_{{mi}}}$ exhibits the photon number of the $m$th photon mode in laser $i$. ${I_i}$ denotes the current injected to laser element $i$, which implies electrical isolation between two laser elements for further detailed examination. $\Gamma$ indicates the optical confinement factor; $\beta$ stands for the coupling efficiency of spontaneously produced photons to the lasing photons, which is considered to be small.

Having parameter ${\tau _d}$ as an indicator of the relaxation lifetime of the unoccupied ground state in QD, the average carrier relaxation lifetime ${\tau _d}$ can be defined by $1/{\tau _d} = \sum_n {\tau _{{dn}}^{- 1}} {G_n} = \sum_n {{\tau _d}^{- 1}} (1 - {P_n})\,{G_n}$. The photon lifetime in cavity $i$ is shown by ${\tau _{{pi}}}$ and can be extracted by ${\tau _{{pi}}}^{- 1} = (c/{n_r}){\alpha _{{toti}}} = (c/{n_r})[{\alpha _{0i}} + (1/2{L_i})\ln (1/{R_1}{R_2})]$ for laser $i$. ${\alpha _{0i}}$ is the total internal loss per unit length of each cavity, which is assumed to be a reasonable number corresponding to each cavity length. For instance, ${\alpha _0}$ is ${{600}}\;{{\rm{m}}^{- 1}}$ for a cavity of length 900 µm [38]. ${R_1}$ and ${R_2}$ represent the power reflectivity coefficients of terminal facets in both cavities. Detailed information about the laser parameters in the coupled system is included in Table 1. It is noteworthy that the cavity lengths variance is small enough, so the difference between ${\tau _{{pa}}}$ and ${\tau _{{pb}}}$ is negligible.

The behavior of a wide range of coupled lasers has previously been studied using coupled-mode theory [11,19,39–41]. Given the fact that the electric field in each laser consists of an amplitude corresponding to the number of photons and a phase component, hereafter we use the following general coupled equations that are modified after Eq. (5) to determine the dynamics of the electric field in each laser under the influence of electric field from the neighboring laser element:

Considering the negligible amount of spontaneous emission in comparison with the lasing photons, ${\gamma _{{mi}}}$ indicates the gain coefficient for laser $i$, which can be expressed by ${\gamma _{{mi}}} = 0.5\,[(c\,\Gamma /{n_r}\sum_n {{g_{{mni}}})} - 1/{\tau _{{pi}}}]$. ${\alpha _H}$ is the linewidth enhancement factor, which is assumed to be zero in our investigations [42]. The parameter ${K_{{ba}}}\;({K_{{ab}}})$ indicates the uniform coupling rate from one mode field in laser B (A) to the corresponding mode field in laser A (B) defined by ${K_{{ba}}} = k/{\tau _{{cb}}}\;({K_{{ab}}} = k/{\tau _{{ca}}})$ [19,40,41,43]. Here, $K$ is the coupling strength, which is a number less than unity and small enough to satisfy the weak coupling condition [14]. The coupling coefficient between the two lasers depends on the lateral separation between the two laser elements, which is denoted as parameter $d$ in Fig. 1. Consequently, $d$ is one of the factors that determines the strength at which two laterally interacting electric fields couple to each other [44]. It is known that small $d$ values are conducive to higher coupling strengths. ${\tau _{{cb}}}({\tau _{{ca}}})$ is the photon cavity roundtrip time in laser B (A). Note that ${\tau _{{ci}}}$ is considered as the reduced photon lifetime (${\tau _{{pi}}}$) by a factor of ${\exp}(- 2{\alpha _{{toti}}}{L_i})$ for a cavity of length ${L_i}$ [15,17]. ${\Omega _{{ma}}}$ and ${\Omega _{{mb}}}$ are the frequency detuning of $m$th longitudinal modes in cavities A and B from a mean optical frequency, which is assumed to be $({\Omega _{{ma}}} + {\Omega _{{mb}}})/2$. If the corresponding modes in laser elements are detuned due to a minute difference in their cavity lengths, according to the equations mentioned earlier for the resonant energy of the $m$th group in each cavity, only the central groups will experience zero detuning. In other words, for two coupled QD lasers with different cavity lengths, only the central mode of each act as the common lasing modes.

## 3. DISCUSSION

In this work, the longitudinal mode spectra for each constituent laser element have been calculated while varying the coupling coefficient in weak coupling regime. Figure 2 contains the energy spectra for a cavity length of ${L_a} = {{600}}\;\unicode{x00B5}{\rm m}$, inhomogeneous broadening of ${\Gamma _0} = {{20}}\;{\rm{meV}}$, and homogeneous broadening value of ${\Gamma _{{cv}}} = {{3}}\;{\rm{meV}}$ in terms of different coupling coefficients and still satisfying weak coupling condition. $k = 0$ represents no-coupling state, where the two laser constituents function independently; therefore, each exhibits a broad spectrum with multiple longitudinal modes. It can be inferred from Fig. 2 that further increase in the coupling coefficient leads to a reduction in the contributive longitudinal modes of the two cavities. For $k = {0.017}$, the contribution of the longitudinal modes other than the common mode between the two constituents, i.e., central mode is diminished, but the resonance of the side modes, which are not common, stands as an obvious sign of no full phase locking, hence no coherent coupling. As clear in Fig. 2, when the coupling coefficient reaches to a critical value of $k = {k_c} = {0.018}$, merely the only common mode, i.e., the central mode contributes to the lasing. The significance of the critical coupling coefficient stems from the fact that, at this point, the system of two coupled lasers happens to be at onset of full phase-locking.

Out of the coupling at critical strength, the single longitudinal mode, which experiences zero-detuning, survives and results in a single longitudinal supermode operation. As obvious as it is, for coupling coefficients above the critical value yet in the weak coupling regime, the system remains in the state of full phase-locking. For instance, at $k = {0.019}$, which corresponds to a coupling strength slightly bigger than the critical one, there is a small increase in the number of photons. However, the state of the coupled system from the phase-locking point of view remains invariant. The same discussion holds for different cavity lengths and homogeneous broadening values. It is worth mentioning that, in all cases, the energy spectra are calculated for a variation in coupling coefficient starting from $k = 0$ with an interval of 0.001 to achieve the onset of full phase-locking. However, out of all the calculations, only four particular states of the coupled system are included in Fig. 2 to present further clarification.

Note that different cavity lengths have been chosen to create the detuning between the noncentral modes and zero detuning (equal energies) between the central modes. To do this, the resonant energies and energy spacing of the both lasers can be easily calculated by $E_{a,b}^k = {k_{a,b}}hc/2n{L_{a,b}}$ and $\Delta {E_{a,b}} = hc/2n{L_{a,b}}$, respectively, where ${K_{a,b}} = {{1}},\;{{2}},\;{{3}}, \ldots$ is the resonant mode number of the laser $a$ or $b$. Considering the same gain medium for the lasers with central energy of ${E_{{cva}}} = {E_{{cvb}}} = {E_{{cv}}}$, the number corresponding to the central mode of each laser is obtained through $k_{a,b}^{{{\rm central}}} = {\rm{round}}({E_{{cv}}}/\Delta {E_{a,b}})$. Thus, having equal central resonant energies, $E_a^{{k_{{\rm central}}}} = E_b^{{k_{{\rm central}}}}$, allowed values for the cavity length of the second laser obtained through $k_b^{{\rm central}}/k_a^{{\rm central}} = {L_b}/{L_a}$. In this equation, ${L_a}$ and $k_a^{{\rm central}}$ are the known parameters. Choosing $k_b^{{\rm central}}$, the cavity length of the second laser is obtained. For simplicity, we have considered $k_b^{{\rm central}} = k_a^{{\rm central}} + 1$, and the cavity length of the second laser can be obtained through ${L_b} = {L_a}(1 + (1/k_a^{{\rm central}}))$. As a result, both lasers have the same central energies and unequal noncentral energies due to different mode spacings of the lasers.

Figure 3 (inset) exhibits the obtained results for laser A (B), considering only two states of no phase-locking (uncoupled regime, i.e., $k = 0$) and the onset of full phase-locking ($k = {k_c}$) for various cavity lengths of 600, 700, 800, and 900 µm. The calculated results for each specific cavity length are also shown in detail for the inhomogeneous broadening of 20 meV and homogeneous broadening values of 3, 5, and 7 meV, thus demonstrating the behavior of laser elements at the intermediate homogeneous broadening regime [45].

At “no-coupling” ($k = {{0}}$) condition, each QD laser shows a broad spectrum at a low temperature, especially for longer cavity lengths corresponding to the large number of longitudinal modes [38]. By further increasing the coupling strength, longitudinal modes that experience detuning disappear while transferring their energy to the surviving modes via homogeneous broadening of the gain [24], and, at a critical value of coupling strength, only the common longitudinal modes, i.e., the central modes in our structure, survive. This indicates that, at critical coupling coefficient (${k_c}$), the whole coupled QD laser system exhibits single-mode operation. At this point, the coupled system is at the onset of full phase-locking, and laser elements are coherently coupled to each other. Because higher values of $k$ ($k \gt {k_c}$) guarantee the stability of temporal oscillations [46], the time margin has been set to be as high as possible in all simulations to achieve stable time responses at the onset of full phase-locking for each laser element.

Comparing the critical values of coupling strength for constant cavity lengths and different values of homogeneous broadening, one can conclude that, for higher values of homogeneous broadening, the energy transfer from detuned longitudinal modes to common modes while increasing the coupling strength takes place drastically; therefore, the critical coupling strength happens to have a lower value. On the contrary, energy transfer accompanied by increasing the coupling strength is inconsiderable for lower values of homogeneous broadening, which leads to higher values of the critical coupling strength. In other words, the onset of full phase-locking occurs at higher values of coupling strength for less homogeneously broadened spectra. Additionally, for a constant value of homogeneous broadening, longer cavities experience the onset of coherent coupling at higher values of critical coupling strength. As can be inferred from earlier equations, longer cavities in QD lasers correspond to a large number of groups and longitudinal modes. The disappearance of a larger number of detuned modes requires higher values of critical coupling strength for a constant homogeneous broadening value. Optical gain spectra have also been depicted in both cases of no-coupling and onset of full phase-locking for various values of the homogeneous broadening and a cavity length of 900 µm. Full phase-locking causes a decrease in the threshold current of the central mode in which the bias current exhibits a huge contribution in the phase-locking of nondetuned modes and leads to a reduction of the modal gain through decreasing the carrier occupation probabilities. Optical gain distributions for laser element A (B) are shown in the Fig. 4 insets, respectively. According to the results shown in Fig. 3, the less homogeneously broadened spectra experience, the more reduction of the modal gain for nondetuned modes.

The coherent coupling based on longitudinal modes between two QD lasers has also been examined under the unbalanced bias conditions.

The injected current to one laser has been set to be 5 mA, while the current injected into the other neighboring laser element slowly varies. Figure 5 exhibits the graphical scheme of the results for a cavity length of 900 µm and three different homogeneous broadening values. Results indicate that the lowest value of critical coupling strength corresponds to the balanced bias condition at which both lasers are almost equally biased. Regardless of the homogeneous broadening value, further increase or decrease in the current injecting into one laser element leads the critical value for coherent coupling to increase. The results also imply that there exists a proper bias condition for an established distance between two neighboring lasers, which allows the coupled system to perform at the onset of full phase-locking.

## 4. CONCLUSION

Coherent coupling of two laterally coupled QD lasers based on longitudinal modes has been investigated as an alternative approach to subdue the broadband multilongitudinal mode nature of the emission spectrum in QD lasers. Due to a minute difference in cavity lengths, while the amount of frequency detuning for central modes in laser elements is zero, other corresponding modes in each element experience a particular amount of frequency detuning with respect to each other. According to our numerical simulation results, in a coupled QD laser system, for even lower values of homogeneous broadening in the intermediate regime (3 meV) and longer QD laser cavities up to 900 µm with a minute difference in cavity lengths, there exists a critical value of coupling strength in which all detuned longitudinal modes disappear and two laser elements lase at their only one common mode, i.e., central mode. Common mode survival, which can be considered as the proof of full phase-locking or in this case coherent coupling, also results in a single-mode operation in our coupled structure. Applying differential bias condition to the coupled structure leads to an increase in the critical coupling strength, meaning that the single-mode operation will also be attainable given that the current difference is compensated by a further increment of the coupling strength. According to our results, two laterally coupled QD laser structure can efficiently be managed to contribute to a wide range of promising optical integration and signal processing applications.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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