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Abstract
Relative intensity noise (RIN) behavior of a quantum dot laser (QDL) at free running and under optical injection locking (OIL) has been reported in detail. Considering the mutual effects of the inhomogeneous and homogeneous broadenings, different and also non-routine RIN characteristics of a QDL at low, intermediate, and high homogeneous broadening regimes have been discussed. Results demonstrate that an injection-locked quantum dot laser exhibit enhanced characteristic over free-running operation along with low-frequency noise behavior in a wide range of homogeneous broadening values. RIN reduction of about 25, 55, and 7 dB/Hz have been achieved for low, intermediate, and high values of homogeneous broadening which implies the spontaneous emission noise suppression even at low injection power. The promising impact of the OIL on side-mode suppression for the intermediate homogeneous broadening regime has also been discussed. Our results guarantee the single-mode and low-noise operation of the OIL-QDL at any homogeneous broadening value.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
QDLs have acquired a great deal of attention in the laser-physics community in the past two decades. With no doubt, QDLs owe this reputation to their promising features including ultra-low threshold current, high modulation bandwidth, and superior temperature stability compared to quantum well and bulk counterparts [1–7]. However, these favorable features are shadowed by a few major drawbacks. The complicated fabrication process of the QDLs can cause inhomogeneity in the size and shape of dots, thus, leading to the broadening of the gain spectrum known as inhomogeneous broadening [1]. Additionally, QDL performance is greatly affected by temperature-dependent homogeneous broadening. As a result of interaction between the homogeneous and inhomogeneous broadening, QDL operation is distinguished by low, intermediate, and high values of homogeneous broadening in which the laser shows different static and dynamic characteristics at each regime [1,8]. The other worth noticing challenges in QDLs from a practical point of view is the degradation of dynamic characteristics along with the inevitable effect of noise in these lasers. The parasitic-free RIN measurements have been applied to assess the dynamic properties and in particular the modulation characteristics of the QDLs [9–11]. Also, as a consequence of the conformity between the direct modulation and noise dynamics, the RIN spectrum analysis is taken into account as an alternative approach to evaluating the modulation characteristics [12].
RIN level of an optical source increases the bit-error-rate of the optical signal and limits the bit rate of the transmitted data [13]. RIN values of ∼ − 160 dB/Hz have been measured in both GaAs and InP-based QDLs while RIN values of ∼ − 120 dB/Hz and − 150 dB/Hz have been reported for QDLs with Ge and Si substrates respectively where the lattice mismatch between the GaAs and the substrate leads to epitaxial defects and consequently results in RIN increment [11,14–16]. It has been discussed that carrier noise arising from the excited and ground states of QDs enhances the RIN of the QDL while increasing the energy interval between the ground state and the excited states modifies the effect of carrier noise [17]. P-doping of the active region can affect the RIN level of QDLs as well. Stable RIN values of about −140 to −150 dB/Hz with temperature variation between 15 to 35°C has been obtained for a GaAs QDL on Si substrate [18]. It has also shown that optical feedback as a beneficial approach can reduce the RIN level of QDLs [19]. OIL as the process of light injection from a master laser into the cavity of an independent slave laser ends in phase-locking between the master and slave lasers as a result of which the slave laser is obliged to follow the master’s phase fluctuations. Experimental and theoretical evidence approve the efficiency of OIL to improve the modulation response, optical output power, and RIN behavior of the semiconductor lasers [20–27]. In various types of semiconductor lasers, OIL has provided simultaneous improvements in laser parameters which pose limitations to the modulation bandwidth under free-running operation [28–31]. However, the promising effects on modulation bandwidth act as an instigator to examine the impact of OIL on the dynamic characteristics of QDLs in terms of noise behavior.
Intensity noise in a free-running laser is the result of carrier generation and recombination processes. This noise component adds up with the intensity and phase noise introduced by the injected light under OIL condition and the overall noise sources contribute to the total RIN of the slave laser. The contribution of the introduced noise sources as a result of OIL is expected to have a considerable improving effect on the overall noise behavior of the slave laser [32].
The relationship between output power and RIN for both interband and intersubband lasers has been illustrated through experimental and theoretical efforts [21,31,33,34]. While there exist experimental reports on bandwidth modulation enhancement in QD lasers employing OIL [31], there are no theoretical contributions to discuss the impact of OIL on the RIN spectrum of QD lasers in detail.
In this manuscript, we provide a comprehensive study on the output characteristics of a self-assembled QDL under OIL condition in comparison to the free-running operation. Additionally, based on the numerical approach introduced in [27], a detailed discussion on the noise behavior of QDLs at free-running and under OIL has been included. The organization of the manuscript is as follows: a set of multimode rate equations describing a free-running QDL are introduced at first. The rate equation model goes through proper modifications considering the presence of OIL and Langevin noise sources. Finally, the obtained results are presented in section 2.
2. Model description
The multimode rate equation model has been used to provide a detailed dynamic description of carriers and photons in self-assembled InGaAs/GaAs QDL [1]. Separate confinement heterostructure (SCH) state, wetting layer (WL), and a single energy level inside each QD are assumed as energy levels that constitute QD active region. Charge neutrality in each QD is guaranteed by exciton approximation. Figure 1 provides a schematic view of each QD active region along with the carrier relaxation, recombination, and re-excitation processes [1]. The multimode rate equation model is based on the following time constants: τs and τd designate the relaxation time constants to the QD ground-state from the SCH layer and the WL, respectively. Also, carrier escape time constant from the QD state to the WL and from the WL to the SCH are represented by τe and τqe, respectively. Finally, τsr, τqr, and τr are indicators for carrier recombination time constants in the SCH region, the WL, and the QD state, respectively. The rate equation model takes into account the polarization dephasing effect and the size fluctuation of QDs as the inhomogeneous broadening of the quantized energies and the homogeneous broadening of the optical gain for every single dot, respectively [1]. The QD ensemble is considered to be quantized into n=1, 2, …, 2M+1 groups, depending on each individual dot’s interband transition energy where the n = M corresponds to the central group at the resonant energy of Ecv [1]. The resonant energy of the nth group can be expressed by En=Ecv+(n-M)×ΔE, where Ecv indicates the transition energy of Ecv=1 eV for the central group. Likewise, the longitudinal modes of the QD laser cavity have been divided into n=1, 2, …, 2M+1 modes, where the resonant energy of each mode happens to have the same energy of the corresponding quantized group. The longitudinal mode energy separation for the QD laser cavity of length L can be calculated through relation: ΔE = hc/2nrL. Here, the parameters h, c, and nr stand 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 size fluctuations of dots on the gain spectrum has been taken into account using the Gaussian distribution function as follows [1]:
The Gaussian distribution function satisfies the relation $\sum\limits_{n = 1}^{2M + 1} {{G_n}} = 1$ and ${\Gamma _0} = 2\sqrt {2\ln 2} {\xi _0}$ is the full width at half maximum (FWHM) for inhomogeneous broadening. The following expresses the linear optical gain impact factor between the nth group and the mth resonant mode [1]:
In the above equation, ${|{P_{cv}^\sigma } |^2}$ is the transition matrix element and the parameters e, $\hbar $, ND, ${\varepsilon _0}$ and m0 represent the electron charge, the reduced Planck’s constant, the volume density of dots, the vacuum permittivity, and the electron rest mass. Pn is the indicator of the occupation probability of the nth QD group and can be calculated through the relation of Pn = Nn / DgNDVGn. Nn is the number of carriers in the nth QD group. Considering QDs with a height of H, the parameter V implies the total active region volume of the QD laser and can be defined as V = HWLNw, where W is the strip cavity width of the QDL and Nw exhibits the assumed number of dot layers in the active region. The Lorentzian-shaped function referring to the homogeneous broadening of each mode or an implicit indicator of the effect of temperature on the gain spectrum in a QD laser is shown by ${B_{cv}}({E_m} - {E_n}) = {{({{{{\Gamma _{cv}}} / {2\pi }}} )} / {({{{({E_m} - {E_n})}^2} + {{({\Gamma _{cv}}/2\pi )}^2}} )}}$ where Гcv gives the FWHM of the homogeneous broadening for each mode in QD laser. The classical carrier and photon rate equations for a QD laser in the presence of Langevin noise sources can be described as follows [1,27,33,35]:
In the above equations, Ns, Nq, and Nn are the number of carriers in the SCH layer, the wetting layer, and the nth QD group. Sm denotes the number of photons in the mth resonant mode in the QD laser. I indicates the bias current injected into the QD laser. Г is an indicator of the optical confinement factor and β represents the coupling efficiency of spontaneously produced photons to the lasing photons. The average carrier relaxation lifetime, $\overline {{\tau _d}} $ can be defined as ${1 / {\overline {{\tau _d}} }} = \sum\limits_n {\tau _{dn}^{ - 1}} {G_n} = \sum\limits_n {{\tau _d}^{ - 1}} (1 - {P_n}){G_n}$. The parameter τd is designated for the relaxation lifetime of the unoccupied ground state. The photon lifetime in the laser cavity is represented by τp and can be extracted through relation ${\tau _p}^{ - 1} = (c/{n_r}){\alpha _{tot}} = (c/{n_r})[{\alpha _0} + (1/2L)\ln (1/{R_1}{R_2})]$. α0 is the total internal loss per unit length and R1, R2 are the power reflectivity coefficients of terminal facets of the cavity. In addition, the output power of the mth mode is calculated through ${P_{ou{t_m}}} = {E_m}c{S_m}\ln ({1/{R_2}} )/({2L{n_r}} )$. To study the impact of OIL on QDL characteristics, the rate equations should be rewritten along with modified temporal variations of photons and relative phase between slave and master lasers by substituting the Eq. (6) with Eq. (7) and (8) [8]:
In the above equations, the noise diffusion coefficient due to phase difference between the slave and master lasers is defined as ${D_{\Delta {\Phi _m}\Delta {\Phi _m}}} = {D_{{S_m}{S_m}}}/4S_m^2$, $\Delta {\Phi _m}(t)$ and $\Delta {\omega _m}$ represent the time-dependent phase difference and the frequency difference between the mth mode of the slave laser (injection-locked mode) and the master laser which is equal to zero for uninjected modes, respectively, ${\alpha _H}$ shows the linewidth enhancement factor, ${S_{in{j_m}}}$ is the number of injected photons into the injection-locked mode, the optical power injection ratio from master to slave is denoted by kinj and 1/τr is the injection rate [8]. Small signal fluctuations due to Langevin noise sources have been taken into account as ${F_i}(t) = {x_i}(t)\sqrt {{{2{D_{ii}}(t)} / {dt}}} $ where dt indicates the step of integration, x is the zero-mean random Gaussian variable, and each Dii determines the diffusion coefficient that can be calculated as follows [27,33]:
To calculate the RIN values of the QDL, coupled Rate equations of the QDL in free-running (Eq. (3)–6) and OIL (Eq. (3)–5 and 7–8) conditions have been solved using the finite difference (FD) method simultaneously in the presence of the Langevin noise sources (Eq. (9)). An integration step value (dt) of 0.5ps has been chosen to guarantee both the convergence of the temporal equations and simulation time management and also, a final time of 100ns has been considered to validate the low-frequency calculations of RIN. Finally, the RIN spectrum of the mth mode is obtained through [27]:
Here ${s_m}(t )= {S_{_m}}(t )- {\overline S _m}$, ${\overline S _m}$ is the average amount of Sm(t). Although the noise power of the master laser is the dominant RIN source in an OIL master-slave system [27], a noiseless master laser has been assumed in this study for simplicity of calculations. Under such condition no Langevin noise sources are associated to the injected current and master laser [20]. According to our recent study, as the αH of the QDL has the near zero value under the OIL, we have considered this parameter equal to zero [8]. Having the random nature of the noise in mind, the current numerical investigation on the noise behavior of QD laser includes an average report of 20 simulations. OIL results in a time-dependent phase relation between master and slave lasers. The consequent relative phase between master and slave introduces a new noise source which shows a significant impact on the noise behavior of the slave laser [23]. As it is described in detail in [8], the modulation response of the QD laser in both cases of free-running and OIL has been calculated numerically. Detailed information about the laser parameters is included in table 1 [1].
Table 1. Parameters and related quantities used in the simulations [1].
3. Results and discussion
The dependence of the lasing spectrum on the homogeneous broadening of optical gain supports the fact that a QDL with low homogeneously broadened modes shows a broad spectrum with low output power for its central mode [1]. Figure 2 exhibits the output power of the central mode in terms of bias current variations in a QDL for different homogeneous broadening values under free-running and OIL conditions at various injecting powers and for Δω=0. It can be inferred from Fig. 2(a) that low values of homogeneous broadening result in low output power under a free-running condition. As it is clear, further increasing the bias current is not capable of making this drawback up. However, rather low values of injection power meet this problem efficiently. Obtained results show that as the injection power is increased, the laser performs with less concern to the homogeneous broadening value. As it is demonstrated in Fig. 2(d), the output power characteristics for different homogeneous broadening values coalesce for higher values of injection power.
Fig. 2. Central mode Output power in terms of bias current variations for different homogeneous broadening values under conditions of free-running and injecting locking with Δω=0.
Figures 3–5 contain the effect of OIL on the temporal evolution of carrier and photon numbers of the central mode (a) and (c), gain spectra (b), and output power spectra (d), for a bias current of 1.5Ith at different homogeneous broadening regimes of low (1.5 meV), intermediate (4 meV) and high (10 meV), respectively. It is well known that higher values of homogeneous broadenings in a QDL under free-running operation cause the output power for the central mode to increase. On the other hand, the same process leads to a reduction in relaxation oscillation frequency and the modulation bandwidth [8,35]. As illustrated in Fig. 3–5, further increasing the homogeneous broadening value under free-running condition accompanies an increment in the turn-on delay which is considered as the time after which the laser output reaches the 10% of its steady-state value in case that the bias current injection time is considered as the reference time [36]. According to the results, the number of photons along with the output power for the central mode increase under OIL condition. OIL gives rise to the relaxation oscillation frequency and as a consequence, the laser happens to have a smaller turn-on delay with respect to free-running operation. In other words, OIL improves the modulation response of the QDL even for higher values of homogeneous broadening [8]. Gain spectra for various homogeneous broadening values under OIL condition indicate a threshold gain reduction for the central mode which means the amplification and higher output power for this mode. Based on this spectrum analysis, it can be inferred that OIL of specified modes can play a potential role in optical spectrum manipulation processes and determines the dominant resonant mode of the cavity. Additionally, OIL causes the threshold current for the central mode to decrease; therefore, the capture rate of the wetting carriers which contribute to the lasing process is increased. It is worth mentioning that all modes other than the central mode show no contribution in the lasing process even for a rather low injection power (Pinj=0.1 mW) and intermediate values of homogeneous broadening (intermediate regime).
Fig. 3. Dynamic behavior for the central mode of the QD laser with Γcv=1.5 meV under both conditions of free-running and OIL with Δω=0.
Fig. 4. Dynamic behavior for the central mode of the QD laser with Γcv=4 meV under both conditions of free-running and OIL with Δω=0.
Fig. 5. Dynamic behavior for the central mode of the QD laser with Γcv=10 meV under both conditions of free-running and OIL with Δω=0.
The rest of the manuscript is dedicated to the analysis of the QD laser RIN behavior under both the free-running and OIL conditions. Results have been extracted through numerically solved rate equations considering the effect of Langevin noise sources. Figures 6 and 7 demonstrate the calculated RIN of the central mode and photon numbers spectra of the free-running QDL under various bias conditions and for homogeneous broadening values of 0.5, 1.5, 4, 5.5, 7, and 10 meV. To study the RIN characteristics of the QDL, we have divided the operation of the QDL into the three regions of the low, intermediate, and high homogeneous broadening values. Homogeneous broadening exhibits the correlation between the QD groups where for low values of the homogeneous broadening, as shown in Fig. 7(a), the correlation between the adjacent modes is negligible and each group starts to lase almost independent above the threshold which leads to a broad emission spectrum. So, the RIN spectra is similar to conventional lasers and RIN reduction is obvious with an increase in the output power. A similar trend is predictable for high values of homogeneous broadening (Γcv > 7 meV), where the carrier transfer between QD groups results in single-mode operation of the QDL. As depicted in Figs 7(c) and (f), the QDL acts as a single-mode laser and the RIN trend can be estimated regarding the bias current and also the output power (photon numbers) of the laser as illustrated in Figs. 6(c) and (f). So, for low and high values of Γcv where there is no correlation between the resonant modes, the RIN values as low as ∼–160 dB/Hz are achievable.
Fig. 6. Central mode RIN spectra of the free-running QDL for various bias currents and homogeneous broadening values.
It is noteworthy to mention that the observed peaks in the RIN spectra correspond to the relaxation oscillation frequency of the laser [21]. However, for intermediate regime the RIN spectra of the QDL exhibits unconventional behavior where several adjacent resonant modes interact with each other and some side peaks appear in the laser spectra as shown in Figs. 7(b) and (e). In this condition, the laser spectrum is not narrow nor broad and changes with the bias current. Figures 6(b) and (e) illustrate the RIN spectra of the central mode for the intermediate homogeneous broadening values at different bias currents. At low bias currents, the device is single-mode (Fig. 7(b) and (e)) and RIN spectrum decreases with the increase in the bias current until the sidemodes reach the lasing threshold where the RIN level of the central mode rises inevitably. The higher the Γcv, the side modes appear at higher bias currents. In this case, two major phenomena affect the RIN characteristics of the laser in the presence of side modes. First, according to Eq. (1)0, RIN increment is predictable with a reduction in the cavity photon numbers of the central mode. However, this RIN enhancement is more than the RIN increase originated from the decrease in photon numbers. Unlike the low and also, high values of homogeneous broadening, the correlation between a number of adjacent resonant modes increases the carrier and photon fluctuations which imposes a high level of noise besides the noise of the central group which strengthens the total noise of the laser as demonstrated in Fig. 6(b) and (e). This behavior is obvious at low frequencies for low side-mode photon numbers (the green curve in Fig. 6(b) and the black one in Fig. 6(e)) and is intensified at higher bias currents (the black curve in Fig. 6(b) and the yellow one in Fig. 6(e)) in a way that turns into the dominant noise power even at higher frequencies. As a result, the peak of the relaxation oscillation frequency is not observable due to its lower amplitude.
In addition, the RIN behavior for the central mode of the QDL at Γcv=1.5 meV has been considered as a low homogeneous broadening value near the intermediate regime in which the photon numbers spectrum has both the broad characteristic observed for low homogeneous broadening values and correlation between adjacent groups for intermediate regime. Therefore, the RIN spectra own the characteristics of both regimes which reduces with bias current and exhibits large low-frequency RIN as depicted in Fig. 6(d). As a result, the relaxation oscillation peak is not observed similar to the behavior reported for the high bias currents at intermediate homogeneous broadening regime arisen from the imposed fluctuations due to correlation of the adjacent resonant modes which is weaker compared with the intermediate homogeneous broadening values.
To evaluate the RIN characteristics of the side modes, the RIN spectra of the QDL for major side modes (second modes next to the central mode) of the laser at different bias current have been depicted in Fig. 8. Modes with identical spacing on the both sides of the central mode have the same amplitude and RIN behavior. For Γcv=0.5 meV, the same behavior similar to the central mode is predictable since all modes lase almost independent from each other which is confirmed in Fig. 8(a). Compared with the central mode, the higher RIN values are due to lower photon numbers of the side modes. A similar trend is observed for side modes of the laser at Γcv=1.5 meV where the RIN of the second modes next to the central mode has been illustrated in Fig. 8(b). Although the photon number of the cavity is almost twice in comparison to Γcv=0.5meV, the higher RIN values are due to the correlation between the adjacent resonant modes of the laser. The RIN characteristics of the major side modes for the intermediate regime (Γcv=4meV) have been depicted in Figs. 8(c) for the second and (d) for the third modes next to the central mode. The abnormal behavior of the RIN spectra for the second modes is associated to the photon number spectra where the adjacent mode appears with increasing the bias current and leads to a reduction in photon numbers of the second mode. So, the RIN level at I=2Ith is the lowest obtained value while the RIN level shows an increasing trend for larger bias currents. On the other hand, the RIN trend happens to decrease with an increase in bias current for the third modes next to the central mode, as illustrated in Fig. 8(d). Apparently, the relaxation oscillation frequency peaks have appeared for I=2Ith and I=3.5Ith in Figs. 8(c) and (d) due to reduced RIN level of the resonant modes.
Fig. 8. The RIN spectra of the major resonant side modes of the QDL for different homogeneous broadening values and bias currents.
Based on our previous discussions, OIL improves the dynamic characteristics of the QD laser and causes the output power for the central mode to increase; therefore, determines the dominant mode of the cavity as obtained from Figs. 2–5. It is also expected that OIL could efficiently improve the RIN behavior of the QDL even for intermediate homogeneous broadening values. Figure 9 exhibits the RIN behavior for various values of homogeneous broadening and injection powers at 1.5Ith for Δω=0. As it has been reported in previous studies, the spontaneous emission noise is the dominant noise component in the interband lasers. Since this dominant noise component saturates for even relatively low values of Pinj, the RIN level for central mode drastically decreases under OIL condition [20,27]. Detailed examinations of results shown in Fig. 9 indicate that there exists a RIN reduction of about 20 dB/Hz for Pinj=0.1 mW with respect to the free-running operation for a homogeneous broadening of 1.5 meV. This relative reduction reaches about 35 dB/Hz for an injecting power of 5 mW. Additionally, the effect of OIL on noise reduction is more prominent for QDLs with intermediate homogeneous broadenings. As for Pinj=0.1 mW and homogeneous broadening of 4 meV, RIN level is ∼−155 dB/Hz and exhibits about 55 dB/Hz reduction with respect to the free-running condition. OIL also leads the RIN to decline for higher homogeneous broadening values, however, it is not as considerable as for lower values of homogeneous broadening. In comparison to the free-running operation, the RIN level declines about 7 and 5 dB/Hz for homogeneous broadening values of 7 and 10 meV for Pinj=0.1 mW, respectively. It worth noting that OIL of a mode suppresses the adjacent resonant modes leads to RIN reduction and advent of the relaxation-oscillation-frequency-dependent peak of the RIN spectra especially for intermediate homogeneous broadening values.
Fig. 9. RIN spectra of the optical injection locked QD laser for various injecting powers and homogeneous broadening values for Δω=0.
4. Conclusion
Output characteristics of an injection-locked QDL in the presence of Langevin noise sources were discussed through a numerical approach. It was shown that the QDL demonstrates enhanced characteristics under OIL condition over the free-running operation. As was expected, the noise behavior of the injection-locked QDL revealed another evidence to support the superiority of OIL on the free-running operation. It was shown that sidemode domination for intermediate homogeneous broadening values at higher bias currents leads to anomaly in the RIN behavior of the resonant modes in the free-running QDL. OIL is a promising solution which guarantees only the central mode contribution to the output spectrum in a wide range of homogeneous broadenings including the intermediate values. Since OIL leads to similar dynamics and statics characteristics as well as RIN values of the QDL at different homogenous broadening regimes, the device demonstrate a predictable behavior compared with a free-running operation. Regarding the considerable RIN reduction of about 55 dB/Hz and enhanced output features, OIL-QDLs are admirable candidates for high-speed and ultra-low threshold optical communications and low-noise sensing applications.
Disclosures
Disclosures: The authors declare no conflicts of interest.
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