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Abstract
This work theoretically investigates the relative intensity noise (RIN) and spectral linewidth characteristics of epitaxial quantum dot (QD) lasers on silicon subject to optical injection. The results show that the RIN of QD lasers can be reduced by optical injection, hence a reduction of 10 dB is achieved which leads to a RIN as low as −167.5 dB/Hz in the stable injection-locked area. Furthermore, the spectral linewidth of the QD laser can be greatly improved through the optical injection locked scheme. It is reduced from 556.5 kHz to 9 kHz with injection ratio of −60 dB and can be further reduced down to 1.5 Hz with injection ratio of 0 dB. This work provides an effective method for designing low intensity noise and ultra-narrow linewidth QD laser sources for photonics integrated circuits on silicon.
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1. Introduction
The emergence of 5G, cloud computing, artificial intelligence and coherent optical transceiver modules are driving the development of photonics integrated circuits (PICs) on silicon, which have inherent advantages of CMOS compatibility, high productivity, low-cost manufacturing, high integration density and high energy efficiency [1]. However, due to lattice mismatch, the epitaxial growth of III-V materials on silicon produces a large number of threading dislocation defects (TDD), which act as nonradiative recombination centers in the Shockley-Read-Hall (SRH) process, thereby affecting the performance of lasers [2]. Owing to the high degree of carrier confinement and the atom-like density of states, compared with traditional quantum well (QW) lasers, quantum dot (QD) lasers are more insensitive to the presence of defects [3,4]. Therefore, epitaxial QD lasers on silicon have been regarded as the most potential on-chip laser sources for PICs on silicon [5,6]. The silicon-based QD lasers have recently achieved record performance with long lasing device lifetime [7], near-zero linewidth enhancement factor ($\alpha _{H}$) [8], high thermal stability [9] and high feedback resistance [10].
Low optical noise light sources are urgently needed not only for PICs but also for optical transmitters of low-cost data transmission in inter/hyper-data centers and metropolitan data links [11]. The applications such as optical atomic clocks and frequency synthesis also require the low-noise oscillators [12,13]. The optical noise in semiconductor lasers includes relative intensity noise (RIN) and frequency noise (FN), the latter one is usually transformed into spectral linewidth. The RIN reduces the signal-to-noise ratio (SNR) and increases the bit error rate (BER) hence degrading the performance of high-speed optical communication systems and Lidar related applications [14–16]. For QD lasers on native substrate, it has been theoretically and experimentally confirmed that the RIN can be as low as −160 dB/Hz [14,15,17]. However, the presence of TDD can affect the performance of epitaxial QD lasers on silicon or germanium such as limited device lifetime [4], high threshold current [18] and high optical noise [19]. The RIN in the range of −120 dB/Hz to −150 dB/Hz has been reported in QD lasers on silicon or germanium [20–22]. Apart from the RIN, the narrow linewidth is also important for high order coherent communications [23,24], optical sensing and metrology [25] as well as free-space optical communication [26]. Owing to the low inversion factor and low $\alpha _{H}$ factor, spectral linewidth below 100 kHz has been achieved in QD lasers [27]. In order to reach a narrower linewidth, external control schemes such as optical feedback or optical injection are considered as effective methods. By using an external cavity configuration, the spectral linewidth of the QD laser can be reduced to less than 100 kHz [28–30]. However, the precise control of feedback strength is required to make the laser operating in the narrow linewidth area, otherwise the optical feedback will lead to laser instability [31]. It has been shown that the optical injection can considerably improve the dynamical performances of QD lasers including the increase of the modulation bandwidth [32], the decrease of the frequency chirp [33] and the improvement of the frequency comb bandwidth [34]. Optical injection is also an effective method to generate photonic microwave in QD lasers [35,36]. Furthermore, optical injection-locked light sources have been widely used in optical communications and signal processing [37], as well as monolithic PICs [38].
This work theoretically investigates the RIN and spectral linewidth characteristics of epitaxial QD lasers on silicon subject to optical injection. We use the single mode laser model [2], which is more efficient than the multimode approach in dealing with the optical noise [30,39]. The results show that the RIN of the QD laser increases due to the presence of nonradiative recombination process which can be suppressed through optical injection. In the stable injection-locked area, the RIN of the QD laser can be reduced from −157.5 dB/Hz to −167.5 dB/Hz by adjusting the injection ratio and frequency detuning. In addition, the spectral linewidth can be greatly reduced by optical injection, thus the laser linewidth can be decreased to 9 kHz with an injection ratio of −60 dB. Particularly, the ultra-narrow linewidth of 1.5 Hz is achieved with the suppression bandwidth of 28.1 GHz when the injection ratio is 0 dB. This work brings new insights for designing on-chip injection-locked QD lasers on silicon for low noise related applications.
2. Rate equation model of injection-locked QD laser
Figure 1(a) illustrates the schematic representation of an optically injection-locked QD laser system where a tunable laser is used as the master laser and the injected light is coupled into the QD laser (slave laser) through an optical circulator. Figure 1(b) shows the structure of ridged QD laser, whose electronic structure and carrier dynamics in QD active region are depicted in Fig. 1(c). The QD laser exciton model consists of a two-dimensional carrier reservoir (RS), excited state (ES) and ground state (GS). This numerical model assumes that there is only one QD ensemble in the active region, and electrons and holes are considered as neutral excitons. The carriers are firstly captured from the RS into the ES with a capture time $\tau ^{RS}_{ES}$ and then relax from the ES down to the GS with a relaxation time $\tau ^{ES}_{GS}$, and the stimulated emission eventually occurs at GS. Due to the thermalization, carriers are re-emitted from the GS to ES with an escape time $\tau ^{GS}_{ES}$, and from the ES to the RS with an escape time $\tau ^{ES}_{RS}$, this process is determined by the Fermi distribution of the quasi-equilibrium state. In addition, carriers also recombine spontaneously in each state with spontaneous emission time $\tau _{RS,ES,GS}^{spon}$, respectively. It is necessary to point out that the effect of SRH recombination lifetime $\tau _{SRH}$ is considered. Thus, the corpuscular rate equations describing the dynamics of the carrier numbers $N_{RS,ES,GS}$, the photon numbers $S_{GS}$, and the phase of the electric field $\phi$ are expressed as:
Through a small signal analysis, we linearize the rate Eqs. (1)–(5), and yield:
Table 1. Material and optical parameters of the QD laser.
3. Result and discussion
3.1 Nonradiative recombination effect
The epitaxial defect density of QD laser on native substrate is typically in the range of $10^3$ - $10^4$ cm$^{-2}$ which corresponds to $\tau _{SRH}$ on the order larger than 10 ns. For silicon-based QD lasers where the defect density is at least two orders of magnitude higher than that in QD lasers on native substrate hence the $\tau _{SRH}$ is less than 10 ns [2]. The explored range of $\tau _{SRH}$ in QD lasers on silicon is from $0.5$ ns to $10$ ns, which correspond to the defect density from $1.3 \times 10^6$ cm$^{-2}$ to $1.3 \times 10^8$ cm$^{-2}$. Figure 2(a) depicts the photon number as a function of the bias current for different $\tau _{SRH}$ ranging from 0.5 ns to 10 ns. The threshold current increases from 52 mA at $\tau _{SRH}=10$ ns to 125 mA at $\tau _{SRH}=0.5$ ns due to the high defect density [18]. As the threshold current changes with the variation of the $\tau _{SRH}$, for fair comparison, the photon number of solitary laser is fixed (3$\times$10$^5$) for different $\tau _{SRH}$ in the optical noise simulations rather than the bias currents due to the fact that the optical noise is closely related to the photon number. Figure 2(b) depicts the RIN spectra of QD laser for different $\tau _{SRH}$. The low frequency RIN level below the relaxation oscillation frequency (ROF) is increased with a short $\tau _{SRH}$ due to the shorten carrier lifetime. The RIN at 10 MHz increases from −161 dB/Hz at $\tau _{SRH}=10$ ns to −157.5 dB/Hz at $\tau _{SRH}=0.5$ ns while the RIN above the ROF remains unchanged. Figure 2(c) shows the FN spectra for different $\tau _{SRH}$ which exhibit a high resonance peak in the frequency range of 1$\sim$10 GHz due to the QD laser is under-damped [40]. The FN maintains stable for different $\tau _{SRH}$ in the whole frequency range except the resonance peak. The spectral linewidth of QD laser is 556.5 kHz at $\tau _{SRH}=0.5$ ns, which is extracted from the FN spectrum at 1 MHz. Figure 2(d) demonstrates the effect of nonradiative recombination on the modulation response through small signal analysis and the corresponding 3 dB modulation bandwidth are shown in the inset of Fig. 2(d). The fast nonradiative recombination shortens the total carrier lifetime, hence significantly enhancing the damping factor and suppressing the resonance peak, which lead to the limited 3 dB modulation bandwidth [19].
Fig. 1. (a) Schematic representation of an optical injection-locked QD laser system. (b) The structure of epitaxial QD laser on silicon. (c) The electronic structure and carrier dynamics in the QD.
Fig. 2. Nonradiative recombination effect on the (a) light-current characteristics, (b) RIN spectra, (c) FN spectra and (d) modulation response. The inset exhibits the nonradiative recombination effect on the 3 dB modulation bandwidth.
3.2 Optical injection effect on the modulation dynamics
The effect of optical injection on the modulation dynamics is now investigated through the small signal analysis. The modulation transfer function of QD laser is expressed as [41] :
Fig. 3. Optical injection ratio dependence of (a) the modulation response and (b) the 3 dB modulation bandwidth with $\tau _{SRH}=10$ ns at zero frequency detuning.
3.3 Optical injection effect on the RIN
The effect of optical injection ratio on the RIN performance of QD laser is investigated. Figure 4(a) compares the RIN spectra in the solitary case and for different injection ratios at zero frequency detuning with $\tau _{SRH}=10$ ns. The RIN level decreases in the whole spectral range with the increase of the injection ratio while the ROF shifts towards a higher frequency along with a reduced peak amplitude. This tendency is consistent with the previous modulation response because the modulation properties and the noise spectra are driven by the same dynamical processes in QD lasers [21]. Particularly, the low-frequency RIN at 10 MHz decreases from −161 dB/Hz for solitary case down to −167 dB/Hz with $R_{inj}=4$. The injection ratio dependence of low-frequency RIN versus $\tau _{SRH}$ from 0.5 ns to 10 ns is shown in the Fig. 4(b). Compared with the solitary case, the RIN in the injection-locked area is reduced by about 6 dB at $\tau _{SRH}=10$ ns, and 10 dB at $\tau _{SRH}=0.5$ ns with $R_{inj}=4$. In addition, it is noted that the RIN is almost constant at −167.5 dB/Hz for different nonradiative recombination lifetime when further increasing the injection ratio, hence the effect of nonradiative recombination on the RIN can be completely offset through the optical injection in silicon-based QD lasers.
Fig. 4. (a) Injection ratio dependence of the RIN spectra with $\tau _{SRH}=10$ ns at zero frequency detuning. (b) Low-frequency RIN at 0.01 GHz versus $\tau _{SRH}$ for different injection ratio at zero frequency detuning.
The effect of frequency detuning on the RIN is analysed through the optical injection mapping. The stable optical injection-locked area of QD laser is determined by analysing the time series of the photon number subject to optical injection [43]. The time series of the photon number in the stable area converges at a certain constant as shown in the Fig. 5(a), while that in the unstable area is always fluctuating (Fig. 5(b)). Figure 5(c) demonstrates the boundaries of the stable injection-locked area for different nonradiative recombination lifetime. The boundary curves of the stable locked area corresponding to different $\tau _{SRH}$ are almost coincident, hence the SRH process has little impact on the stable locked area of QD lasers. The high density defects induced SRH process shortens the carrier lifetime and enhance the damping factor of the slave laser, without greatly affecting the coupling between the master and slave lasers.
Fig. 5. Time series of photon numbers in the (a) stable and (b) unstable area. (c) Stable locked boundaries of QD laser for different nonradiative recombination lifetime.
Figure 6(a) and (b) display the mapping of the RIN with the variation of both injection ratio and frequency detuning at $\tau _{SRH}=10$ ns and $\tau _{SRH}=0.5$ ns, respectively. In the stable injection-locked area, the RIN is reduced down to −167.5 dB/Hz with the increase of the injection ratio and frequency detuning. The boundaries of stable injection-locked area of RIN keep at the same level for different nonradiative recombination lifetime, which is also confirmed by the Fig. 5(c). However, this stable area is asymmetric with respect to zero frequency detuning and shifts to positive frequency detuning which can be attributed to the asymmetry in the gain profile leading to a non-zero $\alpha _{H}$ factor that is fixed at 0.5. Figure 6(c) and (d) depict the mapping of the RIN at $\alpha _{H}=1$ and $\alpha _{H}=2$, indicating that the stable locked area is strongly dependent on the $\alpha _{H}$. The stable injection-locked area moves in the direction of large injection ratio and frequency detuning with the increase of the $\alpha _{H}$. In addition, the gain peak $\alpha _{H}$ is lower at short wavelength than that of long wavelength, hence the negative detuning region is more chaotic than the positive detuning region [44,45].
Fig. 6. The mapping of the RIN with the variation of injection ratio and frequency detuning for (a) $\tau _{SRH}=10$ ns and (b) $\tau _{SRH}=0.5$ ns. The mapping of the RIN with $\tau _{SRH} = 10$ ns for (c) $\alpha _{H}=1$ and (d) $\alpha _{H}=2$.
3.4 Optical injection effect on the spectral linewidth
The effect of optical injection on the FN and spectral linewidth performance of QD laser is now investigated. It is found that a very low injection ratio can greatly reduce the spectral linewidth of the laser hence the injection ratio is converted to dB in this section. Figure 7(a) presents the FN spectra of the injection-locked QD laser for different injection ratios with $\tau _{SRH}=0.5$ ns and at zero frequency detuning. The low-frequency FN below the resonance peak is significantly reduced with optical injection which is due to the strong locking of the low-frequency phase between the slave laser and the master laser. Also, the FN under optical injection exhibits a wider resonance peak than the solitary laser, which is caused by carrier noise disturbance [46]. Figure 7(b) reveals the spectral linewidth extracted from the low-frequency FN at 1 MHz versus the injection ratio for different nonradiative recombination lifetime. For silicon-based QD lasers with $\tau _{SRH}=0.5$ ns, the spectral linewidth is dramatically decreased from 556.5 kHz to 9 kHz for a low injection ratio of −60 dB. When the injection ratio is greater than −10 dB, the entire FN spectrum is completely suppressed and the linewidth is reduced down to 1.5 Hz for injection ratio of 0 dB. In addition, the spectral linewidth decreases at a fixed injection ratio as the nonradiative recombination lifetime increases. It is noted that the spectral linewidth of the injection locked QD laser is controlled only by the FN of the master laser, and no longer follows the intrinsic limit of the laser [43].
Fig. 7. (a) Injection ratio dependence of the FN spectra with $\tau _{SRH}=0.5$ ns. (The dashed line represents the −3 dB of the FN for solitary laser.) (b) Spectral linewidth versus injection ratio for different nonradiative recombination lifetime.
The suppression bandwidth is defined as the bandwidth of FN subject to optical injection which is lower than −3 dB of the FN for solitary laser. Therefore, the FN below the suppression bandwidth is mainly determined by the master laser, while that above the bandwidth is mainly determined by the slave laser [43]. Figure 8(a) and (b) demonstrate the mapping of the FN suppression bandwidth with the variation of both injection ratio and frequency detuning for $\tau _{SRH}=10$ ns and $\tau _{SRH}=0.5$ ns, respectively. The suppression bandwidth increases with the increase of the injection ratio near the zero frequency detuning, while the maximum values are obtained on the negative detuning side. For a strong injection ratio of 0 dB, the maximum suppression bandwidth is 22.7 GHz for $\tau _{SRH}=10$ ns and is further enhanced to 28.1 GHz for $\tau _{SRH}=0.5$ ns, indicating that narrow spectral linewidth can be achieved in silicon-based QD lasers by optical injection. In addition, compared the Fig. 8 with the Fig. 6, in the low injection ratio regime, the stable injection-locked area of the FN is more symmetric than that of the RIN with respect to zero frequency detuning.
Fig. 8. The suppression bandwidth with the variation of injection ratio and frequency detuning for (a) $\tau _{SRH}=10$ ns, (b) $\tau _{SRH}=0.5$ ns.
4. Conclusion
In this paper, we theoretically investigate the effect of optical injection on the optical noise properties of epitaxial QD lasers on silicon considering the presence of SRH recombination. The high defect density in silicon-based QD lasers shortens the SRH lifetime, hence increasing the optical noise and degrading the modulation bandwidth. These results demonstrate that the 3 dB modulation bandwidth, the RIN and the spectral linewidth are greatly improved in epitaxial QD lasers on silicon through the optical injection, and the effect of SRH can be offset by optical injection. The optical injection has been confirmed to reduce the $\alpha _{H}$ and enhance the damping factor [33]. Theorefore, the RIN down to −167.5 dB/Hz is realized with a reduction of 10 dB through optical injection, hence the tolerance to high defect density of $10^8$ cm$^{-2}$ is achieved. The optical injection can also significantly reduce the spectral linewidth from the order of kHz to the order of Hz along with a suppression bandwidth of 28.1 GHz. Overall, this work confirms that optical injection is an effective method for achieving low-noise and narrow linewidth epitaxial QD lasers on silicon, which are meaningful for PICs on silicon.
Funding
National Key Research and Development Program of China (2022YFB2803600); National Natural Science Foundation of China (62105080, 62204072, U22A2093); Basic and Applied Basic Research Foundation of Guangdong Province (2020A1515110328, 2021A1515110076, 2023A1515012304); Shenzhen Science and Technology Innovation Program (GXWD20220811163623002, RCBS20210609103824050).
Acknowledgement
Authors acknowledge Dr. Xingguang Wang and Porf. Cheng Wang from ShanghaiTech University, Shanghai, China for fruitful discussions.
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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