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Abstract
In this paper, we present an investigation of the optical injection-locked parameters which utilize an approach to creating optical complex signals using optically injection-locked (OIL) semiconductor lasers. Subsequently, we discuss the impact of the OIL parameters on the locking map, manufacturing the corresponding complex signal area, pole-zero diagram, and frequency response of the system. The steady-state analysis of the rate equations indicates that changing OIL parameters can result in simultaneous optical AM and PM of the OIL laser while preserving the injection locking state. Based on the simulation results, we found that using strong Rinj and high negative Δfinj (which are mandatory for generating high-quality complex modulation) along with α = 3; can improve the performance in producing complex signal area as well as the frequency response, while using other settings does not lead to the development of both. Therefore, we achieve the maximum generated complex signal area and the -3dB frequency by using an adjusted parameter value for the system, which results in a faster modulation speed and higher quality of the optical complex modulation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Injection locking consists of adequately injecting the light from a master laser (ML) into the slave laser (SL) either via an isolator (transmission type) or a circulator (reflection type). Locking takes place when the ML wavelength is near enough to the free-running lasing SL wavelength. Then the SL is forced to synchronize with the ML, and it will shine at the same frequency with a fixed phase offset. In addition, SL will also maintain a relatively constant output power even if the ML has a slow frequency drift. There are a variety of applications in which laser synchronization is used, such as local oscillators in optical communications [1–4]. In semiconductor lasers, OIL has gained considerable interest owing to its applications in coherent optical communications, frequency chirp suppression, laser spectral narrowing, elimination of noise, and improving frequency response [5–7]. OIL lasers offer many advancements, allowing direct-modulated lasers to be used as transmitters, and can improve their characteristics. Further, to design high-linearity photonics sources, microwave frequency generation, and optical signal processing, many of these developments can be applied to enhance system efficiency [8,9]. Due to performance issues, direct-modulated lasers without injection locking are unsuitable for many applications. As a result of the high-linewidth enhancement factor (α), digital communications suffer from chirp. Because of low linearity in analog communications, signal distortion occurs. Both applications require a high-frequency response that direct-modulated lasers may not be able to deliver [10].
Considering the need to generate high data rates and utilize different optical signal processing technologies, we determine to combine amplitude modulation (AM) with phase modulation (PM) simultaneously to generate complex modulations of optical signals. Photonic applications such as ultra-fast optical communications and photonic integrated circuits can both be addressed using this method. Among the available modulation techniques, AM is commonly preferred for its low complexity compared to PM and frequency modulation (FM). In AM, lasers are directly modulated or externally modulated via electro-optic or electro-absorption modulator technology. Meanwhile, PM has become an essential modulation technique since the emergence of coherent transmission/detection technologies. To generate optical complex signals, external modulators, like electro-optic modulators or piezoelectric transducers, are necessary [11–17]. PM techniques using external modulators suffer from some disadvantages that prevent their widespread use in ultra-fast photonic technologies, and it can be a challenge to combine them with other photonic components. PM technologies based on piezoelectric or acousto-optic transducers, for instance, offer low modulation speeds. Rather than suffer from this limitation, a PM process that uses a LiNbO3-based modulator is already commercially available [13,18,19]. On the other hand, direct modulation of OIL lasers can be implemented for the complex signal generations. Direct modulation in OIL lasers has been predominantly popular thanks to its exquisite performance, including high coherence, low chirp, high-speed modulation and availability of PM. Quadrature amplitude modulation (QAM), which is one of the complex modulation formats, has high-speed operation, however, there are some challenges to use QAM including the appearance of parasitic modes, significant insertion loss and power consumption. Accordingly, to produce high-capacity complex signals for a variety of photonic devices, we need higher bandwidth and improved frequency response using direct modulation in OIL lasers [5,13,20]. Moreover, recent studies show a specific relationship between OIL parameters and complex signal performance. This procedure can conquer the so-called limitations of a conventional direct modulation laser while keeping the ascendancy on OIL laser offers [21].
Even though earlier researchers did not examine the rule of the OIL laser’s parameters on producing the optical complex signal area as well as the frequency response simultaneously, or they have concluded that the smaller the α value, the better performance we have [13]. In contrast, we indicate that not necessarily smaller α leads to improve the performance of the OIL laser, and for obtaining high-quality optical complex modulation, which requires a strong injection ratio (${R_{inj}}$) and a high negative detuning frequency ($\varDelta {f_{inj}}$), it is necessary to moderate the α value. Thereby, we can produce a broader bandwidth and a larger area of complex signals. Based on the simulation results, we suggest the optimum settings for OIL parameters to enhance the modulation speed and quality in optical complex modulation. Thus, in this paper, we initially review the pertinent contents of the existing studies and follow them up with an introduction to the rate equations of the OIL lasers and the extraction of the steady-state solution. Then, we exhibit the correlation between the locking map and two significant variables of the OIL laser; ${R_{inj}}$, and $\varDelta {f_{inj}}$. We clarify the complex equations and obtain the maximum generated complex area. Afterward, we derive the transfer function of the system and investigate the frequency response and pole-zero diagram while sweeping the relevant options, aiming at maximum bandwidth, a process carried out in MATLAB software.
2. Rate equation analysis and locking map
Apart from the typical variables utilized in a direct-modulated free-running laser, two basic parameters of the OIL systems, the ratio of injected power from ML to the output power of free-running SL, or ${R_{inj}}$; and the difference between ML and SL optical lasing frequencies, or $\varDelta {f_{inj}}$; can be defined. We can control the injection locking parameters to realize our desired optical output. Below these two crucial parameters of the OIL laser are described:
Equations (1) and (2) are representing the angular detuning frequency ($\Delta {\omega_{inj}}$), and ${R_{inj}}$, respectively; where ${\omega _{ml}}$ is the angular frequency of the ML, and ${\omega _{sl}}$ is the angular frequency of the SL. ${S_{inj}}$ is the number of photons which is emitted by the ML, and ${S_{fr}}$ is the output photons number from the SL in free-running condition. It can be written as follows:
where ${\tau _n}$ is the carrier lifetime; ${\tau _p}$, the photon lifetime; ${N_{th}}$, the threshold carrier number; and ${I_{bias}}$, the bias current of the SL. The rate equations of the OIL laser are given by:These equations characterize the time-dependence of photon number, S(t); optical phase, $\phi (t)$; and carrier number, N(t); in the SL cavity [13,22]. ${\phi _{inj}}$ is referred as the preliminary phase of the ML. g is describing the gain in net stimulated emissions. ${N_{tr}}$ is the transparency carrier number; κ, rate of coupling between ML and SL; I(t), the injection current; and α, the linewidth enhancement factor in SL. α is defined in Eq. (7), which shows the ratio of variations in the real and imaginary parts of the refractive index, n, into the variation of the carrier density [23].
Figures 1(a)−1(c) illustrate the steady-state condition for photons, carriers, and the corresponding phase in various α using parameter values from Table 1, assuming ${R_{inj}}$=17 dB, and $\varDelta {f_{inj}} = \textrm{ } - 100\textrm{ }GHz$. It can be observed by using smaller α there is more acceleration to reach the steady state. Also, the steady-state relevant phase has been enhanced positively (after deducting 2π). The steady-state carriers and photons have no linear relationship with α; since in α = 3, more photons have been generated from the carriers despite its lower carrier density in comparison with the other α values. This issue refers to the physical nature of the system as in particular α, high negative $\varDelta {f_{inj}}$, and strong ${R_{inj}}$, we have more efficiency in photons generation. By setting the Eqs. (4)−(6) to zero, the steady-state solution of the phase, ${\phi _0}$; photons number, ${S_0}$; and carriers number, ${N_0}$; can form as follows:
Rearranging Eq. (8) provides the locking regime:
Table 1. Parameters for Simulations
Fig. 1. Steady-state condition in OIL laser, (a) Photons density, (b) Carriers density, (c) Photons Phase; in different α values.
Our aim here is to describe the locking conditions that affect the locking range. In the beginning, we find the first constraint in Eq. (8). The arcsine term must not be greater than unity to result in a solution for the phase. Thus:
There is an additional constraint in Eq. (10). The carrier number cannot exceed the threshold; otherwise, the gain will be unstable. Therefore:
The following phase constraint obtains by combining Eqs. (12) and (13):
Putting Eqs. (11) and (14) together, we can derive the detuning frequency boundaries in Eq. (15), which is the same as the locking range [24−26]. As a result, Fig. 2(a) shows the injection-locking map of the OIL laser.
From Fig. 2(a), the upper boundary represents the ${\phi _0} = -\frac{\pi }{2}$, and the bottom shows the ${\phi _0} = \; co{t^{ - 1}}\alpha $. Hence, we can conclude the ${\phi _0}$ value has increased from up to down, and a higher ${\phi _0}$ can be obtained near the bottom boundary in negative $\varDelta {f_{inj}}$. Additionally, $\alpha $ directly affects ${\phi _0}$, and following that, the locking range [5,24].
A combination of the injection-locking map and the intensity of OIL lasers ${S_0}$ is shown in Fig. 2(b). As can be seen, ${S_0}$ is influenced by the ${R_{inj}}$ and $\varDelta {f_{inj}}$ simultaneously. Further, large negative $\varDelta {f_{inj}}$ and strong ${R_{inj}}$ lead to higher ${S_0}$. This allows us to modulate ${S_0}$ and ${\phi _0}$ by adjusting the ${R_{inj}}$ and $\varDelta {f_{inj}}$ in parallel, generating AM and PM, respectively; and subsequently for the complex modulation. Hence, higher speed modulation can be achieved by utilizing a large negative $\varDelta {f_{inj}}$ and a strong ${R_{inj}}$ [13].
3. Optical complex signal
In the previous section, we discussed generating optical complex modulation by combining AM and PM, at the same time. The desired complex signal can be built with ${S_0}$ and ${\phi _0}$.These two outputs are affected by OIL significant parameters, ${R_{inj}}$ and $\varDelta {f_{inj}}$, and other parameters of the OIL laser. The relation between the ${S_0}$ and optical complex intensity Ê(t) has been shown in Eq. (16):
Fig. 2. Injection-locking map of the OIL lasers. (a) 2D view, showing ${\phi _0}$. (b) 3D view, showing ${S_0}$ as a function of ${R_{inj}}\; $ and $\varDelta {f_{inj}}$.
In Eq. (17), ${\textrm{E}_\textrm{r}}(\textrm{t} )$ and ${\textrm{E}_\textrm{i}}(\textrm{t} )$ represent the real part and the imaginary part of the optical complex intensity, accordingly. The accessible area in the complex plane arises from Eqs. (18)–(20):
Fig. 3. Achievable area of the optical complex signal in OIL laser, (a) Related equations, (b) Final complex area; a.u.:arbitrary unit.
Figures 4(a) and 4(b) show the generated complex signal area for different ${R_{inj}}$ and $\varDelta {f_{inj}}$. We can observe how the strong ${R_{inj}}$ and high negative $\varDelta {f_{inj}}$ lead to a higher ${S_0}$ and as a result, a higher ${|\hat{\textrm E } |_{\textrm{max}}}$ in Eq. (18). In Fig. 4(c), we have three different values for ${I_{bias}}$ and fix values for the other parameter. By increasing ${I_{bias}}$, the ${S_0}$ has been improved, but the ${S_{fr}}$ increases, and this case reduce the complex signal area which can be ignored owing to the further increase in ${S_0}$. Figure 4(d) exhibits the various values for the α. Given that both ${S_0}$ and $\varDelta {f_{inj}}$ boundaries are affected by the α, both slope in Eq. (20) and the ${|\hat{\textrm E } |_{\textrm{max}}}$ in Eq. (18) change according to the Figure. The reason for the nonlinear variations in ${|\hat{\textrm E } |_{\textrm{max}}}$ here originates from the outcomes in Fig. 1(a), where ${S_{{0_{|\alpha = 3}}}}$> ${S_{{0_{|\alpha = 1}}}}$> ${S_{{0_{|\alpha = 5}}}}$. Therefore, in α = 3, we have a greater complex area while obtaining a smaller phase range. We can combine the results from Figs. 4(b) and 4(d) in Fig. 4(e). Apparently, in α = 3 and $\varDelta {f_{inj}}$= −70 GHz, we have the maximum complex signal area. If we aggregate the findings from Figs. 4(a) and 4(d) in Fig. 4(f), we can demonstrate that while we utilize a strong ${R_{inj}}$ and smaller α, we can relatively achieve greater complex signal areas in high negative $\varDelta {f_{inj}}$. However, ${R_{inj}}$=17 (dB) cannot practically apply. Consequently, by optimum tuning of OIL laser parameters, we can maximize the optical complex signal area which contributes to a higher modulation speed and modulation quality in the mentioned application discussed in [13].
Fig. 4. Dependance of the complex signal area for: (a) Different ${R_{inj}}$, $\varDelta {f_{inj}}$=−100 GHz, ${I_{bias}}$= 5${I_{th}}$, and α = 5. (b) Different $\varDelta {f_{inj}}$, ${R_{inj}}$=10 dB, ${I_{bias}}$= 5${I_{th}}$, and α = 5. (c) Different ${I_{bias}}$, ${R_{inj}}$=10 dB, $\varDelta {f_{inj}}$=−100 GHz, and α = 5. (d) Different α, ${R_{inj}}$=10 dB, $\varDelta {f_{inj}}$=−70 GHz, and ${I_{bias}}$= 5${I_{th}}$. (e) Different α and $\varDelta {f_{inj}}$, ${R_{inj}}$=10 dB and ${I_{bias}}$=5${I_{th}}$. (f) Different α and ${R_{inj}}$, $\varDelta {f_{inj}}$=−100 GHz and ${I_{bias}}$=5${I_{th}}$; a.u.: arbitrary unit.
4. Transfer function and frequency response of OIL laser
To obtain the pole-zero diagram and frequency response of the OIL laser, we apply a small-signal current and extract the deviations of the S, $\phi$, and N. We can derive the differential rate equations from Eqs. (4)−(6). The small-signal analysis can be represented as follows:
with
The transfer function of the system, is defined below:
The final equation for H($j\omega$) is extracted in Eq. (24):
Fig. 5. Pole-zero diagram (left column) and frequency response (right column) of the OIL laser: (a) and (b) Various ${R_{inj}}$, $\varDelta {f_{inj}}$= −50 GHz, ${I_{bias}}$= 5${I_{th}}$, and α = 3. (c) and (d) Various $\varDelta {f_{inj}}$, ${R_{inj}}$= 9 dB, ${I_{bias}}$= 5${I_{th}}$, and α = 3. (e) and (f) Various ${I_{bias}}$, ${R_{inj}}$= 9 dB, $\varDelta {f_{inj}}$= −50 GHz, and α = 3. (g) and (h) Various α, ${R_{inj}}$= 9 dB, $\varDelta {f_{inj}}$= −70 GHz, and ${I_{bias}}$= 5${I_{th}}$.
Figure 5(d) depicts the influence of the negative $\varDelta {f_{inj}}$ on the frequency response at ${R_{inj}}$= 9 dB, ${I_{bias}}$= 5${I_{th}}$, and α = 3. By raising the negative $\varDelta {f_{inj}}$, the modulation bandwidth starts increasing noticeably and has been calculated 5.67 GHz, 7.66 GHz, and 12.87 GHz, respectively. Considering Fig. 5(c), the main reason behind this enhancement comes from lowering the distance between the real pole and zero; as the zero moves toward the real pole by increasing the negative $\varDelta {f_{inj}}$ and closer zero to the real pole can roughly cancel the real pole roll-off which causes bandwidth improvement. Besides, using these settings has decreased the imaginary part of the complex conjugate poles.
In Figs. 5(e) and 5(f), we can see that using larger ${I_{bias}}$, with ${R_{inj}}$= 9 dB, $\varDelta {f_{inj}}$= −50 GHz, and α = 3, enhance the system bandwidth. In addition, these settings create similar effects with the ${R_{inj}}$ increase in the pole-zero diagram.
Figure 5(h) presents the modulation response of the OIL laser in various α, with ${R_{inj}}$= 9 dB, $\varDelta {f_{inj}}$= −70 GHz, and ${I_{bias}}$= 5${I_{th}}$. The maximum -3dB frequency which is equal to 8.77 GHz can be achieved using α = 3. We have 4.14 GHz and 4.35 GHz bandwidth in α = 5 and α = 1, respectively. According to Fig. 5(g), the sharp decrement of modulation response in α = 1 is the consequence of the zero which lies on the right half of the complex plane. Compared to α = 5, the real pole and zero of α = 3 are closer together; as a result, this setting has a broader bandwidth.
Based on Figs. 5(a) to 5(h), while we swept the $\varDelta {f_{inj}}$ and α, changes have been most noticeable in the pole-zero diagram as well as the modulation frequency; subsequently, we investigate the side effects of these parameters precisely in Fig. 6 to achieve the widest bandwidth. In Figs. 6(a) and 6(b), since more than 10 times injection is not practical, the highest permissible ${R_{inj}}$ is assumed 10 dB. In the following, ${I_{bias}}$= 5.5${I_{th}}$ has been considered for the simulations. Figure 6(a) describes the frequency response while the value −50 GHz is assigned to the $\varDelta {f_{inj}}$. In α = 1, according to the pole-zero diagram, the real pole and zero are almost identical, so they eliminate the effects of each other in the Bode plot, meaning that the frequency response has reached its optimum condition. Thus, we have obtained an extensive bandwidth, equal to 75.6 GHz and 10 times greater than the bandwidth in α = 3. In Fig. 6(b), by using $\varDelta {f_{inj}}$= −100 GHz and α = 3, 17.1GHz bandwidth has been gained. On the contrary, using α = 1 has reduced -3dB frequency to the 2.16GHz value. This is due to the zero being shifted farther from the right side of the imaginary axis in the complex plane compared to the one in Fig. 5(g); consequently, bandwidth is limited here.
Fig. 6. Frequency response of the OIL laser for various α, ${R_{inj}}$= 10 dB, and ${I_{bias}}$= 5.5${I_{th}}$: (a) $\varDelta {f_{inj}}$= −50 GHz. (b) $\varDelta {f_{inj}}$= −100 GHz.
5. Conclusion
As mentioned before, as long as we intend to utilize OIL lasers for generating complex signals, it is necessary to moderate the parameters to have both maximum complex signal area and broad bandwidth. To sum up, according to the simulation outcomes, acquiring the high-quality optical complex modulation which has been discussed in [13], requires strong ${R_{inj}}$ and high negative $\varDelta {f_{inj}}$. Hence, using ${R_{inj}}$=10 dB, $\varDelta {f_{inj}}$=−100 GHz, and α = 3 can produce a desirable complex signal area while keeping the broad bandwidth. However, ${\phi _0}$ availability for the PM has decreased slightly. In contrast, using ${R_{inj}}$=10 dB, $\varDelta {f_{inj}}$=−50 GHz, and α = 1 can dramatically improve the -3dB frequency, but it generates limited complex signal area as well as unsuitable quality for the complex modulation. Eventually, the optimum tuning should be determined to maximize the efficiency and merit of this application.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1 [28], Dataset 2 [29], Dataset 3 [30], Dataset 4 [31], Dataset 5 [32], Dataset 6 [33], Dataset 7 [34], and Dataset 8 [35].
Supplemental document
See Supplement 1 for supporting content.
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