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Abstract
In this paper, we introduce the concept of the cascaded optically injection-locked (OIL) semiconductor laser and present its novel rate equations. Then, the new locking range for this configuration has been obtained by mathematical demonstration. Subsequently, we modified a new adjustment for the detuning frequencies ($\varDelta {f_{inj}}$) of the cascaded OIL system as well as the linewidth enhancement factors (α) values. Utilizing these tunings, improvements in the steady-state photon number and the phase modulation (PM) range become possible. Afterward, we define the generation of the complex optical signal area and extract the transfer function for investigating the frequency response of the cascaded system. The simulations have been performed once with identical α values and once with the various α values in the slave laser (SL) stages. We conclude that these novel proposed adjustments, combined with a strong injection ratio (Rinj) of 15 dB and a high bias current, can significantly broaden the bandwidth near 700 GHz while maintaining the fair gain available up to 180 GHz. Further, the generation of complex optical signal areas has been boosted for high-quality complex modulation applications. Eventually, we exhibit a novel approach for generating different α values in the SL stages by applying managed temperature variations in the experimental setup of the cascaded system, regardless of employing similar SLs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Injection locking is the procedure of injecting the light appropriately from a master laser (ML) into a SL, either via an optic isolation device or a circulator. When the ML wavelength is sufficiently close to the free-running lasing SL wavelength, locking occurs. In this manner, the SL synchronizes with the ML and lases at the same frequency with a fixed phase offset. The SL will also keep a nearly fixed output power even when the ML has a slow frequency drift. Laser synchronization is utilized in many applications, including local oscillators in optical communication [1–5]. OIL semiconductor lasers have received significant attention due to their usage in frequency chirp suppression, coherent optical communications, laser spectral narrowing, frequency response enhancement, and noise elimination [6,7]. OIL lasers allow direct-modulated lasers to be used as transmitters, and can develop their effectiveness for designing high-linear photonics sources, optical signal processing, and microwave frequency generation [8–10]. Besides, direct modulation of OIL lasers can be utilized for producing complex optical signals which are advantageous due to their substantial role including unrivaled coherence, high-speed modulation, small chirp, and availability of PM. In addition to achieving improved bandwidth and frequency response, we can demonstrate how direct modulation in OIL lasers can be a key technology for generating high-capacity complex optical signals for diverse applications in photonic devices. Further, the latest studies indicate a particular correlation between complex signal performance and OIL parameters [11,12]. As a result of this procedure, the existing limitations of the conventional direct modulation laser can be eliminated, while the advantages offered by the OIL lasers are preserved [13].
Modern and next-generation photonics applications need high-speed optical signal processing techniques as the requirements for data capacity increase dramatically [14]. In order to transmit analog and digital signals over optical fibers, very high-speed, low-cost optical transmitters are in high demand; as a result, Lasers and modulators with wide bandwidths have been developed extensively over the past years [15]. For producing a narrow optical beam using optical phase array (OPA) technology, signal processing techniques, and high-speed communication systems, amplitude modulation (AM) and PM are utilized. We can enable high data-rate generation by combining AM with PM simultaneously to produce complex modulations of optical signals. This method is beneficial for ultra-fast optical communications and photonic integrated circuits [11,16]. The technology of AM is well known, and an electro-optic or electro-absorption modulator is utilized to directly or externally modulate laser amplitudes using the AM technique [17]. Aside from that, PM is becoming an encouraging modulation technique since coherent transmission/detection technologies have emerged. External modulators employ PM techniques, which have significant drawbacks that limit their applicability in ultra-fast photonic technology. Also, there are some challenges to the combination of those with the other photonic components. Low modulation speeds are offered by this technology based on piezoelectric or acousto-optic transducers. There is a commercially available PM technology that makes use of a LiNbO3-based modulator to overcome this restriction. For the creation of a complex optical signal, an in-phase and quadrature (IQ) modulator consisting of amplitude and phase modulators is utilized. While this method is high-speed, it still has some disadvantages, which inhibit its widespread use. Hence, as mentioned above, direct modulation of the OIL lasers can provide the PM techniques as well as a complex optical signal generation without these shortcomings [11,18–22].
It has already been reported that cascading OIL configuration can increase the bandwidth and improve the phase range and resonance frequency. In [15], experimental results of direct-modulated vertical-cavity surface-emitting lasers (VCSEL) under OIL with 66 GHz bandwidth have been achieved using cascading of two VCSELs as the SLs, without consideration of theoretical details. Likewise, to conquer some shortcomings of VCSELs, an injection-locking scheme employing a distributed Bragg reflector (DBR) ML monolithically is suggested, which is integrating a unidirectional whistle-geometry micro-ring laser (WRL). Numerical modeling proved the enhancement in fast-speed efficiency of the implemented cascaded injection-locking strategy over a strongly injection-locked WRL [23]. The DC bias control technique is presented in [13] for controlling injection-locking in SLs of the cascaded OIL laser. By utilizing this approach, the cascading system can enhance the phase range and the PM.
Although only these mentioned papers have considered the cascaded configuration, neither the theoretical evaluation of the simple cascaded OIL semiconductor lasers nor its dynamic behavior using numerical modeling has been investigated. Thus, in our current research, not only do we develop the novel rate equations of the cascaded setup and its corresponding steady-state solution, but also we introduce the new locking range of this system along with the illustration of its locking map. Then, we propose a novel setting for the detuning frequencies of the first-stage and second-stage SLs, and we modify the complex equations for the cascaded OIL laser. Subsequently, we perform the small-signal analysis for this new system. The transfer function, pole-zero diagram, and the relevant frequency response have been analyzed. As a next step, we suggest a novel approach of employing different α values in the SL stages. From the simulation outcomes, we depict utilizing this method can result in an ultra-high bandwidth near 700 GHz along with the fair gain available up to 180 GHz, as well as producing more efficient complex optical signals. The simulations have been performed comparatively for both single and cascaded OIL lasers, with identical and non-identical α values in each salve stage, a procedure carried out in MATLAB software. In the last section, we recommend temperature changes using a heatsink in the experimental setup, an innovative process that can practically overcome the challenges of applying different α values in the SL stages. Following that, a technique for calculating the value of the α parameter before and after performing temperature tuning has been briefly reviewed.
2. Developing the rate equations and locking map of the cascaded OIL laser
The schematic of the cascaded configuration of the OIL laser is displayed in Fig. 1(a). The first SL (first stage) is injected and locked by an ML, and the second SL (second stage) is injection-locked by the output of the first stage, meaning the previous SL. Since we intend to gain concurrent control of the two stages, we apply direct modulation to two cascaded SLs simultaneously. Apart from the usual parameters employed in a direct-modulated free-running laser, two significant parameters of the OIL systems, the injection ratio (${R_{inj}}$) and detuning frequency ($\varDelta {f_{inj}}$) are defined. To realize desired optical output, we should manage these parameters. They are described as [11,15,24]:
Fig. 1. Cascaded OIL laser, (a) Schematic, (b) Phasor diagram, locking range, and the output optical intensity of the first-stage and second-stage in the cascaded OIL laser.
Equation (1) depicts ${R_{inj}}$; the ratio of the power injected by the ML, ${S_{inj}}$, to the output power by the free-running first-stage SL, ${S_{fr,s{l_1}}}$. The output power injected by the first-stage SL into the second-stage SL is equal to ${S_{s{l_1}}}(t )$, the time-dependent photon number of the first-stage SL cavity, which will be defined next in the rate equations of the cascaded OIL laser. Therefore, the second-stage injection ratio is the ratio of ${S_{s{l_1}}}(t )$ to the free-running second-stage SL, ${S_{fr,s{l_2}}}$. Equations (2) and (3) show the angular detuning frequency of the first-stage SL, $\varDelta {\omega _{in{j_1}}}$, and the angular detuning frequency of the second-stage SL, $\varDelta {\omega _{in{j_2}}}$, which are equal to the difference between the ML optical lasing frequency, ${\omega _{ml}}$, and the first-stage SL optical lasing frequency, ${\omega _{s{l_1}}}$, and the difference between the ${\omega _{ml}}$ and the second-stage SL optical lasing frequency, ${\omega _{s{l_2}}}$, respectively. ${S_{fr,s{l_1}}}$ and ${S_{fr,s{l_2}}}$ can be written as below [11]:
Equations (6)−(11) identify the time variations of photon number, corresponding phase, and carrier number in the first-stage and second-stage SL cavities. ${S_{s{l_1}}}(t )$, ${S_{s{l_2}}}(t )$, ${\phi _{s{l_1}}}(t )$, ${\phi _{s{l_2}}}(t )$, ${N_{s{l_1}}}(t )$, and ${N_{s{l_2}}}(t )$ are the numbers of photons, related optical phase, and the number of carriers in the first-stage and second-stage SLs, accordingly. ${\phi _{ml}}$ is the preliminary phase of the ML, ${g_{s{l_1}}}$ and ${g_{s{l_2}}}$ are the gain of the net stimulated emissions, ${N_{tr,s{l_1}}}$ and ${N_{tr,s{l_2}}}$ are the transparency carrier number, ${I_{s{l_1}}}$(t) and ${I_{s{l_2}}}$(t) are the injection currents, and ${\alpha _{s{l_1}}}$ and ${\alpha _{s{l_2}}}$ are the linewidth enhancement factors of the first-stage and second-stage SL, respectively. ${\kappa _1}$ describes the coupling rate between ML and first-stage SL, and ${\kappa _2}$ represents the coupling rate between the first-stage and second-stage SLs [23,25].
When Eqs. (6)–(11) are set to zero, the steady-state solution of the optical phase in the first-stage SL cavity, ${\phi _{0,s{l_1}}}$, can be acquired as shown below [11,22]:
Then, in Eq. (13), the steady-state solution of the optical phase in the second-stage SL cavity, ${\phi _{0,s{l_2}}}$, is modified as follows:
Figure 1(b) illustrates the schematic of the locking range and the output optical intensity of the first-stage and second-stage SLs in the cascaded OIL laser. As can be observed, using the cascading system can increase the phase range and optical amplitude concurrently, which leads to further efficiency in utilizing AM and PM to generate complex signals for complex modulation applications.
The steady-state condition of the photons number, carriers number, and optical phase of the single OIL laser compared to the cascaded one are described in Figs. 2(a)−(c) using parameter values from Table 1. According to the figures, the steady-state photons number in a cascaded system can increase to 4.52${\times} {10^5}$ with ${R_{inj}}$=10 dB, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz. Besides, the corresponding steady-state phase has gained −1.23 rad for the simple OIL laser and −1.985 rad (after deducting 2π) for the cascaded OIL laser, which shows the negative enhancement in the value of the steady-state optical phase. We must add this point employing strong ${R_{inj}}$, enhanced bias current, and optimum values of α parameter can produce larger steady-state photons number, for instance, equal to 12.5${\times} {10^5}$ in the cascaded setup. But here, we use values of Table 1 owing to illustrate the variations in the optical complex signal generation more concretely in the next sections [26].
Table 1. Parameters Value for the Simulations
Fig. 2. Steady-state condition for the single OIL laser in comparison with the cascaded OIL laser, (a) Photons number, (b) Carriers number, (c) Photons Phase, (d) Injection-locking maps of the single and cascaded OIL lasers.
Figure 2(d) shows the injection-locking map and detuning frequency boundaries of the single and cascaded OIL lasers. The solid blue and red dashed plots imply the locking range of the single OIL laser and the cascaded one, respectively. We can see the values of both ${\phi _{0,s{l_1}}}$ and ${\phi _{0,s{l_2}}}$ have enhanced positively from up to down, and the locking maps are directly affected by ${\alpha _{s{l_1}}}$ and ${\alpha _{s{l_2}}}$. We should consider this point in Fig. 2(d) that the detuning frequency axis of the solid blue plot is $\varDelta {f_{in{j_1}}}$, and the detuning frequency axis of the red dashed plot is $\varDelta {f_{in{j_2}}}$. Further, $\varDelta {f_{in{j_1}}}$ in the locking map simulation of the cascaded OIL laser is assigned a fixed value of −150 GHz. In this case, using other parameter values for $\varDelta {f_{in{j_1}}}$ changes the graph slope and the beginning point of the boundaries in the detuning frequency axis of the red-dash plot; slightly.
3. Optical complex signal generation
In the previous section, we explained the improvement in the locking range as well as the steady-state photons number by cascading the second SL in the OIL system. Additionally, $R_{i n j}$ and $\Delta f_{i n j}$ simultaneously have notable impacts on the steady-state optical phase and the photons number in both stages. In a single OIL laser, high negative $\varDelta {f_{inj}}$ and strong ${R_{inj}}$ lead to enhanced steady-state optical phase and photons number [26]. As a result, we can modulate them by modifying the ${R_{inj}}$ and $\varDelta {f_{inj}}$, providing AM and PM synchronously, and then performing the complex modulation. Equation (16) depicts the relationship between ${S_{0,s{l_1}}}$ and ${\phi _{0,s{l_1}}}$ with the complex optical intensity of the first-stage SL, ${{\hat{\textrm{E}}}_{s{l_1}\; }}$(t) [11]:
In Eqs. (17)–(19), ${|{\hat{\textrm E} } |_{max,s{l_1}}}$ is the maximum optical intensity, and $|{\hat{\textrm E }_{fr,s{l_1}}}|$ is the free-running optical intensity in the first-stage SL and they can be achieved from the maximum ${S_{0,s{l_1}}}$, and ${S_{fr,s{l_1}}}$, respectively [11,26]. The accessible area of the complex optical signal is calculated based on the real axis (${\textrm{E}_\textrm{}}_{r,s{l_1}}$) and the imaginary axis (${\textrm{E}_\textrm{}}_{i,s{l_1}}$), as shown in Fig. 3(a). Here, two reasons prevent the first-stage SL from reaching all areas inside the red dashed circle. The first limitation is based on Eq. (18), which states that optical injection-locked conditions arise when we have output power higher than a typical laser in its free-running condition. The second constraint concerns the parameter ${\alpha _{s{l_1}}}$. This restriction comes from the locking range of the first-stage SL in Eq. (14), and has been introduced in Eq. (19); thus, Fig. 3(b) exhibits the final accessible complex signal area [11,21].
Fig. 3. The achievable area of the complex optical signal in the single OIL laser, (a) Related equations, (b) Final complex area; a.u.: arbitrary unit.
Similar to Eq. (16), the complex optical intensity of the second-stage SL, ${\hat{\textrm E }_{s{l_2}\; }}$(t), can be obtained. The complex inequalities for the cascaded OIL laser are extended below:
In Eqs. (20)–(22), ${\textrm{E}_\textrm{}}_{r,s{l_2}}(\textrm{t} )$ and ${\textrm{E}_\textrm{}}_{i,s{l_2}}(\textrm{t} )$ are the real part and the imaginary part of the complex optical intensity; ${|\hat{\textrm E } |_{max,s{l_2}}}$ and $|{\hat{\textrm{E} }_{fr,s{l_2}}}|$ are the maximum optical intensity and free-running optical intensity of the second-stage SL, respectively. Based on Eq. (22), the second restriction for creating a complex signal region in a cascaded OIL laser relates to both ${\alpha _{s{l_1}}}$ and ${\alpha _{s{l_2}}}$ items, which have the same value here. This limitation is due to the locking range of the cascaded OIL laser in Eq. (15). Figure 4 depicts the generated complex optical signal of the single OIL laser in comparison with its cascaded one, using ${R_{inj}}$=10 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 5${I_{th}}$, ${\alpha _{s{l_1}}}$=${\alpha _{s{l_2}}}$= 5, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz. In response to the increase in ${S_{0,s{l_2}}}$, according to Fig. 2(a), ${|\hat{\textrm{E}} |_{max,s{l_2}}}$ in Eq. (20) is raised. Likewise, the slope in Eq. (22) is improved; as a result, we can construct a larger complex area by cascading configuration, which can enhance the speed and the quality of the complex modulation.
Fig. 4. The reachable region of the complex optical signal in a single OIL laser in comparison with a cascaded one; ${R_{inj}}$=10 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 5${I_{th}}$, ${\alpha _{s{l_1}}}$=${\alpha _{s{l_2}}}$= 5, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz. a.u.: arbitrary unit.
4. Transfer function and frequency response of the cascaded OIL laser
To attain the cascaded OIL laser's frequency response and pole-zero diagram, the small current deviations, ${i_{s{l_1}}}$= ${i_{s{l_2}}}$ are applied. The deviations of ${N_{s{l_1}}}(t )$, ${S_{s{l_1}}}(t )$, ${\phi _{s{l_1}}}(t )$, ${N_{s{l_2}}}(t )$, ${S_{s{l_2}}}(t )$, and ${\phi _{s{l_2}}}(t )$ are defined as ${n_{s{l_1}}}$, ${s_{s{l_1}}}$, ${\phi _{s{l_1}}}$, ${n_{s{l_2}}}$, ${s_{s{l_2}}}$, and ${\phi _{s{l_2}}}$ respectively. Then we derive the differential rate equations using Eqs. (6)–(11). Below the small-signal analysis is examined:
Fig. 5. Pole-zero diagram (left column) and its corresponding frequency response (right column) for the single and cascaded OIL laser; ${R_{inj}}$=10 dB, ${\alpha _{s{l_1}}}$=${\alpha _{s{l_2}}}$= 2, $\varDelta {f_{in{j_1}}}$= − 50 GHz, and $\varDelta {f_{in{j_2}}}$= − 100 GHz: (a) and (b) ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 5${I_{th}}$, (c) and (d) ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 7${I_{th}}$.
Based on Fig. 5(d), increasing the values of ${I_{bias,s{l_1}}}$ and ${I_{bias,s{l_2}}}$ to 7${I_{th}}$ with holding the previous parameter values constant causes the modulation bandwidth to raise significantly, reaching 17.86 GHz in the single OIL laser and 24.13 GHz in the cascaded one. From Fig. 5(c), this bandwidth increment is the consequence of two variations in the pole-zero diagram; first, the distance between the ${\textrm{Z}_1}$ and ${\textrm{P}_1}$ has been enhanced somewhat; second, the distance between the ${\textrm{P}_1}$ and ${\textrm{P}_2}$ are increased in the complex plane. As a result, these variations allow the Bode plot keeps the state of the ${\textrm{Z}_1}$ roll-one more time and then cause a delay in the dips which happen by the ${\textrm{P}_1}$ and then by the ${\textrm{P}_2}$. Similarly, bias current enhancement in the single OIL laser moves ${\textrm{p}_1}$ farther to the imaginary axis; since the Bode plot reaches ${\textrm{p}_1}$ first, its roll-off happens later; thus, the bandwidth improves here. In addition, raising the bias current in both stages increase the imaginary part of the complex conjugate poles and zeroes.
5. Advantages of using different ${\boldsymbol \alpha }$ values in the slave lasers stages
In this section, we will demonstrate using different $\alpha $ values in the SLs stages can make it possible to noticeably improve the cascaded OIL laser’s performance for both generating the complex optical signal and the modulation frequency. Hence, we repeat the simulations to gain the best outcomes utilizing the optimum values. In Fig. 6, since the output data of Fig. 2(a) was employed to generate the complex optical signal in Fig. 4, we used a similar method for obtaining the complex optical area in the cascaded setup. In this figure, the accessible complex area at ${R_{inj}}$=10 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 5${I_{th}}$, ${\alpha _{s{l_1}}}$= 5, ${\alpha _{s{l_2}}}$=3.5, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz is represented. Compared to Fig. 4, it is evident that using a smaller value of ${\alpha _{s{l_2}}}$, can improve ${S_{0,s{l_2}}}$ and following that ${|\hat{\textrm E } |_{max,s{l_2}}}$. Besides, the slope in Eq. (22) is enhanced, resulting in acquiring the complex optical modulation of better quality.
Fig. 6. The reachable region of the complex optical signal in a single OIL laser in comparison with a cascaded one; ${R_{inj}}$=10 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 5${I_{th}}$, ${\alpha _{s{l_1}}}$= 5, ${\alpha _{s{l_2}}}$= 3.5, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz. a.u.: arbitrary unit.
Figure 7(b) describes the frequency response of the single and cascaded OIL laser with ${R_{inj}}$=10 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 7${I_{th}}$, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz, while the values 2 and 1.5 are assigned to the ${\alpha _{s{l_1}}}$ and ${\alpha _{s{l_2}}}$, respectively. The bandwidths of 17.86 GHz in the single OIL laser and 29.85 GHz in the cascaded one have been gained. Using Fig. 7(a) compared to Fig. 5(c), we can prove the main reason for the bandwidth increment is the distance between the ${\textrm{P}_1}$ and ${\textrm{P}_2}$ that has been enhanced. Consequently, ${\textrm{P}_2}$ dip happens later and bandwidth is broader while we utilize a smaller $\alpha $ value in the second-stage SL. In Figs. 7(d) and (c), ${R_{inj}}$=15 dB has been considered for the simulations. Here, the modulation bandwidths begin developing and have been calculated 70.2 GHz and 115.1 GHz in the single and cascaded systems, respectively. Similarly, in Fig. 7(c), the interval between ${\textrm{Z}_1}$ and ${\textrm{P}_1}$ is remarkably increased, and other poles are shifted to the left side of the complex plane; as a result, the ultra-high bandwidth has been attained. Notwithstanding ${\textrm{Z}_1}$ and ${\textrm{P}_1}$ roughly creating the dominant effects on the frequency response of the cascaded OIL laser, the consequences of the other poles and the dips they produce later in the Bode plot cannot be ignored. According to the results of this section, using cascaded configuration, enhanced bias currents, strong ${R_{inj}}$ along with a smaller $\alpha $ value in the second-stage SL can not only significantly improve the bandwidth to 115.1 GHz but also produce the resonance peak at 19.73 GHz. Thus, a cascaded OIL laser with the proposed optimum tunning can be beneficial for applications that require qualified and high-speed performance in complex optical modulation and ultra-high bandwidth. Moreover, the resonance peak has appeared in the modulation response of the cascaded system; hence, this setup yields a non-linear response as well as a fair gain.
Fig. 7. Pole-zero diagram (left column) and its corresponding frequency response (right column) for the single and cascaded OIL laser; ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 7${I_{th}}$, ${\alpha _{s{l_1}}}$=2, ${\alpha _{s{l_2}}}$= 1.5, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz: (a) and (b) ${R_{inj}}$=10 dB, (c) and (d) ${R_{inj}}$=15 dB.
In addition to the prior results, by the simulation outcomes, we will prove that using the previous parameter values, but this time via ${\alpha _{s{l_1}}}$=2, and ${\alpha _{s{l_2}}}$=3, can produce a tremendously broad bandwidth near 700 GHz in the cascaded setup. Figures 8(a) and (b) illustrate the mentioned consequences. From Fig. 8(a), we can observe the absence of ${\textrm{Z}_1}$ before the ${\textrm{P}_1}$ has led this linear response without any resonance peak. The main cause for the bandwidth extension is ${\textrm{P}_1}$ displacement which is considerably shifted to the left side of the complex plane. Other poles are moved to the left side as well. Further, two pairs of complex zero have appeared in the pole-zero diagram. Although these complex zeroes do not have prominent effects on the modulation frequency, they create a steady response and keep it near the –3 dB border. An important point should deliberate that there is acceptable gain only up to around the frequency of 180 GHz, and we have quite a linear response owing to the lack of the resonance frequency. Also, the accuracy of such created bandwidths should be inspected with the help of laboratory results. We can depict in Fig. 8(b) that the cascaded configuration with the proposed setup can generate a bandwidth 10 times greater than its single one. Hence, the mentioned procedure can be a practical solution for broadening the bandwidth in the OIL system's frequency response. Applications for this setup include radar, high bandwidth telecommunications, and any application that demands a linear response and large amounts of data transfer with a low gain.
Fig. 8. (a) Pole-zero diagram and (b) corresponding frequency response for the single and cascaded OIL laser; ${R_{inj}}$=15 dB, ${I_{bias,s{l_1}}}$= ${I_{bias,s{l_2}}}$= 7${I_{th}}$, ${\alpha _{s{l_1}}}$=2, ${\alpha _{s{l_2}}}$= 3, $\varDelta {f_{in{j_1}}}$= −50 GHz, and $\varDelta {f_{in{j_2}}}$= −100 GHz.
The section ahead should be the presentation of a procedure that overcomes the challenges of using different $\alpha $ values in the cascaded SL stages. Known to us, temperature variations can alter the majority of the laser parameters concerned with the refractive index (n), such as the $\alpha $ value [29]. Also, recent studies have reported that $\alpha $ increase in the Quantum-dash lasers can be arisen due to temperature enhancement [30,31]. Consequently, by temperature tunning, we can vary the value of the $\alpha $ parameter to our desired value while we have utilized identical types of SLs. Given that using similar SLs can reduce the mode profile mismatching between the first-stage SL beam that is reflected into the second-stage SL facet and its guided mode, we can improve the efficient injection power for realizing the ${R_{inj}}$=15 dB. As in the simple OIL semiconductor lasers, we can modify the detuning between the ML and the SL either by adjusting the temperature of the ML or using an external cavity tunable ML [29]. Hence, for producing non-identical $\alpha $ in the SL stages, we can implement the heatsink for the second-stage SL and increase or decrease its $\alpha $ value according to Figs. (7) and (8) adjustments. We are free to employ external cavity tunable lasers in the ML and first-stage SL or apply the temperature variations utilizing heatsinks on these two lasers for fixing the $\varDelta {\omega _{in{j_1}}}$ and $\varDelta {\omega _{in{j_2}}}$ values, respectively. There can be a question about how we can calculate $\alpha $ values before and after carrying out the temperature changes. We should take a glance at the $\alpha $ parameter definition in the lasers, which is described in Eq. (26) and establishes the link between the change of the real and imaginary parts of the effective susceptibility $\chi$, caused by the carrier [32–34]:
Fig. 9. Schematic of the experimental setup for the cascaded OIL laser with different $\alpha $ values in the SL stages.
6. Conclusion
This study introduces the cascaded OIL semiconductor laser's novel rate equations, locking range, corresponding complex optical signal area, and frequency response. In the first section of the paper, we proposed the optimum setup for the detuning frequencies with similar α values for the first-stage and second-stage SLs. In the second section, we employ innovative adjustments for the α values of the SL stages, and different values were assigned to α parameters. Besides, an adequate investigation of the frequency response and pole-zero diagram of the system was performed. Based on simulation outcomes, while we utilize strong ${R_{inj}}$, enhanced bias currents, and mentioned tunings in the cascaded configuration, a frequency bandwidth around 700 GHz (180 GHz applicable bandwidth) has been attained. Finally, we presented an approach (temperature changes) that can enable applying different α values in the cascaded stages in practical work, regardless of using identical SL types in the cascaded setup.
Disclosures
The authors declare no conflicts of interest. The idea, conceptualization, and implementation of this paper have been carried out by Seyyedeh Ladan Talebzadeh. It is under the supervision of Dr. Reza Yadipour and is being reviewed by Dr. Gholamreza Baghersalimi.
Data availability
Data underlying the results presented in this paper are available in Dataset 1 [38], Dataset 2 [39], Dataset 3 [40].
Supplemental document
See Supplement 1 for supporting content.
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