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Abstract
Undamped relaxation oscillation (RO) in a laser self-mixing interferometry (SMI) system may occur in some common application conditions, which may impact the stable operation of the system and degrade its sensing performance. In this work, we proposed to suppress the undamped RO by controlling the system operation parameters in a laser SMI sensing system. By numerically solving the famous Lang Kobayashi equations, the stability of a laser SMI system in a 3-parameter space of external cavity length, injection current and optical feedback factor were investigated. Based on the stability analyses, we determined the system operation conditions required for suppressing the undamped RO and derived an analytical expression for describing the relationship between the operation parameters. An experimental SMI system based on a laser diode (Sanyo, DL4140-001s) was implemented and verified the suppressing method. The experimental results showed that the SMI system in a moderate feedback regime can operate in steady state without undamped RO by setting proper operation parameters. This work provides useful guidance to design a stable SMI sensing system for practical applications.
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1. Introduction
Relaxation oscillation (RO) is one of the important characteristics of semiconductor lasers (SLs), which often occurs when the laser is turned on [1]. The basic physical mechanism of RO is the interaction between the in-cavity laser intensity and the population inversion, i.e., the cavity damping rate of the laser is higher than the carrier damping rate [2]. For an SL without external perturbations, RO is often damped after excitation. It can also be stimulated by perturbations, like optical injection or external optical feedback. With these perturbations, RO may evolve to be undamped [1]. In this case, SLs would experience abundant complex dynamics, e.g., period-one oscillation, quasi-periodical oscillation and chaos, which can be developed into different applications, e.g., chaotic communication [1], random bit generator [3], microwave photonics [4] and sensing of physical quantities [5,6]. On the other hand, undamped RO has great impact on the stable operation of SLs, which may degrade the system performance for those that require stable operation of SLs [7]. One example is the case of self-mixing interferometry (SMI), which is a promising high-precision measurement and sensing technology. The intrinsic mechanism of this technology is the phenomenon known as self-mixing effect, i.e., the laser output power and laser frequency are modulated when a proportion of the emitted laser light reflected by an external object re-injects the laser in-cavity, and mixes with the internal laser beam [8]. A typical SMI sensing system is composed of an SL with a build-in monitor photodiode (PD), a lens and an external target, which represents a minimum-part-count configuration as shown in Fig. 1. As compared with traditional two-beam interferometers, e.g., Michelson interferometer, SMI system has many unique characteristics and merits, such as the structure is simple and compact, the cost of implementation is low, and sensing signals are able to be detected anywhere the laser beam can reach [8]. To date, different SMI-based applications have been proposed, e.g., measurement of displacement [9,10], distance [11–13], speed [14–16], mechanical parameters of materials [17,18], linewidth enhancement factor [19,20], angle [21,22], acoustic emission [23], biomedical signals [24–26], and monitoring plasma accumulation in laser ablation [27,28].
In SMI systems, the SLs should operate in steady state with damped RO. The stable SMI model is the theory basis to analyze the system behavior, which is a set of stationary solutions to the famous Lang Kobayashi (L-K) equations [29], or is derived from the three-mirror model with the assumption that the SL is in steady state [8]. In the SMI model, there is an important parameter called feedback level factor (C). It determines the shape of SMI signals, and its values discriminate between different feedback regimes, i.e., weak feedback regime ($C\textrm{ } \le \textrm{ }1$), moderate/strong feedback regime (C > 1). In weak regime, a typical SMI signal is closely a sinusoidal curve. Whereas it becomes to be sawtooth-like in moderate/strong feedback regimes, and interference fringes may lose due to the existence of multi-solutions to the SMI phase equation and the induced laser mode competition and selection [8,30]. SMI-based applications are usually researched in weak or moderate feedback regime [8]. Strong feedback regime is often avoided in sensing applications because the system may be unstable or even enter chaotic state [8]. As compared with weak feedback regime, moderate feedback regime has unique characteristics and advantages in applications, e.g., SMI signals in moderate feedback regime show hysteresis and displacement can be retrieved by a one-channel interferometric signal without ambiguity [31]. Thus, many applications, e.g., measurement of displacement [31,32], velocity [33], and linewidth enhancement factor [19], are in moderate feedback regime.
However, some works have indicated that an SL with external optical feedback may suffer instability due to the existence of undamped RO with C > 1 [7,34,35]. More recently, we have found the a laser SMI system in moderate feedback regime may exhibit undamped RO and show complex SMI signals instead of conventional SMI signals by numerically solving the Lang Kobayashi equations [36]. Then we analyzed the SMI signal waveform features in details when undamped RO occurs by experiments, which shows SMI signals with undamped RO still contains some features that can be used for sensing potentially [37]. After that, we modeled the SMI signals with period-one oscillation, which is a special case when undamped RO happens [38]. We have found that SMI signals with undamped RO are in complicated forms, which requires extra signal processing to retrieve the required information [36–38]. The widely-accepted stable SMI model would fail to completely describe the behaviors of a laser SMI system [38], and the sensing performance may be degraded [36]. Therefore, it is of great necessity and interest to suppress the undamped RO in a laser SMI sensing system. In [36], we have preliminarily discussed how to achieve a stable SMI signal by considering the external cavity length or the injection current separately but their relationship was not discussed for a certain feedback regime. In fact, we need to consider the feedback level factor, the external cavity length and the injection current simultaneously when building a laser SMI system.
In this work, we study to suppress the undamped RO by controlling the system operation parameters to design a stable SMI sensing system. Firstly, we investigated the stability of a laser SMI system in a 3-parameter space of external cavity length, injection current and optical feedback factor and discussed the combined influence of operation parameters on its stability and SMI signals by numerically solving the L-K equations. Based on the stability analyses, we determined the required operation parameters for suppressing the undamped RO. Then, an analytical expression for the required operation parameters was derived. Finally, an experimental system was built to verify the simulation analyses. The results are useful for designing a stable SMI sensing system with suppression of undamped RO.
2. Theory and simulation
2.1 Fundamentals of SMI
The behavior of a laser SMI system is usually described by the following mathematic model, which is the basis of SMI-based applications [8].
Here, Eq. (1) is called the phase equation. ${\phi _F}(\tau ) = {\omega _F}(\tau ) \cdot \tau$ and ${\phi _0} = {\omega _0}\tau$, where ${\omega _0}$ is the laser angular frequency without perturbation, ${\omega _F}(\tau )$ is that with optical feedback, $\tau = 2L/c$ is the roundtrip time of the laser light within the external cavity, where L is the distance between the laser emitting facet and the target surface, which is also called the external cavity length, c is light speed in the external cavity. $P(\tau )$ and ${P_0}$ are the laser emitting power or intensity with and without optical feedback respectively. m is the modulation index. The feedback level factor C in Eq. (1) is defined as $C = \eta \sqrt {1 + {\alpha ^2}} \kappa L\textrm{/}{L_{in}}{n_{in}}$, where $\kappa $ is the feedback strength which is governed by the reflectivity of the laser emitting facet and target surface, Lin is the length of laser internal cavity, nin is the refractive index of the laser internal cavity, $\alpha $ is the linewidth enhancement factor, $\eta $ is related to the possible loss on re-injection [8,39].The SMI model can be derived from the L-K equations [29], shown as Eqs. (3)– (5), with the assumptions of the system operating in steady state with damped RO, i.e., by setting $dE(t)/dt = 0$, $d\phi (t)/dt = {\omega _F}(\tau ) - {\omega _0}$, and $dN(t)/dt = 0$[36]. Here $E(t)$, $\phi (t)$ is the amplitude and phase of the electric field respectively, and $N(t)$ is the carrier density. ${E^2}(t)$ is considered as the laser intensity or output power in the simulations as commonly treated in the literatures [35,36]. I is the injection current to the SL. The physical meanings of other parameters in L-K equations and their typical values for simulations in this work are in Table 1 [36,40].
Table 1. Typical values of parameters for simulations
2.2 Stability analysis on a laser SMI system
We performed stability analyses on a laser SMI sensing system based on the L-K equations through Hopf bifurcation diagram, which is a useful tool to investigate the evolution of the system dynamics by sampling the local maximum and minimum points of the temporal waveforms. The bifurcation point in the diagram is the critical point that discriminates the steady state and period-one oscillation [1]. The Hopf bifurcation diagram was obtained by numerically solving the L-K equations. Since we focus on the SMI applications, the bifurcation diagram was presented in the plane of laser intensity and optical feedback factor C. Figure 2 shows an example of the Hopf bifurcation diagram of a laser SMI system when $I = 1.3{I_{th}}$, $L = 30.0\textrm{ cm}$, where Ith is the threshold injection current of an SL with ${I_{th}}\textrm{ = eV(}{N_0} + 1/{G_N}{\tau _p})/{\tau _s}$ [1] and it can be calculated as ${I_{th}} = \textrm{14 mA}$ with the values in Table 1. From Fig. 2, we can find a laser SMI system may operate in different states, i.e., steady state, period-one oscillation, quasi-periodical oscillation or chaos. For SMI-based applications, the system should be in steady state. The bifurcation point in Fig. 2 decides the maximum C that ensures the system is stable with damped RO. The RO evolves to be undamped when the feedback level factor C is larger than the value at the bifurcation point. In this case, the SMI system no longer operates at steady state and the SMI model of Eqs. (1) and (2) is inaccurate to describe the complete behavior of SMI system. Complex dynamics would appear in the SMI signals and influence the sensing performance of the SMI system.
A well-accepted method for describing the feedback regimes of a laser SMI system is based on the value of C, i.e., weak feedback regime ($C \le 1$), moderate feedback regime (1 < C < 4.6), and strong feedback regime (C > 4.6) [39,41–44]. Because strong feedback regime is often avoided in sensing applications [8], we focus on the case of a laser SMI system in weak and moderate feedback regime. Additionally, because a laser SMI system in weak feedback regime is unconditionally stable [7,35], we mainly investigate a laser SMI system in moderate feedback regime. Although the work [30] has made some doubts on defining strong feedback regime with C > 4.6 from the aspect of SMI fringe loss, we still adapted the common definition of moderate feedback regime in the community, i.e., with 1 < C < 4.6, because the proposed analyses can be easily to extend to C > 4.6. We denote the C at the bifurcation point by Cbp. We can obtain that Cbp is about 2.85 in Fig. 2, where the dash line is for C = 4.6. It can be predicted that the SMI system in the case as Fig. 2 is not always stable in moderate feedback regime, i.e., it is unstable with 2.85 < C <4.6. In order to confirm it, we then numerically solved the L-K equations to obtain the laser intensity ${E^2}(t)$, i.e., the SMI signals, when a sinusoidal displacement was applied on the external target. Figure 3 presents the simulation results, where Fig. 3(a) is the displacement, Figs. 3(b) - (d) are the corresponding SMI signals when C = 2.80, C = 2.90 and C = 4.00 respectively. We can see that Fig. 3(b) is the typical SMI signal which can be described by the stable SMI model of Eqs. (1) and (2), while Figs. 3(c) and (d) are in complicated patterns containing complex dynamics. As a result, Cbp should be larger than 4.6, i.e., Cbp > 4.6, to keep a moderate feedback SMI system stable. Therefore, the stability of a laser SMI system is described by Cbp. The bifurcation point in the bifurcation diagram as in Fig. 2 should be on the right side of the dashed line to maintain the system stable.
Fig. 3. (a) Displacement of the target; (b), (c) and (d) the corresponding SMI signals with C = 2.80, C = 2.90 and C = 4.00, respectively
We then investigated the influence of external cavity length L and injection current I on Cbp. In this work, we mainly studied a laser SMI system with long external cavity length, i.e., L > c/2fRO, where fRO is the RO frequency of the solitary SL [34,35]. We can calculate with fRO > 2 GHz with the values in Table 1 [1], and thus a long external cavity length corresponds to L >7.5 cm. As a result, we varied the external cavity length with a step of 0.5 cm in the range from 10.0 cm to 100.0 cm, the injection current I from 1.1Ith to 2.0Ith with a step of 0.01Ith, then got $180 \times 90 = 16200$ pairs of L and I. For each pair of L and I, we numerically solved the L-K equations to get the bifurcation point and recorded the Cbp. Figure 4 presents the simulation results, where Fig. 4(a) is the Cbp described by both L and I. It can be seen that Cbp increases with fluctuations when L increases for a fixed injection current. The similar variation trend can be found for Cbp with respect to I for a fixed external cavity length. Figures 4(b) and (c) are two examples for a fixed injection current of I = 1.5Ith and a fixed external cavity length of L = 30.0 cm respectively. It can be concluded that a longer external cavity and a larger injection contribute to a larger Cbp.
Fig. 4. The feedback level factor at bifurcation point, (a) described by external cavity length L and injection current I, (b) described by L with a fixed injection current I = 1.5Ith, (c) described by injection current with a fixed external cavity length L = 30.0 cm
2.3 Operation requirements for suppression of undamped RO
Based on the above simulations, we set Cbp= 4.6 and are able to get the required critical injection current and external cavity length that ensure a moderate feedback SMI sensing system is stable. The simulation results were obtained as shown in Fig. 5. It can be seen that the relationship between the required critical injection current and external cavity length is a hyperbolic curve approximately. The SMI system remains stable when it operates with the parameters in the area above the curve. Note that, for the numerical results in Fig. 5, there is an average uncertainty of about 5% when we use the bifurcation diagram to discriminate the operation states of the SMI system due to the limit of simulation step of time. In fact, if we intend to apply SMI technique in strong feedback regime [45] and maintain the system stable with damped RO, we just need to set Cbp to be equal to the required feedback level factor C. Then the curve for the required external cavity length and injection current in strong feedback regime can be easily obtained with the same procedure.
Fig. 5. The critical injection current and external cavity length for damped RO in moderate feedback regime.
We then try to find an analytical expression for the relationship between the required injection current and external cavity length. Our recent work has found an approximate expression for the critical feedback level factor for a stable SMI system as below [36]:
The critical feedback level factor has been proposed to distinguish the stable and unstable regions of a laser SMI system in the plane of optical feedback factor and optical phase which has the similar meaning with Cbp, but it has been derived under the assumption of ${\omega ^2}_R > > \kappa /({\tau _{in}}{\tau _R})$ and ${\omega ^2}_R > > {(\kappa /{\tau _{in}})^2}$[36]. We take ${C_{bp}} \approx {C_{critical}}$ for approximation. Then, for a long external cavity (i.e., $L > \pi {c / {{\omega _R}}}$), we have $\Omega \approx {\omega _R}$[40]. We insert Eqs. (7)– (9) into Eq. (6), and set Cbp = 4.6 by considering the moderate feedback regime, and then get:
Equations (10) gives the relationship between the required injection current I and external cavity length L for a laser SMI system in moderate feedback regime, where other parameters are constant for a certain SMI system except I and L. The result is also plotted in Fig. 5 for comparison. It can be seen that the curve of Eq. (10) is consistent with the simulation results. The relationship between the required critical injection current and external cavity length is a hyperbolic curve. In order to guarantee a laser SMI system stable in the whole moderate feedback regime, the injection current and external cavity length should be set above the hyperbolic curve.
3. Experiment
Experiments were then carried out to verify the simulation results. Figure 6 is the structure diagram of the experimental setup. The SL is a 790 nm single mode laser diode (Sanyo, DL4140-001s). Its threshold value of injection current is 30 mA. The SL is driven by an SL controller (Thorlabs, ITC4001) with integrated temperature controller. The laser beam transmits through a collimating lens and a 50:50 beam splitter (BS). One proportion of the beam transmits via an optical attenuator (Thorlabs, NDC-50C-2M-B) and is then reflected by a mirror affixed on a piezoelectric transducer (PZT) (Thorlabs, PAS009). The PZT is driven by a PZT controller (Thorlabs, MDT694) and is used to generate a small displacement. The PZT with the mirror assembled on a linear translation stage acts as the external target. Using the linear stage, different external cavity lengths can be obtained. The optical attenuator is used to adjust the feedback strength. By using the attenuator, mirror and the linear stage, different feedback level factors can be achieved. The other part of the light through the BS transmits to a fast external photodetector (Thorlabs, PDA8GS) via an optical isolator (Thorlabs, IO-5-780) and it is detected as the SMI signals. Finally, we use an oscilloscope with an 8 GHz analog bandwidth (Tektronix DSA 70804) to capture and observe the SMI signals.
It is noteworthy that the structure of the experimental setup is slightly different from the typical SMI configuration in Fig. 1. We used an external photodetector with a frequency bandwidth of 9.5 GHz rather than the photodiode packaged in the SL to detect the SMI signals. The reasons are twofold. One is that the RO frequency of an SL is usually in GHz scale. The response speed of the internal packaged photodiode and the common trans-impedance amplifier is not fast enough to detect the complete SMI signals with undamped RO, e.g,. the overall bandwidth of the integrated photodiode with the detection circuit in our experimental system is about 10 MHz. In fact, if we use the integrated photodiode to detect the SMI signals with undamped RO, the detected SMI signals could be like the normal ones but with distortions for different detection bandwidths. In order to keep the detected SMI signals consistent with the simulation results in Fig. 3 and recognize undamped RO, we need a fast photodetector with a wide frequency response bandwidth. The other is that SMI signals can be detected anywhere along the optical path, which means the structure in Fig. 6 can be still treated as a laser SMI configuration [8].
The SL operation temperature was stabilized at $25 \pm 0.01^\circ \textrm{C}$ during the experiments by using the SL controller. According to the simulations, Cbp increases with the injection current and external cavity length. We firstly fixed an external cavity length with a certain value and then carefully adjust the attenuator to set C = 4.6. After that, we applied different injection currents to the SL from large to small and recorded the injection current when undamped RO started to appear in the SMI signals. Then we set another external cavity length, and recorded the corresponding injection current by the same method. Nine pairs of injection currents and external cavity lengths were recorded. Note that we are able to use the variable optical attenuator to adjust the optical feedback strength to set C = 4.6 for a given external cavity length thanks to the ability of continuously adjusting the transmission rate via rotation of the optical attenuator. Although C is related to many paramters, like linewith enhancement factor $\alpha $, laser internal cavity length, ect., it is still possible to set C with the required value, e.g., C =4.6 in this work. That is because C is directly proportional to feedback strength $\kappa $, and $\kappa $ is proportional to the power transmission rate of the attenuator. Once we obtained a reference value for C, we just need to adjust the power transmission rate of the attenuator to get C = 4.6 even though other parameters related to C are unknown but with fixed values. The value of C is determined as follows:
- 1. For each external cavity length, adjust the injection current and get a typical stable moderate feedback SMI signal by observing the waveform.
- 2. Use the method in [19] to estimate the value of C denoted by Cref and record the transmission rate of the optical attenuator denoted by Tref, which can be read from the dial of the attenuator.
- 3. Adjust the transmission rate (T) of the attenuator and obtain the required feedback level factor C with the expression of $T = C{T_{ref}}/{C_{ref}}$, e.g., during the experiments, we obtained a typical SMI signal with ${C_{ref}}\textrm{ = }1.5$ and recorded the corresponding transmission rate with ${T_{ref}}\textrm{ = 10\%}$. Thus we can get the required transmission rate with $T \approx 31\mathrm{\%}$ for C = 4.6.
Fig. 7. The critical injection current and the external cavity length when undamped RO start to appear in SMI signals
We took two pairs of L and I, i.e., point A (L = 12.0 cm, I = 45 mA) and point B (L = 20.0 cm, I = 60 mA) in Fig. 7 for the examples. Figure 8 presents the observed SMI signals with the L and I at points A and B respectively. Here, Fig. 8(a) is the controlling voltage applied on the PZT. Note that each 0.1 V controlling voltage applied on the PZT generates a displacement of 53 nm. The peak-peak amplitude of the PZT controlling voltage in Fig. 8(a) is around 3.8 V, which produces a displacement of about 2024nm. Figures 8(b)-(e) are the corresponding SMI signals, where (b) and (c) are for point A; (d) and (e) are for point B. In addition, Figs. 8(b) and (d) are with C = 1.6; (c) and (e) are with C = 4.6. Based on Fig. 7, we can predict that the SMI system will be unconditional stable with any C≤4.6 for point B. But for point A, we cannot ensure the system is stable or not in the whole moderate feedback regime. From Fig. 8, It can be found that the SMI signal is with typical form with C = 1.6, i.e., Fig. 8(b), but contains complex dynamics with C = 4.6, i.e., Fig. 8(c), at point A. When we set L and I with the values at point B, the SMI signals do not contain any complex dynamics no matter with C = 1.6 or C = 4.6, demonstrating that the SMI system is always stable without undamped RO in the moderate feedback regime. It is noteworthy that Figs. 8(b) and (d) are similar, which is because the system with C = 1.6 is stable when it operates at both point A and point B.
Fig. 8. Experimental SMI signals with different I and L, (a) the controlling voltage of PZT; (b) and (c) the corresponding SMI signals at point A; (d) and (e) the corresponding SMI signals at point B.
4. Conclusion
In this work, we studied to suppress the undamped RO of SLs by controlling the system operation parameters in a laser SMI sensing system, especially in moderate feedback regime. The stability of a laser SMI system in a 3-parameter space of external cavity length, injection current and optical feedback factor was numerically analyzed through Hopf bifurcation diagram. The system is unstable with undamped RO when the feedback level factor is larger than that at the bifurcation point, which leads to the SMI signal waveforms and measurement performance of a laser SMI system are impacted by complex laser dynamics. Based on the stability analyses, we determined the system operation condition required for suppressing the undamped RO and derived an analytical expression for describing the relationship between the operation parameters, which shows that the relationship between the required injection current and external cavity is an approximate hyperbolic curve. We also derived an analytical expression for the required injection current and external cavity length, which is consistent with the numerical simulations. The SMI system is guaranteed to be stable with damped RO when it operates with the parameters above the curve. The theoretical and simulation results were verified by experiments. The experimental results showed that the built SMI system in moderate feedback regime can operate in steady state without undamped RO by setting proper parameters. The results in this work provide useful guidance to design a stable SMI sensing system without effects of complex laser dynamics for practical applications.
Funding
National Natural Science Foundation of China (62005234); Education Department of Hunan Province (20C1791).
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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