Dynamic Stability Analysis for a Self Mixing Interferometry

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Chapter 1 Introduction and Background
In the last decades, the various applications of the Self Mixing Interferometry (SMI) based sensing system have attracted much attention of researchers. An SMI can be used for measuring metrological quantities, such as velocity, absolute distance, displacement and vibration, etc. or even the parameters associated with the laser itself [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. A schematic diagram of the SMI is shown in the following figure ( Fig. 1-1). The core part of an SMI consists of a single longitudinal mode Semiconductor Laser (SL) and a moving target which forms an external cavity of the SL. An SMI works when a small portion of light is back-scattered or reflected by an external target and re-enters into the laser internal active cavity. The re-entered light modulates both amplitude and frequency of the emitted SL power [20,21]. This modulated power is called an SMI signal which can be used to detect metrological quantities associated 2 with the external target and the parameters associated with the SL itself. The advantages of using an SMI-based sensing system to perform measurement have been presented in [3]:  No optical interferometer external to the source is required. This leads to a simple and compact set-up.
 No alignment is needed because the spatial mode that interacts with the cavity mode is filtered out spatially by the laser itself. This means that detection of the diffusive target's movement becomes possible.
 Sensitivity of the scheme is very high (sub-nm sensitivity).
Due to these advantages, the SMI has been intensively investigated both theoretically and experimentally for exploring all kinds of sensing applications. In these applications, it is required that an SMI system operates in a stable mode, in which case the SL biased by constant injection current usually leads to SMI signals with symmetric sinusoidal-like fringes or asymmetric sawtooth-like fringes, depending on the external optical feedback level. However, with the change of operational conditions, such as injection current and parameters associated with the external cavity including the optical feedback level and external cavity length, the SL can also exhibit unstable behavior. In this case, an SMI will degrade or even lose its sensing ability. Therefore it is very important to investigate the stability of an SMI system with respect to its operational conditions, which lead us to the main goal of this thesis, i.e., analyzing the stability of the SMI.
This chapter gives an introduction and background for this thesis. The rest of this chapter is organized into the following sections. Section 1.1 gives detailed derivation of the existing mathematical model for describing the SMI. A literature review for the SMI has been given in Section 1.2 in terms of the stability analysis. Based on the investigation of reviewed articles, the existing problems for investigating the stability of the SMI are presented in Section 1.3. Section 1.4 shows the structure of this thesis. 3

Derived from the Three Mirrors Model
The mathematical model of the SMI can be derived from the classical three mirrors model consisting of a Fabry-Perot (FP) type laser with facet reflection coefficients 1 r and 2 r , and the target with the reflection coefficient of 3 r [22]. Therefore, a simplified arrangement based on the schematic diagram shown in Fig. 1-1 can be used for carrying out the derivation, which is shown in the following figure ( Fig. 1-2): where   ( ), ( ) G N t E t is the modal gain per unit of time and is expressed as [31,32]: and when ignoring the nonlinear effect, the modal gain can be simplified as: Equations (1.10)-(1.13) describe the dynamic behavior of the three variables, namely the electric field amplitude () Et , the electric field phase () t  and the carrier density is the instantaneous optical angular frequency for an SL with EOF.
The dynamics of the SL with an EOF system are governed by the injection current ( J ) to the SL and the parameters associated with the external cavity including  and  .
The other parameters in Eqs. (1.10)-(1.13) are related to the solitary SL itself, and are treated as constants for a certain SL. These parameters are defined in Table 1-1 [31].
Note that the values of the parameters provided in Table 1-1 are adopted from [31].    , thus retrieving the external cavity information. Therefore the knowledge of the theory of generating an SMI signal as well as its waveform is essential to achieve good performance of the SMI.

SMI Waveform
In the SMI model, C (called feedback level factor) is an important parameter as it characterizes the waveform of an SMI signal. When 1 C  (corresponding to weak feedback regime), equation (1.19) presents a unique mapping from 0  and s  . In this situation, the movement of the external target will result in an SMI signal waveform with a fringe structure similar to the traditional interference fringes, and each fringe period corresponds to a phase shift that is equivalent to a displacement of half a wavelength of the external target [35].  In the situation of 1 C  which corresponds to a moderate (1 4 [31,38,39], and thus will never be an oscillating mode of the SMI signal. Similar to the weak feedback regime case discussed above, we present 0 () t   , which is same with Fig. 1-4(a), as well as the corresponding SMI signal () gt in the following figure ( Fig. 1-6) .   . The mechanism of generating an SMI signal as well as predicting its behaviour with respect to C has been well-established and presented in [18,23,33,35,36]. 11 Clearly, the existing SMI model, i.e., Eqs. (1.19) and (1.20), is based on a premise that the system is stable, or in other words, it does not give the condition of stability of the SMI with respect to the SL operational parameters, i.e.,  ,  and J . The goal of this thesis is to achieve a complete and accurate stability analysis for the SMI. Before presenting our approach, a detailed literature review related to the stability analysis of an SMI is presented in the following section.

Literature Review
As an SMI is an SL with EOF by considering a time varying light phase, also called a feedback phase [40][41][42][43][44]   can be varied by minuscule changes of the external cavity length (on the order of one wavelength of the laser) or by altering 0  through slightly adjusting the injection current or temperature [45].
Under the condition of a stable operation, an SL biased by constant current usually emits a laser with a constant intensity when the target is stationary. However, with the change of operational conditions, the laser output can also exhibit unstable behaviour, such as periodic oscillation, quasi-periodic oscillation, low frequency fluctuations (LFFs) or chaos [43,44,[48][49][50][51][52][53][54][55][56]. As the transition from stable to unstable states is caused by the change in the injection current and parameters associated with EOF, it is important to figure out when the transition occurs with respect to the values of these parameters, or the range of these parameters within which the SL is stable or unstable.
Such a range is referred to as the stability limit or the stability boundary [31], or the Hopf bifurcation [57,58].  [45] presented results of an analysis on the stability boundary of an SL with EOF. Three different boundaries were described in [45], namely boundaries A, B and C. boundaries A and B are for the bistability condition described in [51], [59,60]. Boundary C is defined as a dynamic stability boundary, also called a feedback-induced intensity pulsations boundary in [45]. In this thesis, we mainly discuss the boundary C, as it is the boundary identifying the transition from stable to unstable states. In [45] (45) and (47) in [45]) for describing the stability boundary. Here, R  and R  represent the relaxation resonance angular frequency and carrier density relaxation time (also called damping time [31]) respectively for the solitary laser. In [45], the stability boundary was presented as a relationship of the feedback phase in the range of   ,   versus  for a fixed  (see Figure 9 in [45]). In fact, such presentation of the stability boundary, that is, in a two dimensional 13 plane of the feedback phase and one of the operational parameters, is useful for investigating the stability of the SMI in the future, as the SMI is a special case of SL with EOF where the feedback phase is time varying in the range of 2 with the movement of the target (note that 0  is periodic, and so is s  according to Eq. (1.19) [61]). In the figure 9 in [45], it can be seen that below a certain value of  , the SL with EOF is always stable no matter what the values of the feedback phase are, therefore indicating a stable operation range with respect to  for the SMI.
Two years later, in 1986, Olesen, et al. [39] improved the work in [45] in two respects: 1) using the system determinant without the assumptions in [30] and [45], and 2) employing R  and R  of an SL with EOF rather than those of a solitary SL. In [39], the stability boundary was discussed in the same way as in [45] but for three different fixed values of  , showing that the stable operation range of the SMI increases with the increase of  .
In 2002, another work was reported in [46] using the normalized LK equations for the case of short external cavity ( 0.02ns

 
). Due to the use of the normalized LK equations, the numerical analysis on LK equations is significantly simplified [52], [46,62,63]. In [46], starting from the normalized LK equations, a new system determinant with the normalized coefficients was derived. The stability boundary was numerically calculated and presented as a relationship of the feedback phase versus  for a fixed  . The same conclusion with [45] can be drawn from the relationship presented in [46].
In 2009, another set of results were reported in [61] obtained by means of a MATLAB package DDE-BIFTOOL [64]. The stability boundary acquired in [61] was presented as a relationship of the feedback phase versus  and the feedback phase versus J respectively for a fixed  . The normalized LK equations were also used in [61] for the numerical calculation of the stability boundary. The underlying algorithm of the MATLAB package is the linear multi-step method [65,66] which can approximately calculate the locations of zeros of the system determinant. 14 In 2012, Donati and Fathi [67] numerically presented the possible region for achieving a stable SMI for both short and long external cavities by investigating the stability boundary of an SL with EOF in a plane of the feedback phase and  . The results in [67] showed that the SMI can be stable when  is small.
More recently, in 2013, Lenstra [40] found that when the product of R  and  is equal to an integral multiple of 2 , the SL with EOF behaves like a solitary laser which is always stable no matter what the feedback phase is. The stability boundary was presented as a relationship of the feedback phase in the external cavity and J for a fixed  .
Note that in the research conducted in 2009 [61] and 2013 [40], a band stable phenomenon with respect to J was predicated theoretically. The phenomenon indicates that, in some certain regions of J , the stability is free of the influence of the feedback phase, thus leading to a stable SMI, while in other regions of J , the stability strongly depends on the phase. However, the studies were only for short cavities and weak feedback strength (   1 in ).
An outstanding issue associated with the results reported in [30, 39,40,45,46,61,67], is that, none of them considers the effect of nonlinear gain, which was ignored in the process of linearizing the LK equations in [30]. As a matter of fact, nonlinear gain is an important factor in describing the dynamic behaviors of an SL with EOF [32,68,69] and it was reported that the stability boundary of an SL with EOF can be enhanced by the nonlinear gain [68,70]. Hence, in order to better describe the stability, nonlinear gain in the LK equations should be considered. Some preliminary work was reported on this problem [31,[71][72][73]. The work in [31] was based on the expression in [45], but insert a term in the expression of R  to take into account the influence of nonlinear gain (see Equation (19) in [45]). The stability boundary obtained in [45] is a relationship of  versus  for a fixed J . Furthermore, in [31], by analyzing the characteristic points on the stability boundary, a simple analytic 15 expression was derived for the stability boundary described by the critical feedback strength c  . Obviously, [31] considers only partially the influence of nonlinear gain, and it is still based on the approximations used in [45]. In 1993, Ritter and Haug [71] presented another expression Later, in 1994, Tager and Petermann [72] presented similar work to [71], in which the gain in the electric field amplitude equation was considered as nonlinear, but the gain in the electric field phase equation was still linear. Two new coupling equations were derived from the system determinant under assumptions different to [71], that is,  is the contribution of nonlinear gain to R  in a similar way to [31]. The two coupling equations are similar to those derived in [45], but they include the influence of compound cavity mode competition on the stability of an SL with short external cavities. The stability boundary was then presented as a relationship of  versus  and  versus the feedback phase in the external cavity for a fixed J in [72], indicating that (see Fig. 7 in [72]) there is no region where the SL with EOF is always stable for all the values of the feedback phase with a fixed  . That is to say there is no region guaranteeing a stable SMI.
In 1998, Masoller and Abraham [73] investigated the influence of  on the stability limit when the SL is biased well above the threshold. However, the nonlinear gain was still partially considered in the same way as in [31]. 16

Outstanding Issues
The results of analyzing the above existing literature are summarized in Table 1-2 from which the outstanding research issues can be drawn as follows: 1. The existing approaches either ignore the nonlinear gain, which is an important factor in describing the system, or make use of an approximated expression of the system determinant, thus leading to use of incomplete system descriptions and hence inaccurate results.
2. The influence of all parameters associated with an SL with EOF operation, i.e.,  ,  and J on the stability of an SL with EOF has been absent in the literature, as well as the SMI, both theoretically and experimentally.
As the existing SMI model is built on the stationary solutions of LK equations, i.e., Eqs. (1.10)-(1.12), the model is only certainly valid when an SMI system is stable. Or in other words, when the system enters into the unstable state, the premise for deriving the stationary solutions will be no longer valid, i.e., and ( ) 0 dN t dt  , thus leading to the position that actual behaviour of the system cannot be described by the existing SMI model. 4. The existing SMI based applications, such as measurement of displacement as well as the parameters of the SL itself, e.g., the LEF, all assume and require that the system is in a stable status. However, research in Chapter 3 will subsequently show that when the system enters into the unstable states, an unstable SMI can pretend to be a 'stable' SMI, which confuses researchers due to the limited bandwidth of the detection component, thus leading to the degraded system performance.

Thesis Organization
This thesis consists of six chapters: Chapter 1 presents a brief introduction and background to this research project. Firstly, in Section 1.1, the background of the SMI is introduced then its theoretical model originated from both the three mirror and LK models is presented. Based on the SMI model, the characteristics of the SMI waveform as well as its mechanism related to the model are described. Secondly, in Section 1.2, the literature related to the stability analysis of an SMI is reviewed from the viewpoint of their theoretical derivations as well as the manifestation of their results. At the end of Chapter 1, the summarization of the outstanding problems and organization of this thesis are given.
Chapter 2 gives a complete and accurate stability analysis for an SL with EOF with a particular fixed feedback phase value. In Section 2.1, a new and accurate system determinant is derived by removing the assumptions or approximations made in the existing work to ensure a more accurate and complete theoretical analysis. Then by varying  ,  , and J , the trajectory of the zeros of the system determinant is determined by an effective numerical computation developed by this researcher, from which the stability boundary described by these parameters can be obtained. In Section 1.3, the relationships of each pair of the three parameters will be investigated in detail and a list of new and interesting discoveries is presented. 18 Chapter 3 analyzes the stability of the SMI. The stability of an SMI is firstly presented in a two dimensional plane of the feedback phase 0  and the feedback level factor C in Section 3.1, based on which three regions for characterizing the behaviour of an SMI are proposed, and potential applications for the three regions are described.
Furthermore, a critical optical feedback factor critical C , under which the SMI is guaranteed to be stable, is approximately determined in an analytical form. Finally, in is observed from the plane, above which the SMI is always stable. are discovered respectively being proportional and inversely proportional to J and  . 20

System Determinant
The stability of a system is usually analyzed based on the system determinant. For an SL with EOF, its system determinant is obtained based on analysis of the LK equations near the stationary solutions [30, 39,43,45,46,[70][71][72][73][74] where R  is the damping time of the relaxation oscillation of an SL with EOF, 22 13 (1 )

Stability Boundary
With the system determinant in Eq. (2.7), we are able to work out the stability boundary based on the locations of zeros of Eq. (2.7) on the S-plane. The system is 22 stable if all the zeros are located on the left hand side of the S-plane. The zeros on the imaginary axis give the stability boundary of the SL with EOF.
The zeros of () Ds are defined as the roots of ( ) 0 Ds  , which are usually complex numbers, that is, sj     . In order to work out all the zeros, we insert sj     in Eq. (2.7) and rearrange the right hand side by separating the real and imaginary parts and set them to be zero, yielding two equations as follows: In order to have complete knowledge on the stability of the system, we can only utilize numerical computation to work out the zeros with respect to all the possible values of the parameters. In this Chapter, we mainly focus on the influence of  ,  and J , and will discuss the influence of the feedback phases in the following 23 chapters. To achieve numerically calculating the zeros, we considered that the parameters  and J take 200 points equally spaced within the ranges   With the procedures above, we can work out the stability boundary now. Note that all other parameters take the values in Table 1-1.
In order to facilitate the comparison with existing results, parameters are scaled in the same way as in [31], where  ). Considering that the average range of  studied is in the order of 3 10  in existing work [31,39,45,61,[70][71][72] for the stability analysis of an SL with EOF, such a difference is indeed considerable and should not be ignored.  waveform of () It in Fig. 2-3 shows that a constant () It is achieved after the transient dies away. Hence, the system is stable at this chosen point. This result coincides the conclusion reported in [32,70] that nonlinear gain enhances the stability of the SL with EOF.

Feedback Strength
As the locations of zeros of the system determinant change with parameters  , J and  , it is important to investigate the influence of  , J and  respectively on the stability boundary of an SL with EOF system. Because  is directly dependent on the external cavity length L (that is, 2 Lc   , where c is the speed of light), in the following we replace  by its corresponding external cavity length L which provides a more informative physical meaning related to an SL with EOF.

Influence of External Cavity Length
Let us firstly look into the influence of L on the stability boundary. To this end, we extract the relationship between J and  from Fig. 2  From Fig. 2-4, we are able to observe the following features: 1. The stability boundary shows a finger structure underneath the same asymptote for all the different values of L . As area above the asymptote is guaranteed to be stable, a safe choice can be made above the asymptote which gives a guideline for the selection of feedback level and injection current in order to have a stable system. For example, in order to have a stable system, large injection current is required for the case of high feedback strength.
2. Not all areas below the asymptote are unstable. However, with the increasing of L , the number of the fingers also increases, making the unstable area to increase as well and tend to fill the area under the asymptote (that is, the shaded area in Fig. 2-4). For the case of relative long external cavities (e.g., Fig. 2-4(f)), the stability boundary is close to and can be approximately described by the asymptote. However, for the case of a relative short cavity, some areas of considerable size under the asymptotes (e.g., Fig. 2-4(a)) are still stable. 28

Influence of Injection Current
We also examine the influence of injection current on the stability boundary. By setting J to six different values, the relationships of  and  can also be obtained from Fig. 1. In the relationships, parameter  is replaced by L using 2 Lc 


. So, we have the relationships between  and L shown in Fig. 2-5 where the shaded still denotes the unstable region. From Fig. 2-5, we can also find the following features of the stability boundary in terms of  and L . 1. An SL with very short external cavities (e.g., 6.5 L mm  in Fig. 2-5(b)) can always endure a very high feedback strength, which is consistent with the results reported in [31,74,75].
2. The stability boundary also demonstrates a finger structure. The width of all the fingers (denoted as T in Fig. 2-5) is nearly the same with respect to L , but it decreases with the increase of J .
3. The minimum value of  appeared in the unstable region (see the horizontal line adhered to the stability boundary in Fig. 2-5),  increases with the increase of J . The value of c  was also discussed in the previous work [31, 29 70, 72, 76]. Figure 2-6 shows the comparison of our results against the results obtained in [31,70,76] where the parameters are the same as shown in Table   1-1. It can be seen, the values of c  we obtain numerically is higher than those presented by [31,70,76]. This means that the actual stable region is larger than those presented by existing literatures. Note that [72] did not discuss the influence of J on c  .

Influence of Feedback Strength
Similarly, we can discuss the influence of  on the stability by observing the relationship of J and L . The stability described by J and L is presented in We conclude the stability boundary as follows:

Summary
In this Chapter, a comprehensive analysis on the stability boundary of a single mode SL with a feedback phase condition 0 2 arctan( ) The influence of the other parameter, i.e., the feedback phase will be discussed in the following Chapters which will provide a detailed stability analysis for the SMI. 33

Chapter 3 Stability Analysis for an SMI
As mentioned in Chapter one, when the system enters into the unstable state, the premise for deriving the stationary solutions of the Lang and Kobayashi (LK) equations will be no longer valid, thus leading to the actual behavior of the system can not be described by the existing SMI model. As the SMI can be considered as an SL including stable, semi-stable and unstable regions. We found that the existing SMI model is only valid for the stable region, and the semi-stable region has potential applications on sensing and measurement but needs re-modeling the system by considering the bandwidth of the detection components. The actual behavior of s  for a stable SMI signal has been investigated in [35]. It shows that s  will vary along the route of 1  [31,38,39]. These unstable stationary solutions satisfy the following equation: As mentioned above, the stability of a system is usually analyzed based on the system determinant. Based on the system determinant, [31,45] showed that, to a good approximation, the stable condition of the system as well as of the stationary solutions satisfies the following condition: 22 2 sin( ) cos( ) 1 2 2 sin ( 2)

The Stability Boundary Described by
for all the values of  (  is defined as the imaginary part for a complex number in Laplace transform domain. The details can be found in [31,45] When designing a stable SMI, it is important to know how to configure the system in terms of a proper feedback level and suitable movement range for the external target (or the feedback phase 0  ). That is, we need to know the stable boundary for the parameter C and 0  . Hence, we propose to describe the stability of an SMI system in the plane of ( C , 0  ). To achieve this, let us replace  by where the equal sign corresponds to the condition of stability boundary. What we want is to work out the relationship between C and 0  to describe the stability for the SMI. Let us consider the parameters appeared in Eq. (3.5 are two governing parameters that determine the stability boundary of an SMI system described by C and 0  . Therefore, it is very important to investigate how the two parameters influence the stability boundary. 37 In order to work out the stability boundaries in the plane of ( C , 0  ), a specific example is considered here to demonstrate how to determine the stable range of the SMI system using Eq. (3.5). We assume that the external target is located 0 0.25 Lm  away from the SL which is injected with a fixed current of 17  boundary is the stable region. In Fig. 3 Fig. 3-2, the following features of the stability boundary can be found: 1. The stability boundary shows periodic fluctuation with a period of 2 equivalent to a half wavelength movement of the external cavity.
2. The system is always stable at a weak feedback regime and may enter unstable when the feedback level is moderate or high feedback regime.
3. To achieve a stable status at a moderate or high feedback regime, we can either increase the injection current or choose a long external cavity.   Now, let us study the features of the SL output power (below we will call it as an SMI signal) obtained by the LK model at the different region defined in Fig. 3 Fig. 1-4(a), for the purpose of comparison, Fig. 3-4 presents the SMI signals predicted respectively by the LK model shown from Figs. Fig. 3-4, the two SMI signals are obtained under the same operation condition, i.e., the same C value. According to the LK model, obviously, only Fig. 3-4

3-4(b)-(e) and the existing SMI model from Figs. 3-4(g)-(j). In each row of
shows a stable SMI signal which can also be described by using the existing SMI model shown in indicates the SMI is stable in Fig. 3-3. In the region with 41 1.8 8.4 C  , simulations using the LK model shows that the SMI signal contains a high frequency oscillation close to the relaxation oscillation frequency of the solitary laser. Figures 3-4(c) and 3-4(d) give the two SMI signals at the semi-stable region, which are more complicated than Fig. 3-4(b). Hence, the behaviors described by the LK model are different from the ones by the existing SMI model resulting from the stationary solutions of the LK model. It is very interesting to observe that, even for a complicated waveform shown in Figs. 3-4(c) and 3-4(d), the movement information of the target is still visible. This is why we call the region 1.8 8.4 C  as the semi-stable region. With the aid of signal processing technology, the system operating at the semi-stable region can also be used for sensing and measurement.
In order to achieve this, the SMI waveform needs to be investigated to reveal its relationship to the movement of the target. Also, due to the limit in the rising time of the photodiode (PD) packaged at the rear of the SL, it may not be able to detect the details of the high frequency SMI waveform in the semi-stable region, and the SMI signal observed will be a distorted version of the high frequency waveform. A complete theoretical model is required to describe the influence of the limited bandwidth of the PD on the high frequency SMI waveform with the aim to detect the movement of the target from the distorted SMI waveform. Obviously, extensive work is required and could be an interesting topic for future research. When

C 
, it is hard to see the vibration information from the SMI waveform, implying that the SMI system may lose its sensing ability. In this situation, the spectrum of laser is dramatically broadened, which is beyond the scope of this paper. Fig. 3-4(e) shows the SMI signal with 9 C  indicating that the SMI system is not suitable for sensing applications.
Note that the SMI model is derived from the LK equations by letting    and ( ) 0 dN t dt  . These conditions will no longer be valid when the system enters semi-stable or unstable region, e.g., the relaxation oscillation will become undamped [77,78]. In summary, for the system working in the semi-stable 42 or unstable region, the existing SMI model cannot be used, but we can still use the fundamental LK model to describe the system behaviour.

Critical Feedback Level Factor
Furthermore, from the stability boundary shown in Figs. 3-2 and 3 Step5: Repeat Step2 until all the combinations of L and J are chosen. Figure 3-5 shows the complete stability boundary constructed using the above computation procedures.  The shaded area in Fig.3-6 is the unstable area. The area beneath the thin line shown in Fig. 3-6 is guaranteed to be stable. Therefore a safe choice can be made under the thin line. From Fig. 3-6, it is interesting to notice that with the increase of L , the unstable area tends to fill the area above the thin line. Meanwhile, the stability boundary also tends to be described by the thin line.  45 which is the equation for the thin lines shown in Fig. 3-6. Therefore, a safe choice for obtaining a stable SMI signal can be made using (3.9) instead of Eq. (3.7).

Determining Critical Feedback Level Factor
In this section, we present a experimental method to determine critical C , and investigate how L and J influence critical C . When the SMI is in the stable region, the observed optical spectrum is clean showing the SL operating on only one single mode as shown in Fig. 3-8(a). When the system enters into semi-stable region, the relaxation oscillation (RO) of the laser becomes undamped. In this case, a subpeak corresponding to the RO frequency appears near to the main peak of the optical spectrum [77,78]. shows the optical spectrum observed when the system in the semi-stable region. As our spectrum analyzer has a relative low resolution with 0.002nm, it is not able to separate clearly the subpeak from the mean peak. However, it can still tell us the appearance of the RO of the laser with frequency about 2-4GHz, therefore determining the stability of the system changes from stable to semi-stable. One may argue that these subpeaks may be related to the external cavity modes. However, we can confirm that the subpeaks in Fig. 3-8(b) correspond to the RO frequency by the following calculation (The principle for identifying the RO frequency is based on Fig.3(2) in [77]).
As an example, we consider the case with C =4.6 to 7, which corresponds to the strong feedback regime. In this case, there are 5 external cavity modes according to the phase equation but only 3 of them are possible oscillating modes (the other 2 are anti-modes [31,73,79]). The frequency interval between two adjacent modes is  2 f c L [77].
For the case of Fig. 3-8(b . Therefore, the frequency interval is 21 3 . Obviously according to the above calculation, the location of the subspeaks on Fig.   3-8(b) does not correspond to the external cavity mode which distance to the main peak is 1.2GHz . In the following experiments, we varied the feedback level from weak to strong with the aid of the attenuator, the single mode spectrum displayed on the spectrum analyzer will thus change. Once the subpeaks were first observed from the spectrum, the SMI should be at the point of the critically stable. Then, we apply a tiny change to the attenuator by reversely rotating it 2 degrees. A stable SMI signal very close to the critical level can thus be obtained and we used the signal to calculate the parameter C by the method presented in [28]. The C calculated is approximately represented for critical C .
Based on above experimental method for estimating critical C , the influence of J and L respectively on the critical C are also investigated.    Based on the results obtained in Fig. 3-10 (a)-(e), we also simultaneously recorded the SMI signals using the detection circuit which bandwidth is 3MHz. Figure 3-11 shows the SMI signals correspond to the cases shown in Fig. 3-10 Fig. 3-10(b) to (e) and giving the distorted SMI signals or say untrue SMI signals shown in Fig. 3-11(b)-(e) which will negatively affect the sensing and measurement performance of the SMI, e.g., SMI based  measurement. Therefore, as we mentioned before in Section 3.

Summary
The stability of an SMI is investigated in this Chapter. It is found that, to achieve a stable SMI signal for sensing purpose under moderate or strong feedback level, we can either increase the initial external cavity length or the injection current to the laser. By monitoring the spectrum of the SMI, a critical optical feedback factor critical C can be determined approximately. Under the critical C , an SMI is guaranteed to be stable and the existing SMI model can exactly describe the waveform of an SMI signal. Furthermore, we presented another two regions on the plane of ( C , 0  ) called semi-stable and unstable with boundaries corresponding to the undamped relaxation oscillation and the chaos status respectively. We found that semi-stable region has potential applications on sensing and measurement but may require further signal processing technology. The results presented in this Chapter provide useful guidance for designing various SMI based sensing and instrumentations. 52

Chapter 4 Influence of the Injection Current on the Stability of an SMI
In this Chapter, the influence of another SMI parameter, i.e., the injection current, on the stability of an SMI is both numerically and experimentally explore. In fact, the idea of work presented in the chapter is originally inspired by the two recently published papers [61] and [40]. In [61], Green found that, with the variation of 0  within the range of 2 , the system is always stable for some values of the injection current J (or called band stable phenomenon) when L is short and  is weak, but no explanation was given in [61]. Later in 2013, [40] predicted that the always stable region found in [61] is due to the interaction between the excitation of relaxation oscillation (RO) of the lasers and L , and discovered that the region exists when the product of RO frequency ( RO v ) and  equals an integer. However, we notice that both [61] and [40] ignore the effect of nonlinear gain which is very important for describing and modelling an SL, and can provide a good agreement between numerical results and experimental results. It is also well known that the nonlinear gain has the effect of stabilizing the dynamics of an SL with an external target. 53 The results presented in this chapter show that the stable region is significantly wider than the region predicted previously, and such phenomenon is caused by the nonlinear gain inherently existed in the lasers. Furthermore, the relationship between the critical injection strength ( c J ), above which guarantees a stable SMI, and the other two important system parameters, i.e., C and L , is investigated. The results presented in this chapter are also useful guidance for designing a feedback phase independent stable semiconductor lasers with optical feedback.

Stability Boundary Described by J and 0 
In this chapter, , beginning with a revisit of the result obtained in [40], we perform the simulation with the same parameters' values adopted in [40] by using the complete system determinant derived in Chapter 2 of this thesis, where the nonlinear gain was included. Figure 4-1 (a) presents the result obtained in [40] which does not consider the nonlinear gain (see Fig. 1 in [40]). In Fig. 4-1(a), the gray region is the always stable operation region for an SMI system where 1, 2,...

RO v  
, and the dark shaded region is the unstable region. The vertical axis P is the injection strength which is defined as () th th P J J J  [40]. From Fig. 4-1(a), we can see that to achieve a stable SMI system, we need to carefully choose P to meet the condition of 1, 2,...

RO v  
. Figure 4-1 (b) shows the stability region obtained using the complete system determinant derived in Chapter 2 by considering the nonlinear gain. Clearly, comparing with Fig. 4-1(a), the unstable region is significantly suppressed, and the critical pump strength c P (dotted line in Fig. 4-1(b)), above which guarantees an stable SMI system, is slightly lower than the condition obtained in [40], i.e., 1 To verify to our result, i.e., Fig. 4-1 (b), we intentionally choose three values of P on Fig. 4-1 (b), respectively are 2 P  , 0.8 P  , and 0.1 P  , which correspond to the case that SMI is unstable, stable and unstable in Fig. 4-1(a). Using these three values of P and keeping all the other parameters as the same as used in Fig. 4-1, we numerically solve the LK equations by letting the feedback phase varying as the same trace as shown in Fig. 1-4(a), and Fig. 4-2 shows the simulation results. From Fig.   4-2, we can see that when 2 P  , the SMI is stable which proves that our result is correct.
With the result shown in Fig. 4-1(b), it is naturally to think about if and how c P is associated with other two system parameters, i.e., C and L , because, for an SMI system, it is usually preferred to be operated under a relatively high feedback strength and long external cavity, i.e., can see that the nonlinear gain plays an important role in describing the dynamics of the SMI, and it can greatly suppress the unstable region of the SMI. Therefore, such important factor should not be ignored and should be paid attention. To illustrate the importance of the nonlinear gain, we here investigate how the nonlinear gain will influence the stability of the SL with EOF and thus revealing it importance. Let us firstly review some background knowledge of how to determine to stability of the system as discussed in Chapter 2. With the system determinant in Eq. (2.7), we are able to work out the locations of zeros of () Ds on the S-plane. The zeros of () Ds are defined as the roots of ( ) 0 Ds  , which are usually complex numbers and can be found using various techniques. Note that for a fixed set of parameters, Equation (2.7) has multiple roots. Among the roots, the right most root (denoted as 0 s ) can be used to determine the stability of the system. The system is stable if 0 s lies in the left side of the S-plane. The stability limit is reached when 0 s is on the imaginary axis. The variation range of  is from 0.00 to 1.50 with 31 samples. All the other parameters are adopted from Table 1  With the locus of 0 s , we are able to present the stability limit in a two dimensional plane of  and  (shown as in Fig. 4-4)  . For each of the parameters, we take 200 samples by equally-spaced sampling. From Fig.4-4, we can see that, with the increase of  , the value of  for guaranteeing a stable system also increases. This phenomenon just coincides with the results show in the previous chapter as well as in [70]. When  increases, the relationship of  and  changes from linear to distorted while keeps the size of the unstable region almost unchanged.

Critical Injection Current
In this section, we firstly investigate the band stable phenomena for the case with a fixed initial external cavity length. Figure 4-5 shows the stability region on the plane of ( P , 0  ), which is obtained by using () Ds derived in Chapter 2 with six different 58 values of C but for a fixed 0.15 Lm  . Note that, as 0  is directly related to the movement of the external target for an SMI sensing system, we are more interested in how 0  influences the stability. Hence we choose to use 0  to represent the x-axis in the plane rather than s  according to Eq. (1.19).
From Fig. 4-5, it can be seen that the band stable phenomena occurs when the system operates at moderate or high feedback levels (e.g., 1 C  ). Let's call the band stable related region as the feedback-phase-dependent stable areas. Obviously, such region occupies more and more space in the plane of ( P , 0  ) with the increase of C .
Moreover, comparing with Fig. 4-1(a), the unstable region is significantly suppressed due to the inclusion of the nonlinear gain in () Ds presented in chapter 2. Then, we study the cases with a fixed C but varying L . Figure 4

Determining Critical Injection Current
We make the following experiments to verify the simulation results shown in Fig. 4-7.
The experiments are carried out with the setup shown in Fig. 3-7. The SL used in the experiment is still HL8325G and its temperature is maintained at and maximum operating injection current of max 120 J mA  . The laser focused by a lens hits a mirror surface glued on a loudspeaker. The loudspeaker is driven by a sinusoidal signal with 220Hz and peak-peak voltage of 400mV. This harmonic vibrating target can cause the feedback phase varying from 0 to 20 . The stability of the system is monitored by an optical spectrum analyzer (OSA). The detail approach for monitoring the stability can refer to Section 3.2 in this thesis. An attenuator is inserted in between the beam splitter (BS) and the loudspeaker in order to adjust the feedback level and thus C , and a translation stage is used to hold the loudspeaker and vary the external cavity length L .
The first group of experiments for investigating the relationship between c P and C for a fixed 0.15 Lm  follows the procedures below:  Step1: Set the injection current to a high value 100 J mA  , that is 1.5 P  . 61  Step2: While keeping the SMI system stable, adjust the attenuator so that C reaches 4.5 (note that the value of C can be obtained from the waveform of a SMI signal using the method reported in [17,35]).
 Step3: While keeping C unchanged based on the observed SMI signal waveform, gradually decrease the injection current until the system just enter the critical stable state, then record the injection current at this moment, c P .
 Step4: Adjust the attenuator to decrease the C by a step of about 0.2, and then repeat the steps from 1-3.
With the procedures above, we are able to obtain a number of pairs of

Summary
In this chapter, we investigated the influence of the injection current on the stability of an SMI with a moving external cavity. Numerical calculations on the system determinant of the LK equations are performed. We found that the relaxation 62 oscillation has a strong dependence on the feedback phase when the injection current is low and the external cavity is short, especially for the cases when an SMI is operated at moderate or high feedback cases. Both simulation and experiment show that there is a critical injection current c P above which the SMI can be feedback-phase-independent stable. This critical c P is determined by C and L .
The results and the method presented in this letter are helpful for designing a stable SMI sensing system. 63

Measuring the Linewidth Enhancement Factor
It is well known that semiconductor lasers (SLs) play a key role in the emerging field of optoelectronics, such as optical sensors, optical communication and optical disc systems. For these applications the linewidth enhancement factor (LEF), also known as the alpha factor or  -parameter, is a fundamental descriptive parameter of the SL that describes the characteristics of SLs, such as the spectral effects, the modulation response, the injection locking and the response to the external optical feedback [24,26]. Therefore, the knowledge of the value of the LEF is of great importance for SL based applications.
It has been proved that LEF exhibits a strong dependence on the combination of the refractive index  , gain G and the injected carrier density j N , and is defined by the following equation [27,[80][81][82]: where  is the complex electric susceptibility, superscripts "R" and "I" denote the real and imaginary parts of  , and  and c are the angular optical frequency of the SL and the speed of light respectively. 64 Over the past three decades, various techniques were developed for measuring the alpha factor. These techniques can be mainly classified [80]  were found by [24,26] for Fabry-Perot (FP) lasers which vary from 4.6 to 8.2.
Besides the above basic laser structures, the values of 3.9 and 4.4 were reported respectively by [86] and [37] for channeled substrate planar (CSP) lasers.
3) Using the optical injection technique: For DFB lasers, the value of 5.5 0.6  for the alpha factor was obtained by [98]. In 2003, also for DFB lasers, [99] calculated the values of the alpha factor as 4 obtained for DFB lasers [27]. A wide range from 1.8~6.8 was obtained for FP lasers [27,102].
The above techniques can also be classified into two categories based on the amount of injection current to the SL. For the first category, the injection current is below the threshold and in this situation, the LEF is regarded as a material parameter and is measured according to the definition of the LEF in Eq. (5.1). In the second category the injection current is above or close to the threshold and a mathematical model for measuring the LEF was developed from the rate equations of the SL. In this situation, the LEF is considered as a model parameter or effective parameter which is detached from its physical origin to a certain extent [29,80]. Among the techniques in the second category, the optical feedback method, i.e., the SMI, is an emerging and promising technique which does not require high radio frequency or optical spectrum measurements, thus providing ease of implementation and simplicity in the system structure [4,27].
Based on the SMI, various methods were proposed for measuring the LEF. In 2004, Yu et al. [27] proposed an approach which can obtain LEF by geometrically measuring the SMI signals' waveform. However, this approach requires the SMI signal to have zero crossing points, which means the optical feedback level C falls within a small range, i.e., 13 C  which is difficult to achieve for some types of lasers. Additionally, the movement trace of the target must be away from and back to the SL at a constant speed, which is also difficult in practice. In subsequent years, several approaches [12,15,19,103] for measuring the LEF were developed, and these approaches are mainly based on the numerical optimization for minimizing the cost functions in parameters. Similarly to [27], these methods are also restricted to certain feedback levels, e.g., approaches in [15,19,103] require a weak feedback level, i.e., 1 C  , and the method in [12] requires a moderate feedback level, i.e., 1 4.6 C


. Furthermore, these methods are quite time consuming due to the large data samples to be processed. Recently, two different approaches [18,28] were developed for measuring the LEF over a large range of C , but they still face the problem of requiring a large amount of computation time.
Note that the above methods [12,15,18,19,27,28,103] all assume that the SMI signals they used for measuring the LEF are "true", or in other words stable. However, Chapter 3 (Section 3.3) has shown that when the SMI operates in the semi-stable or 66 even unstable (chaos) region, the SMI signals detected by the PD packaged at the rear of the SL still look like "stable" SMI signals. Such a phenomenon is caused by the limit in the rising time (also known as cut-off frequency, or bandwidth) of the PD packaged at the rear of the SL, as well as the circuit used for detection. The PD may not be able to detect the details of the high frequency (close to the relaxation oscillation frequency of the SL) SMI waveform in the semi-stable or unstable regions, and the SMI signal observed is actually a distorted version of the high frequency waveform. Therefore, if the above methods [12,15,18,19,27,28,103] are applied to the distorted SMI signals to measure the LEF, it may induce significant measurement error. Furthermore, the methods mentioned above are either restricted to a certain optical feedback level or are quite time consuming, which will hinder the use of SMI for embedded and industrial applications.
In this chapter, starting from investigating the influence of the bandwidth of the detection components (including PD and the detection circuit) on the SMI signals, we firstly study the measurement error of the LEF at the semi-stable region, to determine whether it has a significant impact on the results. Then based on our previous stability theory (explained in Chapters 3 and 4), a set of external parameters is determined for designing a stable SMI system to measure the LEF. Finally, using the stable SMI system we designed, a simple method for measuring the LEF is proposed and demonstrated in order to lift the above mentioned limitations of previously proposed methods. This simple method is based on the relationship between the light feedback phase and the output power from the well known LK equations. It was found that the LEF can quite simply be measured by the power value overlapped by two stable SLs' output power, i.e., an SMI signal, under two different optical feedback strengths. This high frequency is close to the relaxation oscillation frequency of the SL (usually 1.5-4GHz). Interestingly, the slow-varying envelopes are similar to the SMI signal characterized by the same fringe structure. It can be seen from Fig. 5-1(b), that there are nearly 6 fringes corresponding to the peak-peak displacement ( 0 3 ) of the target. That is, each fringe in the semi-stable SMI signal also corresponds to a target displacement of 0 2  , and hence the semi-stable SMI signal can also be used to measure the displacement with the same resolution as the normal SMI operating in the stable region.

Semi-stable Region
However, taking into account of a practical SMI system, the bandwidth (BW) of PD packaged at the rear of the SL for detection is usually under 1GHz, which is in the range of about 200MHz to 800MHz. Furthermore, the BW of the detection circuit is also 68 usually less than 1GHz. For example, the BW of our detection circuit shown in Fig. 3-7 is 3MHz. Therefore, obviously, the detection part of a practical SMI system is actually equivalent to a low pass filter on the real SMI signal, as shown in Fig. 5-1(b). This will dramatically influence the waveform of the SMI signal. Figures 5-2(b)-(f) show the filtered results by applying the detection circuit with different BWs. That is, a low pass filter is used with different cut-off frequencies, denoted as cut off f  , to process the SMI signal in Fig. 5-1(b). The range of cut off f  is from 600MHz to 3MHz, which should cover most practical situations. Most  measurement approaches [12,15,18,19,27,28,103] are performed based on the waveform of the SMI signal. Thus, the distorted or filtered SMI signals due to different BWs of the detection components will work out different results of  . Here, we adopt the approach in [27] as an example to see how the filtered SMI signals 69 introduce the measurement error. The approach in [27] can obtain  by geometrically measuring the SMI signals' waveform, and this approach requires the characteristic points of the SMI signal, i.e., zero crossing points, to perform the measurement. Figure 5-3 shows a typical experimental SMI signal, where A and B are zero crossing points.  We also measured the LEF for other filtered SMI signals as shown in Fig. 5-2 using the method described in [27]. The estimated LEF are respectively 4.41, 6.89, 5.51 for . Therefore, it is very important to design a practical SMI system working in the stable region.

Designing a Method for Achieving a Stable SMI System
A practical stable SMI system can be designed based on the methods described in the 71 previous chapters. Here, a specific example is presented, which applies the method described in Chapter 3, to illustrate how to design a stable SMI system and thus to acquire stable SMI signals. A practical SMI system can be represented by the experimental setup shown in Fig. 3-7, which we re-draw here as a simplified version shown in Fig. 5-5.     Step4: Once the system enters into the semi-stable region, the relaxation oscillation (RO) of the laser becomes undamped. In this case, a subpeak corresponding to the RO frequency appears near the main peak of the optical spectrum, where an example is shown in Fig. 5-6.  the LEF. Note that the value of C can be obtained from the waveform of an SMI signal using the method reported in [17,35].

A New Method for the LEF Measurement
Based on the stable SMI signals acquired using the above stable SMI system we designed, we now develop a new method for the LEF measurement. The proposed method is simple and eliminates the feedback level restrictions used in previous methods. another important parameter i.e., optical feedback level C , determines the shape of g . For a target subjected to a simple harmonic vibration, g is symmetrical with sinusoidal-like fringes when 1 C  . When 1 C  , g shows asymmetrical hysteresis sawtooth-like fringes.  To verify our proposed approach, we firstly carry out the computer simulations to generate SMSs. Without loss of generality, the external target is assumed to be subjected to a simple harmonic vibration, i.e.,    Table 5-1. In Table 5-1, we also present the  values calculated using the approach in [27]. Note that the approach in [27] is only valid when the feedback level is moderate, i.e., 13 C  and the value we choose for the results in Table 5-1 is 77 2.5 C  . Also from Table 5-1, we can see that even for the value of C falling in the range of 13 C  , the approach in [27] is still not valid when  is equal to or less than 1.0 because there is no zero crossing point of SMI signals, which is essential for utilizing the approach in [27]. From Table 5-1, it can be seen that our proposed method is more accurate and has wider practical utility. Our proposed approach is also verified by the experiment using the two stable SMI signals ( Fig. 5-8) obtained in Section 5.2 of this Chapter. Similarly to the measurement method described previously, we are able to obtain the measured  as

and
 as 4.05%, whereas  and  are respectively calculated as 3.01 and 6.8% using the approach in [27].

Summary
In this chapter, a stable SMI system is designed in order to measure the LEF. Then, a new and simple method for measuring the LEF by investigating the relationship between the light phase and power is demonstrated. The proposed method is superior over the existing methods due to the following two aspects: (i) the proposed method eliminates the feedback level restrictions used in previous methods, thus allowing the experimental design to be simpler, (ii). the LEF can be simply determined from the intersection point of two different SMI signals' waveforms, thus providing a fast and easy measurement technique for the LEF. 78

Chapter 6 Conclusion and Future Work
Self-Mixing Interferometry (SMI) is an emerging non-contact sensing technique for the measurement of various metrological quantities, such as absolute distance, angle, displacement, and velocity. An SMI system is composed of a semiconductor laser (SL) with a photodiode (PD) packaged at the rear of the SL, a lens, an external target and a data processing unit. When the external target moves, a small portion of light reflected re-enters the internal cavity of the SL, leading to the modulation in both the amplitude and frequency of the SL output power. The modulated power is detected by the PD as an SMI signal, which is fed to the data processing unit for extracting useful information related to both the external target and the SL itself [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. However, when the system enters into the unstable state, the performance of the above mentioned applications will be severely degraded as the SMI signal is contaminated by a fast oscillation component. Meanwhile the existing SMI model will be no longer valid.
In this thesis, starting from the well known Lang and Kobayashi (LK) equations, and by examining the root locus of the system determinant with respect to all the system parameters, including the feedback strength, external cavity length and injection current, a comprehensive stability analysis for the SMI was presented. In addition, a simple method for measuring the linewidth enhancement factor (LEF) is proposed in this thesis based on the stable SMI signals. 79

Research Contributions
According to the extensive computer simulations and experimental results presented in this thesis, the following contributions can be stated:

Suggested Future Research Topics
Subsequent to the investigations described in this thesis, conducting research into the following topics would be helpful for enhancing the stability of the SMI: 1. The stability boundaries obtained in Chapter 2 are based on minimum linewidth conditions, i.e., the feedback phase 0 arctan( ) s       , and hence they can only distinguish "the possible stable" and "the unstable" areas.
However, when increasing the injection current, a semiconductor laser experiences a red shift in wavelength corresponding to ~0.5GHz/mA [105].
One can conclude that the above feedback phase is extremely sensitive to, and wildly increases as a function of the injection current, encompassing several intervals of 2 . In practice, control over the precise feedback phase is therefore impossible and stability should always be checked for all possible phase values. As a task for future work, studies are required to investigate how the stability limits presented in Chapter 2 vary with respect to changes of feedback phase over the entire range of 2 , based on which a set of limits guaranteeing the stability of laser operation can be obtained.
2. It is shown in Chapter 3 and 5 that due to the bandwidth limit of the detection components, it may not be possible to detect the details of the high frequency SMI waveform in the semi-stable region, and the SMI signal observed will be a distorted version of the high frequency waveform which will strongly reduce the accuracy of classical measurement systems based on an SMI, e.g., displacement, vibration, speed and absolute distance measurements.
Therefore, a complete theoretical model is required to describe the influence of the limited bandwidth of the detection components on the high frequency SMI waveform with the aid of the advanced signal processing technique, which apparently requires extensive work and could be an interesting topic for 81 future research.
3. Finally, it should be pointed out that the LK model itself is also derived under certain approximations, which may lead to discrepancies between the experimental results and numerical results. The work presented in this thesis is based on the LK model and its effectiveness is surely dependent on the accuracy of the LK model. Therefore, the limitations of the LK model due to the approximations in the process of its derivation should be investigated in the future.