Abstract

An optical feedback laser diode (OFLD) operating in period-one oscillation (POO) with a moving external target is investigated by exploring its potential sensing capability. First, the modeling of an OFLD-POO sensing system is presented. An analytical expression is derived for OFLD-POO sensing signal, from which a new displacement measurement method is developed. The proposed sensing model is verified by the well-known Lang-Kobayashi equations used to describe the dynamic behavior of a laser with optical feedback. Then, an experimental OFLD-POO system is built in order to demonstrate an application example for displacement sensing. The measurement results show that the OFLD-POO sensing system can achieve displacement measurement with large measurement range, high sensitivity, and resolution.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical feedback interferometry (OFI) or Self-mixing interferometry (SMI) is a promising non-contact sensing technology, which uses the self-mixing effect that occurs when a fraction of the light back-reflected or back-scattered by an external target reenters the laser cavity [1,2]. In this case, the steady-state intensity of the lasing light is modulated by the external optical feedback. A typical OFI system consists of a laser diode (LD), a photodiode (PD) packaged at the rear of LD, a lens and an external target [3–6]. We call the laser diode in this case as optical feedback laser diode (OFLD) and the modulated intensity as OFLD signal. As a preferred, minimum part-count scheme useful for engineering implementation, various OFLD-based applications have been developed in the industrial and laboratory environment, such as measurement of displacement [7,8], vibration [9], velocity [10], alpha factor [11,12], Young’s modulus [13], strain [14], acoustic emission [15], acoustic field imaging [16], etc.

Most of the OFLD-based applications are based on the stable OFI mathematic model, which is derived from the steady-state solution of the Lang and Kobayashi (L-K) equations [17], or from the classical three-mirror model [2], by assuming that the OFLD system operates in stable mode, i.e., both the electric field and carrier density in an LD with a stationary external cavity can reach a constant state after a transient period. However, many works [18–20] have demonstrated that an OFLD system with a stationary external target may be unstable under some certain operation conditions, such as in periodic, quasi-periodic or chaotic oscillation state. We studied OFLD signals with a moving target in different conditions and our recent works [21,22] shows that such sensing system may produce different types of OFLD signals which are classified into three regions defined in [21], i.e. stable, semi-stable and unstable region under different feedback level factors C. C is defined as: C=κτ1+α2/τin, where κis the optical feedback strength, τand τin is the laser round-trip time in the external and internal cavity respectively, and αis the linewidth enhancement factor [2]. In stable region, for a stationary target, the laser intensity will be constant after the transient state, whereas in semi-stable and unstable region, relaxation oscillation will be undamped and the laser may enter period-one, quasiperiodic or chaotic oscillation under different feedback strength. The work [21,22] discovers that an OFLD can easily and sometimes inevitably enter the period-one oscillation, e.g. for the case when the feedback level factor C is around 2.5, which is quite commonly reported for the moderate feedback regime in the literature. In this case, the OFLD signals (laser intensity) generated by an OFLD system cannot be described by the existing OFI model. The laser intensity is modulated in a quite complicated form and contains very high-frequency components. In order to differentiate the conventional OFLD signals, we name the laser intensity signals generated by the OFLD system operating in period-one oscillation (POO) as OFLD-POO signals. The work in [23] also depicts some high-frequency components in the sawtooth-like OFLD signals, but the LD used in [23] still operates in conventional self-mixing conditions rather than in POO state. Usually, the PD packaged inside an LD is used to detect the laser intensity, but it has limited bandwidth less than 1GHz. However, the frequency components contained in an OFLD-POO signal can even higher than several GHz [21,22]. Hence, using existing OFI configuration for detection of OFLD-POO signal, many frequency components are missing. The work in [22] has experimentally revealed the features of such signals and indicated the possible sensing application but it did not make analytic expression on the signal waveform. As an OFLD-POO signal contains rich information associated with the laser and its external target, it is of great interest and significance to investigate the signal features by modelling the signal waveform, from which we can develop a suitable signal processing algorithm to retrieve the useful information contained in the signals.

In this work, an OFLD operating in POO with a moving external target is investigated by exploring its potential sensing capability. Starting from the L-K equations, we firstly derive an approximate analytical model to describe the OFLD-POO signals. The numerical simulations for original L-K equations are then conducted to verify the derived analytical expression. Finally, a displacement sensing method by simultaneously using the time-domain OFLD-POO waveform and relaxation oscillation (RO) frequency contained in this signal is proposed. From both simulation and experiment, the feasibility of proposed displacement sensing method has been verified.

2. Modelling and analysis

2.1 Theoretical modelling

The behavior of an LD with optical feedback can be described by the well-known L-K equations [17], as shown in Eqs. (1)-(3). Here, E(t) is the amplitude of the electric filed, ϕ(t) is the phase of the electric field, and N(t) is the carrier density. E2(t) is usually considered as the laser intensity or power, denoted by P(t)=E2(t).

dE(t)dt=12{G[N(t),E(t)]1τp}E(t)+κτinE(tτ)cos[ω0τ+ϕ(t)ϕ(tτ)].
dϕ(t)dt=12α{G[N(t),E(t)]1τp}κτinE(tτ)E(t)sin[ω0τ+ϕ(t)ϕ(tτ)].
dN(t)dt=IN(t)τsG[N(t),E(t)]E2(t).
where, G[N(t),E(t)]=GN[N(t)N0][1εΓE2(t)] is the modal gain per unit time. In the following derivation, the non-linear gain is ignored [18]. There are 3 controllable parameters for a specific OFLD, i.e. injection current (I), feedback strength (κ) and external cavity laser round-trip time (τ). The physical meanings and typical values of the other symbols appearing in Eqs. (1)-(3) are listed in Table 1, which are adopted from [19,24].

Tables Icon

Table 1. Meanings of the symbols in the L-K equations

Starting from the L-K equations, we aim to get an approximate analytical expression for OFLD-POO signal (i.e.,E2(t)). Based on the numerical simulation of L-K equations and the work in [18,20], when an OFLD operates in POO, E(t), N(t) and ϕ(t) can be written in the form of below:

E(t)=E¯+ΔEcos(ωrt+θe).
N(t)=N¯+ΔNcos(ωrt+θn).
ϕ(t)=(ω¯ω0)t+(Δϕ/2)cosωrt.
where, E¯, N¯, and ω¯ are respectively the mean values of E(t), N(t) and optical frequency, ΔE, ΔN and Δϕ/2 are the modulation amplitudes, ωris the RO angular frequency when the OFLD is in POO, θe andθnare the initial phase of E(t)andN(t) respectively. We will need to express the mean values and the modulation amplitudes as well as the initial phases using the parameters related to the LD and its external cavity listed in Table 1.

RegardingE¯andN¯, they can be determined by solving Eqs. (1)-(3) through setting dE(t)/t=0,dN(t)/t=0,dϕ(t)/dt=ωsω0 [18], as shown below:

E¯2=IN¯/τsGN(N¯N0).
N¯=N0+1GNτp2kcos(ωsτ)τinGN.
ωs=ω0k1+α2τinsin(ωsτ+arctanα).
where ωsis the optical frequency when the OFLD is stable. Next, we look into the analytical expressions for ΔE, ΔN, Δϕ, θe and θn. With a similar treatment as done in [21], we setωrτ=(2m+1)π. Note that, in the case of ωrτ=2mπ, the OFLD will not be in POO [25].

Substituting E(t) andN(t) in Eq. (3) by Eq. (4) and Eq. (5), and neglecting the terms withΔEΔN, ΔE2 and ΔN2(as ΔE/E¯<<1 and ΔN/N¯<<1) [18], we have:

2GN(N¯N0)E¯ΔEcos(ωrt+θe)=IN¯/τsGN(N¯N0)E¯2+ΔNωr2+(1/τs+GNE¯2)2cos{ωrt+θnarctan[(1/τs+GNE¯2)/ωr]π/2}.
As Eq. (10) is an identical equation, letting the corresponding terms in both sides are equal, we have:
2GN(N¯N0)E¯ΔE=ΔNωr2+(1/τs+GNE¯2)2.
θe=θnarctan[(1/τs+GNE¯2)/ωr]π/2.
IN¯/τsGN(N¯N0)E¯2=0.
Interestingly, it can be found that Eq. (13) is the same as Eq. (7). In a similar way, substituting E(t), N(t) and ϕ(t)in Eq. (1) by Eqs. (4), (5) and (6), we can get:
-ωrΔEsin(ωrt+θe)={12[GN(N¯+ΔNcos(ωrt+θn)-N0)1τp]}[E¯+ΔEcos(ωrt+θe)]+κτin[E¯-ΔEcos(ωrt+θe)]cos[ωsτ+Δϕcos(ωrt)].
After expansion and reorganization of Eq. (14), we have:
-ωrΔEsin(ωrt+θe)=12[GN(N¯-N0)1/τp]E¯+κτinE¯cosωsτJ0(Δϕ)+12GNE¯ΔNcos(ωrt+θn)+12[GN(Ns-N0)1τp]ΔEcos(ωrt+θe)2κτinE¯sinωsτJ1(Δϕ)cosωrt.
where J0(Δϕ)and J1(Δϕ)are the 0th-order and 1st-order Bessel function respectively. Note that higher harmonic components of ωr are neglected [18]. Then, we have:
12[GN(N¯-N0)1/τp]+κτincosωsτJ0(Δϕ)=0.
ωrΔEsin(ωrt+θe)=2κτinE¯sinωsτJ1(Δϕ)cosωrt12[GN(N¯-N0)1/τp]ΔEcos(ωrt+θe)12GNE¯ΔNcos(ωrt+θn).
After expansion and reorganization of Eq. (17), we have:

ωrΔEcosθesinωrt+ωrΔEsinθecosωrt={12[GN(N¯-N0)1/τp]ΔEsinθe+12GNE¯ΔNsinθn}sinωrt+{12[GN(N¯-N0)1/τp]ΔEcosθe12GNE¯ΔNcosθn+2κτinE¯sinωsτJ1(Δϕ)}cosωrt.

As Eq. (18) is an identical equation as well, letting the corresponding terms in both sides are equal, we have:

ωrΔEcosθe=12[GN(N¯N0)1/τp]ΔEsinθe+12GNE¯ΔNsinθn.
Inserting Eq. (12) into Eq. (19) yields:
cotθn=1ωrωr2GN2E¯2(NsN0)(1/τs+GNE¯2)(κ/τin)J0(Δϕ)cosωsτ1/τs+GNE¯2+(κ/τin)J0(Δϕ)cosωsτ.
Similarly, substituting E(t), N(t) and ϕ(t)in Eq. (2) with Eqs. (4), (5) and (6), we get:
ω¯ω0=ακτincosω¯τJ0(Δϕ)κτinsinω¯τJ0(Δϕ).
12αGNΔNcosθn2κτincosω¯τJ1(Δϕ)=0.
Δϕωr=αGNΔNsinθn.
Combining Eqs. (16) and (8), we have:
cosω¯τJ0(Δϕ)=cosωsτ.
With the Taylor expansion of the Bessel functions, we have the approximations: J0(Δϕ)1Δϕ2/4, and J1(Δϕ)Δϕ/2Δϕ3/16.Solving Eqs. (20), (22), (23) and (24), we get:
cotθn=1ωrωr2GN2E¯2(NsN0)(GNE¯2+1τs)κτincosωsτGNE¯2+1τs+κτincosωsτ.
Δϕ=2ωrτincotθn2κcosωsτωrτincotθnκcosωsτ.
ΔN=2ωrαGNsinθnωrτincotθn2κcosωsτωrτincotθnκcosωsτ.
Inserting Eq. (27) into Eq. (11) leads to
ΔE=ωrαGNsinθnωrτincotθn2κcosωsτωrτincotθnκcosωsτωr2+(GNE¯2+1/τs)2GN(N¯N0)E¯.
Sinceωr>>1/τs+GNE¯2 [20], Eq. (28) can be simplified as:
ΔE=ωr2αGNsinθnωrτincotθn2κcosωsτωrτincotθnκcosωsτ1GN(N¯N0)E¯.
Now, we can have the expression for E2(t)orP(t)=E2(t) as below:
P(t)=E2(t)=E¯2+2E¯ΔEcos(ωrt+θe)+ΔE2cos2(ωrt+θe).
From Eq. (8), we have GN(N¯N0)=1/τp2(κ/τin)cosωsτ. In the case of κ<0.01, neglecting the terms with ΔE2, Eq. (30) can be approximated as follows:
P(t)=E¯2+2τpωr2αGNsinθnωrτincotθn2κcosωsτωrτincotθnκcosωsτ(1+2κτpτincosωsτ)cos(ωrt+θe).
where E¯2 is shown in Eq. (7). Inserting Eq. (8) into Eq. (7), in the case of κ<0.01, it can be rewritten as [1]:
E¯2=ES02+2(ES02+1GNτs)κτpτincos(ωsτ).
where ES02 is the laser intensity of the LD without optical feedback, which is determined by the injection current. Inserting Eqs. (7) and (8) into Eq. (25), we have
cotθn=τs(ωr2ωr02)κτincosωsτωr.
where ωr0 is the RO frequency of the LD without optical feedback, which is determined by the injection current. Inserting Eqs. (32) and (33), into Eq. (31) and considering the case of κ<0.01, we obtain:

P(t)=ES02+2(ES02+1GNτs)κτpτincos(ωsτ)+2τpωrαGNωr2+[τs(ωr2ωr02)κτincosωsτ]2cos(ωrt+θe).

Therefore, Eq. (34) is the laser intensity when the OFLD system is in period-one oscillation. When the external cavity is moving, it will cause a varying feedback phase ϕF (ϕF=ωsτ=2ωsL/c, where c is the vacuum light speed) [2,11–13] as well as ωr [24]. We re-write Eq. (34) as below to include displacement sensing:

ΔP(L,t)=P(t)ES02=ΔPOFLD(L)+ΔPOFLDPOO-Ep(L)cos[ωr(L)t+θe].
where,

ΔPOFLD(L)=2(ES02+1GNτs)κτpτincos[ϕF(L)].
ΔPOFILDPOOEp(L)=2τpωr(L)αGNωr2(L)+{τs[ωr2(L)ωr02]κτincos[ϕF(L)]}2.

Equation (36) is the conventional OFLD signal reported in the literature [2,11,12]. This signal is in the form of sinusoidal (at weak feedback case) or sawtooth-like (moderate or above feedback level) waveform when the external cavity length Lchanges. Each periodical variation (called a fringe) corresponds to a displacement with half laser wavelength (λ0/2) [2]. For a varying external target, the external cavity length can be expressed as L=L0+ΔL, where L0 is the initial external cavity length, and ΔL is the displacement.

We are more interested in the last term appeared in Eq. (35). Since this term also contains the information of displacement ΔL, we can make use of it to achieve displacement sensing. The RO frequency of an OFLD has been investigated and an approximate analytical expression for RO frequency has been derived for an OFLD is in steady state [20]. The analysis on the RO frequency and its application for measuring linewidth enhancement factor of the LD has been discussed in [26]. When the OFLD operates in POO, the expression is invalid and the RO frequency is different from that of the OFLD in steady state. In this case, the frequency of POO can be treated as the RO frequency, i.e. ωr contained in Eqs. (35) and (37), as in our previous work [24], which shows that ωrchanges with displacement ΔL in the form of a saw-tooth waveform with a period of λ0/2 when the OFLD system is in period-one oscillation. Thus, we have ωr(ΔL)=ωr(ΔL+λ0/2). Denoting the maximum and minimum value of ωrwithin a period by ωrmax and ωrminrespectively, the expression for ωrwithin one period (0<ΔL<λ0/2) can be expressed as:

ωr(ΔL+Nλ02)=ωr0+ωr-offset+2(ωrmaxωrmin)λ0ΔL.
where Nis an integer, ωr0is the angular RO frequency of the solitary laser diode which is determined by injection current density, ωr-offset is the offset value which is determined by the initial external cavity length and feedback level, and ωr0and ωr-offsetare constant when the operation conditions of an LD are fixed. Equation (37) deserves a close examination. Since both ωrand κτincos[ϕF(L)] are periodical with same period of λ0/2for a varying external cavity length, we have:

ΔPOFLD-POOEp(L+Nλ02)=2τpωr(L+Nλ02)αGNωr2(L+Nλ02)+[τs(ωr2(L+Nλ02)ωr02)κτincos(ϕF(L+Nλ02)]2.=ΔPOFLD-POOEp(L)

It can be seen that ΔPOFLDPOOEp(L) also periodically change with displacement ΔL byλ0/2 . Figure 1(a) shows an example of the relationship between the angular RO frequency and displacement of external target when L0=24cm, I=1.3Ith and C=4.5, where Ithis the threshold injection current. Figure 1(b) shows the waveform of ΔPOFLD-POO-Ep(L) for a varying external cavity with the same operation conditions.

 figure: Fig. 1

Fig. 1 (a) Relationship between RO frequency and displacement of external cavity, (b) waveform of ΔPOFLDPOO-Epfor a change of external cavity length when L0=24cm, I=1.3Ith andC=4.5.

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In the following, we only consider the last term in Eq. (35), which is called OFLD-POO signal denoted byΔPOFLDPOO(L,t), and re-write it as below:

ΔPOFLDPOO(L,t)=ΔPOFLD-POOEp(L)cos[ωr(L)t+θe].

Now, we can describe that the amplitude and frequency of an OFLD-POO signal is modulated by the displacement to be measured. Both the envelope (ΔPOFLD-POOEp(L)) and frequency of the signal contain displacement information. So we can explore a new sensing approach by using this signal. We denote the maximum and minimum values of ΔPOFLDPOOEp as ΔPOFLD-POOEp-max andΔPOFLDPOOEpmin, respectively. Based on the simulation and the work in [24], ωr-offset and ωrmaxωrmin is much smaller than ωr0 and we haveωrωr0 . In the case of κ<0.01, we get:

ΔPOFLDPOOEpmin2τpωr02αGN.
ΔPOFLDPOOEpmax2τpωr02αGN1+[2τs(ωr-offset+ωrmaxωrmin)]2.
Thus the peak-peak variation in ΔPOFLDPOOEp(L) (denoted by ΔPOFLDPOOEppp) is expressed as:
ΔPOFLDPOOEppp=2τpωr02αGN(1+[2τs(ωr-offset+ωrmaxωrmin)]2-1).
Making a comparison on the magnitude of ΔPOFILDPOOEp and ΔPOFLD as below:

ΔPOFLDPOOEpppΔPOFLDpp=2τpωr02αGN(1+[2τs(ωr-offset+ωrmaxωrmin)]2-1)4(ES02+1GNτs)κτpτinτin2ακτp(1+[2τs(ωr-offset+ωrmaxωrmin)]2-1).

Based on the simulation and the work in [24], ωr-offsetand ωrmaxωrmin are usually in the order of 108rad/s. e.g. in the case in Fig. 1, ωr-offset+ωrmaxωrmin6×108rad/s. Substituting the typical values in Table1 into Eq. (44), i.e.τin=8.0×1012s, τp=2.0×1012s, α=3τs=2.0×109s and κ=0.007 in the case in Fig. 1, we get ΔPOFLDPOOEppp/ΔPOFLDpp154 . This reveals that the envelope is much more sensitive to the displacement compared to using conventional OFLD signal, which motivates the use of the envelop for displacement detection.

Since the frequency of the OFLD-POO signal is also modulated by the displacement, we can detect a micro-displacement within λ0/2 through frequency variation. This enables us to achieve very high sensing resolution for displacement. Regarding this point, our work in [24] has demonstrated that the sensing resolution can be as high as λ0/1280 when the spectrum analyzer has a resolution bandwidth of 62.5 KHz. If we use envelop for large range displacement by fringe-counting with a resolution of λ0/2 and cooperate it with frequency sensing, a new displacement sensor can be designed with large measurement range, high sensitivity and resolution. The restriction on the speed of the target movement (denoted by ν) depends on the value of the RO frequency. Each OFLD-POO fringe should at least cover a few RO periods. Thus we can evaluate that the speed meet: ν<λ0fRO/2K, where K is number of the RO periods within one OFLD-POO fringe. Based on our simulation, to get a clear envelop of OFLD-POO signal, we need to have K>20. Theoretically, the dynamic measurement range is the maximum measurable displacement that guarantees OFLD always operating in POO during the displacement measurement. This limits the displacement measurement range. The work in [27] has investigated how POO region change with external target position in a certain injection current, from which we can estimate the dynamic displacement range is around several centimeters.

2.2 Verification for the proposed sensing model

Figure 2 shows the simulation results by numerically solving the original L-K Eqs. (1)-(3) and the approximate analytical expression of Eq. (40) when the external cavity length has a linear displacement of 3λ0, the initial external cavity length L0=24cm, and the injection current I=1.3Ith. Note that ωrτ23π in this case, which satisfies the requirement for Eq. (40). Figure 2(a) is the displacement; Figs. 2(b) and 2(c) show the numerically simulation results of L-K equations with C=2.8 and C=4.5; Figs. 2(d) and 2(e) are the simulation results of the Eq. (40) with the same operation conditions as those in Figs. 2(b) and 2(c). For the simulation of Eq. (40), we firstly obtain the relationship between ωrand ΔLby numerically solving the L-K equations as shown in Fig. 1(a). From Fig. 2, it can be concluded that the simulation results from the L-K equations and the approximate analytical expression match closely, demonstrating that the approximate expression is valid when the OFLD is in period-one oscillation. Note that, in Figs. 2(b) and 2(d), there are regions between the fringes where the OFLD is stable. Therefore, it is essential to ensure that the OFLD system is always in period-one oscillation with the change of external cavity length when using RO frequency for displacement sensing, as shown in Figs. 2(c) and 2(e). To guarantee the robustness of the so-called OFLD-POO regime, we need to choose suitable injection current the initial external cavity length. The method of how to choose them is detailed in our previous work [24].

 figure: Fig. 2

Fig. 2 Verification of OFLD-POO signal expressed in Eq. (40) by the numerical results from L-K equations. (a) displacement of the external target; (b) and (c) results from L-K equations with C = 2.8 and C = 4.5 respectively; (d) and (e) results from Eq. (40) when C = 2.8 and C = 4.5 respectively.

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3. Displacement sensing

As discussed in Section 2, the displacement of the external target can be retrieved by using both time-domain waveform (OFLD-POO signal) and the RO frequency contained in the signal. Let us express this displacement to be measured as below:

ΔL=ΔLt+ΔLf.
where ΔLt is measured by fringe counting with λ0/2 resolution by using the envelop in OFLD-POO signal, and ΔLfis measured by the RO frequency for the fractional displacement within λ0/2. As an example, Fig. 3 shows an OFLD-POO signal obtained by Eq. (40) as in Fig. 2(e) and the corresponding displacement information is contained in it.

 figure: Fig. 3

Fig. 3 OFLD-POO signal and displacement corresponding to ΔLtandΔLf.

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It can be seen that an OFLD-POO signal contains a few complete fringes and two incomplete fringes (or called fractional fringes) at the beginning and end part of the waveform. The measurement of the fractional fringe enable a displacement measurement with resolution higher than λ0/2. As described in Fig. 1, there is a certain linear relationship between ωr and the displacement within each fringe. As in Eq. (38), the maximum and minimum RO frequency within λ0/2 are ωrmax and ωrmin.We denote the RO angular frequency at the start and end point in a fractional fringe by ωr1 and ωr2respectively. Then the displacement corresponding to fractional fringes can be obtained as:

ΔLf=λ02ωr2ωr1ωrmaxωrmin.
where, for the first fractional fringe, ωr2=ωrmax, and ωr1is the RO angular frequency at the start point of the displacement, for the last fractional fringe, ωr1=ωrminand ωr2is the RO angular frequency at the end point of the displacement. Because ωrmax and ωrminare fixed for a LD with certain operation conditions, the displacement corresponding to fractional fringes can be determined by only measuring the RO frequency of the OFLD when displacement starts and ends. Note that the first and last fringes are always considered as fractional fringes.

A detailed procedure of retrieving the displacement from an OFLD-POO signal is presented as below. Assuming that the external target has a linear movement with ΔL=3λ0 as shown in Fig. 4(a), the corresponding OFLD-POO signal is shown in Fig. 4(b) which can be obtained by Eq. (40). The upper envelope of the OFLD-POO signal can be retrieved by upper-peak detection shown in Fig. 4(c). Then, applying differentiation on Fig. 4 (c), we can get pulse train shown in Fig. 4(d). By counting the number of pulses, we can obtain ΔLt=(N1)λ0/2 (here N is the pulse train number). It can be seen from Fig. 4(d) that there are 6 pulses. Thus ΔLt=2.5λ0. Then we need to obtain the fractional displacement determined by the first fractional fringe and last fractional fringe denoted by ΔLf1and ΔLf2 as shown in Fig. 4(b). In order to get ΔLf1and ΔLf2, the RO angular frequency at the start and end of the displacement should be obtained firstly. Before the displacement starts and after the displacement ends, for the stationary target, we get the laser intensity and then the RO angular frequency by applying FFT on the intensity as done in Fig. 1(a). Using this method, we get that the RO angular frequencies at these two time points are the same, which are1.399×1010rad/s. According to Fig. 1(a), ωrmax=1.436×1010rad/s and ωrmin=1.394×1010rad/s. Hence, based on Eq. (42), for ΔLf1, ωr2=1.436×1010rad/s, and ωr1=1.399×1010rad/s, whereas for ΔLf2, ωr2=1.399×1010rad/s, and ωr1=1.394×1010rad/s . Accordingly, we can get ΔLf1=0.441λ0, andΔLf2=0.059λ0. Thus, ΔL=ΔLt+ΔLf1+ΔLf2=3λ0, which is the same as the pre-set value.

 figure: Fig. 4

Fig. 4 Displacement reconstruction procedure from the OFLD-POO signal, (a) the pre-set displacement, (b) corresponding OFLD-POO signal, (c) upper envelope of OFLD-POO signal, (d) envelope differentiation.

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In the following, an experimental system is built and an application example of displacement sensing by using the proposed method is presented. Figure 5 shows the experimental system. The laser source is a single mode laser diode (Hitachi, HL8325G) with a wavelength of 830 nm, maximum output power of 40 mW, and threshold current of 35 mA. The laser diode is driven and temperature-stabilized by an LD controller (Thorlabs, ITC4001). The temperature of the LD is stabilized at 21 ± 0.01Co. A mirror is used as the external cavity which is affixed on the surface of a piezoelectric transducer (PZT) (PI, P-841.20) in order to provide sufficient optical feedback to the LD. The PZT actuator with an integrated displacement sensor is used to generate the displacement, and it has a displacement resolution of 9 nm with a traveling range up to 30 μm, which is driven by a PZT controller (PI, E-625). A high-speed digital oscilloscope (Tektronix DSA70804) with sampling rate 25 GS/s is used to observe OFLD-POO signals. An attenuator is used to adjust the optical feedback level of the OFLD system. As the OFLD-POO signals contain high-frequency components, high speed detection devices are needed. A beam splitter (BS) with splitting ratio of 50:50 is used to direct a part of the self-mixing light into the external fast PD, with a bandwidth of 9.5GHz, through the fiber port coupler. The detected OFLD/OFLD-POO signals are then collected and recorded by the digital oscilloscope.

 figure: Fig. 5

Fig. 5 Experiment setup, where LD, BS and PZT represent laser diode, beam splitter, and piezoelectric transducer.

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To make the system operate in period-one oscillation, the initial external cavity length and injection current should be properly chosen, which is referred to the work in [24]. We choose I=50mA,and L0=18cm in this work. Before staring the measurement, we firstly need to get the relationship between RO frequency (ωr) and external target displacement (ΔL) when the external cavity length changing by λ0/2.

we apply 30 nm displacement on the PZT each time, then use the Tektronix DSA 70804 oscilloscope to record the laser intensity waveform and thus obtain the corresponding peak frequency (denoted by fRO=ωr/2π) by using the FFT function provided by the oscilloscope [24]. Figure 6 shows an example of the laser intensity waveform and the corresponding spectrum. Thus, in each half wavelength, we will get about 13 points. Figure 7 illustrates the experimental results for the relationship between fROand displacement ΔL.

 figure: Fig. 6

Fig. 6 Experimental results of laser intensity when I=50mA, L0=18cm,and ΔL=300nm.

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 figure: Fig. 7

Fig. 7 RO frequency variation for different displacements when I=50mA, andL0=18cm.

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Then we apply a dynamic linear displacement on the PZT by applying a controlling voltage through the PZT driver. Figure 8 (a) is the PZT controlling signal applied on the PZT, Fig. 8(b) is the corresponding OFLD-POO signal. The displacement generated by the PZT (PI, P-841.20) can be read from its internal integrated sensor which indicates that the displacement is ΔL=2304nm.

 figure: Fig. 8

Fig. 8 (a) PZT controlling signal, (b) corresponding OFLD-POO signal.

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The experimental OFLD-POO signal in Fig. 8(b) is processed according to the procedure described in Fig. 4 to get the displacement. 5 complete fringes can be found which corresponds to ΔLt=5λ0/2. In order to obtain the displacement of the first and last fractional fringes, we need to obtain the RO frequency at the start and end points. By using the method in Fig. 6, the RO frequency at these two points are measured as 2.634GHz and 2.595GHz respectively. From Fig. 7, it can be seen the variation range of fRO in each period is from 2.579GHz to 2.645GHz. Thus, we can get ΔLf1=0.083λ0and ΔLf2=0.121λ0 based on Eq. (46). So ΔL=ΔLt+ΔLf1+ΔLf2=2.704λ0 (2244 nm forλ0=830nm), which matches the value from the integrated sensor in the PZT. We then repeat the experiments under the same laser operation conditions for different displacements. The results are shown in Table 2, from which, it can be found that the results from these two methods are consistent and the average difference is about 60nm. The difference between them may come from the accuracy of the PZT integrated sensor, laser wavelength shift induced by external optical feedback, and the accuracy of the measured RO frequencies.

Tables Icon

Table 2. Displacement results by PZT sensor and OFLD-POO

The resolution of this method is determined by the measured RO frequency and its variation range within λ0/2. In this experiment, the frequency resolution of the oscilloscope in the experiments is 62.5 KHz and the variation of fRO in each period is measured as 66 MHz. Hence, in theory, if a perfect linearity can be guaranteed, the proposed experimental system can achieve a resolution of λ0/2112 for the displacement measurement, which is 0.39 nm for λ0 = 830 nm. Therefore, the proposed method can be used for displacement measurement with high resolution.

4. Conclusion

Precision displacement measurement and positioning have become increasingly important in many areas, e.g. precision manufacturing, semiconductor industry, biotechnology, etc. With the progress of these fields, there is great demand for development of displacement sensing technique with high sensitivity, resolution and relatively simple configuration. In this work, a sensing system with an OFLD operating in POO is implemented for measuring displacement of a target. An analytical expression is presented for describing the sensing signals generated by the OFLD-POO system. It can be seen that both the amplitude and frequency of an OFLD-POO sensing signal are modulated by the displacement to be measured. Hence, we developed a new algorithm for measuring the displacement by simultaneously using the time-domain OFLD-POO signal waveform and RO frequency, which enables us to achieve displacement sensing with large measurement range, high sensitivity and resolution.

References

1. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]  

2. T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015). [CrossRef]  

3. D. Guo, M. Wang, and S. Tan, “Self-mixing interferometer based on sinusoidal phase modulating technique,” Opt. Express 13(5), 1537–1543 (2005). [CrossRef]   [PubMed]  

4. M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004). [CrossRef]  

5. S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Fiber self-mixing interferometer with orthogonally polarized light compensation,” Opt. Express 24(23), 26558–26564 (2016). [CrossRef]   [PubMed]  

6. M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012). [CrossRef]  

7. U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010). [CrossRef]  

8. U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013). [CrossRef]  

9. L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002). [CrossRef]  

10. M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013). [CrossRef]   [PubMed]  

11. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004). [CrossRef]  

12. Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19(10), 9582–9593 (2011). [CrossRef]   [PubMed]  

13. K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016). [CrossRef]   [PubMed]  

14. M. Dabbicco, A. Intermite, and G. Scamarcio, “Laser-self-mixing fiber sensor for integral strain measurement,” J. Lightwave Technol. 29(3), 335–340 (2011). [CrossRef]  

15. B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018). [CrossRef]   [PubMed]  

16. K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22(24), 30346–30356 (2014). [CrossRef]   [PubMed]  

17. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]  

18. A. Ritter and H. Haug, “Theory of laser diodes with weak optical feedback. II. Limit-cycle behavior, quasi-periodicity, frequency locking, and route to chaos,” J. Opt. Soc. Am. B 10(1), 145–154 (1993). [CrossRef]  

19. B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984). [CrossRef]  

20. A. Uchida, Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (John Wiley & Sons, 2012).

21. Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014). [CrossRef]   [PubMed]  

22. B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016). [CrossRef]   [PubMed]  

23. R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37(18), 3771–3773 (2012). [CrossRef]   [PubMed]  

24. B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017). [CrossRef]   [PubMed]  

25. D. Lenstra, “Relaxation oscillation dynamics in semiconductor diode lasers with optical feedback,” IEEE Photonics Technol. Lett. 25(6), 591–593 (2013). [CrossRef]  

26. Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018). [CrossRef]   [PubMed]  

27. J. Ohtsubo, Semiconductor lasers: stability, instability and chaos (Springer, 2012), Vol. 111.

References

  • View by:
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  • |
  • |

  1. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
    [Crossref]
  2. T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
    [Crossref]
  3. D. Guo, M. Wang, and S. Tan, “Self-mixing interferometer based on sinusoidal phase modulating technique,” Opt. Express 13(5), 1537–1543 (2005).
    [Crossref] [PubMed]
  4. M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004).
    [Crossref]
  5. S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Fiber self-mixing interferometer with orthogonally polarized light compensation,” Opt. Express 24(23), 26558–26564 (2016).
    [Crossref] [PubMed]
  6. M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
    [Crossref]
  7. U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
    [Crossref]
  8. U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
    [Crossref]
  9. L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002).
    [Crossref]
  10. M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013).
    [Crossref] [PubMed]
  11. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
    [Crossref]
  12. Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19(10), 9582–9593 (2011).
    [Crossref] [PubMed]
  13. K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
    [Crossref] [PubMed]
  14. M. Dabbicco, A. Intermite, and G. Scamarcio, “Laser-self-mixing fiber sensor for integral strain measurement,” J. Lightwave Technol. 29(3), 335–340 (2011).
    [Crossref]
  15. B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
    [Crossref] [PubMed]
  16. K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22(24), 30346–30356 (2014).
    [Crossref] [PubMed]
  17. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  18. A. Ritter and H. Haug, “Theory of laser diodes with weak optical feedback. II. Limit-cycle behavior, quasi-periodicity, frequency locking, and route to chaos,” J. Opt. Soc. Am. B 10(1), 145–154 (1993).
    [Crossref]
  19. B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
    [Crossref]
  20. A. Uchida, Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (John Wiley & Sons, 2012).
  21. Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014).
    [Crossref] [PubMed]
  22. B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
    [Crossref] [PubMed]
  23. R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37(18), 3771–3773 (2012).
    [Crossref] [PubMed]
  24. B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
    [Crossref] [PubMed]
  25. D. Lenstra, “Relaxation oscillation dynamics in semiconductor diode lasers with optical feedback,” IEEE Photonics Technol. Lett. 25(6), 591–593 (2013).
    [Crossref]
  26. Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
    [Crossref] [PubMed]
  27. J. Ohtsubo, Semiconductor lasers: stability, instability and chaos (Springer, 2012), Vol. 111.

2018 (2)

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

2017 (1)

2016 (3)

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

S. Zhang, S. Zhang, L. Sun, and Y. Tan, “Fiber self-mixing interferometer with orthogonally polarized light compensation,” Opt. Express 24(23), 26558–26564 (2016).
[Crossref] [PubMed]

2015 (1)

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

2014 (2)

2013 (3)

D. Lenstra, “Relaxation oscillation dynamics in semiconductor diode lasers with optical feedback,” IEEE Photonics Technol. Lett. 25(6), 591–593 (2013).
[Crossref]

U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
[Crossref]

M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013).
[Crossref] [PubMed]

2012 (2)

M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
[Crossref]

R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37(18), 3771–3773 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (1)

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

2005 (1)

2004 (2)

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004).
[Crossref]

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

2002 (1)

L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002).
[Crossref]

1995 (1)

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[Crossref]

1993 (1)

1984 (1)

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Bernal, O. D.

U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
[Crossref]

Bertling, K.

Bony, F.

R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37(18), 3771–3773 (2012).
[Crossref] [PubMed]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

Bosch, T.

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22(24), 30346–30356 (2014).
[Crossref] [PubMed]

U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
[Crossref]

R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37(18), 3771–3773 (2012).
[Crossref] [PubMed]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

Chicharo, J. F.

Dabbicco, M.

Donati, S.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[Crossref]

Fan, Y.

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014).
[Crossref] [PubMed]

Giuliani, G.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[Crossref]

Guo, D.

Guo, Q.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014).
[Crossref] [PubMed]

Haug, H.

Hicks, E.

Intermite, A.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Lai, G.

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Lenstra, D.

D. Lenstra, “Relaxation oscillation dynamics in semiconductor diode lasers with optical feedback,” IEEE Photonics Technol. Lett. 25(6), 591–593 (2013).
[Crossref]

Lewis, R. A.

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

Li, H.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Lim, Y. L.

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013).
[Crossref] [PubMed]

Lin, K.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Liu, B.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

Malkin, R.

Merlo, S.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[Crossref]

Nikolic, M.

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013).
[Crossref] [PubMed]

Norgia, M.

M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
[Crossref]

Olesen, H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
[Crossref]

Osmundsen, J. H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
[Crossref]

Paone, N.

L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002).
[Crossref]

Perchoux, J.

Pesatori, A.

M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
[Crossref]

Rajan, G.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Rakic, A. D.

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22(24), 30346–30356 (2014).
[Crossref] [PubMed]

M. Nikolić, E. Hicks, Y. L. Lim, K. Bertling, and A. D. Rakić, “Self-mixing laser Doppler flow sensor: an optofluidic implementation,” Appl. Opt. 52(33), 8128–8133 (2013).
[Crossref] [PubMed]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

Ritter, A.

Robert, D.

Rovati, L.

M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
[Crossref]

Ruan, Y.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

Scalise, L.

L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002).
[Crossref]

Scamarcio, G.

Su, L.

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Sun, L.

Taimre, T.

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22(24), 30346–30356 (2014).
[Crossref] [PubMed]

Tan, S.

Tan, Y.

Teysseyre, R.

Tong, J.

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Tromborg, B.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
[Crossref]

Wang, M.

D. Guo, M. Wang, and S. Tan, “Self-mixing interferometer based on sinusoidal phase modulating technique,” Opt. Express 13(5), 1537–1543 (2005).
[Crossref] [PubMed]

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004).
[Crossref]

Xi, J.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014).
[Crossref] [PubMed]

Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19(10), 9582–9593 (2011).
[Crossref] [PubMed]

Yu, Y.

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, and R. A. Lewis, “Displacement sensing using the relaxation oscillation frequency of a laser diode with optical feedback,” Appl. Opt. 56(24), 6962–6966 (2017).
[Crossref] [PubMed]

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

Y. Fan, Y. Yu, J. Xi, and Q. Guo, “Dynamic stability analysis for a self-mixing interferometry system,” Opt. Express 22(23), 29260–29269 (2014).
[Crossref] [PubMed]

Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19(10), 9582–9593 (2011).
[Crossref] [PubMed]

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

Zabit, U.

U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
[Crossref]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

Zhang, S.

Adv. Opt. Photonics (1)

T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (3)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
[Crossref]

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[Crossref]

IEEE J. Sensors (2)

M. Norgia, A. Pesatori, and L. Rovati, “Self-mixing laser Doppler spectra of extracorporeal blood flow: a theoretical and experimental study,” IEEE J. Sensors 12(3), 552–557 (2012).
[Crossref]

U. Zabit, O. D. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE J. Sensors 13(2), 824–831 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (3)

U. Zabit, F. Bony, T. Bosch, and A. D. Rakic, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photonics Technol. Lett. 22(6), 410–412 (2010).
[Crossref]

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photonics Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

D. Lenstra, “Relaxation oscillation dynamics in semiconductor diode lasers with optical feedback,” IEEE Photonics Technol. Lett. 25(6), 591–593 (2013).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of microprofile,” Opt. Commun. 238(4–6), 237–244 (2004).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

L. Scalise and N. Paone, “Laser Doppler vibrometry based on self-mixing effect,” Opt. Lasers Eng. 38(3–4), 173–184 (2002).
[Crossref]

Opt. Lett. (1)

Sensors (Basel) (4)

B. Liu, Y. Yu, J. Xi, Y. Fan, Q. Guo, J. Tong, and R. A. Lewis, “Features of a self-mixing laser diode operating near relaxation oscillation,” Sensors (Basel) 16(9), 1546 (2016).
[Crossref] [PubMed]

Y. Ruan, B. Liu, Y. Yu, J. Xi, Q. Guo, and J. Tong, “Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback,” Sensors (Basel) 18(11), 4004 (2018).
[Crossref] [PubMed]

K. Lin, Y. Yu, J. Xi, H. Li, Q. Guo, J. Tong, and L. Su, “A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus,” Sensors (Basel) 16(6), 928 (2016).
[Crossref] [PubMed]

B. Liu, Y. Ruan, Y. Yu, J. Xi, Q. Guo, J. Tong, and G. Rajan, “Laser self-mixing fiber Bragg grating sensor for acoustic emission measurement,” Sensors (Basel) 18(6), 1956 (2018).
[Crossref] [PubMed]

Other (2)

J. Ohtsubo, Semiconductor lasers: stability, instability and chaos (Springer, 2012), Vol. 111.

A. Uchida, Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (John Wiley & Sons, 2012).

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Figures (8)

Fig. 1
Fig. 1 (a) Relationship between RO frequency and displacement of external cavity, (b) waveform of Δ P OFLDPOO-Ep for a change of external cavity length when L 0 =24cm, I=1.3 I th and C=4.5.
Fig. 2
Fig. 2 Verification of OFLD-POO signal expressed in Eq. (40) by the numerical results from L-K equations. (a) displacement of the external target; (b) and (c) results from L-K equations with C = 2.8 and C = 4.5 respectively; (d) and (e) results from Eq. (40) when C = 2.8 and C = 4.5 respectively.
Fig. 3
Fig. 3 OFLD-POO signal and displacement corresponding to Δ L t and Δ L f .
Fig. 4
Fig. 4 Displacement reconstruction procedure from the OFLD-POO signal, (a) the pre-set displacement, (b) corresponding OFLD-POO signal, (c) upper envelope of OFLD-POO signal, (d) envelope differentiation.
Fig. 5
Fig. 5 Experiment setup, where LD, BS and PZT represent laser diode, beam splitter, and piezoelectric transducer.
Fig. 6
Fig. 6 Experimental results of laser intensity when I=50 mA, L 0 =18 cm,and ΔL=300 nm.
Fig. 7
Fig. 7 RO frequency variation for different displacements when I=50 mA, and L 0 =18 cm.
Fig. 8
Fig. 8 (a) PZT controlling signal, (b) corresponding OFLD-POO signal.

Tables (2)

Tables Icon

Table 1 Meanings of the symbols in the L-K equations

Tables Icon

Table 2 Displacement results by PZT sensor and OFLD-POO

Equations (47)

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dE(t) dt = 1 2 { G[ N(t),E(t) ] 1 τ p }E(t)+ κ τ in E(tτ)cos[ ω 0 τ+ϕ(t)ϕ(tτ) ].
dϕ(t) dt = 1 2 α{ G[ N(t),E(t) ] 1 τ p } κ τ in E(tτ) E(t) sin[ ω 0 τ+ϕ(t)ϕ(tτ) ].
dN(t) dt =I N(t) τ s G[ N(t),E(t) ] E 2 (t).
E(t)= E ¯ +ΔEcos( ω r t+ θ e ).
N(t)= N ¯ +ΔNcos( ω r t+ θ n ).
ϕ(t)=( ω ¯ ω 0 )t+(Δϕ/2)cos ω r t.
E ¯ 2 = I N ¯ / τ s G N ( N ¯ N 0 ) .
N ¯ = N 0 + 1 G N τ p 2kcos( ω s τ) τ in G N .
ω s = ω 0 k 1+ α 2 τ in sin( ω s τ+arctanα).
2 G N ( N ¯ N 0 ) E ¯ ΔEcos( ω r t+ θ e ) =I N ¯ / τ s G N ( N ¯ N 0 ) E ¯ 2 +ΔN ω r 2 + (1/ τ s + G N E ¯ 2 ) 2 cos{ ω r t+ θ n arctan[(1/ τ s + G N E ¯ 2 )/ ω r ]π/2}.
2 G N ( N ¯ N 0 ) E ¯ ΔE=ΔN ω r 2 + (1/ τ s + G N E ¯ 2 ) 2 .
θ e = θ n arctan[(1/ τ s + G N E ¯ 2 )/ ω r ]π/2.
I N ¯ / τ s G N ( N ¯ N 0 ) E ¯ 2 =0.
- ω r ΔEsin( ω r t+ θ e ) ={ 1 2 [ G N ( N ¯ +ΔNcos( ω r t+ θ n )- N 0 ) 1 τ p ] }[ E ¯ +ΔEcos( ω r t+ θ e )] + κ τ in [ E ¯ -ΔEcos( ω r t+ θ e )]cos[ ω s τ+Δϕcos( ω r t)].
- ω r ΔEsin( ω r t+ θ e )= 1 2 [ G N ( N ¯ - N 0 )1/ τ p ] E ¯ + κ τ in E ¯ cos ω s τ J 0 (Δϕ) + 1 2 G N E ¯ ΔNcos( ω r t+ θ n )+ 1 2 [ G N ( N s - N 0 ) 1 τ p ]ΔEcos( ω r t+ θ e ) 2 κ τ in E ¯ sin ω s τ J 1 (Δϕ)cos ω r t.
1 2 [ G N ( N ¯ - N 0 )1/ τ p ]+ κ τ in cos ω s τ J 0 (Δϕ)=0.
ω r ΔEsin( ω r t+ θ e )=2 κ τ in E ¯ sin ω s τ J 1 (Δϕ)cos ω r t 1 2 [ G N ( N ¯ - N 0 )1/ τ p ]ΔEcos( ω r t+ θ e ) 1 2 G N E ¯ ΔNcos( ω r t+ θ n ).
ω r ΔEcos θ e sin ω r t+ ω r ΔEsin θ e cos ω r t= { 1 2 [ G N ( N ¯ - N 0 )1/ τ p ]ΔEsin θ e + 1 2 G N E ¯ ΔNsin θ n }sin ω r t +{ 1 2 [ G N ( N ¯ - N 0 )1/ τ p ]ΔEcos θ e 1 2 G N E ¯ ΔNcos θ n +2 κ τ in E ¯ sin ω s τ J 1 (Δϕ) }cos ω r t.
ω r ΔEcos θ e = 1 2 [ G N ( N ¯ N 0 )1/ τ p ]ΔEsin θ e + 1 2 G N E ¯ ΔNsin θ n .
cot θ n = 1 ω r ω r 2 G N 2 E ¯ 2 ( N s N 0 )(1/ τ s + G N E ¯ 2 )(κ/ τ in ) J 0 (Δϕ)cos ω s τ 1/ τ s + G N E ¯ 2 +(κ/ τ in ) J 0 (Δϕ)cos ω s τ .
ω ¯ ω 0 =α κ τ in cos ω ¯ τ J 0 (Δϕ) κ τ in sin ω ¯ τ J 0 (Δϕ).
1 2 α G N ΔNcos θ n 2 κ τ in cos ω ¯ τ J 1 (Δϕ)=0.
Δϕ ω r =α G N ΔNsin θ n .
cos ω ¯ τ J 0 (Δϕ)=cos ω s τ.
cot θ n = 1 ω r ω r 2 G N 2 E ¯ 2 ( N s N 0 )( G N E ¯ 2 + 1 τ s ) κ τ in cos ω s τ G N E ¯ 2 + 1 τ s + κ τ in cos ω s τ .
Δϕ=2 ω r τ in cot θ n 2κcos ω s τ ω r τ in cot θ n κcos ω s τ .
ΔN= 2 ω r α G N sin θ n ω r τ in cot θ n 2κcos ω s τ ω r τ in cot θ n κcos ω s τ .
ΔE= ω r α G N sin θ n ω r τ in cot θ n 2κcos ω s τ ω r τ in cot θ n κcos ω s τ ω r 2 + ( G N E ¯ 2 +1/ τ s ) 2 G N ( N ¯ N 0 ) E ¯ .
ΔE= ω r 2 α G N sin θ n ω r τ in cot θ n 2κcos ω s τ ω r τ in cot θ n κcos ω s τ 1 G N ( N ¯ N 0 ) E ¯ .
P(t)= E 2 (t) = E ¯ 2 +2 E ¯ ΔEcos( ω r t+ θ e )+Δ E 2 cos 2 ( ω r t+ θ e ).
P(t) = E ¯ 2 +2 τ p ω r 2 α G N sin θ n ω r τ in cot θ n 2κcos ω s τ ω r τ in cot θ n κcos ω s τ (1+ 2κ τ p τ in cos ω s τ)cos( ω r t+ θ e ).
E ¯ 2 = E S0 2 +2( E S0 2 + 1 G N τ s ) κ τ p τ in cos( ω s τ).
cot θ n = τ s ( ω r 2 ω r0 2 ) κ τ in cos ω s τ ω r .
P(t)= E S0 2 +2( E S0 2 + 1 G N τ s ) κ τ p τ in cos( ω s τ) +2 τ p ω r α G N ω r 2 + [ τ s ( ω r 2 ω r0 2 ) κ τ in cos ω s τ] 2 cos( ω r t+ θ e ).
ΔP(L,t)=P(t) E S0 2 =Δ P OFLD (L)+Δ P OFLDPOO-Ep (L)cos[ ω r (L)t+ θ e ].
Δ P OFLD (L)=2( E S0 2 + 1 G N τ s ) κ τ p τ in cos[ ϕ F (L)].
Δ P OFILDPOOEp (L)=2 τ p ω r (L) α G N ω r 2 (L)+ { τ s [ ω r 2 (L) ω r0 2 ] κ τ in cos[ ϕ F (L)]} 2 .
ω r (ΔL+N λ 0 2 )= ω r0 + ω r-offset + 2( ω rmax ω rmin ) λ 0 ΔL.
Δ P OFLD-POOEp (L+N λ 0 2 ) =2 τ p ω r (L+N λ 0 2 ) α G N ω r 2 (L+N λ 0 2 )+[ τ s ( ω r 2 (L+N λ 0 2 ) ω r0 2 ) κ τ in cos ( ϕ F (L+N λ 0 2 )] 2 . =Δ P OFLD-POOEp (L)
Δ P OFLDPOO (L,t)=Δ P OFLD-POOEp (L)cos[ ω r (L)t+ θ e ].
Δ P OFLDPOOEpmin 2 τ p ω r0 2 α G N .
Δ P OFLDPOOEpmax 2 τ p ω r 0 2 α G N 1+ [2 τ s ( ω r-offset + ω rmax ω rmin )] 2 .
Δ P OFLDPOOEppp =2 τ p ω r 0 2 α G N ( 1+[2 τ s ( ω r-offset + ω rmax ω rmin ) ] 2 -1).
Δ P OFLDPOOEppp Δ P OFLDpp = 2 τ p ω r 0 2 α G N ( 1+ [2 τ s ( ω r-offset + ω rmax ω rmin )] 2 -1) 4( E S0 2 + 1 G N τ s ) κ τ p τ in τ in 2ακ τ p ( 1+ [2 τ s ( ω r-offset + ω rmax ω rmin )] 2 -1).
ΔL=Δ L t +Δ L f .
Δ L f = λ 0 2 ω r2 ω r1 ω rmax ω rmin .

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