## Abstract

A self-mixing birefringent dual-frequency laser Doppler velocimeter (SBD-LDV) for high-resolution velocity measurements is presented in this paper. The velocity information of the object can be accurately extracted from the self-mixing Doppler frequency shift of the birefringent light-carried microwave signal. We generate a virtual stable light-carried microwave by using a birefringent dual-frequency He-Ne laser which further simplifies the structure of the light source. Moreover, the optical configuration based on the laser self-mixing interference brings benefits of compact optical setup, self-alignment, and direction discriminability. Experimentally, we extracted the Doppler beat frequency signal by the low-frequency (millihertz) phase lock-in amplifier, measured the beat frequency precisely in time-domain with a low sampling rate and calculated the magnitude of velocity. Compared with the previous self-mixing LDV, the average velocity resolution of SBD-LDV is improved to 0.030 mm/s for a target with longitudinal velocity, benefiting from the high stability of light-carried microwave. It is of great meaning and necessity because it helps to provide an available velocimeter with high stability and an extremely compact configuration, making a potential contribution to the velocimetry in practical engineering application.

© 2017 Optical Society of America

## 1. Introduction

Benefiting from the depth-in study on the dynamics of such kinds of lasers with self-mixing interference effect, self-mixing laser Doppler velocimetry has been well established in last decades [1–4]. Self-mixing interference effect generally occurs when a portion of light emitted from a laser is backscattered from a movable target and reenters into the laser cavity; the feedback optical field interacts with the gain medium and mixes with the light field inside the cavity, generating a modulation of both the amplitude and the frequency of the lasing field. With the advantages of similar phase sensitivity, modulation depth ratio, self-alignment, and compact optical setup in an optical feedback configuration, self-mixing laser Doppler velocimeter (SLDV) is always simpler and more excellent than the conventional laser Doppler velocimeter. In addition, the SLDV can intuitionally discriminate the direction of the velocity by determining the orientation of the asymmetric interferogram when the laser is working in the moderate feedback regime [5–7]. So it has great potential in many kinds of practical industry domains.

As same as the conventional laser Doppler velocimeter, the measurement resolution of the SLDV further relies on the frequency stability of the laser. Usually, the optical frequency is used as the measurement basis of these laser Doppler velocimeters. However, the laser frequency is modulated by the optical feedback in SLDV which introduces unavoidable error source and disturbs the measurement basis. To overcome the problems caused by the deterioration of the optical frequency, an optically-injected semiconductor laser system operated in a period-one dynamical state with two optical frequencies in its spectrum is used as the dual-frequency light source recently, instead of using a conventional single frequency laser [8–11]. A stable microwave beat signal is generated as the measurement basis with much narrow width and high stability carried by the light of a slave and master lasers. So far, the velocity resolution is improved to 0.42mm/s for a target with a longitudinal velocity of 2 cm/s, due to its good directionality, low speckle noise and good coherence [9]. However, in an optically-injected semiconductor laser system, numbers of optical polarization components and electronic devices, such as polarizer, isolator, wave plate, microwave frequency synthesizer and avalanche photo-detector involve in the generation of light-carried microwave, which makes the system very complicated and expensive. Hence, it seems to a SLDV that generation of light-carried microwave based on optically-injected semiconductor laser system displays the fact of high cost and complicated structure, which would fade the merits of the SLDV configuration. In our recent works, an alternative mechanism to generate a light-carried microwave to LDV is optical beat from a birefringent dual-frequency laser with the same stable laser frequency difference and the potential lower cost compared to the optically injected semiconductor laser [12]. The simplicity of the birefringent dual-frequency laser comes to a complete agreement on the merits of the SLDV.

Therefore, we report on a self-mixing birefringent dual-frequency laser Doppler velocimeter (SBD-LDV) for high resolution longitudinal velocity measurement in this paper. It is of great meaning and necessity because it helps to make full use of merits from both a birefringent light-carried microwave generator and a SLDV. Differently from the studies mentioned above, this work deals with the situation that both ordinary light and extraordinary light are involved in the optical feedback effect in a birefringent dual-frequency He-Ne laser. Previous researches on birefringent laser self-mixing interference indicate that the phase difference behaviors of the two birefringent modes are independent when the frequency difference of the birefringent dual-frequency He-Ne laser is adjusted to a reasonable range [13]. Moreover, the light-carried microwave signal provides better immunity against environmental disturbances compared with the laser frequency. To our knowledge, the SBD-LDV can be extremely compact and with a considerable resolution simultaneously. In principle, the proposed SBD-LDV is derived to be feasible and can improve the velocity measurement accuracy a lot. Experimentally, a new signal-processing scheme of DSP low-frequency (millihertz) phase lock-in amplifier is designed to precisely extract the Doppler frequency signals, which greatly reduces the complexity of the electrical signal processing part and improves the velocity measurement resolution. The average velocity resolution of the SBD-LDV is improved from 0.42mm/s to 0.030mm/s for a target with longitudinal velocities. It provides an available velocimeter with high stability and an extremely compact configuration, which is significant for the practical engineering application.

## 2. Principle

In SBD-LDV, a birefringent dual-frequency He-Ne laser with a half-intracavity is employed as the light source to generate a virtual stable light-carried microwave. As we know, light entering and passing through a quartz crystal will be decomposed into two linearly-polarized birefringent components, namely the ordinary and extraordinary beams (o light and e light). And these two components propagate with different transmission velocities, due to the anisotropy property of quartz crystal. The two polarization directions and phase difference are related to the angle and the projection of the activity vector on to the light traveling direction. The existing of the phase difference and the laser resonant half-intracavity make sure that a single mode of the intracavity laser is split into two orthogonally polarized modes ${\nu}_{o}$and ${\nu}_{e}$ [14–16]. Taking no account of the optical activity of quartz crystal, the output frequency difference $\Delta \nu $between the two modes can be expressed as [14]

Here $\Lambda $is the longitudinal mode interval, $\lambda $ is the central wavelength of the laser, *h* is the thickness of the quartz crystal, ${n}_{o}$and ${n}_{e}$are the refractive indexes of o-light and e-light respectively, and $\theta $is the angle between the crystalline axis of quartz crystal and the light traveling direction. From Eq. (1), the frequency difference further correlates with thickness *h* and the angle$\theta $. Perturbation of$\lambda $introduce only extremely tiny change of$\Delta \nu $. And the frequency difference between the orthogonally-polarized lasing modes can be precisely controlled by adjusting the angle$\theta $, which allows an experimental frequency difference to be set in a range from 40 MHz to one longitudinal mode interval.

As mentioned earlier, if the frequency difference between two modes is greater than the line width of homogeneous broadening gain curve of laser (about 100-300MHz), the two modes don’t scramble the glowing atoms with same velocity on Maxwell distribution curve. Hole-burning curves for the two lasing modes have been separated very far so that the optical feedback effect of one mode does not confuse the other. On the other hand, considering the inhomogeneous broadening line width of a He-Ne laser (about 1.5GHz), we cannot adjust the frequency difference to much more than 1GHz because of the possibility of resonance quenching. The change of frequency difference should be limited in a reasonable range. So we can simply theoretically describe the optical feedback effect of a birefringent dual-frequency He-Ne laser in this case.

In the basic theory of optical feedback modeled as a three-mirror Fabry–Perot etalon [17], the oscillating condition of one mode in an birefringent dual-frequency He-Ne laser with optical feedback can be given as${r}_{1}{r}_{3}\left[1+r\left(1-{r}_{1}^{2}\right){e}^{i\omega {\tau}_{E}}/{r}_{1}\right]{e}^{\left(G-a\right)}{e}^{i\omega {\tau}_{I}}=1$ [13], where *G* is the laser total normalized gain, *a* is the total normalized internal loss,${\tau}_{I}$ is the laser beam round-trip time in internal cavity.${\tau}_{E}$ represents the laser beam round-trip time in external cavity related to the instantaneous distance$L={L}_{0}+\upsilon t$ between the laser and the feedback mirror, as${\tau}_{E}=2({L}_{0}+\upsilon t)/c$, where$\upsilon $ is the velocity of the moving object including its directional information, and *c* is the speed of light. Here we apply the model to the birefringent dual-frequency He-Ne laser. In a SBD-LDV as shown in Fig. 1, the laser beam, with two orthogonally-polarized dual-frequency components, is attenuated by a ND and then focused onto a moving object M_{T}. A portion of light, which is reflected by the object, propagated back into the lasing cavity and interacts with the gain medium and the light field there. The interactions generate modulations of both the amplitude and the frequency of the two modes of the laser, thus introducing in the Doppler frequency shifts. Since *r _{1}*,

*r*>

_{3}*r*, both of

*r*and

_{1}*r*are approximate 1 and$r(1-{r}_{1}{}^{2})/{r}_{1}<<1$ the normalized threshold gains change of the two modes can be transformed as

_{3}Where${\alpha}_{o}$and${\alpha}_{e}$are optical feedback factors, ${\varphi}_{o}$and${\varphi}_{e}$are the initial phase which can be written as${\varphi}_{o}=4\pi {\nu}_{o}{L}_{o}/c$ and ${\varphi}_{e}=4\pi {\nu}_{e}{L}_{0}/c$. The orthogonally-polarized dual-frequency components of the beam, emitted from the rear facet of lasing cavity, are separated by a PBS and monitored by a pair of photo detectors (PD1 and PD2) independently. The variations of laser intensities are proportional to$\Delta {G}_{o}$ and$\Delta {G}_{e}$, so the output intensities of two orthogonally polarized lights with optical feedback are modulated by two different Doppler frequencies. Supposing that the birefringent dual-frequency He-Ne laser operates in the weak optical feedback regime, the frequency of laser is unperturbed [5–7]. The output signals of two orthogonally polarized modes received by the photodiodes can be simplified as

Where${P}_{o0}$ and${P}_{e0}$ represent the average intensities of two orthogonally polarized lights without optical feedback, ${\beta}_{o}$and${\beta}_{e}$are the undulation coefficient depending on intrinsic laser parameters, optical feedback intensity and the reflectivity of the mirrors, ${f}_{D1}$and${f}_{D2}$denote the Doppler frequencies which can be expressed as ${f}_{D1}=\frac{2\upsilon}{c}{\nu}_{o}$ and${f}_{D2}=\frac{2\upsilon}{c}{\nu}_{e}$. Multiplying${P}_{o0}$ by${P}_{e0}$ in the signal processing module, a mixed Doppler-shifted signal is obtained as

Selecting a proper filter bandwidth, the difference-frequency term of the mixed signal is extracted by a low-pass filter as

Where *M* represents the intensity coefficient and$\Delta \varphi ={\varphi}_{o}-{\varphi}_{e}$is the initial phase of this signal. The process is simulated in Figs. 2(a)-2(c). The undulation coefficients${\beta}_{o}$and${\beta}_{e}$ are assumed as fixed constant functions in the weak optical feedback regime and the moving velocity of the target is set as 1mm/s here. Figure 2(d) depicts the simulated Doppler beat frequency signal when the object is moving with typical velocities. We assume an ideal multiplier without any electrical noise in these simulations.

Note that, the differential frequency can be expressed as

Obviously, the signal quality of ${S}_{-}(t)$strongly depends on by the laser frequency difference$\Delta \nu $. Here, the laser frequency difference can be considered as the frequency of a virtual light-carried microwave. So the beat frequency$\Delta {f}_{D}$can be regarded as the Doppler frequency shift of this virtual light-carried microwave. The results that we have heretofore discussed for frequency difference of the birefringent dual-frequency He-Ne laser show that the virtual light-carried microwave frequency $\Delta \nu $is of great stability, and concerns with the angle$\theta $mainly. Although the modulation of the birefringent feedback light field for the frequency should be taken into consideration, the variation of angular frequencies here is much smaller compared with the optical frequencies of the two orthogonally polarized modes in the case of weak feedback. So$\Delta {f}_{D}$should be firm stable, and signal quality could be excellent. The frequency information of${S}_{-}(t)$can be precisely measured by a low-frequency (millihertz) phase lock-in algorithmic program so the magnitude of the velocity can be obtained with high resolution by Eq. (6).

## 3. Experiment

The experimental setup and measurement process for SBD-LDV-based measurement system demonstration is shown in Fig. 3. The feedback mirror M_{T} is mounted on a linear stage (M-521.DD, Physik Instrument Co., Germany) with maximum speed 50 mm/s and resolution 0.1$\text{\mu m/s}$. The birefringent dual-frequency He-Ne laser operates at a center wavelength of 632.8 nm, accompanying with the following parameters. The split frequency difference is set as about 1GHz by alerting the angle$\theta $. The rations of gaseous pressure in laser are He: Ne = 7:1 and Ne^{20}: Ne^{22} = 1:1. The internal cavity length is 135 mm, and the reflectivities *r _{1}* = 0.994 and

*r*= 0.999.

_{3}In the signal processing module, the power signals ${P}_{o}(t)$and${P}_{e}(t)$detected by photodiodes are sent to a low-noise amplifier and a low-pass filter with a cutoff frequency of 150 kHz to amplify the signal and filter out the high-frequency electronic noise firstly, and then a DSP lock-in amplifier is introduced to mix the power signals and obtain the Doppler beat frequency shift signal${S}_{-}(t)$. The DSP lock-in amplifier contains two input channels, measurement channel and reference channel. In the measurement channel, the signal is processed consecutively by a low-noise amplifier and filter, a low-noise differential amplifier, a 50/60Hz notch filter, a 100/120Hz notch filter and a gain module. In the reference channel, the signal is sent successively into a low-noise amplifier and filter, a frequency discriminator, a phase lock loop, and an internal oscillator. Then the signal from the measurement channel mixed with the signal from reference channel, generating an output signal. The output signal of DSP lock-in amplifier are sampled and converted into digital signals by a commercial 16-bit data acquisition board (USB-6361, National Instruments Co., USA). The sampling rate is set as 100 kHz and the total sampling points is 10000. At last, the velocity of the object can be computed by an application program in LabVIEW.

The discussion heretofore indicates that velocity measuring method using a SBD-LDV is feasible. So we need to test and verify the performance of the proposed SBD-LDV experimentally. The experimental steps are as follow. Firstly, we initialized the program and observed inclined direction of the power signal to discriminate the positive and negative of the velocity. An algorithmic program was utilized to measure the frequency of ${S}_{-}(t)$. The points-in-time at the peak and zero of the signal were obtained in the algorithmic program. The accurate frequency value of the signal was derived then. Afterwards, the magnitude of velocity was calculated by the Eq. (6) as described in Section 2. At the front panel of the LabVIEW program, the measurement results of velocity were displayed clearly. In the end, the velocity measurement results and waveforms of the sampled signals are all saved in the computer to facilitate the further experimental analyses. For distinctness, the velocity measurement experiment is divided into two parts: (A) discrimination of positive and negative of the velocity. (B) Velocity Measurement.

#### 3.1 Discrimination of positive and negative of the velocity

In order to discriminate the direction of velocity, we should consider the modulation of the birefringent feedback light field for the output power solution of self-mixing interference. The undulation coefficient${\beta}_{o}$and${\beta}_{e}$ in Eq. (3) cannot be consider as a constant. In fact,${\beta}_{o}$and${\beta}_{e}$are modulation functions related to the optical feedback strength *C* and intrinsic laser parameters. When the value of *C* is very small (*C*<0.01), the self-mixing interference signal is a sinusoidal waveform, but for larger values of *C* (0.1<*C*<1), the signal becomes sawtooth-like waveform. The power signals can be distorted from a sinusoidal waveform to an asymmetric saw-tooth waveform depending on the feedback strength. Moreover, the sinusoidal wave should incline left when the velocity is positive (target approaching). In the contrary, the sinusoidal wave will incline right. Once we need observe the incline of the sinusoidal wave and discriminate the direction of the velocity, we rotate the neutral density filter to increase the optical feedback strength.

So as to test the performance of the proposed SBD-LDV for discrimination of positive and negative of the velocity, Figs. 4(a) and 4(b) show the filtered Doppler signals (o light) obtained from the SBD-LDV when the measured object is moving toward and away from the laser with $\upsilon $ = + 1 mm/s and $\upsilon $ = −1 mm/s. The same as the heretofore analysis, when the measured target is moving toward, the asymmetric saw-tooth waveform inclines left. While the target moves in the opposite direction, the asymmetric saw-tooth waveform inclines right. The SBD-LDV can give the discrimination of positive and negative of velocity simply and intuitively as same as a conventional SLDV.

#### 3.2 Velocity measurement

Velocity measurements with a range of 30 mm/s have been conducted and the experimental results are shown in this subsection. Figures 5(a)-5(e) show the normalized Doppler beat frequency signal ${S}_{-}(t)$obtained from the SBD-LDV with typical velocities of target mirror, which validates the feasibility of the SBD-LDV. As the speed increases, the signal frequency increases, but the number of fringes of ${S}_{-}(t)$is always fixed, determined by the range of the linear stage. The filtered Doppler beat frequency signal is sinusoidal waveforms with frequencies at the millihertz scale. Its frequency spectrum (FFT) is demonstrated in Fig. 5(f). Since the measured frequency interval is too small and the resolution is not enough in the frequency domain, we use a time domain method to measure the frequency precisely. Figure 6 illustrates the dependence of the measured Doppler beat frequency on the actual speed of M_{T}. The slope of 7.16179, obtained by linear fitting to the experimental data, indicates that the measured Doppler beat frequency has a linear relationship with the actual speed of target, which is in conformity with the Eq. (6). For the current electric circuit, the minimum detectable velocity is about 0.2 mm/s. If the moving velocity was lower, the signal would be unlocked in the DSP lock-in amplifier at the reference channel. On one hand, the minimum detectable velocity is mainly attributed to the stability of laser frequencies, the signal-to-noise ratio of electrical signals and the resolution of DSP lock-in amplifier. On the other hand, since the velocity is linear with the Doppler frequency in this work, the maximum detectable velocity is about 30 mm/s, which is limited by the bandwidth (102 kHz) of the DSP lock-in amplifier.

In Fig. 7, we plot the comparison experimental result between measured speed and the actual speed of the target on M-521.DD stage. The deviations denote the differences between the actual velocities of the M-521.DD stage and the measurement results of the SBD-LDV. We measured the every velocity for 20 times. A typical group of data shows that the maximum deviation is about 0.076 mm/s. The linear fitting of the measurement results is shown in Fig. 7 and the linear coefficient is 0.9999, indicating that the proposed SBD-LDV is capable of significant measuring dynamic range of velocities with high resolution. The average deviation is about 0.030 mm/s with a standard deviation of 0.006 mm/s. These results demonstrate the excellent repeatability, reliability and robustness of the SBD-LDV, which is favorable for practical applications.

## 4. Discussions

In the velocity measurement experiment, the speed equation can be transformed from Eq. (6):

Considering errors caused by the stability of virtual light-carried microwave and the actual measuring precision of the Doppler beat frequency, the uncertainty of this system [18] is expressed as

Where $\delta \Delta \nu $ is the uncertainty of virtual light-carried microwave frequency and$\delta \Delta {f}_{D}$denotes uncertainty of the actual measuring error of the Doppler beat frequency.

The virtual light-carried microwave frequency, indeed, is the frequency difference of a birefringent dual-frequency He-Ne laser. When the laser is in thermal equilibrium, from Eq. (1), $\delta \Delta \nu $can be written as

*h*= 3mm,

*l*= 135mm,${n}_{o}$ = 1.54263,${n}_{e}$ = 1.55169.$\theta $is set as a certain angle. We can see that$\delta \Delta \nu <<\delta \nu $. Compared with the optical frequency, the frequency difference of the birefringent dual-frequency He-Ne laser is further more stable. The short term frequency stability of the optical frequency of the laser here is 1.5 ppm, so$\delta \Delta \nu $in the Free State should be less than10kHz. The uncertainty of the velocity from the virtual light-carried microwave frequency is at the level of${10}^{-5}$~${10}^{\text{-}4}$mm/s. The light-carried microwave frequency is measured by MSA (FSV30, Rohde & Schwarz Technology Co., Germany) with the auto sweep mode as shown in Fig. 8(a). We can see the microwave frequency is stable at 1.075GHz with full width at half maximum (FWHM) of 20.3MHz. In addition, taking account of laser feedback effect, we can give an evaluation of the periodical fluctuation of the frequency difference from Eq. (10).

*C*can be altered by rotating the ND in this work, and the fluctuation of optical frequency difference is simulated in Fig. 8(b). Obviously, the optical frequency difference fluctuates periodically over the external cavity phase so that the spectrum should be broadened. However, the central frequency cannot be changed because the integration of periodical frequencies fluctuation is zero. Furthermore, the measurement experiment for magnitude of the velocity is conducted with the weak optical feedback strength as$C<0.1$. Hence, compared with the optical frequency, the periodical frequencies fluctuation is insignificant which can be ignored in this work.

The characteristics of the frequency difference can be observed on a scanning Fabry-Perot interferometer (SA200, Thorlabs). We can evaluate the optical frequency shift speed via observation of the intensity variation of the laser modes. In the case of self-mixing interference, the optical frequency shifts over time obviously as shown in Figs. 9(a) and 9(b). In contrast, Figs. 9(c) and 9(d) shows that the optical frequency shifts more slowly in the absence of optical feedback. However, the observed the frequency difference in all the figures seems to be always fixed. The phenomenon indicates that the frequency difference always keep stable whether the optical frequency shifts or not. Furthermore, the FWHM of the optical spectrum displays little variation with the self-mixing interference from our observation.

The uncertainty of the actual measuring precision of the Doppler beat frequency$\delta \Delta {f}_{D}$ can be tracked back to the errors from the LabVIEW-based frequency measurement program and noises in our system. The measuring accuracy of the Doppler beat frequency can be expressed as

where$\Delta t=1/\left(2\cdot \Delta {f}_{D}\right)$represents the time interval between the consecutive maximum points, $\Delta n$denotes number deviation of the sampling point and ${F}_{s}$ is the sampling rate of the DAQ board. In our present system, the sampling rate is set as 10 kHz. Meanwhile,$\Delta t$should be more than 2.326s in our experiment. Assuming$\Delta n=10$, the measuring accuracy of the Doppler beat frequency is about$4.299\times {10}^{-4}$.We give an illustration of the comprehensive uncertainty analysis in Fig. 10. As seen in Fig. 10 (a), the uncertainty of the system decreases as a hyperbola when the sampling rate is raised. If the sampling rate is set greater than 7 kHz, the uncertainty can be controlled within the limits of 0.01 mm/s. The actual uncertain of the velocity caused by the measured Doppler beat frequency is in conformity with the theoretical analysis as shown in Fig. 10(b). Overall, the maximum measurement error is theoretical about 0.01mm/s with the same level of the experimental results. For a number of industrial applications such as machine-tool control and velocity measurement sensor, the velocity measurement accuracy of the SBD-LDV is considerable.

## 5. Summary

In conclusion, we have developed a self-mixing birefringent dual-frequency laser Doppler velocimeter (SBD-LDV) to measure the longitudinal velocity of a moving target with high measurement resolution. In principle, the proposed SBD-LDV is derived to be feasible and can further improve the velocity measurement accuracy. Experimentally, we proposed a method to generate a virtual light-carried microwave with a much stable frequency by a simple birefringent dual-frequency laser, and designed a scheme of DSP low-frequency (millihertz) phase lock-in amplifier to precisely extract Doppler beat frequency shift of this virtual light-carried microwave. So far, compared with the previous self-mixing LDV, the average velocity resolution of the SBD-LDV is improved from 0.42mm/s to 0.030 mm/s for a target with longitudinal velocities. The proposed velocimeter combines the merits of a birefringent light-carried microwave generator and a SLDV, which can be utilized industrially in more extensive fields. It is of great meaning and necessity because it helps to provide an available velocimeter with high stability and an extremely compact configuration, making a potential contribution to the velocimetry in practical engineering application.

## Funding

National Natural Science Foundation of China (NSFC) (51405240, 91123015); Natural Science Foundation of Jiangsu Province of China (BK20161559); Natural Science Foundation of the Higher Education Institutions of Jiangsu Province of China (16KJB510018); University Postgraduate Research and Innovation Project of Jiangsu Province (KYLX16_1289).

## Acknowledgments

We acknowledge the support by Jiangsu Key Laboratory on Opto-electronic Technology.

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