Review Papers

Laser feedback interferometry and applications: a review

[+] Author Affiliations
Jiyang Li, Haisha Niu, Yanxiong Niu

Beihang University, School of Instrumentation Science and Optoelectronics Engineering, Beijing, China

Opt. Eng. 56(5), 050901 (May 09, 2017). doi:10.1117/1.OE.56.5.050901
History: Received January 17, 2017; Accepted April 6, 2017
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Abstract.  The progress on laser feedback interferometry technology is reviewed. Laser feedback interferometry is a demonstration of interferometry technology applying a laser reflected from an external surface, which has features including simple structure, easy alignment, and high sensitivity. Theoretical analysis including the Lang–Kobayashi model and three-mirror model are conducted to explain the modulation of the laser output properties under the feedback effect. In particular, the effect of frequency and polarization shift feedback effects are analyzed and discussed. Various applications on various types of lasers are introduced. The application fields range from metrology, to physical quantities, to laser parameters and other applications. The typical applications of laser feedback technology in industrial and research fields are discussed. Laser feedback interferometry has great potential to be further exploited and applied.

Laser feedback interferometry, or self-mixing interferometry, is a demonstration of interferometry technology applying a laser reflected from an external surface.1 Unlike traditional laser interferometry, laser feedback interferometry does not need additional optical elements for inducing interference outside the laser cavity.2 Instead, self-mixing interference occurs inside the laser cavity. The laser feedback phenomena can easily be caused by any surface in the light paths of the optical systems. Previously, laser feedback phenomena was regarded as damaging to the laser's performance because it induced instability in the laser power and frequency or caused strong quantum noises and coherent collapse.3 However, since its application by King and Steward as a displacement sensor in 1963,4,5 a massive amount of both theoretical and experimental work has been done in the field of laser feedback interferometry. According to laser feedback theory, laser intensities, polarization states, and phase behavior of lasers can be modified by introducing coherent optical feedback from external surfaces. Also, the bandwidth of the lasers can be enhanced or compressed under the laser feedback effect.

As long as the electromagnetic wave emitted is reinjected back into the laser cavity, self-mixing interference will be caused.6 The laser feedback effect is a remarkably universal phenomenon that can be observed in lasers of all different types. Among the work reported before, the laser feedback effect has been observed in gas lasers,7 semiconductor diode lasers,8 solid-state lasers,9,10 vertical-cavity surface-emitting lasers (VCSELs),11 mid-infrared lasers,12 and terahertz quantum cascade lasers (THZ QCLs),13 as well as in interband cascade lasers,14 fiber15 and fiber ring lasers,16 micro ring lasers,17 and quantum dot lasers.18

Compared to traditional laser interferometry, laser feedback interferometry has inherent advantages. The structure is more compact with fewer optical elements, the optical setup is easier for alignment, the laser feedback signal is detectable everywhere on the beam, and the beam at the target side can be exploited in special applications. Since laser feedback interferometers have no limits on the types of lasers, a variety of lasing wavelengths and bandwidths can be realized. The laser feedback interferometer can operate on a normal diffusing target surface due to its high sensitivity. Above all, with the growing need for precise measurement under various conditions, laser feedback interferometry is becoming more and more widely applied.19

This paper aims to provide an overall view of laser feedback interferometry. The framework of the paper is structured as follows. Section 2 reveals the theoretical analysis of laser feedback interferometry. Together with traditional laser feedback, the scheme and principles of the frequency-shifted and polarization-shifted laser feedback are proposed. Section 3 is an overview of the laser feedback effect on applications in metrology, including displacement, vibration, distance, velocity, angle, and thickness measurement. Section 4 reveals laser feedback interferometry used for laser parameters measurement, such as linewidth, line width enhancement factor, and polarization cross-saturation coefficient. In Sec. 5, the measurement of physical quantities is discussed, including refractive index, internal stress, thermal expansion coefficient, evaporation rate of liquid, and acoustic field distribution. In Sec. 6, applications including confocal tomography and imaging technology, sound rehabilitation, random bit generator, and chaos generator are explored. A summary and promising future directions are discussed in Sec. 7.

The laser feedback effects in all kinds of lasers are always following the same principle. Light is emitted from the laser cavity and then transmitted to an external target. From the surface of the target, the light is partially reflected and transmitted back into the laser cavity in the same optical path, where a portion of the light re-enters the laser cavity. Then, the reinjected light interacts with the resonant mode of the laser, and the property of the laser is modulated. Since the laser has a self-coherent nature, laser feedback systems are only sensitive to the laser reflected back to the laser cavity, suppressing most of the external radiation entering the laser cavity, which is regarded as noise in laser feedback systems. Due to the fact that all kinds of laser feedback interferometers share the same theoretical basis, the theoretical analyses are of great significance in revealing the phenomena in laser feedback effects and bringing to light potential applications.

In the transmission process, the feedback light passes through the optical path twice and is reflected by the surface of the target. Then, the reinjected light is embedded with the information of the external optical path or the property of the target. The feedback process brings the information back into the laser cavity. The interaction or the interference between the feedback light and the laser cavity causes modulations in the laser output properties, such as intensity, polarization state, working frequency, and phase position. Beam properties can be monitored by various kinds of detectors, and the information can be demodulated. Therefore, information about the target or external optical path can be easily obtained in a highly sensitive way. To fulfill precise demodulation of the information in the feedback light, theoretical analyses need to be carried out. Below are the two most commonly used methods for analyzing the laser feedback effect. Also, the effects of frequency and polarization shift feedback are discussed as important research foci to be explored further.

Lang–Kobayashi Model and Three-mirror Model

Lang and Kobayashi20 reported the laser feedback effect in laser diodes. They analyzed the multistable and hysteresis phenomena in laser diodes and put forward a theoretical solution by defining the degree of matching between the external feedback cavity and the laser resonant cavity. A component representing the effect of the external feedback in the form of a complex number has been added to the rate equations of diode lasers, which is known as the famous Lang–Kobayashi model. The proposal of this model has laid a solid foundation for the following studies.

The space between the output mirror of the laser cavity and the surface of the object is called the laser external feedback cavity. If the coupling strength between the laser cavity and the external cavity is κ, the standard Lang–Kobayashi equation can be expressed as follows:2125Display Formula

ddt[E(t)ejωt]={jωm+12(ΓG1τp)}×E(t)ejωt+κ˜E(tτext)ejω(tτext),(1)
where E(t) is the scaled, slowly varying complex envelope of the electric field; ω is the laser mode angular velocity; t is the time; ωm is the cavity resonance angular frequency; Γ is the optical confinement factor; G is the gain in the laser cavity; τp is the photon lifetime in the laser cavity; τext is the external feedback cavity round-trip time; and κ˜ is the feedback coupling rate, meaning the reinjected light is coupled into the laser cavity at this rate, which can be expressed as follows: Display Formula
κ˜=κ1τin.(2)

The gain term G is also related to another variation, which is the carrier density N. It can be expressed as follows: Display Formula

dN(t)dt=ηiI(t)qVN(t)τnGS(t),(3)
where ηi is the current injection efficiency of the laser diodes, I is the laser driving current, q is the charge of an electron, V is the cavity volume, τn is the carrier lifetime, and S is the photon density in the laser cavity. Equation (1) together with Eq. (3) describes the laser diodes dynamics under optical feedback.

Performing the differentiation of the product on the left side of Eq. (1), it can be deduced as follows: Display Formula

dE(t)dt={j(ωmω)+12(ΓG1τp)}×E(t)+κ˜E(tτext)ejωτext.(4)

In order to obtain the phase property through the rate equations, the phase φ(t) of the scaled, slowly varying complex envelope of the electric filed E(t) is defined as follows: Display Formula

φ(t)=arctan{Im[E(t)]Re[E(t)]}.(5)

Combining Eqs. (4) and (5), the phase can be expressed as follows: Display Formula

dφ(t)dt=(ωmω)κ˜[S(tτext)S(t)]12×sin[ωτext+φ(t)φ(tτext)].(6)

Using the linewidth enhancement factor α, which is defined as the ratio of change in the real part of the laser refractive index to the change in the imaginary part of the laser refractive index, Eq. (6) can be deduced as follows: Display Formula

dφ(t)dt=12α(ΓG1τp)κ˜[S(tτext)S(t)]12×sin[ωτext+φ(t)φ(tτext)].(7)

The Lang–Kobayashi model is time-dependent and can be used to describe the dynamic properties of active materials. It is found to give a remarkably accurate modeling of both the weak-level feedback phenomena and the high-level feedback chaos-related dynamics. Although it is deduced on the basis of the performance on laser diodes, it predicts many complex behaviors over short-time scales that have been observed in practice. In contrast to the dynamic Lang–Kobayashi model, there also exists another way of analyzing the laser feedback effect, which employs the geometry of the laser feedback cavity and effective reflectivity. Unlike the dynamic description obtained by adding the feedback effect component in the rate equations as in the Lang–Kobayashi model, the three-mirror model employs the static analysis method. The three-mirror model is equivalent to the Lang–Kobayashi model in analyzing the phenomena in the laser feedback technology, but the analysis processes of the two models focus on different aspects. de Groot26 proposed the three-mirror model, which explains the feedback effect in the aspect of the laser output intensity. The key point in the three-mirror model is to treat the reflective target as the third reflective mirror. This reflective mirror reflects the laser back into the resonant cavity, equivalent to changing the effective reflectivity of the laser output mirror, which induces modulation of the optical field inside the laser cavity. The schematic diagram is shown in Fig. 1.

Graphic Jump Location
Fig. 1
F1 :

The schematic diagram of three-mirror model in the laser feedback system. (a) The standard three-mirror model. (b) The simplified model of the compound cavity. M1, M2, mirrors of the resonant cavity; M3, the equivalent reflective mirror of the target; Meff, the compound reflective mirror of M2 and M3.

Suppose the electric field inside the laser cavity is E1(t); the electric field reflected by the laser cavity M2 is E2(t)=E1(t)r2, whereas the electric field output of the laser cavity is Eext(t)=E1(t)1r22. The electric field transmitting through the external feedback cavity and back to the laser cavity can be expressed as27,28Display Formula

Eext(tτext)=E1(tτext)1r22×exp(j2πντext)×κ,(8)
where ν is the laser frequency; since the external feedback cavity round-trip time τext is in a small scale, approximation can be done with E1(tτext)E1(t). The compound reflected electric field can be expressed as Display Formula
Eeff(t)=E1(t)r2+Eext(t)=E1(t)r2+E1(t)1r22×exp(j2πντext)×κ=E1(t)[r2+(1r22)×exp(j2πντext)×κ]=E1(t)r2[1+κextexp(jϕ)],(9)
where κext=(1r22)κ/r2 is the equivalent electric field feedback coefficient and ϕ=2πντext is the phase shift under the feedback effect. From Eq. (9), the equivalent reflective index can be expressed as follows: Display Formula
reff=EeffE1=r2[1+κextexp(jϕ)].(10)

In ordinary lasers, |r1|2=1 and |r2|21. The optical field loss rate γc can be expressed as follows: Display Formula

γc=c2nLln(1|r1|2|r2|2)c2nL(1|r2|2).(11)

Under the laser feedback effect, the reflective index r2 can be replaced by reff, such that the optical field loss rate with laser feedback influence γceff can be expressed as follows: Display Formula

γceffc2nL(1|reff|2)γc2γcκexp(jϕ).(12)

The initial rate equations in the laser can be expressed as Display Formula

dNdt=γ(N0N)BN|E|2,dE(t)dt=[j(ωmω)+12(BNγc)]E(t),(13)
where N0 is the inversion particle number under a small signal, γ is the inversion particle loss rate, and B is the stimulated radiation coefficient. The rate equations under the laser feedback effect can be expressed as follows: Display Formula
dNdt=γ(N0N)BN|E|2,dE(t)dt=[12(BNγc)+j(ωmω)]E(t)+γcκcos(ϕ)E(t).(14)

Comparing Eqs. (13) and (14), it can be seen that, under the effect of laser feedback, the intensity of the laser output has been added with a modulation component, while the phase ϕ can be defined as Display Formula

tanϕ=κextsin(4πLext/λ)1+κextcos(4πLext/λ),(15)
where Lext is the external cavity length. With Eqs. (9) and (15), by the approximation method, the intensity of the laser output under the laser feedback effect can be expressed as follows: Display Formula
I=I0κextcos(4πLextλ).(16)

From Eq. (16), I0 is the laser intensity without the laser feedback effect. Under the laser feedback effect, a cosinoidal modulation is added in the intensity. Every half of the wavelength change in the external cavity length corresponds to one cycle in the intensity modulation. The three-mirror model is especially appropriate for analyzing the intensity modulation in the laser feedback field. Both the Lang–Kobayashi model and the three-mirror model can explain the feedback phenomena in various types of lasers, e.g., gas lasers, semiconductor diode lasers, solid-state lasers, and VCSELs. Therefore, both models are effective tools in the research of laser feedback interferometry.

Laser Feedback Effect with Frequency Shift and Polarization Shift

In the above section, the effect of single-frequency and single-polarization feedback is discussed in the two models. However, to improve the performance of lasers under the laser feedback effect and expand its applications, the feedback effects with different properties are discussed in this section.

Frequency-shifted feedback

In the laser feedback system, frequency shift technology can also be applied to improve the performance of the laser feedback interferometers. When the frequency shift modulator is added inside the external feedback cavity of the laser, the model of the frequency-shifted laser feedback system is shown in Fig. 2. The frequency of the laser output is ν, and the frequency shift of the modulator is Ω. Since the light reinjected into the laser cavity transmits through the frequency shifter twice, the feedback laser has the frequency of υ+2Ω.

Graphic Jump Location
Fig. 2
F2 :

The model of the frequency-shifted laser feedback system. ν, the frequency of the laser output; Ω, the frequency shift of the modulator.

Under the effect of the frequency-shifted feedback, Eq. (8) can be rewritten as follows: Display Formula

Eext(tτext)=E1(tτext)1r22×exp[j2π(ν+2Ω)τext]×κ.(17)

The rate equations under frequency-shifted feedback can be expressed as follows:29Display Formula

dNdt=γ(N0N)BN|E|2,dE(t)dt=[12(BNγc)+j(ωmω)]E(t)+γcκexp(j2Ω)exp[j(ω+2πΩ)τext]E(t).(18)

With Eqs. (17) and (18), the relative modulation of the laser intensity is expressed as Display Formula

ΔI(2Ω)Is=κG(2Ω)cos(2Ωtϕ+ϕs),(19)
where ΔI(2Ω) is the modulation under the frequency-shifted laser feedback effect, Is is the stable laser output intensity, ϕs is the fixed additional phase, and G(2Ω) is the gain factor. In some types of lasers, such as solid-state lasers or semiconductor lasers, there exists the relaxation oscillation phenomenon. In these continuous-functioning lasers, the output light is made up of spike pulses instead of smooth pulses.30 The relaxation oscillation phenomenon is caused by the dynamic interaction between the radiation inside the laser resonant cavity and energy storage in the laser medium. If the relaxation oscillation frequency of the laser is υR, the gain factor G(2Ω) in the solid-state laser is a function of the oscillation phenomenon frequency υR, which can be expressed as Display Formula
G(2Ω)=2γc[η2γ2+(2Ω)2]12{η2γ2(2Ω)2+[υR2(2Ω)2]2}12,(20)
where η is the pumping level. This gain factor is a unique property in the frequency-shifted feedback systems. This gain factor equals the amplification in the laser feedback signal. From Eq. (20), when 2Ω equals υR, the gain factor G(2Ω) reaches its maximum value. Then, Eq. (20) can be deduced as follows: Display Formula
G(2Ω)=2γc[η2γ2+(νR)2]12[η2γ2(νR)2]12=2γcηγ·[η2γ2+(νR)2]12νR.(21)

Since the relaxation oscillation frequency of the laser is relatively large in amount, it can be seen that η2γ2νR2, [η2γ2+(νR)2]12νR, then Eq. (21) can be simplified as Display Formula

G(2Ω)2η(γc/γ).(22)

For the solid-state lasers, this maximum gain factor value can reach to a scale as high as 106, indicating that the solid-state laser frequency-shifted feedback system has ultrahigh sensitivity for the feedback light. Once the proper shift frequency is set, the feedback light can be greatly amplified, which makes the solid-state laser frequency-shifted feedback system extremely appropriate for weak signal detection.

Polarization-shifted feedback

By adding a birefringence component into the laser feedback cavity, the polarization states of the feedback light can be shifted and the laser output intensities in the two orthogonal directions with the phase difference can be modulated.3134 The birefringence components are usually in the form of wave plates or quartz crystals. When the birefringence component is put into the external cavity, the original external cavity forms two different physical external cavities. Supposing the two axes in the laser are x and y, the birefringence axes in the birefringent component are in the same directions. Also suppose the amplitude reflection coefficients of both laser cavity mirrors are r1 and r2; the transmission coefficient of the laser is T=1r22; r3 is the reflection coefficient of the feedback mirror; l is the length of the external cavity; and in the corresponding axes, the external cavity lengths are lo and le, d is the length of laser cavity, n is the refractive index of the laser material, and ξ is the feedback coefficient. Then, the gain of the laser in the x and y axes is as follows: Display Formula

{gx=1nd[ln(r1r2)+β2cosφ],gy=1nd[ln(r1r2)+β2cos(φ+2δ)],(23)
where β=Tr3ξ/r2 and φ=4πυlo/c, δ is the phase difference of the birefringence component. Under the effect of the polarization-shifted feedback, the laser intensities in two orthogonal axes can be expressed as Display Formula
{Ix=I0x[1+βK2ndcosφ]Iy=I0y[1+βK2ndcos(φ+2δ)],(24)
where I0 is the initial intensity without feedback and K is the constant. From Eq. (24), the intensities have a phase difference that is twice the phase difference of the birefringence component. As a result, the polarization-shifted feedback effect can be utilized to measure the birefringence, optic axis azimuth,35 or displacement.36

The above models and analyses give theoretical explanations of the corresponding relationship between the external feedback cavity and the variations in the parameters inside the laser cavity. By applying this correspondence relationship, various kinds of applications can be fulfilled with laser feedback interferometry. Below are some typical applications of the laser feedback effect. In order to simplify the interferometer structure and reduce the cost, the laser sources in these self-mixing laser interferometers are mostly commercial semiconductor lasers with photodetectors sealed inside. In some special occasions, the lasers are solid-state lasers, gas lasers, or other lasers. In fields needing precision measurement, the stability of the laser frequency is required, since the cycles of the laser feedback signals are directly traced back to the laser wavelength. Instabilities in laser power and frequency will bring errors in the measurement results. Therefore, the laser stability and robustness are required for laser feedback interferometers.

Metrology is the science of measurement, embracing both experimental and theoretical determinations at any level of uncertainty in any field of science and technology. As the basic research in the measurement field, precise measurement is significant and can be applied in a large variety of fields. On the basis of the theoretical analyses above, six typical aspects of laser feedback effect applications in metrology are described in the following sections.

Displacement Measurement

In the fields of metrology, the precise measurement of displacement has always been a research focus. Since the displacement measurement is widely applied in various industrial situations and scientific research, noncontact measurements are in strong demand. Compared with traditional laser interferometers, the laser feedback interferometer has great potential because it does not need a retroreflector and can be applied to a nonmatching target with a diffuse or rough surface. In the above analyses of laser feedback theory, according to Eq. (16), every change of half the wavelength in the length of the external cavity corresponds to a cycle in the laser intensity modulation. The congruent relationship makes it easy to measure displacement by counting the total cycles of modulation.

With the rapid development of displacement measurement based on laser feedback technology, various methods emerge, such as fringe counting technology with its simple structure and high sensitivity.37 The method makes use of fringe counting technology since half of the wavelength distance change in the external cavity corresponds to a cycle of intensity modulation in the laser feedback effect. The directions of dips in the sawtooth-shaped feedback waveforms indicate the direction of the displacement. The method is capable of measuring displacement in half of the wavelength scale on a substantial distance up to 1 m without any optical adjustment aside from the initial pointing on the target.

A research focus of laser feedback technology applied in the displacement measurement field is the heterodyne microchip solid-state laser employing external frequency-shifted feedback.38,39 As mentioned above, the relaxation oscillation phenomenon in solid-state lasers causes a highly sensitive response as large as 106 to the frequency-shifted laser feedback. The schematic diagram is shown in Fig. 3.

Graphic Jump Location
Fig. 3
F3 :

Schematic configuration of the frequency-shifted laser feedback interferometer. LD, laser diode; GL, grin lens; ML, microchip laser; BS, beam splitter; D, detector; AOMs, acousto-optic modulators; L, lens; MR, reference mirror; T, target.

As shown in Fig. 3, the microchip laser is a laser diode pumped Nd:YAG or Nd:YVO4 laser and is frequency stabilized by precise control of the temperature.40 The feedback levels are studied and ensured for the performance of the interferometers. The acousto-optic modulators are used to produce the shift in the frequency of the laser reinjected back into the laser cavity. Since the working frequency of the acousto-optic modulator is usually high, two acousto-optic modulators are employed, and the frequency difference between them is used as the feedback frequency shift. Suppose the working frequency of the AOM1 and AOM2 is Ω1 and Ω2, respectively, Ω is the difference between Ω1 and Ω2. The schematic diagram of the optical path is shown in Fig. 4.

Graphic Jump Location
Fig. 4
F4 :

Schematic diagram of the optical path in the frequency-shifted feedback interferometer. ML, microchip laser; BS, beam splitter; PD, photodetector; AOM, acousto-optic modulators; L, lens; MR, reference mirror; T, target.

Due to the frequency shift characteristics of the acousto-optic modulators, the laser reflected by the target can obtain a frequency shift of 2Ω in the measurement signal, while the laser reflected by the reference mirror can obtain a frequency shift of Ω acting as the reference signal. The analyses are as follows: The optical path of the reference beam is shown in hollow arrows while the measurement beam is in solid arrows. If the frequency of the initial light emitted is υ, the reference beam passes the acousto-optic modulators AOM1 and AOM2 without diffraction, which means the frequency remains υ. The measurement beam passes through AOM1 and is diffracted at 1 order, which means the frequency is υΩ1; then it passes through AOM2 and is diffracted at +1 order such that the frequency now is υΩ1+Ω2=υ+Ω. The reference beam is focused by lens L and reflected back to the acousto-optic modulators, and then it is diffracted at +1 order in AOM2 and at 1 order in AOM1, while the measurement beam is reflected at the target surface and is diffracted again at +1 order in AOM2 and at 1 order in AOM1. As a result, the feedback reference beam has the frequency of υΩ1+Ω2=υ+Ω, whereas the feedback measurement beam has the frequency of υΩ1+Ω2+Ω2Ω1=υ+2Ω. The displacement information in both the measurement signal and reference signal is demodulated, and the displacement detected in the reference signal is used for compensation. Therefore, the frequency-shifted laser feedback interferometer can achieve high resolution and accurate displacement measurement with little zero drift. The resolution of the frequency-shifted laser feedback interferometer is 1 nm by phase discrimination method, and the zero drift within 3 min is less than 8 nm.39 The performance of the laser feedback frequency shift interferometer is calibrated, compared with the commercial dual-frequency laser interferometer Agilent 5529A. The correlation coefficient of the linear fit differs from 1 by 6×104, and the residual error is <1.5  μm.

The method has relatively high resolution and sensitivity due to the frequency-shifted feedback characteristics in solid-state lasers, as analyzed in Sec. 2. Therefore, the frequency-shifted laser feedback increases the sensitivity in the laser feedback interferometer, making it especially appropriate for measuring objects with rough and black surfaces or irregular shapes. The sensitivity is improved at the cost of a more complicated structure and increased expense. However, the measurement range is limited to about 1 m since the measurement process needs a proper feedback level,39 and the method for determining the direction of displacement is difficult and complex to realize. Due to the fact that the common part of the measuring optical path and the referring optical path is limited, with the increasing measurement range, the accuracy will be affected. Another method is a laser feedback interferometer based on the phase difference of orthogonally polarized lights in the external birefringence cavity.41,42 The method determines the direction of the displacement easily by the sequences of two orthogonally polarized lights. The measuring range is large, and the measurement is not limited by the feedback level. The schematic diagram is shown in Fig. 5. The laser applied is a laser diode pumped Nd:YVO4 laser.

Graphic Jump Location
Fig. 5
F5 :

Schematic diagram of laser feedback interferometer based on phase difference of orthogonally polarized lights in external birefringence cavity. LD, laser diode; GL, grin lens; ML, microchip laser; NPBS, nonpolarizing beam splitter; ATT, attenuation plate; WP, wave plate; ME, feedback mirror; W, Wollaston prism; D1, D2, detectors.

The theory of this method is as follows: when linearly polarized light passes through the birefringent external cavity and then is reflected back into the laser resonator by an external object, a phase difference is generated between the laser sinusoidal modulated intensities in the two orthogonal directions. The birefringence in the external cavity is produced by the wave plate, the phase retardation of which is 45 deg. A phase difference of 90 deg is induced between the two orthogonally polarized components. By applying fringe counting technology and fourfold subdivision, displacement can be easily calculated with high precision. The linearity of this system is 2.57×105 over a 7-mm range, and the standard deviation is 0.34  μm. The resolution is 53.2 nm and can be further improved by applying multifold subdivision methods adopted in the grating encoders.

In addition to the three methods mentioned above, there also exist various other methods for displacement measurement applying laser feedback technology. Guo et al.43 reported a method based on the laser feedback grating interferometer, which applies a double-diffraction system. The operating frequency is 100 MHz, and the error is <2.9  nm. Bernal et al.44 proposed a robust fringe detection method based on biwavelet transform. Wavelet transform is applied, and it enables the measurement of arbitrarily shaped displacements based on the pattern recognition capability. Donati et al.45 reported a method for simultaneous measurement of displacement and tilt and yaw angles of the target, taking advantage of two orthogonal modulations of the beam. Zeng et al.46 proposed laser feedback interferometer based on high-density cosine-like intensity fringes with phase quasi quadrature; the system has a resolution of 0.51 nm in 850  μm and the displacement measurement accuracy was 5 nm.47 Jha et al.48 presented a method making efficient use of direct laser injection current modulation to induce continuous wave frequency modulation and nonlinear dynamics effects in a laser diode subjected to optical feedback to measure nanometric amplitude displacements. The error is 2.4 nm. Chen et al.49 proposed synthetic-wavelength self-mixing interferometry for displacement measurement. The virtual synthetic wavelength is 106 times larger than the operating wavelength, so the subnanometer displacement can be obtained.

Vibration Measurement

When the displacement to be measured is less than half the wavelength in the amplitude with a relatively high frequency range, the fringe counting method is not appropriate and not accurate enough. In the case of vibration, the amplitude is relatively small and the frequency is high, which requires different techniques from displacement measurement.

Signal processing proposed by Tao et al.50 synthesizing wavelet transform and Hilbert transform is employed to measure uniform or nonuniform vibrations in the laser feedback interferometer based on a semiconductor laser diode with quantum well. Background noise and fringe inclination are solved by the decomposing effect, fringe counting is adopted to automatically determine the decomposing level, and a couple of exact quadrature signals are produced by Hilbert transform to extract vibration. The continuous wavelet transform is applied to weaken the inclination of the fringe. From the results, it can be seen that the reconstructed phase curve is in high accordance with the assumptive phase curve, indicating that this method has excellent performance in vibration measurement. This method has the following advantages. First, the frequency or phase modulation is deleted from the system, so the bandwidth limitation from modulation is removed; the size and cost both decrease sharply. Second, the feedback level is not limited, making it more applicable in general environments. Third, the high computational efficiency provides potential in real-time monitoring, and noise is greatly restrained.

When self-mixing interferometry occurs in an asymmetrical cavity, the light and the vibration target are not strictly vertical. The light will experience multiple reflections between the vibration target and the laser. In such cases, the number of interference fringes doubles or even increases by three or four times. This phenomenon is called multiple self-mixing interferometry.51 Jiang et al.52 proposed a demodulation algorithm based on the power spectral analysis for multiple self-mixing interferometry. Its precision can reach as high as λ/47.6, and the absolute error of amplitude is 0.075  μm. Magnani et al.53 presented a method on diode lasers including an electronic feedback loop for increasing the measurement resolution. This system is able to measure vibrations up to 110 mm peak-to-peak at a distance of 1 m, with bandwidth between 5 Hz and 15 kHz and resolution of about 60 nm. Wu et al.54 constructed a single-channel laser feedback interferometer that measures the magnitude of the vibration and the angle of the position at the same time. The curve of the peak frequency versus the vibration amplitude follows the linear distribution, whereas the curve of the difference between the peak power versus the angle follows a Gaussian distribution. Lukashkin et al.55 measured basilar membrane vibrations by laser feedback interferometer without opening the cochlea. Giuliani et al.56 demonstrated a laser vibrometer based on the self-mixing interferometric effect in a laser diode. The system has features such as better than 100  pmHz1/2 noise equivalent vibration, 180  μm peak-to-peak maximum measurable vibration, larger than 100-dB dynamic range, 70-kHz bandwidth, and successful operation on most rough surfaces. Huang et al.57 proposed a vibration system extreme points model based on the laser feedback effect; the piezoelectric coefficient of a piezoelectric transducer in which d33 is 0.66034  nm/V was obtained. Chen et al.58 investigated a simple damping microvibration measuring method that can accurately obtain the damping factor. The damping factor is solved by recording the period and counting the fringe number of the self-mixing signal. The damping factor of 0.0483  s1 with a standard deviation of 0.0013 and the coefficient of variation of 2.69% were experimentally achieved.

Distance Measurement

Distance measurement with optical methods is widely used. However, the traditional methods have a distance measurement range limited by the power of the laser, and the accuracy is hard to ensure. Due to the high sensitivity in laser feedback interferometry and its simplicity in structure, which causes less loss to laser power, laser feedback interferometry is applied in the field of distance measurement. Shinohara et al.59 put forward the method of distance measurement using a self-mixing laser diode. The interferometer measures the averaged mode hop time interval of successive external mode hops by the backscattered light from a target. The interferometer has precision of ±15% in the wide range of 0.2 to 1 m. Since then, the laser feedback effect in diode lasers has been widely utilized for distance measurement.

Laser feedback interferometers based on laser diodes have a feature that a small variation in the laser driving current will produce an approximately linear variation in the laser operating frequency.60 By utilizing this feature, a periodic modulation in the driving current will lead to frequency sweeping in the output of laser diodes. With the method of injected current reshaping in a laser self-mixing interferometer based on VCSEL proposed by Kou et al.,61 a precise distance measurement can be achieved. A decrease in wavelength has a similar effect as a scattering object moving away from the laser source, whereas an increase in wavelength mimics an object moving toward the source. A sawtooth-shaped tuning current is utilized to drive the VCSEL to demonstrate the change of the beat frequency. The relationship between the laser wavelength and the frequency versus the injected current is researched. Each value of the divided frequency is used to deduce backward to the corresponding injected current, which eliminates the nonlinearity. Therefore, the injected current is reshaped. The reshaped injected current technique is effective for reducing the nonlinearity in current tuning, and the resolution of the distance measurement is improved to be 20  μm in the range from 2.4 to 20.4 cm.

Moench et al.,62 Michalzik,63 and Gouaux et al.64 also applied the VCSEL laser feedback interferometer based on the modulation of the working current. Guo et al.65 proposed a method based on a double-modulation technique on a laser diode for absolute distance measurement. The intensity of the laser and the amplitude of phase are both modulated to raise the resolution. The absolute distance has been measured with a resolution of ±0.3  mm over the range of 277 to 477 mm.

Velocity Measurement

Velocity measurement is another important field in metrology. Velocity is usually measured by the laser Doppler velocimeter employing the Doppler effect. Since 1968, when Rudd66 applied the laser feedback effect in velocity measurement, laser feedback velocimeters have been the research foci.67,68 The signal curve of the velocimeter is in sawtooth shape and the discrimination of the movement directions is researched.69,70 In the laser feedback field, the laser feedback Doppler velocimeter is a type of compact and high cost-effective Doppler velocimeter with high precision.

One of the most attractive velocimeters uses the three perpendicular beams fiber irradiation scheme. Initially, Mikami and Fujikawa71 proposed a two-beam irradiation scheme. In the two-beam irradiation structure, the two beams are irradiated at one spot on the target. The offset angle between the two beams is set to be Δθ. In this configuration, the interference takes place and the Doppler shift frequencies can be obtained by the spectrum analyzer. Then, the velocity of the target V can be expressed as follows: Display Formula

V=λ2sinΔθf12+f222f1f2cosΔθ.(25)

It can be seen that the velocity of the target can be obtained only with prior knowledge of the offset Δθ. However, the two-beam scheme demands that the velocity vector of the target is on the same plane with two irradiation beams. To solve the issue in the two-beam velocimeter, the three-beam scheme is proposed. The beam from a laser diode having a single longitudinal mode is divided into three beams by the fiber couplers. The three irradiation beams of the sensor head are set perpendicular to one another. Therefore, each beam direction is considered the axis of the orthogonal coordinate system. The configuration and experimental setup are shown in Fig. 6. By this method, the measurement of arbitrary velocity can be fulfilled.

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Fig. 6
F6 :

Configuration and experimental setup of the three-beam self-mixing velocimeter. f, frequency of the beam; V, velocity of the target.

Scalise et al.72 developed a new model and a suitable processing algorithm for the analysis of the signal in the case of relevant speckle effects. Bosch et al.73 employed multimode VCSELs for velocity measurements, which can be of great interest due to its low price, high stability, and low threshold. In the applications of the laser feedback velocimeters, the scattering of the laser emission from a rough surface will lead to a speckle effect that models the Doppler signal, causing broadening of the signal spectrum and adding uncertainty to the velocity measurement. To eliminate this uncertainty induced by the speckle effect, various research has been done and different methods, such as analytic equation prediction74 and fractal analysis with box-counting,75 are employed. Huang et al.76 studied the results of the speckle self-mixing velocimeter concerning the angles of incidence. The relative error drops dramatically with the incident angle increasing from 5 to 30 deg. Özdemir et al.77 proposed the autocorrelation data processing method for speckle self-mixing velocimeter; effects of various parameters have been analyzed and the system has been optimized with fewer measurement errors. The linear relation between the reciprocal of the autocorrelation time of the speckle signal obtained and the velocity of the target has been analyzed.78 By employing another laser diode, simultaneous measurement of velocity and length of moving targets with homogeneous rough plane surfaces can be realized.79 Shinohara et al.80 have also applied the speckle effect of the self-mixing interferometer for surface classification, making the method inexpensive and highly sensitive.

In addition to the measurement of the velocities in solid targets, the velocity of flowing liquid can also be measured by applying a laser diode self-mixing technique.81 Compared with the traditional method using microparticle imaging velocimetry, which has disadvantages such as being complex, time-consuming, and bulky, laser feedback interferometry has a simpler structure and faster response in flow measurement. Fluid velocity measurements of water seeded with titanium dioxide have been performed using a diode to measure the effect of the seeding particle concentration and the pump speed of the flow. The laser light was focused at the center of the fluid discharging from the nozzle. The measured velocity linearly increases with the pump speed as expected. The system has demonstrated accuracy better than 10% for liquid flow velocities up to 1.5  m/s with a concentration of scattering particles in the range of 0.8% to 0.03%. By using different types of seeders, higher velocity measurement can be realized. Campagnolo et al.82 reported the work of flow profile measurement in microchannels applying a VCSEL laser feedback interferometer. In this work, the measurement of local velocity in fluids with spatial resolution in the micrometer range is achieved by minimizing the laser spot size with the diameter being 32.26  μm. Ramírez-Miquet et al.83 utilized a blue–violet laser diode with a wavelength of 405 nm, which is capable of measuring very slow velocities. The system is found to be a good representation of the liquid–liquid two-phase system’s hydrodynamics.

With the growing need for accurate diagnosis and disease prevention in biology and the medical fields, precise measurement of blood flow in vessels and tissues is attracting greater research attention. The invasive measurement of skin, capillary, or cutaneous blood flow is considered to be significant and can be used for the assessment of vasopasm, ischemia, wound and ulcer healing, skin diseases, therapeutic trials, and various pathological conditions.84 A very active research focus in this field is laser feedback interferometry. Since de Mul et al.,85 Slot et al.,86 and Mito et al.87 first utilized the self-mixing Doppler phenomenon for blood flow measurements, multiple studies have been done on this subject. Shinohara et al.88,89 utilized the self-mixing speckle signal of backscattered light from the red blood cells and the speckle autocorrelation method to realize blood flow measurement. The experimental setup is shown in Fig. 7. The system has the capability of measuring blood flow at speeds from 30 to 125  mm/s. Norgia et al.90 applied the fast Fourier transformation method and logarithm-weighted mean to optimize the linearity of the system. The accuracy was better than 0.1  L/min in the large range of 0 to 6  L/min of clinical interest. Figueiras et al.91 applied the laser self-mixing velocimeter as a microprobe for monitoring microvascular perfusion in rat brain. The mean detection depth was 0.15 mm, and the size of the probe was as small as 785 nm. The authors also introduced three signal processing methods, including the counting method, autocorrelation method and power spectrum method, which supplied efficient methods for data processing.

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Fig. 7
F7 :

Experimental setup of laser self-mixing velocimeter for flow measurement. D, detector; LD, laser diode.

In addition, Zhao et al.92 presented a method of velocity measurement using a self-mixing velocimeter with an orthogonal beam incident system, which enables velocity measurement without knowing the incident angle information. The fiber ring laser acts as a stable and narrow-width laser light source, which could enhance the stability and the signal-to-noise ratio of the self-mixing fiber ring laser velocimeter. The relative error rates of this velocimeter system are up to 1.258%.

Angle Measurement

The noncontact precise measurement of angles is strongly needed in industrial and research situations. The traditional methods, such as autocollimator or laser interferometers, usually need a reflective mirror, which limits the application. Since the laser feedback effect has high sensitivity, it can be applied in adjusting the remote mirror or angle measurement.93 One solution to precise angle measurement proposed by Zhang et al.94 is the parallel multiplex laser feedback interferometer. The system setup is similar to Fig. 3; however, the laser in the parallel multiplex laser feedback interferometer is produced by two parallel laser diodes pumping one microchip. The system also applies frequency-shifted feedback interferometry. From the configuration, it can be seen that the laser emits two parallel beams at different positions on the same target. The displacements of these positions are measured at the same time. Therefore, the displacement difference reflects the angle variation of the target. This scheme does not generate any signal crosstalk for several reasons. First, only when the laser beams go back exactly along the same path injecting to the target can the frequency of the beam be shifted by 2Ω and demodulated by the heterodyne phase measurement, which effectively restrains the external noise. Second, although the two beams are generated in the same microchip, the frequencies of the beams are different from each other. Thus, there is no cross interference. The beat frequency of the two beams can be filtered. Third, the two optical paths are very close, so the displacement difference, which is the subtraction of the two measured displacements, can eliminate the environmental disturbance and thus accurately reflect the angle variation. The measurement results are compared with the laser interferometer Agilent 5529A. The parallel multiplex laser feedback interferometer exhibits good stability with the maximum nonlinear error of 8′′ in the range of 1400′′.

Another method for angle measurement is applying the external-cavity birefringence feedback effects of a microchip laser proposed by Ren et al.95 The wave plate inside the external feedback cavity has the role of producing the external-cavity birefringence feedback. The cross-section of the refractive index ellipsoid of the wave plate is an ellipse. It can be seen that, based on the external birefringence feedback, the information of the roll angle in the birefringence element placed inside the external cavity can be transferred into the phase difference between the two orthogonally polarized beams. Giuliani et al.96 proposed a technique to measure the angle of a remote flat surface with respect to the propagation direction of the laser beam, based on injection detection in a laser diode. The technique enables the measurement of angles with a sensitivity of 5×107  rad. Zhong et al.97 presented a signal defining method that places the refractive mirror on the surface of the object for measuring a small angle. The rotation angles are measured with an accuracy of 106  rad. The angle measurement range is approximately 0.0007 to +0.0007  rad. As mentioned above in Ref. 53, the simultaneous measurement of vibration and the angle of the position of the target relative to the incident beam can be realized. The rotation angle over a range of 0.075 deg can be measured, with the error <11.7%. Tan and Zhang98 also proposed a system including two orthogonally linearly polarized modes produced by a tunable frequency difference that exists in microchip lasers with two quarter-wave plates in the laser inner cavity. The frequency difference relates to the orientation of one quarter-wave plate. The sensitivity can be as high as 0.3 arc sec, which is 1.6×106  rad.

Thickness Measurement

Precise measurement of the thickness of optical elements plays a significant role in optical industrial applications. However, there is a lack of effective testing methods other than contacting methods, which have low accuracy and cause damage to the optical elements. The laser feedback effect has been applied in thickness measurement by Fathi and Donati99 Considering the high sensitivity of frequency shift feedback, the frequency-shifted laser feedback interferometer has been used for thickness measurement.100 The configuration is shown in Fig. 8.

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Fig. 8
F8 :

Configuration of frequency shift feedback thickness measurement system. LD, laser diode; GL, grin lens; ML, microchip laser; BS, beam splitter; D, detector; AOM, acousto-optic modulators; L, lens; MR, reference mirror; S, sample; ME, optical wedge as the measurement mirror.

By rotating the sample inserted into the external feedback cavity, the optical path difference induced can be measured by the laser feedback interferometer. The optical path change in the rotating sample can be derived as Display Formula

ΔL=(ΔφmΔφr)λ2π=d(n2n02sin2θ2n0cosθ2n2n02sin2θ1+n0cosθ1),(26)
where ΔL denotes the optical path change induced by the rotation of the sample, λ is the laser wavelength, d is the thickness of the sample, n is the refractive index of the sample, n0 is the refractive index of the air, and θ1 and θ2 are the angles between the laser beam and the normal direction of the sample surface before and after the rotation. By measuring θ2 and ΔL at multiple angles, the overdetermined equation can be solved and the thickness of the sample can be obtained together with the refractive index. Through the experiments, the measurement uncertainty of the thickness is better than 0.0006 mm. However, in measuring the thickness, this method requires the sample to be flat and parallel. The parallelism is better than 2″ in the reference. Due to the fact that the sample needs to be rotated, if the sample is not parallel or flat, after the rotation, the thickness d in Eq. (26) is hard to maintain, which causes a large error in measurement. Also, the measurement procedure needs to be combined with the rotation of the sample at high precision angles. Another method proposed by Tan et al.101,102 combining the confocal effect and frequency shift feedback, which is noncontact, nondestruction, and highly sensitive, can be applied in measurement not only of the thickness but also between the air gap. The schematic setup of the system is shown in Fig. 9.

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Fig. 9
F9 :

Schematic diagram of the laser frequency shifted confocal system. LD, laser diode; GL, grin lens; ML, microchip laser; BS, beam splitter; D, detector; AOM, acousto-optic modulators; BE, beam expander; AP, annular pupil; Obj, objective lens.

The laser is separated into two beams by the beam splitter. The measuring light selected is expanded and collimated via the beam expander. The annular pupil is inserted into the path before the objective lens to create the annual beam. There exists a conjugation relationship between the focus point of the objective lens and the laser beam waist. The light intensity modulation under the confocal effect is expressed as Display Formula

I(u)=|sin[(1ϵ2)u/2](1ϵ2)u/2|2,u=8πλzsin2(α/2),(27)
where z is the defocus distance, α is the numerical aperture angle of the object lens, and ϵ is the ratio of the inner and outer diameters caused by the annular pupil. The intensity modulation in the frequency-shifted confocal feedback can be derived as follows: Display Formula
ΔI(2Ω)Is=|sin[(1ϵ2)u/2](1ϵ2)u/2|2·κG(2Ω)cos(2Ωtϕ+ϕs).(28)
The simulation and experimental results are shown in the Figs. 10(a) and 10(b).

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Fig. 10
F10 :

Results of laser frequency-shifted confocal feedback system. (a) Simulation results and (b) measured results.