1. Introduction

Measurements of physical parameters with optical techniques play an important role in science and technology [1]. Possibility of non-contact measurements is a significant advantage of optical techniques, which makes it possible to determine distances to objects, as well as register their displacement, speed and vibrations. The most direct technique for measuring the distance (most often used for remote objects) is to determine the time of flight taken for pulsed radiation to travel to the target and back [2]. In this case, the measurements spatial resolution is limited by the pulse duration. Researchers face a significant complication of the measuring system when applying this approach to distances less than 1 m [3] because of the need to use high-end modulators and detectors. For example, low noise superconducting nanowire single-photon detector is used to obtain centimeter resolution at stand-off distances of the order of one kilometer [2].

Interferometric techniques, as well as their modifications, are an alternative to the “time of flight” technique. The interferometric techniques allow one to measure subwavelength displacements of objects [3–6]. For this application either low-coherence with broad generation spectrum or highly coherent widely tunable radiation sources are used. It is necessary to match with high accuracy the distances of the reference and test arms in the former case, and one does not need to make this precise adjustment in the later one. It should be noted that interferometric distance sensors based on low-coherence sources with scanning of reference position (or “white-light interferometry”) are mostly successfully commercialized due to cheaper sources [6]. The second type of interferometry is refer to “optical coherence tomography” (OCT) and can be associated with frequency domain interferometry. For example, OCT based technique with rapidly-tuned wavelength-swept laser allows to obtain cross-sectional imaging of biological samples with axial resolution of 13.5 µm [7]. Another interferometric technique of a target displacement measurement is self-mixing interferometry based on miniature semiconductor lasers (see, for example, reviews [8,9]).

It should be also noted about an evident opportunity to calculate speed and/or vibration of an object based on its distance measurements, if its displacement during the measurements can be neglected. Besides, the velocity of an object can be directly measured by registering the optical frequency shift Δν_D associated with the Doppler effect when radiation is reflected/scattered by the moving object having velocity projection V along the light propagation direction: Δν_D/ν₀ = 2V/c, where ν₀ and c are the optical frequency and speed of light, respectively (see, for example, [3,10]). The Doppler frequency shift measurements can also be used to obtain information about vibrations of targets (see, for example, [3,10–11]), since the velocity is a derivative of the coordinate with respect to the time. Currently, Doppler vibrometers have already been applied for routine quality control in mass production of various mechanisms.

It should be noted that scattering of a coherent laser beam by surfaces, which are optically rough and often encountered in traditional engineering structures, produces speckle patterns. It is known that the speckles can significantly influence results of optical backscattered signals measurements [8,10]. A technique of spatial filtering of the laser speckle pattern, can be applied to detect in-plane movements of a diffusing object [12]. It should be also mentioned that there are some techniques for vibration measurements based on analysis of the speckle patterns (see, for example, [13]).

Among the huge number of works devoted to the development of optical measurement techniques, one should single out those that can be used for simultaneous distance to a target and its speed determination. For this purpose, the mentioned above interferometric systems with swept laser sources are also applied [14–15]. The optical frequency of the probe laser is scanned in upward and downward directions to measure the intermode spacing of an interferometer containing target under test as a weak reflector. For a stationary target, the result of the intermode spacing (frequency) measurements is independent of the scanning direction. The movement of the target leads to the fact that the intermode spacing becomes dependent on the scanning direction. It increases for one direction and decreases by the same value for the other one. The average value and difference of these two measurements contains information about the distance to the target and its speed, correspondingly. A frequency-modulated continuous-wave LiDAR with a Si photonic crystal beam scanner demonstrated in [16] allows to measure the distributions of distance, velocity, and vibration frequency, simultaneously. Reported distance and velocity resolutions were less than 15 mm and 19 mm/s, respectively. The detection limit of the vibration amplitude determined by the signal-to-noise ratio was 2.5 nm.

In our work, we also propose to use an interferometric method for measuring distances, but instead of speed, we propose to measure simultaneously the amplitude and frequency of small vibrations. A fiber Mach-Zehnder interferometer (MZI) is used to measure distance to a target and its vibrations. A key element of the vibrometer-rangefinder is a self-sweeping ytterbium-doped fiber laser, which does not require any spectrally selective elements and drivers for wavelength tuning. The laser is used as a tunable source of probe radiation with a sweeping range of 1056-1074 nm [17]. Here we present measurements and modeling results of the main characteristics of the developed vibrometer-rangefinder. We demonstrate the possibility of measuring target vibrations in the frequency range from 2 Hz to 5 kHz with an amplitude of down to ∼5 nm at a distance of up to ∼13 m. The maximum measurable vibration frequency is limited by the instability of the self-sweeping laser parameters in the time domain and is estimated as ∼7.5 kHz. The results obtained show that characteristics of vibrometer-rangefinder is comparable with similar range and vibration measurement systems mentioned above. The advantages of the proposed system are design simplicity of the tunable probe laser and high spatial resolution.

2. Experiment

The scheme of a proposed laser rangefinder-vibrometer is shown in Fig. 1, which is similar to one for a rangefinder presented in [18]. The optical scheme is based on polarization maintaining components. A probe radiation source is a linearly polarized Yb-doped fiber laser with wavelength sweeping in a range of ∼1056-1074 nm, due to the self-sweeping effect taking place without any external wavelength control [19]. The self-sweeping laser generates normally a sequence of microsecond pulses during its operation. Each pulse consists of a single longitudinal mode with a spectral linewidth below 1 MHz [17]. The pulse-to-pulse optical frequency change value δν ∼5.5 MHz is determined by intermode beating frequency of the self-sweeping fiber laser cavity. The laser average output power is constant at fixed pump, but average peak power of the generated pulses changes gradually along with wavelength. This change leads to corresponding drift of the average pulse repetition period τ_rep from 22 to 25 µs during every wavelength scan from 1056 to 1074 nm. To keep the repetition period closer to a certain value for different experiments, the measurement time τ_meas was restricted to 2 (or 4) s and all of the measurements are carried out approximately near the same wavelength in corresponding sweeping ranges of 2 (or 4) nm. In these conditions, the average pulse repetition period τ_rep changes in a narrower interval from 23 to 23.5 µs.

 

Fig. 1. Schematic of a laser rangefinder-vibrometer based on a self-sweeping fiber laser.

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A fiber Mach-Zehnder interferometer (MZI) is used to measure the distance to a target under test, as well as its vibrations (periodic changes of the optical path difference). The MZI is formed with two 10/90 fiber couplers of the same type at the input and the output (C2 and C4, respectively). The most of the probe laser power is directed into the test arm of MZI. An additional 5/95 fiber coupler (C3) is installed into the reference arm to reduce power and improve the interference visibility at output of the MZI during the probe laser tuning. A 50/50 fiber coupler (C5) and a collimator are inserted into the MZI test arm for output coupling of probe radiation from the fiber part of the MZI and receive part of it back after scattering by the target under test. The effective focal length and numerical aperture of the fiber collimator (F220APC-1064, Thorlabs) are 11.17 mm and 0.25, respectively. Optical signals pass through the reference and the test (containing the target) arms of the MZI and are mixed at the output coupler, where an interference signal is formed. An additional 10/90 coupler (C1) is installed in front of the MZI to record incident power with a photodetector PD1 (hereinafter referred to as the input photodetector/signal) and to normalize the interference power measured with another photodetector PD2. The normalization allows us to reduce noise in the interference signal originated from the laser peak power fluctuations. The signals are detected with two fast InGaAs photodetectors with a bandwidth of ∼1 GHz and are digitized with a sampling rate of 5 MHz using an oscilloscope (LeCroy, WavePro 725Zi-A). Both input and interference signals consist of pulses. For simplicity of consideration, we can assume that the pulses are strictly equidistant both in the frequency and in the time domains with discrete δν and τ_rep, respectively. The interference signal amplitude changes with the probe laser tuning according to the harmonic law. Period of the harmonic modulation is associated with the MZI arms difference and contains information about the reflectors locations in the line under the test [18]. Measured interference signal is pulse-by-pulse normalized to the input one to construct a dependence of the normalized amplitude on the pulse number. Relative detuning of the self-sweeping laser optical frequency Δν_i can be easily obtained from the pulse number i with multiplication by the laser intermode frequency Δν_i = i · δν, since the optical frequencies are exactly equidistant for each pulses in our laser. The spectral transmission function of the MZI, formed between two couplers with splitting ratios of k₁ and k₂ and having the arms difference of 2L (here we include the factor of 2 to describe a double pass in the MZI test arm between the 50/50 coupler and the target, Fig. 1), has the form [20]:

 

Fig. 2. Interference signal at the output of the MZI (left) and the corresponding reflectogram (right) for a stationary (a) and vibrating (b) target.

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A black anodized polished aluminum surface of a standard mirror mount is used as target in the range and vibrations measurements experiments. A folding mirror mounted on a piezoelectric transducer is used for the optical path modulation (see Fig. 1). Vibration amplitude of the mirror is calibrated with a scanning Fabry-Perot interferometer (FPI). The mirror was used as a part of the FPI for the calibration. Control voltage with a sinusoidal waveform is supplied with a signal generator (DG 4162, Rigol) to produce vibrations. It is well known that the FPI transmission varies periodically with the distance between the mirrors, with a period equal to half the wavelength. In the case of a single-frequency Nd:YAG laser used in our experiments the period is equal to 532 nm. The vibration frequency was varied in the range from 1 to 1500 Hz. A photodetector and an oscilloscope were used to measure required voltage to change the FPI transmission by one period at each vibration frequency. The calibration curve demonstrating the voltage dependence on the vibration frequency is shown in Fig. 3 (black squares). An inverse value corresponding to longitudinal displacement per unit voltage can be also calculated from this data (blue curve in Fig. 3). One can see a sharp amplitude increase in the curve near frequency of ∼1 kHz corresponding to mechanical resonance of the mirror and mount system. It should be also noted that the modulation amplitude for the optical path is twice as large, because the folding mirror vibrates instead of the target in our experiments (see Fig. 1). Further experiments are carried out at following vibration parameters of the folding mirror: modulation frequency is varied from 2 Hz to 5 kHz and the voltage amplitude is increased up to 6 V.

 

Fig. 3. Dependence of the half wavelength voltage (black graph) and the efficiency of mirror vibrations (blue graph) on the modulation frequency in a scanning FPI.

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Let's consider a case when a target (reflection point in the MZI) vibrates in longitudinal direction. These vibrations lead to modulation of the optical path in the test arm of the MZI. The transmission function of the MZI (1) in the case of harmonic modulation of the target position with frequency f and amplitude A:

To predict a result of the Fourier transform of this function, it is convenient to assume again that the optical frequency varies strictly linear in time: ν_i= c/λ + δν·t_i /τ_rep, where τ_rep is a period of time between measurements (pulse repetition period for the self-sweeping laser). One the one hand, it can be seen from Eq. (3), that the transmitted interference signal is modulated at frequency f₀= 2Lδν/(cτ_rep) in time if there are no vibrations (A = 0). One the other hand, it is known that weak sidebands located near the main modulation frequency f₀ and separated by the optical path modulation frequency f appear in the Fourier spectrum if the modulation amplitude A is relatively weak. Thus the Fourier transform of expression (3) at small amplitude of A results in three peaks in frequency space: the central one at frequency f₀ corresponding to the reflection from the target and a pair of symmetrically located additional peaks separated by frequency f from the main one corresponding to the vibrations. For space coordinate corresponding to the reflectogram the distance between the central and side peaks is equal to:

It is also known that the number of the side peaks corresponding to harmonics of the vibration frequency mf (m = 1, 2…) increases with vibration amplitude A [21]. Corresponding locations of the peaks in the reflectogram are L ± mΔL [see right graph in Fig. 2(b)]. Let's move on now to describing the distance and vibrations measurements results.

3. Results

3.1. Vibration frequency variation at fixed vibration amplitude

Figure 4 shows examples of reflectograms measured at fixed vibration amplitude of a folding mirror (of about 30 nm) shown in Fig. 1 at following modulation frequencies of 2, 400, 1400 and 5000 Hz. It should be noted that modulation voltage of the piezo actuator for three former frequencies is selected in accordance with calibration curve of Fig. 3. Unfortunately, we do not have calibration data for the last one. So, in this case the modulation amplitude is set in accordance with the calibration data for the highest available frequency of 1500 Hz. The longitudinal coordinate of the reflectograms corresponds to the optical length nL, where the refractive index n is equal either to 1 or to 1.45 for air (n_air) and for fiber (n_fib), respectively. The first peak at a distance of z₀ ∼1.4 m corresponds to the reflection from the FC/APC connector inserted into the collimator. Thus, the geometric length difference between the MZI arms at the collimator point is ∼1.4/n_fib = 0.96 m. In fact, this value defines a starting point of the free space part of the line under test, where the target and the vibrating folding mirror are located. Thus, the maximum distance measurable in free space with these fiber optical components is estimated to be z_max- z₀= 12.24 m. It should be noted that the reflectograms also show positions of peaks corresponding to the folding mirror and the target at points with coordinates 2.66 and 3.27 m, respectively. The later peak corresponding to the target will be further referred to as the central one. Additional peaks symmetrically located relative to the central one appear in the reflectogram, when the folding mirror starts to vibrate. It should be noted that positions of these peaks do not correspond to locations of real objects (mirrors, collimators, etc.) in contrast to the central one. These peaks describe oscillation of the optical path to the target. Positions of the side peaks are marked in Figs. 4(b)-(d) by arrows directed from the central one. Detailed views of the central and right side peaks are presented in Fig. 5. It can be seen in the Figs. 4–5 that the distance to the side peaks increases with vibration frequency. It should be noted that the distance of ∼1.2 mm to the side peaks corresponding to the vibration frequency of 2 Hz is comparable with the width of the peaks. Therefore, the central and both side peaks are shown on the same graph in Fig. 5(a).

 

Fig. 4. Examples of reflectograms measured at fixed effective vibration amplitude of ∼60  nm and four vibration frequencies of (a) 2, (b) 400, (c) 1400 and (d) 5000 Hz.

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Fig. 5. Central and right side peaks in the reflectogram at modulation frequencies of (a) 2, (b) 400, (c) 1400, (d) 5000 Hz are shown in left and right columns, correspondingly.

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Similar reflectograms are also measured at other vibration frequencies in the range from 2 Hz to 5 kHz. A linear increase of the distance between the central and side peaks with vibration frequency is observed [see Fig. 6(a)]. This growth is well described with Eq. (4). It should be also noted that at the vibration frequencies of above 3 kHz, the left side peak is poorly resolved because of the noise level increase in the region of the reflectogram corresponding to the fiber part in the MZI test arm (range from 0 to 1.5 m in Fig. 4). The range can be minimized with proper selection of the fiber lengths of the MZI arms. The side peaks merge with the central one and become unresolvable at low vibration frequencies, if the vibration period becomes comparable or lower than the measurement time τ_meas (2 s for these experiments). That's why we kept the vibration frequency in our experiments at relatively high level f ≥ 2 Hz (f >> 1/τ_meas = 0.5 Hz).

 

Fig. 6. Dependencies of (a) the distance to the side peaks and (b) the amplitude and (c) the width of the side peaks on the vibration frequency.

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It can be seen from the reflectograms that the side peaks become increasingly inhomogeneous obtaining multiple peaks structure with the vibration frequency increase (right column in Fig. 5). To carry out a qualitative analysis of the changes in the width and shape of the side peaks, they were approximated by a Gaussian function (blue curves in Fig. 5). Dependence of the approximation parameters on the vibration frequency is shown in Figs. 6(b) and (c) for side peaks amplitude and width, respectively. The observed reduction of the side peaks amplitude with vibration frequency f can be described with a rational hyperbolic function I_s(f) ≃ 1/(1 + 5·10⁻³·f) [Fig. 6 (b)]. As the vibration frequency increases (up to ∼7.5 kHz), the amplitude of the side peaks decreases down to the noise level (I_n≃ 0.026) shown in Fig. 6(b) with a green line. This value, in fact, determines the maximum modulation frequency that can be measured at effective MZI arms length difference modulation amplitude A ≃ 60 nm used in the experiments. The width of the side peaks increases almost linearly with increasing modulation frequency [Fig. 6(c)].

3.2. Vibration amplitude variation at a fixed vibration frequency

Further, a series of reflectograms is measured at constant vibration frequency of 100 Hz with a vibration amplitude varying from 0.5 to 6 V (corresponding to the optical path modulation amplitude variation from 30 to 350 nm). Figures 7(a) and (b) show examples of the reflectograms near the target point position located at distance of ∼3.286 m for the maximum and the minimum vibration amplitudes, respectively. The central peak in the Figs. 7(a) and (b) corresponds to the reflection from the target. Figure 7(b) demonstrates that an increase of the vibration amplitude leads to an increase in the number of side peaks, while the distance between the peaks (corresponding to the vibration frequency) remains unchanged.

 

Fig. 7. Reflectograms at vibration frequency of 100 Hz and modulation voltage for the piezoelectric transducer of (a) 0.5 V and (b) 6 V.

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The dependence of the central I₀ and the nearest side I_s peaks amplitude on the vibration amplitude A (modulation voltage for piezoelectric transducer) is shown in Figs. 8(a) and (b), respectively. Such a behavior of central and side components is already well known for communication systems [21], where the effects of frequency and/or phase modulation of a signal with a carrier frequency were considered in detail. In the case of the vibrometer, half of the MZI arms difference L corresponds to the carrier frequency of the signal, and the doubled value of the wave vector 4πν_i/c of the tunable laser corresponds to time coordinate in communication systems. The phase change in the MZI (4πν_i/c)dL caused by vibrations corresponds to phase modulation in communication systems. It is natural to describe amplitude of modulation in such systems using a modulation index M, which in our case, in accordance with the definition of dL (expression (2)) is equal to M = (4πν_i/c)A. The optical frequency ν_i used in the expression for the modulation index M depends on time in our case, but it can be substituted with the average one ν₀, because the tuning range is several orders of magnitude narrower than the average value. It follows from the theory (see, for example, [21]) that the amplitude of the side peaks, depending on the modulation index M, can be described by the Bessel functions J_j(M) of order j, where integer parameter j corresponds to the index number for side component (j = 0 corresponds to the central peak). It is known that zero J₀(M) and the first J₁(M) order Bessel functions have the first zeros (for M > 0) at M ≃ 2.4 and 3.83, correspondingly. These two zeros correspond to two minima observed in experimental results in Figs. 8(a) and (b) for central and the first side peaks intensities, respectively. Positions of the first maxima of the Bessel functions are also known: M = 0 and M ≃ 1.84 for J₀(M) and J₁(M), correspondingly. Based on the data we can estimate vibrations amplitude, corresponding to the first maximum of the first peak using average generation wavelength λ₀ = 1064 nm as follows: A_max1 ≃ 1.84λ₀/4π ≃ 155 nm. This estimation shows that 1) the scheme is very sensitive to vibrations, 2) position of the first maximum is in good agreement with experimental results [see upper axis in Fig. 8(b)]. It should be noted that the blue lines in Fig. 8 show results of numerical modeling. The details of the modeling, as well as some reasons for the side peaks intensity decrease in the experiments, will be described in the next section.

 

Fig. 8. Dependence of the central (a) and side (b) peaks amplitudes on modulation voltage amplitude at fixed modulation frequency of 100 Hz. Points and lines correspond to experiment and modeling, respectively.

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4. Theoretical modeling

Experiments have shown that the side peaks become increasingly inhomogeneous with multiple peaks structure with the vibration frequency increase. A modeling of the vibrometer is carried out to explain the results. We use Eqs. (2-3) to model the MZI transmission. A set of optical frequencies ν_i = c/λ₀ + i·δν contains equally spaced values separated by intermode beating frequency (δν = 5.5 MHz) of the self-sweeping laser. At first, we generated an equally spaced sequence of times corresponding to moments of the pulses generation t_i = i·τ_rep. The time vector elements are separated with an average pulse repetition period in the self-sweeping laser τ_rep = τ_meas/(N-1), where N is total number of pulses. These two sets are substituted into Eq. (3) and the FFT is applied to construct reflection amplitudes of the reflectogram. Longitudinal coordinate z_i in the reflectogram is calculated according to the FFT properties as follows: z_i = (i·c)/(2Nδν).

The calculation results in reflectograms with relatively narrow peaks. The width of the peaks is inversely proportional to the number of points N and does not depend on the vibration frequency f, which contradicts the measurement results [Fig. 6(c)]. Figure 9 shows an example of comparing this modeling (blue line) with experimental results (black line) for a vibration frequency of 400  Hz. The central and right peaks are shown in Fig. 9(a) and (b), respectively. Values of visibility F₁ and the target position L in Eq. (3) are varied in the model to fit the central peak amplitude and position, respectively. The calculated peak width agrees well with the measured one [Fig. 9(a)]. On the one hand, position of the side peak depends on vibration frequency f and it is also agrees well with the experimental results [Fig. 9(b)]. On the other hand, the width of the side peak in the modeling is comparable with the width for the central one and cannot be fitted to the experimental data.

 

Fig. 9. Comparison of experimental reflectogram (black line) and modeling (blue line) at vibration frequency of 400 Hz for the central (a) and side (b) peaks, when fluctuations of the probe laser pulse repetition period are ignored or modeled with a normal distribution.

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We assumed that the side peak broadening is associated with fluctuations of the pulse repetition period in the self-sweeping laser. Indeed, our experiments show that the distance between adjacent pulses t_i – t_i-1 fluctuates around an average value of 23 µs with a standard deviation of ∼1.7 µs [black line in Fig. 10(a)].

 

Fig. 10. Dependence of the interpulse distance on time (black line), smoothing over 1000 consecutive pulses (blue line): a) experimental data, b) modeling with normal distribution around τ_rep.

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At first, we added in our model random time pulse shifts δτ_i having the normal distribution with zero average to initial linear dependence: t_i = i·τ_rep + δτ_i to take into account some influence of the pulse repetition period fluctuations. It is found that this operation practically does not affect the width of the side peaks, and the modeling results practically coincide with those presented in Fig. 9.

To improve the modeling accuracy, we use the pulses generation time counts t_i measured directly during the experiments with the vibrometer. The agreement between experimental reflectograms and modeling is dramatically improved in this case. Some examples of the experimental data and the modeling result comparison at three mirror vibration frequencies of 0.4, 1.4 and 5 kHz are shown in upper, middle and bottom rows of Fig. 11, correspondingly.

 

Fig. 11. Reflectograms near central and right side peaks and zoomed views corresponding to left, middle and right columns, respectively. The vibration frequencies of 0.4, 1.4 and 5 kHz correspond to rows increasing from up to bottom. The experimental data, modeling with the pulse positions taken from the experiments, and modeling with smoothing of the pulse period over 1000 points are shown in black, red and blue lines, correspondingly.

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Central and right peaks as well as zoomed view of the right peaks are shown in left middle and right columns, correspondingly. It can be seen from the graphs that the agreement between the modeling and experiments is much better now for the side peaks widths as well as for their internal structures.

Thus, we can see that modeling the pulse repetition period fluctuations with normal distribution and with data taken from experiments results in completely different shapes of the side peaks in the reflectograms. In order to determine the main type of fluctuations producing the main contribution to the measured side peaks distortions, a similar modeling is carried out using the experimentally measured interpulse distance Δτ_i= t_i - t_i-1 between adjacent pulses with numbers i and i-1, when this distance is smoothed over 1000 points Δτ_i → Δτ_i^(sm) (blue line in Fig. 10). It can be seen from the Fig. 10(a) that some residual fluctuations are still visible for the blue line even for smoothed dependence Δτ_i^(sm). New positions of the pulses can be constructed from this smoothed dependence in a usual way: t_i^(sm)= t_i-1^(sm)+ Δτ_i^(sm). It turns out that the modeling with smoothed pulse positions t_i^(sm) also results in good agreement with experimental reflectograms. Results of modeling with original time dependencies t_i and smoothed ones t_i^(sm) are shown with blue and red lines in Fig. 11. Zoomed view of Fig. 11(h) shown in Fig. 11(i) demonstrates that the results of modeling practically coincide at this level of smoothing. Moreover, the difference is even not noticeable at lower vibration frequencies of 400 and 1400 Hz shown in upper and middle rows of Figs. 11, correspondingly. At a large degree of smoothing, the results of modeling with and without the smoothing will start to deviate noticeably from each other. Taking these results into account, we can suppose that slow pulse repetition period fluctuations are responsible for observed broadening of the side peaks in measured reflectograms. On the one hand, the distance between the side peaks according to Eq. (4) is proportional indeed to the pulse repetition period τ_rep, and slow fluctuations of pulse repetition period lead to the observed side peaks broadening. On the other hand, fast pulse repetition period fluctuations do not affect the side peaks position and width, because our previous result shows that addition of random time pulse shifts δτ_i having the normal distribution with zero average do not result in any noticeable side peaks broadening. Similar smoothing procedure over 1000 points applied to interpulse distance do not result in any noticeable long-term repetition period fluctuations [blue line in Fig. 9(b)]. Thus, we can conclude that a linear growth of the side peaks widths with vibration frequency, observed experimentally at a rate of ∼1 cm/kHz [see Fig. 6(c)], is associated with a slow drift of the pulse repetition period within δτ_rep ∼0.4 µs. An uncertainty of the repetition period leads according to Eq. (4) to a corresponding uncertainty of the side peak position ∼cδτ_repf/(2δν), which in its turn can be associated with observed broadening of about cδτ_rep/(2δν) ≃ 1 cm/kHz. This estimation is in good agreement with the experimental results.

The described above modeling is performed in four cases: 1) absence of fluctuations which can be expected in ideal conditions, 2) random and fast frequency fluctuations without slow drift, 3) pulse time spacing fluctuations obtained from experimental data and 4) slow drift of the pulse time spacing. It should be noted that the first and the second cases leads to similar results which are both inconsistent with experimental observations. The modeling of the other cases shows that a slow pulse time spacing drift is much more important for proper operation of the proposed system as compared with the fast fluctuations.

We should note that the modeling results presented in Fig. 8 are obtained with a constant pulse repetition period τ_rep (without using the experimental positions of the generated pulses). On the one hand, the modeling in this case gives an overestimated amplitude for side resonances [see Fig. 8(b)]. On the other hand, the difference from experiment is not very significant, since for the used vibration frequency of 100 Hz, broadening of the peaks (according to the above estimate is 1 mm only) is comparable with the width of the peaks obtained without taking into account pulse positions fluctuations. In addition, according to the experimental data of Fig. 6(b), the amplitude decrease for the side peaks is not very large (∼30%) at vibration frequencies of ∼100 Hz. That is why the Fig. 8(b) shows a relatively good agreement between the amplitude of the measured side resonances and the results of the simplified modeling.

The system was tested at fixed target positions and a fixed set of vibration frequencies. Let’s discuss the more complicated cases of time-varying frequencies or target positions. The side peaks broadening is expected, if the target position is fixed and the vibration frequency is changed in time because of mentioned above proportionality of their detuning form the central peak position. The side peaks broadening is proportional to the vibration frequency change. The situation becomes more complicated in the case of non-stationary target position. The central peak position is calculated accordingly to oscillations frequency of the transmission function (3). The oscillation frequency depends not only on probe laser tuning direction and frequency but on the target velocity V_t as well (in the similar way, which has been shown for the other interferometric techniques demonstrated in [14–15]). Analysis of the Eq. (3) shows that the central peak position in reflectogram will be shifted by a value of $\Delta {L_\nu }\textrm{ = (}{\nu _\textrm{i}} \cdot {\tau _{\textrm{rep}}}\textrm{/}\delta \nu \textrm{)}{V_\textrm{t}}.$ Thus, in our case, noticeable shift of ΔL_v = 1 mm will be observed in the system at the target velocity V_t of about 1 µm/s. Measurements of the shift will require possibility of the probe laser tuning in both upward and downward directions.

5. Conclusion

As a result, the proposed system based on a self-sweeping ytterbium-doped fiber laser allows measuring not only the distances to individual elements in an optical line, but also vibrations that appear themselves in fluctuations of the optical path. The self-sweeping laser, which does not require any spectrally selective elements and drivers for wavelength tuning, with a sweeping range of 1056-1074 nm is a key element of the system. We describe here in detail a technique for extracting the amplitude of the optical path fluctuations and the optical elements vibration frequency. The sensitivity level of ∼5 nm can be estimated from vibration peaks amplitudes clearly visible above a noise floor in Fig. 7(a) at vibration amplitudes of 30 nm. In addition, measurements of vibration frequencies in the range from 2 Hz to 5 kHz are experimentally demonstrated. The maximum measurable vibration frequency is limited by the instability of the self-sweeping laser parameters in the time domain and can be roughly estimated as ∼7.5  kHz. The results obtained show that characteristics of vibrometer-rangefinder is comparable with similar range and vibration measurement systems. For example, the sensitivity of the frequency-modulated LIDAR (of 2.5 nm) [16] is comparable with obtained one (about 5 nm). At the same time the spatial resolution in our system (500 µm) is sufficiently smaller than in [16] (15 mm). The spatial resolution of about 1 mm at distances of several meters is also demonstrated in the self-mixing interferometry based systems [8]. The advantages of the proposed system are design simplicity of the tunable probe laser and high spatial resolution. Spatial resolution of ∼500 µm is demonstrated in this work and even better resolution (down to 160 µm) has been demonstrated recently in [18] with larger sweeping spectral range. Maximum measurable distance of 13.64 m demonstrated with the self-sweeping laser is limited by the magnitude of the frequency jump of ∼5.5 MHz occurring between subsequent generation of pulses and corresponding to the laser intermode beating frequency. A possibility of decreasing the frequency jump down to ∼1 MHz was demonstrated in [22], which can potentially increase the maximum target distance in the vibrometer-rangefinder up to ∼75 m. In this case, it may be necessary to increase the output power of the source as well. The power increase in the self-sweeping lasers can be achieved with application of corresponding amplifiers or large mode area fibers (see, for example, [23]).

Moreover, we obtained excellent agreement between results of numerical modeling and experimental measurements. In particular, the performed numerical modeling and comparison with experimental results demonstrate that the effect of the slow drift of the self-sweeping laser average pulse repetition period considerably influences the amplitude as well as the width of the vibration peaks in the reflectograms. To take into account the influence of the drift on the vibrations measurements, it is necessary to measure not only the MZI transmission dependence on the optical frequency, but also simultaneous variation of the average pulse repetition period. It can be expected that the maximum measurable vibration frequency can be increased by decreasing fluctuations of the pulse repetition period. However, the issue of the repetition period stabilization in self-sweeping lasers has not been studied yet and requires additional research.