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Distances to multiple targets are measured simultaneously using a single femtosecond pulse laser split through a diffractive optical element. Pulse arrival from each target is detected by means of balanced cross-correlation of second harmonics generated using a PPKTP crystal. Time-of-flight of each returning pulse is counted by dual-comb interferometry with 0.01 ps timing resolution at a 2 kHz update rate. This multi-target ranging capability is demonstrated by performing multi-degree of freedom (m-DOF) sensing of a rigid-body motion simulating a satellite operating in orbit. This method is applicable to diverse terrestrial and space applications requiring concurrent multiple distance measurements with high precision.
© 2015 Optical Society of America
1. Introduction
Multi-degree of freedom (multi-DOF) sensing of an object in 3-D space is intended to identify its location as well as orientation simultaneously with respect to a unified reference frame. Such multi-DOF sensing is essential for diverse tasks such as attitude control of satellites in orbit [1,2], beam-line alignment of particle accelerators [3], and thermal stabilization of large machinery [4]. In practice, multi-DOF sensing is accomplished by means of triangulation [5] or multilateration [6,7] which employs multiple sensors so that each one measures distance or angle within a pre-set geometrical relation with others. Various sensors of diverse principles are available for the purpose, among which gravity-referenced inclinometers or gyroscopes are preferably used for angle measurements, while laser interferometers or optical time-of-flight instruments are well suited for distance measurements in free space.
Laser distance interferometers are grouped into either incremental or absolute type, depending on the non-ambiguity range (NAR) of the measured distance. Incremental interferometers basically measure distance by means of continuous accumulation of instant interferometric phases, so the NAR is limited to half the wavelength of the source laser [8]. Incremental distance measurement offers sub-wavelength resolutions but suffers difficulty with setting up the zero fiducial point with precise geometrical relation to other sensors for multi-DOF sensing. In contrast, absolute interferometers are able to determine distance with a single measurement within an extended non-ambiguity range. Absolute distance determination is achieved usually by modulating the intensity or frequency of continuous-wave lasers [9,10]. As a result, absolute interferometers provide a well-defined zero reference for multi-DOF sensing, but their measurement accuracy is relatively lesser compared to those of incremental interferometers.
In the last decade, femtosecond lasers have been investigated as a new light source for absolute distance measurement. A single femtosecond laser is able to offer a large number of evenly spaced radio-frequency harmonics or optical wavelengths, enabling absolute distance measurement beyond the capabilities of traditional lasers [11–13]. As results, quite a few advanced interferometric techniques have been demonstrated; radio-frequency synthetic wavelength interferometry [14,15], pulse to pulse cross-correlation [16], dispersive interferometry [17], dual-comb multi-heterodyne interferometry [18–20], multi-wavelength interferometry [21–23] and time-of-flight measurement using nonlinear optical cross-correlation [24,25]. These techniques pursue the common aim to achieve the sub-wavelength precision in long-distance ranging by making use of unique time and/or frequency domain characteristics of femtosecond lasers.
In this study, we propose an extended scheme of absolute distance interferometer for multi-DOF sensing using femtosecond laser pulses. The proposed interferometer implements simultaneous time-of-flight measurements of short light pulses sent out to multiple targets from a single pulse train source. Absolute distances are determined in parallel by combining the nonlinear optical cross-correlation technique [24,25] with dual-comb interferometry [18–20] which allows for 0.01 ps pulse timing resolution at a 2 kHz update rate. The sub-picosecond timing capability leads to sub-micrometer accuracy in measuring distances up to several meters. Finally, an uncertainty analysis is conducted to systematically validate the achieved measurement accuracy in consideration of major error sources involved in the measurement.
2. Measurement method
As illustrated in Fig. 1, the measurement system proposed in this study is based on a dual-comb interferometer employing two femtosecond lasers referred to respectively as the signal laser and the local laser. The signal laser is used as the main light source of time-of-flight measurement, while the local laser of different repetition rate is needed for frequency down-conversion sampling. The signal laser is an Er-doped fiber laser (1560 nm center wavelength) emitting 90 fs duration pulses with a 10 mW average power. A diffractive optical element (DOE) (MS-354-G-Y-A, Holo-Or) is adopted such that the output beam of the signal laser is split with a separation angle of 0.73° by means of diffraction towards multiple targets of retro-reflector type. The front-surface of the DOE is coated partially reflective, which acts as the reference target mirror reflecting the pulse Mref at every pulse repetition cycle as the zero-point datum of distance measurement. Every pulse emanated from the signal laser reaches multiple targets and returns as a sequence of discrete pulses; Mref in front followed by other pulses M1, M2, … and Mn in the time domain.
Fig. 1 Measurement system for absolute ranging to multiple targets. C: circulator, CL: collimator, DOE: diffractive optical element, HWP: half-wave plate, PBS: polarizing beam splitter, fr: pulse repetition rate, Δfr: repetition rate difference between the signal and local lasers, L: lens, M: target mirror, PD: photo-detector, DM: dichroic mirror, and PPKTP: periodically poled KTP crystal.
The target pulse sequence is converted to an electric signal by means of nonlinear optical cross-correlation as depicted in Fig. 1. The balanced cross-correlator (BCC) installed in the receiving part of the measurement system utilizes a PPKTP (periodically poled KTP) crystal. This type-II crystal is a second harmonic (SH) generator reacting to the cross-correlation intensity between two orthogonal polarization components [24,25]. The target pulse sequence of linear polarization is fed to one direction of the PPKTP crystal, while the other orthogonal direction is aligned to take another pulse train of linear polarization emitted from the local laser. The SH signal resulting from the PPKTP crystal is separated using a dichroic mirror (DM). For balanced detection, two separate SH signals are produced with a temporal offset by reciprocating the dual-comb interference beam to pass through the PPKPT crystal twice. The SH signals are detected using a balanced photo-detector (PD) with suppression of optical and electric background noise. The BCC signal shows that each target pulse is converted to an S-shaped signal, of which the zero-crossing point designates the target location.
For experimental validation of measurement performance, four targets (M1-M4) were prepared at different distances of about 3 m along the first-order diffraction lines of the DOE as illustrated in Fig. 1. Target M4 was set on an aerostatic stage, of which the position was monitored with an incremental HeNe laser interferometer for comparative evaluation of time-of-flight measurement. The received target pulse sequence was amplified to maintain a 100 mW average power for effective generation of SH signals in the balanced cross correlator (BCC). Pulse broadening due to long fiber delivery was compensated by inserting a dispersion compensating fiber so that the duration for each target pulse was kept to be ~510 fs. The repetition rate of the signal as well as the local laser was stabilized by phase-locking control to the Rb clock with a 10−12 stability at 1 s averaging.
The distance d to each target is determined as d = cΔt/(2N) with c being the vacuum speed of light, N the group refractive index of air, and Δt the time-of-flight of the pulse for return travel between the DOE and the target. Time-of-flight Δt may be measured directly in the real time domain by counting the elapse time of each target pulse using relevant electronic instruments, but this conventional way of pulse timing offers no benefit of using femtosecond pulses. This is because even state-of-the-art photodetectors are still not fast enough to resolve femtosecond pulses, thus the timing resolution remains a few picoseconds at best, equivalent to several hundreds of micrometers in distance. Instead, the BCC scheme of Fig. 1 combined with dual-comb interferometry is able to enhance the timing resolution and also accuracy of time-of-flight measurement intended using femtosecond pulses. For dual-comb interferometry, the signal laser is operated at a repetition rate of fr, and the local laser is set to a slightly different repetition rate with an offset of Δfr from fr. Then, collective beating of all the optical modes in dual-comb interferometry makes the BCC signal repeat at Δfr, permitting detection of the time-of-flight Δt in a slowed time scale down-converted by the factor of fr/Δfr [26,27].
In our experiment, Δfr was set at 2 kHz while fr was 100 MHz with the ratio fr/Δfr being 5 × 104. Figure 2(a) shows an actual BCC signal which was digitized at a fast sampling rate (fs) of 200 MHz using a 14 bit digitizer so as to minimize quantization error [20]. A low-pass filter with a cutoff frequency of 35 MHz was incorporated to suppress both the repetition rate signals of fr and fr-Δfr and also unwanted high frequency noise. The sampled BCC signal is displayed in the ‘effective time’ scale by up-converting the sampling rate as fs’ = fs × (fr /Δfr) with the effective sampling rate fs’ being worked out to be 10.0 THz. This infers that the sampling resolution in the effective time scale is equivalent to 0.1 ps. Now, the time-of-flight Δt is quantified by locating the zero-crossing point of the S-shaped BCC pulse signal from its cross-correlation with a standard BCC signal pattern (Fig. 2(b)). The peak location of the resulting cross-correlation curve (Fig. 2(c) and 2(d)) is recognized as the zero-crossing position. This electronic signal processing leads to a drastic improvement in the pulse timing resolution of determining Δt, far better than the pulse sampling resolution of 0.1 ps, which will be discussed in detail with experimental results in the next section.
Fig. 2 BCC signal sampling and cross-correlation processing. (a) An exemplary BCC signal sampled from three targets. (b) Reference BCC signal pattern for cross-correlation. (c) Cross-correlation between the reference and the measurement BCC signal. (d) Enlarged views of cross-correlation curves for peak detection. Yellow lines indicate original sampled BCC signals and red curves are calculated cross-correlation data.
3. Performance evaluation
Figure 3 shows experimental results obtained to evaluate the repeatability, linearity and speed of the measurement system described so far. During the experiment, the environmental condition was controlled by regulating the ambient temperature, pressure, humidity and CO2 concentration as listed in Table 1. Firstly, the measurement repeatability was observed in terms of the Allan deviation for two distances selected at d1 = 0.015 m and d2 = 1.6 m (Fig. 3(a)). Note that the Allan deviation was given as a function of the averaging time, of which the shortest time was 0.5 ms corresponding to a single measurement time as the distance reading was updated at 2 kHz rate. Figure 3(a) shows that the repeatability for d1 was 0.936 μm for 0.5 ms averaging. The single-measurement repeatability lies below 0.01 ps which is in fact one order of magnitude less than the sampling resolution of 0.1 ps. The repeatability improves further to 17 nm for 0.5 s averaging, of which the timing resolution reaches the range of 0.1 fs with increased averaging to cover 1,000 distance readings. It is worthwhile to note that the test distance d1 is short enough to neglect the effects of external disturbance such as vibration, temperature variation and air fluctuation. In other words, the repeatability for d1 can be considered as the standard uncertainty (k = 1) of the measured time-of-flight Δt itself. As such, the repeatability for d2, which is longer than d1, appeared to be worse than that of d1, regardless of the averaging time, being attributed to increased external disturbance.
Fig. 3 Absolute multi-target ranging performance. (a) Repeatability in Allan deviation at two distances. (b) Linearity test result compared with an incremental HeNe laser interferometer. (c) Speed of absolute ranging of multiple target objects. Three targets were modulated at 10, 20 and 30 Hz, respectively, using PZTs. During all the tests, the environment was controlled as specified in Table 1 in terms of temperature, pressure, humidity and CO2 concentration.
Table 1. Uncertainty evaluation of distance measurement.
Secondly, the linearity test of the measurement system was conducted in comparison with an incremental HeNe laser interferometer. The target mirror of M4 in Fig. 1 was moved over a 1.0 m travel, from 1.6 m to 2.6 m in distance, on an aerostatic stage in 100 mm steps (Fig. 3(b)). The maximum deviation between the two measurements was 279 nm at 0.5 s averaging (79 nm in standard deviation), while no cyclic error was observed. Thirdly, the distances to three targets of M1, M2 and M3 of Fig. 1 were modulated with a same ~10 μm amplitude but with different frequencies of 10, 20, and 30 Hz, respectively, using piezoelectric actuators (PZTs) installed underneath the targets to provide dynamic distance variations. As shown in Fig. 3(c), the sinusoidal motions of three targets were precisely monitored and reconstructed; Fourier analysis confirmed that absolute distances to multiple targets are clearly resolved with a high signal-to-noise ratio in the frequency domain.
4. Multi-DOF sensing
Multi-DOF sensing capability of our measurement system was tested with an experimental apparatus configured with a rigid body simulating the satellite motion as illustrated in Fig. 4. Four target mirrors M1-M4 were installed on the rigid body with a 90 degree apart along the ± 1st-order diffraction lines of the DOE. The mirrors were positioned with a ~1.0 mm distance offset from each other, so they can be distinguished by the pulse arrival order in the BCC signal. On the rigid body, the local xyz-coordinates are set up so that the x-axis connects M1 and M3 while the y-axis is aligned from M2 to M4. The normal vector was set along the z-axis at the origin of the xy plane, of which the orientation was defined by the yaw angle θx and the pitch angle θy with respect to the 0th-order diffraction line of the DOE. Let d1, d2, d3, and d4 be the distances measured to the target mirrors from the DOE, then the nominal distance d to the rigid body is calculated by the mean of the four measured distances; d = (d1 + d2 + d3 + d4)/4. At the same time, the yaw and pitch angles are calculated instantly as θx = sin−1[(d1-d3)/A] and θy = sin−1[(d2-d4)/A].
Fig. 4 Distance and angle measurement for multi-DOF sensing of a rigid body motion. (a) Experimental configuration. (b) Measurement geometry. C: circulator, CL: collimator, DOE: diffractive optical element, PBS: polarizing beam splitter, and HWP: half wave plate.
Figure 5 shows a repeatability result for angular motions obtained for the rigid body located at a nominal distance of ~3.7 m. When the rigid body was stationary, the Allan deviation was 5.289 arcsec at 0.5 ms averaging, decreasing further to 0.073 arcsec with increasing the averaging time to 0.5 s. Next, for testing of dynamic measurements, the rigid body was given continuous tilt motions modulated at 1 Hz excitation using PZTs with different modulation amplitudes. Four distances (d1, d2, d3, and d4) were measured, from which yaw and pitch angles were calculated. Fourier-transformed signals clearly showed individual modulation peaks at 1 Hz with different amplitudes for yaw and pitch angles. Note that 2nd and 3rd harmonic peaks due to imperfect excitation were also observed at 2 Hz and 3 Hz.
Fig. 5 Angle measurement test result. (a) Repeatability in terms of Allan deviation. (b) Distances (d1, d2, d3, and d4) to four targets under 1 Hz excitation. (c) Reconstructed angular motions. (d) Fourier transformed spectra of (c).
5. Uncertainty evaluation for absolute distance measurement
As summarized in Table 1, the measurement uncertainty was evaluated in accordance with the guideline provided by the international organization for standardization (ISO) [28]. Our systematic analysis starts with the target distance expressed as
where m is an integer (m = 0, 1, 2, …) and Λ denotes the non-ambiguity range (NAR) given by Λ = c/(2frN). Equation (1) indicates that there are four major uncertainty sources; the timing accuracy ∆t, the signal laser repetition rate fr, the repetition rate offset ∆fr and the group refractive index of air N. The combined standard uncertainty uc(d) is defined aswith sensitivity coefficients; c∆t = ∂d/∂∆t, cfr = ∂d/∂fr, c∆fr = ∂d/∂∆fr, and cN = ∂d/∂N. The standard uncertainty of each source (k = 1) is determined based on relevant experimental data. First, the uncertainty for ∆t was estimated from the result of Fig. 3(a) obtained at a 0.015 m distance, which is worked out to be u(Δt) = 1.1 × 10−16 s, corresponding to 17 nm in distance. Second, the uncertainty for fr was 3.7 × 10−12 d with d being the nominal distance given in meter, which was estimated from the Allan deviation at 0.5 s averaging. Being in proportion to d, this uncertainty contribution becomes significant for long distances. Third, the uncertainty for ∆fr originates from the relative phase jitter between the signal laser and the local laser. This contribution appears to be dominant for short distances because the timing jitter is amplified by the factor of fr/Δfr in the effective time scale. The uncertainty for repetition rate offset Δfr is estimated at 0.5 s averaging to be 188 nm. Fourth, the uncertainty for N is estimated to be 1.8 × 10−8 d, in consideration of the uncertainty of the empirical Ciddor’s equation as well as the environmental parameters of temperature, pressure, relative humidity, and CO2 concentration. Finally, the combined uncertainty is worked out to be [(1.7 × 10−8 d)2 + (189 nm)2]1/2 for the nominal distance d. The uncertainty evaluation result reveals that the repetition rate offset Δfr turns out to be the most influencing uncertainty source for absolute ranging within a few meters. For longer distances, the air refractive index becomes significant. It should be noted that the uncertainty evaluation presented in Table 1 was based on the experimental data obtained at 0.5 s averaging time in a well-controlled environmental condition. The uncertainty level can be enhanced by increasing the averaging time to an extent where the atomic clock provides better accuracy for stabilizing Δfr. However, in reality, the ultimate limit in achieving high level of combined uncertainty is imposed by the refractive index of air particularly for long distances to be measured in open air conditions.6. Conclusions
Parallel determination of absolute distances to multiple targets was demonstrated by time-of-flight measurement using femtosecond light pulses. The timing resolution was 0.01 ps at a 2 kHz update rate, which was attained by combining the principle of dual-comb interferometry with nonlinear balanced cross-correlation of time-of-flight measurement. This sub-ps multi-target ranging capability was tested by performing multi-degree of freedom sensing of a rigid-body object simulating a satellite operating in orbit. The ranging repeatability was 17 nm in terms of the Allan deviation at 0.5 s averaging, while a 0.073 arcsec repeatability was obtained for angle measurement. This result is in agreement with the uncertainty analysis conducted in consideration of all dominant error contributions. This method has the potential for diverse terrestrial and space applications requiring concurrent multiple distance and angle measurements.
Acknowledgments
This work was supported by the National Honor Scientist Program funded by the National Research Foundation of the Republic of Korea (NRF-2012R1A3A1050386). Y.-J. Kim acknowledges support from the Singapore National Research Foundation (NRF-NRFF2015-02) and thanks N. R. Newbury, I. Coddington, and W. C. Swann at the National Institute of Standards and Technology (NIST) for fruitful discussions and technological supports.
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