Synthetic-wavelength-based dual-comb ... - OSA Publishing

Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5747

Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement ZEBIN ZHU,1 GUANGYAO XU,1 KAI NI,2 QIAN ZHOU,2 AND GUANHAO WU1,2,* 1State

Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China 2Division of Advanced Manufacturing, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China * [email protected]

Abstract: We present an absolute distance measurement system using a phase-stable dualcomb system with 56.09 MHz repetition rate and 2 kHz repetition rate difference. A relative phase stability of 0.1 rad in 0.5 ms between two combs is achieved using a mutual locking scheme. The dual-comb ranging system combines the time-of-flight (TOF) method, syntheticwavelength interferometry (SWI), and carrier wave interferometry (CWI). Each method provides a particular ambiguity range and resolution, and they can be applied simultaneously and linked to enhance the precision and measurement rate of the ranging system. The experimental results demonstrate that a precision of 1.2 μm is obtained without time averaging, and the precision can be improved to 3 nm with only 10 ms averaging time using the SWI method described in this study. The precision reaches a sub-nanometer when the averaging time exceeds 0.1 s. A system with high accuracy and short averaging time would enhance fast measurement performance in various industrial applications. The ambiguity range is about 2.67 m in our system, we test the performance of the system with 1.5 mm range at 1.5 m distance. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (320.7100) Ultrafast measurements; (120.3930) Metrological instrumentation; (140.3425) Laser stabilization; (140.4050) Mode-locked lasers; (120.3180) Interferometry.

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#318552 Journal © 2018

https://doi.org/10.1364/OE.26.005747 Received 28 Dec 2017; revised 12 Feb 2018; accepted 20 Feb 2018; published 26 Feb 2018


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1. Introduction An efficient and dynamic distance measurement method plays an important role in satellite ranging, LIDAR, and industrial applications [1, 2]. In traditional ranging methods, a continuous wavelength (CW) laser with Michelson interferometer measures the interferometric phase to achieve a high resolution, even detecting gravitational-wave strain [3]; however, the ambiguity range is only half the wavelength of the CW laser. In contrast, the LIDAR measures distance through pulsed or radio frequency (RF)-modulated waveforms with a large ambiguity range but only tens of micron resolution [4]. The advent of the optical frequency comb (OFC) as a light source has led to revolutionary progress in absolute distance measurement and other applications [1, 5]. A phase locked OFC emits a stable pulse train, so a non-interferometric approach based on the time-of-flight (TOF) principle using an optical cross-correlation technique achieves a measurement with an approximate one-kilometer range and several nanometers’ precision [6]. Other methods are based on interferometry, such as synthetic-wavelength interferometry using the harmonics of an OFC’s repetition rate [7, 8], combining measurements from a series of optical wavelengths referenced to a fully stabilized comb [9–11], or dispersive interferometry that detects the interferometric phase of the optical spectrum [12–14]. In particular, the dual-comb-based ranging method using a pair of OFCs with slightly different repetition rates can achieve a large dynamic range measurement of absolute distance [15]. This dual-comb-based concept has been modified and widely used in distance measurement [16–20], but these systems have not met their full potential. To the best of our knowledge, only the work reported in Ref [15]. has realized a nanometer-level dual-comb ranging system, presumably due to the complexity of enhancing two comb lasers’ accuracy and mutual coherence [21–26]. Moreover, we need a long averaging time to improve TOF results’ precision until TOF is sufficiently stable compared to a quarter of the carrier wavelength to link with the carrier phase. In some dynamic measurement cases, such as precise satellite formation, high-speed and high-precision distance measurement is required. Thus, it is necessary to shorten the averaging time in the dual-comb ranging system while retaining high measurement precision. In this paper, we present a synthetic-wavelength interferometry (SWI) in a phase-stable dual-comb interferometer where the two combs are mutually locked. Time-resolved interferograms (IGMs) and a frequency-resolved spectrum are obtained; thus, information from spectral lines can be utilized without the help of additional CW lasers as light sources


[10]. Further, similar to the frontier work of a combined method in single-comb ranging system [13], the synthetic-wavelength interferometry can be directly employed in the dualcomb system, which serves as a bridge between the TOF method and carrier wave interferometry (CWI). As a result, we realize an absolute distance measurement with an approximate 2.67 m ambiguity range and nanometer precision with several milliseconds’ averaging time. 2. Experiment set-up and principle Each ideal value of the spectral line of an OFC can be expressed as f(n) = nfrep + fceo, where frep is the repetition rate and fceo is the carrier-envelope offset frequency. The dual-comb system using two such combs with a slight repetition rate difference has appeared in the literature as linear optical sampling and multi-heterodyne spectroscopy. As Fig. 1(a) shows, a pulse train from Comb 1 (frep1 = 56.090 MHz) passes through a Michelson interferometer and then interferes with Comb 2 (frep2 = 56.092 MHz). The two separated pulse trains are sampled by the pulse train from Comb 2 with an effective time shift ΔT = Δfrep/(frep1∙frep2), generating interference signals (IR and IM) with a certain update time Tupdate = 1/Δfrep. The distance can be calculated from TOF information, expressed as DTOF =

υ∆t ∆f rep 2

f rep1

,

(1)

where υ is the velocity of the optical pulses, and Δt is the time delay between reference interferograms IR and measurement interferograms IM. Two optical filters with different center wavelengths are used to generate the synthetic wavelength. To enhance the phase stability of the dual-comb interferometer, we developed a system to suppress the relative optical linewidth of the two combs as shown in Fig. 1(b). The fceo values of both combs are stabilized by the f-2f interferometers. The repetition rate of Comb 1 is directly stabilized using a piezoelectric transducer and active temperature control. The repetition rate of Comb 2 is stabilized by locking the relative frequency of the two combs generated from an intermediary free-running CW laser (fCW = 1565 nm, RIO). The two beats (fbeat1 and fbeat2) between the two combs and the CW laser are extracted by filters and are electronically mixed, generating a beat signal frelative equal to f1 −f2 that is immune to the noise of the intermediary CW laser. To obtain a narrow relative linewidth between the two combs, an intra-cavity electro-optic modulator is used to realize rapid control of the effective cavity length of Comb 2. The repetition rate of Comb 2 is monitored by a frequency counter to determine the exact value of repetition rate difference Δfrep. In the frequency domain, the two combs with slightly different repetition rates generate a sub-comb in the RF domain as depicted in Fig. 1(c). This RF comb contains the amplitude and the relevant phase of two combs. The optical frequency value of the comb lines is traceable using the frequency information of the sub-comb. The linewidth of the sub-comb reaches a sub-hertz value that is much narrower than that of the employed OFCs due to the locking scheme. The details of the locking scheme are given in our previous work [27].


Fig. 1. (a) The principle of a dual-comb ranging interferometer. Two Erbium doped fiber combs are used. The distance between the measurement mirror and reference mirror is D. Two optical bandpass filters with λc1 (1578 nm) and λc1 (1546 nm) central wavelength are used for SWI. The bandwidth is 3 nm to avoid aliasing. The IGMs and a detailed one are shown. (b) The mutually locked principle. frep1 = 56.090 MHz, frep2 = 56.092 MHz, fceo1 = fceo2 = 10.56 MHz. The relative beat signal to be locked between f1 and f2 is extracted using an intermediary CW laser. (c) Two coherent combs with a repetition rate difference of Δfrep beat with each other, yielding an RF comb with spacing Δfrep = 2 kHz. The mapping factor (the ratio of optical comb’s spacing to RF comb’s spacing) is frep1 / Δfrep.

Then, the mutually locked dual-comb interferometer is applied to the ranging system. The ambiguity range (AR) and precision of the three different methods are shown in Table 1. The time delay between IR and IM provides TOF information about ranging with an ambiguity range ARTOF = υ ⁄ (2·frep1) ≈2.7 m and 8 μm precision (Allan deviation). The interferometric information is determined by the carrier phase difference Δφc between IR and IM. The two carrier phase differences Δφc1 and Δφc2 are obtained via detecting interferograms at two different optical wavelengths. Two carrier wave signals can be expressed as: cos(2πfc1t−Δφc1) and cos(2πfc2t−Δφc2), where fc1 = υ/λc1 and fc2 = υ/λc2. The synthetic signal is constructed from the heterodyne process between two carrier waves: cos[2π(fc1−fc2)t−(Δφc1−Δφc2)]. Thus, the synthetic wavelength can be expressed as

λsyn =

λc1λc2 . λc1 − λc2

(2)

It is about 76.2 μm, and the corresponding phase φsyn = Δφc1−Δφc2. The Allan deviation of the carrier phase of either IR or IM is 0.1 rad without averaging, yielding 1.2 μm precision of SWI and 19 nm precision of CWI. Detailed results are presented in the next section. Table 1. The three methods’ ambiguity range and precision Method TOF SWI CWI

Ambiguity range 2.67 m 38.1 μm 0.773 μm

Precision (without averaging) 8 μm 1.2 μm 19 nm


The three methods are combined to measure a distance D; the principle is shown in Fig. 2. The TOF method provides approximate results of DTOF, which determines the multiple integer Nsyn of the half synthetic wavelength λsyn/2:  D  N syn = INT  TOF  . λ 2  syn 

(3)

INT represents the integer conversion identifier. Thus, a more precise result can be calculated with the SWI method, expressed as = Dsyn ( N syn +

ϕsyn λsyn ) , 2π 2

(4)

where φsyn is the phase of the synthetic wavelength introduced above. Furthermore, Dsyn determines the multiple integer Nc of the half carrier wavelength λc/2. In this process, the SWI results should be averaged until the precision is below λc/4. The multiple integer Nc can be expressed as  Dsyn  N c = INT  .  λc 2 

(5)

Finally, a highly precise result can be calculated via the CWI method, expressed as Dc ( N c + =

∆ϕc λc ) . 2π 2

(6)

Here, Δφc is the interferometric phase of carrier wave λc; either Δφc1 and λc1 or Δφc2 and λc2 can be used. As a result, precision is enhanced and the ambiguity range of the TOF method is maintained. Note that in our experiment, the measured distance D is smaller than ARTOF, which is sufficient for most precise metrology. If a longer distance were to be measured, other approaches should be applied to determine the multiple integer of ARTOF. The successful combination of different ranging methods requires a specific relationship between the unambiguity range and resolution, the detailed process of which appears in the next section. DTOF =

υg ∆t ∆f rep 2

f rep1

O

ϕ

Dsyn ( N syn + =

2π

λsyn 2

O

ϕ

= Dc ( N c +

2π

O

λc 2

ARTOF

ϕsyn λsyn ) 2π 2

ϕc λc

2π

)

2

D

Distance

Fig. 2. The principle of combined ranging method. The TOF result (DTOF) determines the multiple integer Nsyn of the half synthetic wavelength (λsyn/2). The synthetic wavelength result (Dsyn) determines the multiple integer Nc of the half carrier wavelength (λc/2).


3. Results All the TOF or phase information described above is calculated through fast Fourier transform (FFT) of IGMs. The frequency-domain information of a pair of IR and IM is shown in Fig. 3. The phase difference Δφ between the measurement phase and the reference phase is used to calculate the time delay between IR and IM, expressed as 1 d ∆ϕ ∆t = . 2π df RF

(7)

Meanwhile, the carrier frequency fc and carrier phase Δφc are obtained. Because the value of fceo and the frep of both combs are known, the corresponding optical carrier wavelength λc (its rough value is determined by the optical bandpass filter) can be calculated based on the principle shown in Fig. 1(c). Note that the Δφc should be adjusted to 0~2π if applied to Eq. (6). Two optical bandpass filters with different center wavelengths are used, and Δφc1, Δφc2, λc1, and λc2 are obtained.

Phase (rad)

1.0

40

0.8

30

0.6

20

0.4

10

0.2

0

10

15 Radio Frequency (MHz)

20

Amplitude (a.u.)

Ref. Mea. Phase difference

50

0.0

Fig. 3. FFT results of a pair of IR and IM. Both phase-frequency curves (orange and blue) and an amplitude-frequency curve (black) are shown. The purple curve depicts the difference between the two phase-frequency curves, and its slope is calculated to determine the time delay Δt. The carrier phase Δφc is pointed in the gray circle.

Taking the IGMs filtered by the filter with center frequency λc1 as an example, the carrier phase of IR and IM and their phase difference Δφc1 within 4 seconds are shown in Fig. 4. The respective carrier phases of IR and IM drift steadily. Because we process the IGMs for every period with frep1 ⁄Δfrep (28045) sampling points, there is little difference between the real value of update rate Δfrep and ideal value of 2 kHz, which leads to uniform phase (time) accumulation in every period. The carrier phase difference is stable because it depends on the measured distance rather than the update rate. The precision (Allan deviation) of carrier phase σφ of either the reference IGMs (IR) or measurement IGMs (IM) is about 0.1 rad (without averaging), and the precision of the carrier phase difference Δφc1 is about 0.15 rad. The highphase stability benefits from the locking method used in this study. Similarly, we can obtain the carrier phase difference Δφc2 using the IGMs at wavelength λc2 and build the synthetic wavelength described in Eq. (2); then, the precision of φsyn is about 0.2 rad.


30 16.0

Ref.

Mea.

15.5

Phase (rad)

20 15.0 14.5

2.00

2.05

10

12.0 11.5 11.0

0 4.5

2.00

2.05

Carrier Phase Difference

4.0 3.5 3.0

0

1

2 Time (s)

3

4

Fig. 4. The carrier phase of both IR and IM and their phase difference. The data length is 4 s with 8000 pairs of IGMs. The inset present detailed conditions at 2 s with about 50 ms length.

Figure 5 shows the precision (Allan deviation) of the TOF, SWI, and CWI measurements versus averaging time. The measured distance is about 0.5 m in air. The precision is computed according to the 4 s of data recorded. The precision of the TOF distance is σTOF ≈8(Tupdate /T)1/2 μm, where T is the averaging time. This TOF instability results from approximately 1.5 ns timing jitter of IGMs and the value of Δfrep/frep1 in Eq. (1). The precision of the SWI distance is σSWI ≈1.2(Tupdate /T)1/2 μm, reaching ~300 nm at 10 ms. The precision of the CWI distance is σCWI ≈19(Tupdate /T)1/2 nm, reaching 3 nm at 10 ms and a sub-nanometer when longer than 0.1 s. Because the precision of the TOF measurement without averaging is sufficiently stable compared to λsyn/4, the Nsyn in Eq. (3) can be determined and the Dsyn can be calculated using Eq. (4). When combined with the carrier-phase interferometric results, about ~10 ms averaging of synthetic-wavelength interferometric results is necessary. Without the SWI method, ~100 ms averaging of TOF results is necessary as shown in Fig. 5. Finally, the three methods are combined, and the precision is enhanced from several microns to a nanometer with just 10 ms averaging time. 10-4 TOF SWI CWI

-5

Allan Deviation (m)

10

10-6 10-7 10-8 10-9 10-10

0.001

0.01

0.1

1

Averaging time (s) Fig. 5. The precision (Allan deviation) of the distance measurement versus averaging time, computed from 4 s length data. Three different measurement methods are shown: TOF, SWI, and CWI. The precision is enhanced from the TOF measurement to the SWI measurement


without any averaging. It is further enhanced around ~10 ms when the SWI measurement is sufficiently stable compared to λc /4 (horizontal gray line) because the carrier phase information can be combined. The CWI results under 10 ms averaging time are not applicable for absolute ranging, but results are available for relative displacement within an unambiguity range of λc/2.

To evaluate the measurement accuracy of the present system, we measured a moving target corner mirror set at ~0.5 m with ~100 µm discrete moving step and 1.5 mm range. In the measurement, we combined data obtained from the TOF, SWI, and CWI methods. Figure 6 shows the measurement accuracy of the present system compared with a commercial laser interferometer. Both ranging results are calculated with identical environmental parameters including temperature, humidity, and air pressure, which induce ~10−7 uncertainty. By applying linear fitting [Fig. 6(a)], the slope is 0.9999989660 and the correlation coefficient (R2) is 0.9999999933. The residuals are kept within ± 50 nm with ~30 nm standard deviation, mainly caused by environmental disturbance. (a)

(b)

Fig. 6. Experimental evaluation of the measurement accuracy compared with a commercial laser interferometer. (a) Comparison of incremental distance measured by the dual-comb ranging system (vertical axis) with the displacements measured by a commercial laser interferometer (horizontal axis) at a distance of ∼0.5 m with ∼100 μm moving step. (b) The residuals shown are within ± 50 nm.

4. Discussion and conclusion The experimental results show three ranging methods realized and combined based on a dualcomb interferometer. The two mutually locked combs yield a stable carrier phase wherein precision is improved from the TOF measurement to the CWI measurement. SWI is used as an intermediary ranging method that decreases the averaging time from ~100 ms to ~10 ms. Essentially, the measurement rate is determined by the repetition rate difference Δfrep, which is limited by the optical bandwidth BW and repetition rate Δfrep< frep2/(2·BW) as deduced from Fig. 1(c). Results indicate that the long ambiguity range of the TOF method by low repetition rate combs would decrease the measured optical bandwidth if the Δfrep were maintained. However, TOF measurement precision improves (averaging time decreases) by increasing the bandwidth and repetition rate [15, 19]. The SWI method breaks through the contradiction of the measured optical bandwidth and the limitation of the repetition rate of the two combs. In our experiment, this method provides 1.2 µm precision and 38 µm ambiguity range and can be combined with the TOF results without averaging. Only ~10 ms is needed to combine the SWI results with CWI results. The different averaging time in the two combined processes also indicates we have not realized the SWI method’s full potential. Because we only utilize a


32-nm wavelength difference to generate a synthetic wavelength, results are limited by the optical bandwidth of the OFCs we used. For the optimal condition (shortest averaging time), about a 100-nm wavelength difference can be used so that only 1 ms averaging time is needed. In this condition, the ambiguity range of SWI is about 27 µm, and the precision of SWI and TOF is about 300 nm and 5.6 µm, respectively. Thus, the combination processes among the TOF, SWI, and CWI results are realized in only ~1 ms averaging time. The ultimate precision determined by the CWI method is about 13 nm, which is still precise enough for most applications. In conclusion, synthetic-wavelength-based dual-comb interferometry provides nanometer precision, several meters’ ambiguity range, and ~100 Hz measurement rate. The syntheticwavelength interferometry we employed in the dual-comb ranging system can greatly enhance the measurement rate. The present system proves promising for distance measurement requiring both high precision and a fast measurement rate. Funding National Natural Science Foundations of China (61575105, 61611140125); Shenzhen fundamental research funding (Grant No. JCYJ20170412171535171); Tsinghua University Initiative Scientific Research Program.