## Abstract

A dual-comb nonlinear asynchronous optical sampling method is proposed to simplify determination of the time interval and extend the non-ambiguity range in absolute length measurements. Type II second harmonic generation facilitates curve fitting in determining the time interval between adjacent pulses. Meanwhile, the non-ambiguity range is extended by adjusting the repetition rate of the signal laser. The performance of the proposed method is compared with a heterodyne interferometer. Results show that the system achieves a maximum residual of 100.6 nm and an uncertainty of 1.48 μm in a 0.5 ms acquisition time. With longer acquisition time, the uncertainty can be reduced to 166.6 nm for 50 ms and 82.9 nm for 500 ms. Moreover, the extension of the non-ambiguity range is demonstrated by measuring an absolute distance beyond the inherent range determined by the fixed repetition rate.

© 2014 Optical Society of America

## 1. Introduction

An optical frequency comb offers a large number of discrete optical frequencies which can be stabilized collectively with direct traceability to atomic clocks [1,2]. During the past decades, it has been employed as a light source in various absolute distance measurement methods [3,4]. These methods can be categorized into three groups based on their principles. In the first place, the absolute distance is calculated by synthetic wavelength method with continuous wave (CW) lasers as the light source. The optical frequency comb is adopted as frequency standard for CW wavelength calibration [5–8]. In a simplified measurement, synthetic wavelengths are considered to be directly generated by heterodyne beatings among optical frequency components within the comb bandwidth and the absolute distance is measured through phase analysis of these heterodyne frequencies [9]. Nevertheless, an estimated distance serves as the prerequisite for the establishment of an effective synthetic wavelength chain. To overcome this disadvantage, the second approach is proposed to measure absolute distances without auxiliary estimations. Based on time-of-flight principle, periodic train of pulses emitted by the frequency comb can be utilized for absolute length measurement. Thus, an idea was proposed by Ye [10], that time intervals between pulses were used to determine the estimated distance based on time-of-flight principle and interference fringes were analyzed to obtain an enhanced resolution. A few experiments were conducted to corroborate Ye’s method with precision comparable to theoretical predictions [11–14]. Furthermore, an incoherent scheme, named as balanced cross-correlation [15,16], was also put forward merely using time-of-flight principle with a compact design. However, the second approach only worked at discrete ranges required by pulses overlap. To realize absolute distances measurement with continuous range and high resolution, dual-comb interferometry is employed as the third method [17]. It scans the entire range quickly using two combs with slight difference in repetition rates. Meanwhile, dead zone can be eliminated by an orthogonal polarization approach [18]. The distance calculation in this method is similar to that in dispersive interferometry [19–22]. The differences are that the phase demodulation of the dual-comb interferometry is in a relatively low frequency region and a large tunable range can be achieved by changing repetition rates. Albeit time-of-flight measurement is introduced into dual-comb technology [23] for simplified distance calculation, the curve fitting for temporal pulses is still complicated since the carrier frequency need be eliminated. Comparing the three groups of methods, dual-comb technology is superior for absolute length measurement at arbitrary distances. However, it still suffers from complex distance calculation whether utilizing Fourier transform for phase demodulation or Hilbert transform for carrier frequency elimination.

In this paper, an incoherent scheme based on asynchronous optical sampling (ASOPS) [24,25] is proposed with the particular aim of performing absolute length measurement and simplifying distance calculation. This scheme can be divided into two parts: dual-comb temporal optical scanning and second harmonic generation (SHG). Dual combs, with a tiny difference in their repetition rates, are involved to fulfill temporal optical scanning among pulses and guarantee the measurement of arbitrary distances. In addition, type II SHG [26], is adopted to detect the cross-correlation generated by temporal optical scanning. Since the cross-correlation deriving from type II phase matching is an envelope of interference fringes, SHG detection will simplify the curve fitting and pledge a high update rate. Moreover, type II phase matching, as an incoherent method, can release offset frequencies of the two combs from active control, which considerably simplifies the laser structure. The results obtained from our scheme demonstrate advantages when compared with those from a conventional heterodyne interferometer.

## 2. Principle

The schematic structure of the ASOPS for absolute length measurement is shown in Fig. 1. The structure mainly comprises of two parts: dual-comb temporal optical scanning which confirms overlap among pulses and the SHG which plots the peak positions of pulses [15,27].

In temporal optical scanning, a femtosecond fiber laser is the signal laser (SL) with a repetition rate of *f*_{r} + Δ*f*_{r}. The light pulses emitted from the SL are incident on both the reference mirror (M_{ref}) and the target mirror (M_{tar}). The optical path difference of the two arms is recorded as a function of time intervals between adjacent pulses. In order to spot the time intervals, another femtosecond fiber laser with a slightly different repetition rate *f*_{r} serves as the local oscillator (LO). Pulses from the SL and the LO are sent to the SHG structure for cross-correlation measurement. Taking pulses from the SL as static reference, pulses from the LO walk through with a step of

The SHG structure is based on type II phase matching with input pulses of orthogonal polarization. The SHG intensity, *I*_{2}* _{ω}*, can be expressed as [26]

*I*

_{ω,}_{SL}and

*I*

_{ω,}_{LO}are the intensities of fundamental pulses from the SL and the LO and

*τ*is the temporal offset between pulses. According to Eq. (2),

*I*

_{2}

*reflects the temporal offset since the overlap between pulses is quantified and the maximum of SHG intensity is obtained when the LO and SL pulses completely overlap in time.*

_{ω}Optical scanning, deriving from the difference in repetition rate of the two combs, generates the temporal offset *τ* while the SHG generates the relevant *I*_{2}* _{ω}*. The LO repetition rate

*f*

_{r}is taken as the sampling rate to record

*I*

_{2}

*. Pulses from the LO can serve as a probe to depict the pulses from the SL. By judging the peak position of*

_{ω}*I*

_{2}

*, the time interval between adjacent pulses from the SL can be obtained. However, because of the fixed time step of Δ*

_{ω}*T*

_{r}, the SHG intensities

*I*

_{2}

*are discrete data points with a step size of 1/*

_{ω}*f*

_{r}, which corresponds to the sampling rate. It will lead to a quantization error as peak positions are found by the maximum SHG intensities. For an enhanced resolution, a sech

^{2}function is used to fit the detected

*I*

_{2}

*. The time interval between adjacent pulses from the SL,*

_{ω}*τ*, can be calculated by

_{d}*t*

_{ref}and

*t*

_{tar}are the peak positions in time domain determined by curve fitting,

*f*

_{r}is the sampling rate and Δ

*T*

_{r}is the time step in Eq. (1).

The absolute distance *L* is given by

*c*is the speed of light in vacuum,

*n*

_{g}is the group refractive index of air and

*τ*is the time interval in Eq. (3). But the non-ambiguity range (NAR) of the absolute distance is limited by the repetition rate of the SL, i.e. Λ

_{d}_{NAR}=

*c*/[2(

*f*

_{r}+ Δ

*f*

_{r})]. In order to extend the NAR, the repetition rate of the SL is changed from

*f*

_{r}+ Δ

*f*

_{r}to

*f*

_{r}+ Δ

*f*

_{r}’, corresponding to a variation of the NAR from Λ

_{NAR}=

*c*/[2(

*f*

_{r}+ Δ

*f*

_{r})] to Λ’

_{NAR}=

*c*/[2(

*f*

_{r}+ Δ

*f*

_{r}’)], while the repetition rate of the LO is fixed. With Λ

_{NAR}and Λ’

_{NAR}, the absolute length

*L*

_{abs}can be expressed by where

*m*and

*m*’ are positive integers while

*L*and

*L*’ are the ‘wrapped’ distances with the given NARs [18]. Since the change of the repetition rate of the SL is tiny, the relation between

*m*and

*m*’ can be forced to be

*m*=

*m*’ under active control. With Eqs. (5) and (6), the absolute distance

*L*

_{abs}can be obtained as illustrated in Fig. 2.

## 3. Experimental setup

The experimental setup is shown in Fig. 3. The light sources are two Er-doped fiber lasers (M-comb, MenloSystems): one for the SL and the other for the LO. The fiber laser for the SL emits pulses with a spectral bandwidth of 30 nm centered at 1550 nm. The repetition rate can be tuned in the range of 250 ± 2.3 MHz using a PZT actuator coupled to a motorized stage inserted in the laser cavity. The pulses are amplified to 200 mW and compressed to ~70 fs by an Er-doped fiber amplifier (EDFA). The LO fiber laser shares the same characters with the SL except the fixed repetition rate of 250 MHz. The repetition rates of the two femtosecond lasers are locked to an Rb atomic clock (8040C, Symmetricom) with a difference of 2 kHz and the offset frequencies of the two lasers are not actively controlled.

The pulses from the SL are split into the reference and measurement arms of a Michelson interferometer by a polarization beam splitter (PBS). The retro-reflector in the reference arm is fixed while that in the measurement arm is on a 200 mm linear translation stage (M-521.DD, PI). The exit beam of the reference retro-reflector propagates through the PBS because of the quarter wave plate (QWP). Meanwhile, the reflector in the Michelson interferometer is used to reflect the exit beam of the target retro-reflector. It is represented by a dotted line since the input and output beams are at different heights. The pulses reflected by the two retro-reflectors are combined with the LO pulses respectively and focused onto two type II barium borate (BBO) crystals for the SHG. The exit beams of the two retro-reflectors are separated to minimize polarization crosstalk caused by the low extinction ratio of one PBS. The resulting signals from the two BBO are digitized and acquired individually by a 14 bit 250 MHz data acquisition card (7965R, National Instruments) synchronized to the repetition rate of the LO. Meanwhile, curve-fitting algorithm, using 9 points near the maximum value, is employed to get peak positions of the resulting signals. The waveform in Fig. 3 is a screen shot of the resulting signals from the two BBO, acquired by an oscilloscope (DSO9254A, Agilent). A third retro-reflector for the heterodyne interferometer (5519, Agilent) is also mounted on the translation stage for comparison.

## 4. Experimental results

The performance of the system is tested by measuring an actual distance in air and compared with the heterodyne interferometer. The stability of the dual-comb system is evaluated with different averaging times while the target fixed at ~39.2 mm. Figure 4(a) illustrates that the standard deviation varies with different averaging time: 1.48 μm for 0.5 ms, 166.6 nm for 50 ms and 82.9 nm for 500 ms. Moreover, the precision of the dual-comb system is appraised by comparison with the heterodyne interferometer. The target retro-reflector moves along the translation stage with a step of 1 μm for a short distance of ~39.2 mm. 20 displacement points are recorded by the dual-comb system and the heterodyne interferometer independently with an averaging time of 500 ms for comparison. As plotted in Fig. 4(b), the residuals range from −74.1 nm to 100.6 nm.

According to Eqs. (1), (3) and (4), the absolute length *L* can be expressed as

*t*equals to |

*t*

_{ref}-

*t*

_{tar}|, representing the time interval between the peak position

*t*

_{ref}and

*t*

_{tar}. From Eq. (7), the uncertainty of the

*L*,

*U*, can be calculated by

_{L}*U*

_{Δ}

*is the uncertainty of Δ*

_{t}*t*and

*U*

_{f}_{r}is that of

*f*

_{r}. Noting

*U*

_{f}_{r}is 3 mHz and

*f*

_{r}is 250 MHz in our experiment, the third term on the right of Eq. (8) is much smaller than the first and the second terms on the right of Eq. (8) and is hence neglected. During the measurement, the temperature varies from 23.624 °C to 23.728 °C. Meanwhile, the humidity is kept at 21.1% and the pressure increases from 102.284 kPa to 102.291 kPa. According to Ciddor equation and its uncertainty, the fourth term in the Eq. (8) is no larger than 10

^{−7}. Compared with the repetition rate stability and the curve fitting accuracy,

*n*

_{g}can be assumed as a constant. Equation (8) can be further represented as

*U*

_{Δ}

*stands for the curve fitting accuracy and*

_{t}*U*

_{f}_{r}illustrates the repetition rate stability. In Eq. (9) Δ

*f*

_{r}is 2 kHz,

*L*is 39.2 mm,

*c*is 299792458 m/s,

*n*

_{g}is 1.00027 calculated through Ciddor equation based on measured temperature, pressure and humidity, and

*U*

_{Δ}

*is 1 ns estimated by introducing functions with similar signal to noise ratio in experiment into curve fitting algorithm.*

_{t}*U*is no larger than 1.2 μm, which matches the standard deviation for 0.5 ms illustrated in Fig. 4(a). According to Eq. (9), the curve fitting accuracy dominates the uncertainty of a single measurement when the measurement is restricted to short distances. In this case, a clear pulse in the SHG measurement is in need to pursue a higher accuracy. When a long distance is measured, the uncertainty of a single measurement is mainly decided by the uncertainty of the repetition rate, since pulse shapes can be controlled by adjusting fiber length of the EDFA and

_{L}*U*

_{Δ}

*is basically maintained. Taking current parameters into account,*

_{t}*U*

_{f}_{r}is more important than

*U*

_{Δ}

*when the absolute length*

_{t}*L*is larger than 0.56 m. In addition, fluctuation of refractive index ought to be considered when measured distance is large.

The extension of the NAR is substantiated by measuring step movements from 493.7 mm to 693.7 mm, including the NAR of 600 mm, corresponding to the 250 MHz repetition rate and folded optical length in Michelson interferometer. The test range includes *m* = 0 and *m* = 1 because with the same variation of the repetition rate, the change from *L* to *L*’ is minimum in this case. The results are shown in Fig. 5, compared with results from the heterodyne interferometer. In Fig. 5(a), distances are measured without the integer *m*. In comparison, in Fig. 5(b) the integer *m* is judged by the method proposed above with a change of the repetition rate of 4 kHz from 250.002 MHz to 249.998 MHz. Residuals and standard deviations of Fig. 5(b) is shown in Fig. 5(c). The residuals range from −141.1 nm to 175.8 nm while the standard deviations vary from 87.5 nm to 194.0 nm for 500 ms, which are slightly larger than those in the short distance measurement caused by turbulence of atmosphere. The results prove that changing the repetition rate can extend the NAR of the system.

## 5. Conclusion

In this paper, absolute distances are measured by the proposed nonlinear ASOPS utilizing dual-comb optical scanning and type II SHG. Dual combs, with a tiny difference in their repetition rates, are involved to fulfill pulses overlap at arbitrary distances. Meanwhile, the type II SHG is adopted to obtain the cross-correlation envelope directly for simplified distance calculation. The basic principle for the system is based on time-of-flight measurement, which implements a single measurement within 1 millisecond. Several experiments are conducted and the results are compared with a heterodyne interferometer. The residuals are scattered from −74.1 nm to 100.6 nm and the stability of the system is demonstrated with an uncertainty of 1.48 μm for a single measurement, corresponding to an averaging time of 0.5 ms. Moreover, the uncertainty can be reduced to 166.6 nm for 50 ms and 82.9 nm for 500 ms. The NAR is also extended by adjusting the repetition rate of the signal laser and a mechanical range of 693.7 mm is proved by a series of tests.

## Acknowledgments

This work was supported by Tsinghua University Initiative Scientific Research Program, the State Key Lab of Precision Measurement Technology & Instruments of Tsinghua University and the National Natural Science Foundation of China (Grant No. 61205147).

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