## Abstract

We propose here a method for absolute distance measurement by chirped pulse interferometry using frequency comb. The principle is introduced, and the distance can be measured via the shift of the widest fringe. The experimental results show an agreement within 26 μm in a range up to 65 m, corresponding to a relative precision of 4 × 10^{−7}, compared with a reference distance meter.

© 2015 Optical Society of America

## 1. Introduction

High-accuracy and long-range absolute distance measurement is of key importance in many fields, such as large scale manufacturing, remote sensing, and missions in space. As a coherent light source, frequency comb emits an ultra-short pulse train with a stable temporal interval, consisting of hundreds of thousands of narrow modes in the optical domain, where these discrete and uniform lines can be well traced to a time/frequency standard, truly uniting the services of time keeping and distance ranging [1]. It is excited that scientists all over the world have been keeping on working hard to develop more and more accurate clock, and the state of the art can achieve 10^{−18} uncertainty [2,3
]. The distance metrology could be improved in future when such a precise clock can be universally used as the time reference.

Frequency comb has already performed distance ranging in air since 2000 [4], with extremely high accuracy (below 10^{−6} relative precision) [5], large dynamical range (up to 0.7 km) [6], high up-date rate (fast responding speed) [7,8
], and compact system set up [9]. We would like to summary these various methods here, by roughly categorizing them into several groups. Frequency comb can work as an indirect light source to calibrate the traditional continuous wave lasers [10], whose frequency can be well stabilized and traceable to a time clock, dramatically improving the measurement precision, but involving a complex structure of the system. Directly, fs frequency comb can perform the distance measurement via the mode phases, which can be obtained by the inter-mode beat [4,5
], the dual comb interferometry [11], and multi-heterodyne using frequency comb and a tunable laser [12]. Technique of pulse cross correlation can determine distances by evaluating the peak position of the fringe packets [13,14
], interfered intensity [15,16
], measuring the peak shift of the cross correlation patterns [17], and Fourier transform to pick up the stationary phases of the chosen wavelengths [18–20
], while a constantly scanning stage is required to adjust the time delay between the pulses, leading to a system with movable elements and relatively long measuring time. In the spectral domain, the distances can be measured by dispersive interferometry [21–23
], through the slope of the unwrapped spectral phase, but with a multi-steps data process. Fortunately, dual-comb technique can solve most of the limitations mentioned above, based on which the two comb lasers with slightly tuned repetition frequencies can evenly and finely sample each other, and we can easily obtain the interested information, such as the flight time of the pulses [24], the peak position of the cross correlation patterns, the exact phases of the specific wavelengths [25], and the slope of the unwrapped spectral phase [26], without any movable devices. However, complex phase locking circuits are required, making the system expensive and difficult to satisfy the applications out of the lab due to the large size. In addition, pulse-to-pulse alignment [6,14,27,28
] by changing the repetition frequency of the comb has also been reported, where the distances can be determined by the current repetition rates. We try hard to propose some another new methods for our distance metrology using this powerful light source.

In this paper, we introduce a method of the chirped pulse interferometry for absolute distance measurement, where the unknown distances can be determined by the shift of the widest fringe. It is important to recognize that the basic concept of time-of-flight distance measurement using the chirp of the ultrashort pulse has been demonstrated by Minoshima et al. in Ref [29]. Comparing to the conventional spectral interferometry, this method can directly determine the relative position between the measurement pulse and the reference pulse. The effective measuring range can be also dramatically extended by using the high-order diffraction light of the grating with a simple spectrometer. We analyze the measurement principle, and do the experiments in long distance. Our experimental results show an agreement within 26 μm in a range up to 65 m on an optical rail, indicating a relative precision of 4 × 10^{−7}.

## 2. Chirped pulse interferometry based distance measurement

Generally, the chirped pulse can be obtained by severe dispersion, often using a long fiber link, a piece of dispersion glass, or a pair of gratings [30]. After propagating such a pulse stretcher, the pulse frequency will be chirped, and we will get a linearly chirped pulse train, whose center frequency can be expressed as *ω _{c}* =

*ω*

_{0}+

*bt*,

*ω*is the current center frequency,

_{c}*ω*

_{0}is the initial center frequency,

*b*is the chirp rate, and

*t*is the time delay.

Figure 1 shows the schematic of the chirped pulse interferometry. Frequency comb emits a pulse train into an unbalanced Michelson interferometer, and is split at BS. One part goes into the reference beam with a pair of gratings as the chirped pulse generator, and is reflected by the reference mirror. The other part travels through various unknown distances, and is reflected by the target mirror. Finally, these two parts of the light pulse are combined at BS, and are detected by a dispersive spectrometer.

Assuming that in Fig. 1, one single pulse emitted by the comb source can be expressed as:

*E*

_{0}is the electrical field,

*a*is the attenuation factor,

*a*= 2ln2/

*τ*

_{0}

^{2},

*τ*

_{0}is the initial pulse width,

*ω*is the center frequency of the comb source,

_{c}*φ*

_{0}is the initial phase, and

*φ*is the phase shift rate due to the difference between the phase velocity and the group velocity. For convenience of calculation, we suppose both the initial phase and the phase shift are equal to zero, which means

_{ce}*φ*

_{0}=

*φ*= 0, and consider the case of equal arm position. The reference chirped pulse and the measurement pulse with a time delay

_{ce}*τ*can be respectively denoted as:

*α*and

*β*are the power factors, showing the power difference of the pulse trains. Please note that, these two factors can affect the modulation depth of the spectral interferograms, and a deeper depth is preferred in the data process [22].

*a*

_{1}= 2ln2/

*τ*

_{chirp}^{2},

*τ*is the chirped pulse width,

_{chirp}*b*is the chirp rate, and

*b*≈∆

*ω*/(2

*τ*), ∆

_{chirp}*ω*is the frequency bandwidth of the pulse source.

We Fourier transform Eqs. (2) and (3) , and get:

The spectrums of the reference chirped pulse and the measurement pulse interfere, and are detected by a dispersive spectrometer. The measured spectral intensity can be calculated as:

*ϕ*(

*ω*) equals to

*ϕ*(

_{ref}*ω*)-

*ϕ*(

_{pro}*ω*),

*ϕ*(

_{ref}*ω*) and

*ϕ*(

_{pro}*ω*) are the spectral phases of the reference pulse and the measurement pulse, respectively.

*ϕ*(*ω*) can be calculated to be:

We find that in Eq. (7), when *b* equals to zero, the spectral phase is exactly the same as that of the dispersive interferometry [21,23
], where the unwrapped phase in the case of dispersive interferometry is a strictly linear function. In our case of chirped pulse interferometry, the spectral phase is a quadratic function, which means the interfered spectral fringes are not the stable oscillation with a single frequency any more, but with a modulated frequency. The widest fringe appears when the oscillation frequency achieves the extremum, i.e., the lowest frequency. To deduce the derivation of Eq. (7), we get:

Therefore, the frequency location of the widest fringe can be expressed as:

*n*is the pulse group refraction of air,

_{g}*L*is the unknown distance, and

*c*is the light speed in vacuum. In our cases, we consider that

*b*(10

^{24}) is much larger than

*a*

_{1}(10

^{19}). The unknown distance can be thus denoted as:

*ω*is the shift of the center frequency of the comb source. We find that we can determine distances by the shift amount of the center frequency, i.e., the shift of the widest fringe in the spectrograms.

_{shift}To make this more intuitive, we give a short simulation in Fig. 2
. In the simulations, the center wavelength of the light source is 1560 nm, and the spectral width is 4.9 THz. The chirped pulse width is 20 ps, and the measurement pulse width is 200 fs. The chirp rate is 1.1 × 10^{24} rad/s^{2}, and the power factors *α* and *β* are 0.3 and 0.7, respectively. We find that due to the chirped process the fringe frequency is not a constant, but with a modulation where there will be a widest fringe with the lowest oscillation frequency. The position of the widest fringe changes with tuned time delays between the reference pulse and the measurement pulse, where a forward time difference corresponds to a left shifted frequency amount (to lower optical frequency), and a backward time delay moves the widest fringe to the right side (to higher optical frequency), relative to the center frequency.

In the long-range cases, the measured distance can be expressed as:

*N*is an integer,

*L*is the pulse-to-pulse length, and

_{pp}*f*is the repetition frequency. Based on Eq. (11), we find that the distances can be determined by measuring the integer

_{rep}*N*and the shift of the widest fringe

*f*, with the given group refractive index

_{shift}*n*, the chirp rate

_{g}*b*, and the locked repetition frequency

*f*.

_{rep}## 3 Experiments

#### 3.1 Experimental set up

We do the experiments on an optical rail underground in National Institute of Metrology. Figure 3 is the experimental photograph, and the experimental set up is shown in Fig. 4 . The pulse source (Onefive Origami-15, 250.012 MHz, 80 mW), which is well locked to a Rb clock (Stanford FS725), is split at BS. Subsequently, the reference beam is chirped by a grating pair, and the measurement pulses travelling through a beam expander sample different distances. These two beams are combined at BS. A simply dispersive spectrometer (YOKOGAWA AD6370D-20) is used to detect and record the spectrograms. An incremental distance meter (Agilent 5519B) is applied to be a reference, to verify the measurement results. Both the measuring beam of our system and the beam of the reference He-Ne laser are aligned to be strictly parallel with the long optical rail. The environmental conditions are well controlled, which are 22.4°C,1004.7hPa, and 48.1% humidity. The stability of the He-Ne interferometer is below 0.2 μm for 10 s, and below 0.8 μm for 10 min with a distance of 80 m, and below 70 nm at 2 m distance.

Figure 5 shows the spectrum of the light source, with the center wavelength of 1560 nm and the bandwidth of about 55 nm. The group refractive index of air is correspondingly calculated to be 1.0002649 based on Ciddor formula [31].

#### 3.2 The system self-calibration

From Eq. (11), we find that the chirp rate *b* should be determined precisely to measure the distances. As mentioned above, *b* can be roughly expressed as *b*≈∆*ω*/(2*τ _{chirp}*), where

*b*is inversely proportional to the width of the chirped pulse

*τ*with a fixed spectral width ∆

_{chirp}*ω*. In our cases using a pair of gratings as the pulse stretcher,

*τ*is related to several factors, including the grating interval

_{chirp}*D*, the incidence angle of the grating

*i*, the grating constant

*d*, the diffraction order

*m*, the spectral width ∆

*ω*, and the center wavelength of the light source

*λ*. However, it is not easy to determine all the factors accurately, which means a self-calibration of the measurement system is necessary at the initial position when we start to perform the measurement. For a completed system (i.e., all the factors have been well fixed), based on Eq. (10),

_{c}*b*can be simply calculated as:

*L*is a shifted displacement which is given by the reference distance meter, and

*f*is the corresponding frequency shift of the widest fringe. At the initial position (around the equal-arm position of the Michelson interferometer), we first measure the spectrogram as shown in Fig. 6(a) . The target mirror is then moved by ∆

_{shift}ʹ*L*, which is measured and displayed to be 0.301 mm by the reference cw counting interferometer, and the corresponding interferogram is shown in Fig. 6(b).

As shown in Figs. 6(a) and 6(b), the widest fringes can be obviously observed, whose position can be measured by:

Please note that, the widest fringe could be not sufficiently distinct in the case of the edge of the spectrogram. Some possible spectrum distortions can also cause the measurement error. It is preferred to locate the widest fringe near the center of the spectrogram, and the stable environmental conditions are needed to suppress the spectrum distortion. The positions of the widest fringes before and after shifting the target mirror can be thus determined to be 193.780 THz (in Fig. 6(a)) and 191.678 THz (in Fig. 6(b)). The chirp rate *b* turns to be:

#### 3.3 Experimental results

We measured the distances up to 65 m, and Fig. 7 shows the spectrograms at about 13 m and 60 m. It can be found that the spectral intensity decreases when increasing the distances due to the power attenuation caused by the air absorption.

Based on Eq. (13), the positions of the widest fringes are measured to be 191.525 THz and 193.894 THz, corresponding to Figs. 7(a) and 7(b), respectively. The initial position shown in Fig. 6(a) is 193.780 THz. Hence, the shift amounts of the widest fringes *f _{shift}* are −2.255 THz and 0.114 THz.

As discussed in [6,14,18,22
], the integer number *N* can be determined precisely by using the reference distance meter or changing the repetition frequency of the comb source. In our experiments, we determine the integer number *N* with the help of the reference distance meter. In the cases of Fig. 7, the distance values given by the reference are 13186.736 mm and 59939.818 mm, respectively. The integer number *N* can be calculated by *N* = *round*(2 × *L _{ref}*/

*L*), where

_{pp}*round*denotes the nearest integer,

*L*is the reference distance, and

_{ref}*L*is the pulse-to-pulse length. Considering the spectrograms shown in Fig. 7,

_{pp}*N*can be measured as:

The uncertainty of the reference distance value is below 0.8 μm. The uncertainty of the group refractive index is at 10^{−8} level, and the uncertainty of the repetition frequency is below 10^{−11}. Therefore, *N* can be precisely determined with uncertainty well below 0.5. The measured distances can be thus determined using Eq. (11) as:

Compared with the reference values given by the cw counting interferometer, the measurement results indicate the differences of 6 μm and 17 μm, respectively.

Figure 8
shows the experimental results, where the midpoints are the average values for five single measurement, and the errorbars are the standard deviations with 15 s integration time. The final experimental results indicate an agreement within 26 μm in a distance range up to 65 m, compared with a cw counting interferometer, corresponding to a relative precision of 4 × 10^{−7} (below 10^{−6}).

## 4 Uncertainty evaluation

Based on Eq. (11), we find that the uncertainty is mainly related to the shift of the widest fringe *f _{shift}*, the repetition frequency of the light source

*f*, the chirp rate

_{rep}*b*, and the air group refractive index

*n*. The uncertainty can be thus expressed as:

_{g}In the experiments, the laser source is well locked to the Rb clock with 2 × 10^{−11} uncertainty. Due to the phase locking loop and some another instabilities, it is reasonable to estimate the uncertainty related to the repetition frequency to be 10^{−10}·*L*. The term related to the air group refractive index is also distance-dependent. In our experiments, the refractive index of air for both the reference distance meter and the dispersive interferometer is corrected by Ciddor formula based on the same environment sensors, and this term turns to be 6.8 × 10^{−8}·*L*. The uncertainty contributed by the measurement of the shift of the widest fringe dominants the combined uncertainty. In general, the air turbulence, random vibration, the environment fluctuation, the intensity of the laser beam, and the system set-up can all affect this term [22], which is measured to be 25.1 μm. In addition, the uncertainty related to the chirp rate is found to be 1.1 μm, which is measured at the initial position. It is important to clarify that the target beam has been also chirped in long distance measurement, which can make contributions to the total uncertainty. In general, several reasons can lead to the spatial chirp of the femtosecond pulse laser, such as the beam divergence and the air dispersion, etc. In our experiment, the beam size is about 25 mm while the diameter of the transmitting-receiving lens (Thorlabs AL7560-C) is 75 mm, lager than 25 mm. It is reasonable to consider that most of the returned beam from the target is detected. On the other hand, due to the air dispersion the ultrashort pulse can be stretched after travelling a long distance, and correspondingly the optical frequency of the pulses can be chirped. In our cases, the initial pulse width is about 150 fs, and the spectrum width is about 55 nm (1530 nm ~1590 nm). Assuming that the distance is 65 m (130 m optical path), the pulse width can be broadened by 20 fs based on Ciddor formula, which means the pulse duration changes into about 170 fs. With respect to the chirped pulse width of about 20 ps, we consider that the pulse width of 170 fs is not very considerable to introduce strong chirp. However, we can analyze this distance-dependent part by using the method of pulse cross correlation. In Ref [32], we find the uncertainty of about 0.8 μm caused by the spatial chirp in 50 m range. Hence, it may be reasonable to evaluate this part to be below 1 μm in a range of 65 m. The uncertainty of the optical rail itself is measured to be below 0.8 μm in 80 m range. Overall, the combined uncertainty with a coverage factor of k = 1 can be represented as $\sqrt{{\left(25.2\mu m\right)}^{2}+{\left(6.8\times {10}^{-8}\cdot L\right)}^{2}}$, corresponding to a relative precision of 4 × 10^{−7} in a range of 65 m. The sources of the total uncertainty discussed above are summarized in Table 1
.

## 5 Conclusion

In this paper, we propose a new method for absolute distance measurement by chirped pulse interferometry in the optical domain using a femtosecond pulse laser. We introduce the measurement principle, and the unknown distance can be determined via the shift of the widest fringe. In the experiments, long-range distance measurement, up to 65 m, has been performed. The experimental results show an agreement well within 26 μm, corresponding to a relative precision of 4 × 10^{−7}, compared with a reference distance meter.

## Acknowledgments

We would like to thank National Institute of Metrology for great support of the long optical rail. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 51105274, 51327006), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120032130002), and Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 15JCZDJC39300).

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