## Abstract

Dual-comb system parameters have significant impacts on the ranging accuracy. We present a theoretical model and a numerical simulation method for the parameter optimization of a dual-comb ranging system. With this method we investigate the impacts of repetition rate difference, repetition rate, and carrier-envelope-offset frequency on the ranging accuracy. Firstly, the simulation results suggest a series of discrete zones of repetition rate difference in an optimal range, which are consistent with the experimental results. Secondly, the simulation results of the repetition rate indicate that a higher repetition rate is very favorable to improve the ranging accuracy. Finally, the simulation results suggest a series of discrete optimal ranges of the carrier-envelope-offset frequency for the dual-comb system. The simulated results were verified by our experiments.

© 2015 Optical Society of America

## 1. Introduction

A mode-lock laser provides a series of optical pulses separated by the travel time of the laser ring cavity. When its repetition rate and carrier-envelope-offset frequency are stabilized referencing to a frequency standard (e.g. an atom clock), it becomes an optical frequency comb that can be used as an ultra-stable ruler in the space, time and frequency domains [1]. In the past 15 years, optical frequency combs have enabled revolutionary progress in distance metrology [2]. Generally, there are two ways to use optical frequency combs for distance measurement. One is to use the comb as a wavelength or frequency standard for CW lasers [3–7 ]. The other is to use the comb as a light source directly. Due to the inherent advantages of the comb itself, various principles have been developed for distance measurement, e.g. using the inter-mode beat signals of the comb [8, 9 ], using dispersive interferometry [10–12 ], using the pulse separation distance as a ruler [13–21 ], and dual-comb method [22–27 ]. In particular, the dual-comb method which can realize large dynamic range measurements of absolute distance with high precision at high speed has received more and more attention. This concept is also widely used in Fourier transform spectroscopy [28–33 ].

In the dual-comb system, several parameters have been demonstrated to have significant impacts on the final results, e.g., timing jitter [33, 34
] and repetition rate difference. In a previous work [35], we investigated the influence of repetition rate difference (Δ*f*
_{r}) on the ranging accuracy of a dual-comb system by changing experimental parameters. We showed an optimal range of Δ*f*
_{r}, and demonstrated the impacts of Δ*f*
_{r} on sub-sampling and the ranging results. However, such experimental optimization is laborious and time consuming. Furthermore, some important parameters are difficult to optimize experimentally. For example, the repetition rate is another factor of the equivalent sampling rate. However, due to limited tunable range of repetition rate, it is impractical to prepare a series of laser pairs with different repetition rates to investigate its impacts on measurement results.

In order to resolve the above problem, we developed a theoretical model and numerical simulation method for dual-comb system in the present study. Firstly, we simulated the influences of Δ*f*
_{r} and made a comparison with our previous experimental results to demonstrate the efficacy of the model and simulation method. Secondly, we investigated the impacts of repetition rate on the ranging accuracy by using this simulation method. Thirdly, we simulated the impacts of carrier-envelope-offset frequency which was normally ignored in dual-comb distance measurement based on time-of-flight method [24–26
]. Finally, we did an experiment to verify the simulation results of carrier-envelope-offset frequency.

## 2. Principle and theoretical model of dual-comb distance measurement

#### 2.1 Principle

Figure 1
shows the schematic view of the dual-comb ranging system. One comb with a repetition rate of *f*
_{r} serves as the signal laser, and the other with a repetition rate of (*f*
_{r} + Δ*f*
_{r}) serves as the local oscillator (LO) laser. The pulse train from the signal laser is introduced into a Michelson interferometer. In the output beam of the interferometer, there are two staggered co-propagating pulse trains separated by a time delay (Δ*t*) caused by the path length difference (*D*) of the two arms in the interferometer [Fig. 1(a)]. Due to the minor difference of repetition rate, the pulse train from the LO laser can realize optical sampling of the two staggered pulse trains from the interferometer with a time slip of Δ*T*
_{r} ≈Δ*f*
_{r}/*f*
_{r}
^{2}, yielding a pair of cross-correlation interferograms in each updating period of *T*
_{update} = 1/Δ*f*
_{r} [Fig. 1(b)].

To extract the target distance from the cross-correlation interferograms obtained above, one solution is to fast Fourier transform (FFT) the cross-correlation interferograms in the RF domain, and obtain the time delay Δ*τ* between the two interferograms in real time scale according to the phase shift in frequency domain. Another solution is to Hilbert transform the signal to get the envelopes of the interferograms, and then use Gaussian fitting to locate the peak centers of the interferograms to get Δ*τ*. By dividing the scaling factor of *f*
_{r}
*/*Δ*f*
_{r}, the time delay (Δ*t*) between reference and target pulses in effective time scale can be obtained as Δ*t* = Δ*τ*Δ*f*
_{r}/*f*
_{r} [22,35
]. Consequently, the target distance *D* can be calculated by

*c*is the speed of light in vacuum,

*n*

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

#### 2.2 Theoretical model

In order to simulate the dual-comb ranging system, we set up a theoretical model first. The electric field of each ultrashort pulse can be treated as a carrier signal multiplied by the envelope function [36]:

where $\widehat{E}(t)=$ ${a}_{0}\mathrm{exp}[-{(\frac{t}{{c}_{0}})}^{2}]$ is the envelope function,*ω*is the angular frequency of the carrier and

*ϕ*

_{ce}is the carrier-envelope phase. In this system, Gauss function is used to describe the envelope of pulse, the parameters

*a*

_{0}and

*c*

_{0}can be determined according to the parameters of the real pulse from the laser.

Then let us consider an infinite ultrashort pulse train with a period of *T*, it can expressed by

*ϕ*

_{ce}evolves from successive pulses by a fixed amount Δ

*ϕ*

_{ce}. For the ease of calculation, we designate

*ϕ*

_{ce}=

*n*Δ

*ϕ*

_{ce}. The repetition rate of this pulse train is

*f*

_{r}= 1/

*T*, and the carrier-envelope-offset frequency is

*f*

_{ceo}= Δ

*ϕ*

_{ce}

*f*

_{r}/2π. When the pulse train travels a distance

*x*, it can be expressed by

*T*and Δ

_{i}*ϕ*

_{ce}

*, respectively, where*

_{i}*i*= 1 for the signal laser and

*i*= 2 for the LO laser. We also denote the path lengths of reference arm and measurement arm by

*x*

_{1}and

*x*

_{2}, respectively. The signals arriving at the photodetector can be given by:

According to this equation we can use a numerical method to simulate the cross-correlation interferograms generated by the dual-comb system. Note that the refractive index of air is not considered in this model for simplicity.

## 3. Simulation of dual-comb distance measurement

In order to investigate the impacts of the dual-comb system parameters on the ranging performance, we use a numerical method to generate the cross-correlation interferograms according to the theoretical model described above. In the simulation, the general parameters are the same as in our experimental system [35], i.e., the repetition rate of LO laser is 56.17 MHz while that of signal laser is tunable from 56.16 to 56.185MHz, the center wavelength is 1560 nm, the pulse width is 3.06 ps (after a band-pass-filter with 200 GHz bandwidth), and the target distance is set to 0.5 m. Figure 2 shows samples of simulated cross-correlation interferograms of the dual-comb system. In Fig. 2(a), there are two peaks that stand for the cross-correlation interferograms. The left one is generated by the reference pulses and LO pulses, and the right one is generated by the measurement pulses and the LO pulses. Figure 2(b) shows an expanded view of the right peak. The simulated cross-correlation interferograms are very similar to that of the real experimental system shown in Figs. 1(a) and 1(b).

Consider the timing jitter effect of real experimental system, we set a random noise within [−1.75, 1.75] ps on the repetition period *T* of the signal laser in the simulation. By FFT of the simulated interferograms, we can get the path length difference (*x*
_{2}−*x*
_{1}), i.e. target distance *D*. The difference between the preset distance and the result obtained by FFT method can be used to represent the ranging accuracy (*δ*
_{D}) in simulation.

#### 3.1 Impacts of repetition rate difference

In the dual-comb system, when the repetition rate of one laser (*f*
_{r}) is fixed, the repetition rate difference (Δ*f*
_{r}) decides the equivalent sampling rate (*f*
_{s} = *f*
_{r}
^{2}/Δ*f*
_{r}) in optical frequency domain. According to Eq. (1), the relative uncertainty of ranging result *U*
_{D}/*D* can be given by

*f*

_{r}is helpful to decrease the ranging uncertainty caused by the fluctuation of

*f*

_{r}. On the other hand, a larger Δ

*f*

_{r}leads to a lower equivalent sampling rate, which is not good for recover signals in sampling. Additionally, too low sampling rate will increase the uncertainty of time measurement in FFT [35]. Therefore, optimization of Δ

*f*

_{r}is essential in the dual-comb system.

In our previous study, we have demonstrated experimental optimization of the repetition rate difference in a dual-comb ranging system [35]. In the present simulation, we set the system parameters the same as in the experimental system. When we tune the repetition rate difference from 100 Hz to more than 6000 Hz, there is an optimal range around 3400 Hz for distance measurement [Fig. 3(a) ]. The simulation results agree with the experimental results reported in [35] very well. The wide band of simulation results is resulted from the random noise setting of the repetition rate.

Similar to the experimental optimization, we also tuned the repetition rate difference finely from 2440 to 2480 Hz at an increment of 0.1 Hz. In this short range of tuning, the ranging accuracy varies over several cycles sharply [Fig. 3(b)]. This phenomenon also agrees with the experimental results exactly [35]. In the present system, the equivalent sampling rate *f*
_{s} is about 1.3 THz, but the frequency spectrum of a cross-correlation interferogram between two ultra-short laser pulses corresponds to the optical frequency (192 THz). This is why we use a band-pass-filter to limit the bandwidth of interferograms to be smaller than *f*
_{s}/2, so that it is still possible to reconstruct the signals under a specific condition according to the bandpass sampling theory. The specific condition is that a proper *f*
_{s} should be selected to make the spectrum of the sampled interferogram locate in the baseband (from 0 to *f*
_{s}/2) without overlapping with 0 Hz or *f*
_{s}/2 to avoid phase distortion [35]. Only a series of discrete range of *f*
_{s} as well as Δ*f*
_{r} can fulfill this requirement. This is the reason that the ranging accuracy varies sharply and periodically while tuning Δ*f*
_{r} finely. Note that in the simulation of wide tuning of Δ*f*
_{r} [Fig. 3(a)], it is necessary to set Δ*f*
_{r} to fulfill the requirements of bandpass sampling. The high consistency between the simulation and experimental results indicates the theoretical model and the simulation method developed in the present study are effective and reliable.

#### 3.2 Impacts of repetition rate

The equivalent sampling rate of a dual-comb system is determined by *f*
_{r} and Δ*f*
_{r}. Therefore, it is also attractive to investigate the impacts of *f*
_{r} on the ranging performance. In the simulation, we set *f*
_{r} at 50 MHz, 100 MHz, 150 MHz and 200 MHz successively, and investigated the relationship between ranging accuracy and the repetition rate for each case (Fig. 4
). All the other parameters are the same as those described above. For each repetition rate, there is an optimal range of the repetition rate difference. With the increase of repetition rate, the repetition rate difference in the optimal range also increases and the corresponding ranging accuracy gets better. When the repetition rate is increased from 50 MHz to 200 MHz, the optimal repetition rate difference is changed from 3 kHz to 40 kHz, and the corresponding ranging accuracy is improved by two orders of magnitude. Therefore, a higher repetition rate is very favorable to improve the ranging accuracy of a dual-comb system. In the practical dual-comb system, the upper bound of repetition rate can be limited by the laser itself or the sampling and data processing electronics.

The data shown in Fig. 4 for the four repetition rates were obtained with the same optical bandwidth (Δ*υ*
_{comb} = 200 GHz) of the interferograms. It is necessary to point out that the optimal range of repetition rate difference is also related to the optical bandwidth of the interferograms. For example, if we change the optical bandwidth from 200 GHz to 400 GHz (corresponding 1.5 nm to 3 nm) in case of 100 MHz repetition rate, the upper bound of Δ*f*
_{r} (*f*
_{r}
^{2}/2Δ*υ*
_{comb}) limited by the Nyquist condition changes from 25 kHz to 12.5 kHz. The center of optimal range of repetition rate difference shifts from 10 kHz to 5.5 kHz accordingly (Fig. 5
). This result indicates that it is reasonable to select a repetition rate difference of 5 kHz for a dual-comb ranging system with a repetiton rate of 100 MHz and a bandwidth of 3 nm [22].

#### 3.3 Impacts of carrier-envelope-offset frequency

In most dual-comb ranging systems, *f*
_{ceo} is free running to reduce the system complexity [24–26
]. The investigators used Hilbert transform [24] or nonlinear crystal [26] to remove the carrier and got the envelope of the interferogram. However, whether the drift of *f*
_{ceo} will still impact the ranging accuracy has not been investigated yet. Here, we reveal this impact by simulation using the same parameters described in Section 3.1 (except Δ*f*
_{r} is fixed at 2500 Hz). In the simulation, we set *f*
_{ceo} of the LO laser to zero for simplicity and tuned *f*
_{ceo} of the signal laser by 112 MHz (nearly twice of *f*
_{rep}). The simulation results (Fig. 6
) show that the ranging accuracy varies four cycles sharply. There are several discrete optimal ranges of *f*
_{ceo} for the dual-comb system, which is similar to that obtained by tuning Δ*f*
_{r}. In order to investigate whether removing the carrier of interferogram can remove the impacts of *f*
_{ceo} on ranging accuracy, we also calculated the ranging results by using the Hilbert transform method described in Section 2.1. Nevertheless, the results obtained by FFT method and Hilbert transform method do not show significant differences (Fig. 6). Therefore, it is necessary to select and maintain *f*
_{ceo} in these discrete ranges in dual-comb system even only considering the envelope of the interferogram in data processing.

## 4. Experimental verification

In order to verify the simulated ranging results of tuning *f*
_{ceo}, we built an experimental setup. The schematic of this setup is shown is Fig. 1. The parameters of the experimental system are the same as in simulation, i.e., *f*
_{r} = 56.17 MHz, Δ*f*
_{r} = 2500 Hz, the center wavelengths of both lasers are 1560 nm, the bandwidth of the band-pass-filter is 200 GHz, and the target distance is 0.5 m. The repetition rates of the signal and the LO lasers were both stabilized referencing to a rubidium atomic clock (SIM940, Stanford Research Systems). To extract *f*
_{ceo} signals, we built *f*-to-*2f* interferometers [36] for both lasers. During the experiment, *f*
_{ceo} of the LO laser was stabilized while *f*
_{ceo} of the signal laser was changed by 103 MHz in total by tuning its pump current. Here, we use the variation of measured distance and the corresponding standard deviation to indicate the ranging accuracy. Both of them vary over several cycles sharply during the *f*
_{ceo} tuning (Fig. 7
). In each cycle, there is an optimal range of *f*
_{ceo}. The changing period is nearly 28 MHz, which is highly consistent with the simulated results (Fig. 6). Such good accordance between simulation and experimental results shows our simulation method is also applicable for optimizing *f*
_{ceo} in the dual-comb ranging system.

To explain the impacts of *f*
_{ceo} on the ranging accuracy, it is necessary to analyze the spectrum of the interferograms generated by the signal and LO lasers. In optical frequency domain, the *a*
^{th} to (*a* + *k*)^{th} modes of the LO laser and the *b*
^{th} to (*b* + *k*)^{th} modes of the signal laser were filtered out by the bandpass filter [Fig. 8(a)
]. The beat frequency between the *a*
^{th} mode of the LO laser and the *b*
^{th} mode of the signal laser can be expressed as

*f*

_{ceo_S}and

*f*

_{ceo_L}are the carrier-envelope-offset frequencies of the signal and LO lasers, respectively. When the interferograms are received by the photodetector (PD), the RF spectrum of the PD output contains multi-heterodyne beats including a set of frequencies:

*f*

_{set}= {

*f*

_{b},

*f*

_{b}+ Δ

*f*

_{r},

*f*

_{b}+ 2Δ

*f*

_{r}, …,

*f*

_{b}+

*k*Δ

*f*

_{r}} [Fig. 8(b)]. The output of PD was digitized synchronously with the LO pulses and low pass filtered at 28 MHz (

*f*

_{r}/2). To meet Nyquist sampling condition, the bandwidth of

*f*

_{set}, i.e.,

*k*Δ

*f*

_{r}should be smaller than

*f*

_{r}/2. Therefore, the bandwidth of the bandpass filter used for the interferogram (Δ

*υ*

_{comb}=

*kf*

_{r}) should be smaller than

*f*

_{r}

^{2}/(2Δ

*f*

_{r}). This conclusion is the same as that obtained from the optical frequency domain analysis [35].

To avoid aliasing in sampling, *f*
_{set} should locate at the Nyquist band (0, *f*
_{r}/2) without overlapping with 0Hz or *f*
_{r}/2. Unfortunately, when *f*
_{ceo_S} or *f*
_{ceo_L} is changing, *f*
_{b} as well as *f*
_{set} is shifting. It will cause phase distortion in sampling periodically, and the changing period is exactly *f*
_{r}/2. Therefore, in the dual-comb distance measurement, even only use the time-of-flight method based on the envelope of the interferogram, aliasing problem in sampling cannot be avoided if *f*
_{ceo} is shifting. It is still necessary to select and maintain *f*
_{ceo} in discrete optimal ranges during measurement. On the other hand, when *f*
_{ceo_S} and *f*
_{ceo_L} are stabilized, changing Δ*f*
_{r} will also shift *f*
_{b} and *f*
_{set}. This will also cause phase distortion periodically with a period of *f*
_{r}/*b* (8.2 Hz for the present system). This conclusion is in accord with the simulation results shown in Fig. 3(b) and the experimental results [35]. According to Eq. (7), it is also easy to derive that tuning Δ*f*
_{r} or tuning *f*
_{ceo} can compensate each other to avoid aliasing in sampling.

## 5. Conclusion

We have presented a theoretical model and a numerical simulation method for the dual-comb ranging system. With this simulation method, we firstly investigated the impacts of repetition rate difference on the dual-comb ranging accuracy. The simulation results suggest a series of discrete zones of repetition rate difference in an optimal range for the dual-comb system, which is consistent with the conclusion drawn in experiments. Secondly, the simulation results of the repetition rate indicate that a higher repetition rate is very favorable to improve the ranging accuracy of a dual-comb system. Finally, we investigated the impacts of carrier-envelope-offset frequency on the dual-comb ranging accuracy. The simulation results suggest a series of discrete optimal ranges of the carrier-envelope-offset frequency for the dual-comb system. The simulation results were confirmed by our experiments. The analysis of frequency domain reveals that shifting carrier-envelope-offset frequency causes aliasing in sampling periodically, no matter the carrier of the interferogram is removed or not in data processing. Therefore, it is also essential to select carrier-envelope-offset frequency in dual-comb system. The conclusions of parameter optimization presented here are also potentially useful for the dual-comb spectroscopy.

## Acknowledgments

This work was supported by National Natural Science Foundations of China (61575105 and 61377103), the Beijing Higher Education Young Elite Teacher Project (YETP0085) and the Special-funded Program on National Key Scientific Instruments and Equipment Development of China (2011YQ120022).

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