## Abstract

We propose an interferometric method that enables to measure a distance by the intensity measurement using the scanning of the interferometer reference arm and the recording of the interference fringes including the brightest fringe. With the consideration of the dispersion and absorption
of the pulse laser in a dispersive and absorptive medium, we investigate the cross-correlation function between two femtosecond laser pulses in the time domain. We also introduce the measurement principle. We study the relationship between the position of the brightest fringe and the
distance measured, which can contribute to the distance measurement. In the experiments, we measure distances using the method of the intensity detection while the reference arm of Michelson interferometer is scanned and the fringes including the brightest fringe is recorded. Firstly we
measure a distance in a range of 10 µm. The experimental results show that the maximum deviation is 45 nm with the method of light intensity detection. Secondly, an interference system using three Michelson interferometers is developed, which combines the methods of light intensity
detection and time-of-flight. This system can extend the non-ambiguity range of the method of light intensity detection. We can determine a distance uniquely with a larger non-ambiguity range. It is shown that this method and system can realize absolute distance measurement, and the
measurement range is a few micrometers in the vicinity of N*l*_{pp}, where N is an integer, and *l _{pp}* is the pulse-to-pulse length.

© 2014 Optical Society of America

## 1. Introduction

Precise ranging system is of key importance in fields like multiple satellites flying formation, future space-based sciences, planets spatial positioning, and large scale manufacture [1–8]. The invention of
the optical frequency comb has led to the significant development of the applications mentioned above. The optical frequency comb is a consecutive pulse train in the time domain and has a frequency spectrum that consists of discrete, regularly modes in the frequency domain known as a comb.
In this case the repetition frequency and the carrier-envelope-offset frequency can be locked to a frequency standard, like Rb clock (10^{−12}) or Cs clock (10^{−15}), hence the optical frequency comb has the frequency stability and accuracy same as the
frequency standard [9]. Due to the super-stability and high accuracy of the frequency, the optical frequency comb can be used as an optical source in a distance measurement system with nm or even pm precision [10].

Over past decade scientists have developed various methods for absolute distance measurement using optical frequency comb in both time domain and frequency domain. In 2000, Minoshima and Matsumoto proposed a high-accuracy optical distance meter with a mode-locked femtosecond laser, the
8-ppm accuracy in a distance of 240 m was obtained by use of a series of beat frequencies [11]. In 2004, Ye reported a scheme taking advantages of both the incoherent method and the coherent method, and the distance measured can be up to 10^{6}
m with an accuracy of subwavelength theoretically [12]. In 2006, Joo and Kim described a way of dispersive interferometry, which can lead a resolution of 7 nm over 0.89 m, and the maximum distance measured can reach 0.89 m [13]. In 2006, N. Schuhler and Y. Salvadé proposed a multiple-wavelength interferometer stabilized by the optical comb. This method can expand the non-ambiguity range and obtain an accuracy better than 0.2 part in 10^{6} m [14]. In 2011, W. Dong proposed a multiple pulse train interference-based time-of-flight method. The highlight is the fraction part of *l*_{pp} (*l*_{pp} is the pulse-to-pulse length) can be directly measured as the distance between
temporal coherence peaks of the fringes, and the accuracy is 1 µm over 1.5 m [15]. In 2011, X. Wang developed a heterodyne interference system for position measurement with optical frequency comb, and the absolute distance up to 22.478 m can be
measured by fringe scanning and frequency-shifting [16]. In 2012, P. Balling described a method of the stationary phase evaluation which enables 10 nm uncertainties for distance measurement, and their system can be used as a kind of interferometric
multimeter for measurement of long distance, air absorption, e.g [17]. In 2012, S. A. van den Berg demonstrated a technique for absolute distance measurement with an optical frequency comb based on unraveling the output of an interferometer to distinct
comb modes with 1 GHz spacing, and this technique combined the spectral and homodyne interferometry, with a high accuracy far with an optical fringe and a large non-ambiguity range [18].

Over recent years, large scale distance measurement using optical frequency comb has aroused the interest of researchers. In 2009, I. Coddington proposed a coherent laser ranging system combining the advantages of time-of-flight and interferometric approaches with two coherent broadband fibre-laser frequency comb sources, and the precision is better than 5 nm at 60 ms in a distance up to 1.14 km [19]. In 2010, J. Lee improved the time-of-flight precision to the nanometer regime with femtosecond light pulses, and the experiment results show an Allan deviation of 117 nm in measuring a 0.7 km distance in air [20]. Long range highly accurate distance measurement will play an important role in future space-based sciences, planets spatial positioning, and satellites flying formation. Precise distance measurement at long range is the future development direction of the technique of absolute distance measurement.

In this paper, we measure a distance by using intensity detection and recording the interference fringes. This paper is organised as follows. In Section II, we analyze the temporal coherence function of the pulses in the dispersive and absorptive medium. In Section III, we demonstrate the
principle of absolute distance measurement. Here two key parameters are determined by the intensity evaluation based on detection of the interference fringes. In Section IV, we introduce the influence of the carrier phase slip rate Δ*φ _{ce}* on the
position of the brightest fringe of the correlation patterns. This is important for time-of-flight method and heterodyne interference method. In Section V, we measure a distance in a range of 10 µm based on a Michelson interferometer. The experimental results show that the maximum
deviation is 45 nm with a small non-ambiguity range using the model of asymmetric sech

^{2}pulse. To extend the range of non-ambiguity, we design an interference system combining the methods of light intensity detection and time-of-flight, which is composed of three Michelson interferometers in Section VI, and the system is not complex. Finally, the main conclusions and future plan of this work are summarized in Section VII.

## 2. Analysis of pulse temporal coherence function

In this section, we focus on a model of the temporal coherence function of the femtosecond pulses emitted from the optical frequency comb. In 2009, W. Dong deduced the function in the frequency domain roughly, with no consideration of the dispersion and absorption of the light wave when propagating in dispersive and absorptive medium like air [21]. In 2010, M. G. Zeitouny proposed a pulse cross-correlation function mostly based on Fourier transform using Parseval’s formula in non-absorptive medium [22]. However, both the dispersion and the absorption should be taken into account to create a model of the temporal coherence function.

First, we give a description of a distance metrology setup using a Michelson interferometer. The basic elements of this scheme are shown in Fig. 1. The pulse train from optical frequency comb is split into two identical parts at the beam splitter BS. One part of the pulse train goes into the reference arm and is reflected by the reference mirror
M_{R}. The other part of the pulse train goes into the measurement arm and is reflected by the target mirror M_{T}. These two parts of the pulse train are finally recombined at the beam splitter BS. The reference arm is scanned over a fixed range using a piezoelectric
transducer, while the measurement arm is displaced over a distance to be determined. In this work, the distance determined is a tiny displacement in a range of 10 µm, another piezoelectric transducer is needed as a length standard. When the two parts of the pulse train overlap in
space, the interference fringes can be observed on the oscilloscope OS by scanning the reference arm.

In vacuum, the speed of light was defined to be exactly equal to c = 299792458 m/s since 1983. In dispersive and absorptive medium, the speed of light is c_{n} = c/n = c/(n_{R} + *i*n_{I}), where n, n_{R} and n_{I} denote the complex
refractive index, the real part and imaginary part, respectively. n_{R} is the refractive index, and n_{I} characterizes the wave absorption when traveling through the medium. For the case of temporal coherence function of the light pulses, all the proposed models were
created with neglection of absorption. However, the limited power of the laser gets lower and lower when propagating in absorptive medium like air because of the absorption, that is the reason why the distance measured cannot be infinity, thus it is necessary to consider the absorption of
the medium. In this section, we develop a model with the consideration of both the dispersion and absorption. The Ciddor formula [23] is applied to correct the real part n_{R} of complex refractive index to make the model more comprehensive.

The two key parameters of optical frequency comb are the repetition frequency *f*_{rep} and the carrier-envelope-offset frequency *f*_{ceo}. *ω*_{m} = m*ω*_{rep} +
*ω*_{ceo} = 2π (m*f*_{rep} + *f*_{ceo}). m is a positive integer. The spectrum of the optical frequency comb consists of hundreds of thousands of discrete and single lines with the equal space of
*ω*_{rep}. The shorter the pulse is, the wider the spectrum is. A pulse train from optical frequency comb can be expressed as:

_{train}(t,z) is the electric field of the pulse train in the time domain, ∑E

_{z1,m}(t,z

_{1})exp[-

*i*(

*ω*

_{m}

*t*-

*k*

_{m}z

_{1}) +

*i*(

*φ*

_{0}+ Δ

*φ*

_{ce}

*t*)] is the field of the pulse, propagating in the direction of positive z, at z = z

_{1},

*E*

_{z1,m}(

*t*,z

_{1}) is a real amplitude,

*k*

_{m}is the propagation vector of the pulse,

*k*

_{m}= 2π/

*λ*

_{m}=

*ω*

_{m}/

*c*

_{m}=

*n*

_{m}

*ω*

_{m}/

*c*= (

*n*

_{Rm}+

*in*

_{Im})

*ω*

_{m}/

*c*.

*λ*

_{m}is the wavelength corresponding to n

_{R}(

*ω*

_{m}).

*φ*

_{0}is an initial phase of the carrier pulse. Δ

*φ*

_{ce}is carrier phase slip rate because of the difference between the group and phase velocities.

*h*is an integer. It is significant that

*l*

_{pp}is a function of the wavelength

*λ*

_{m}because the wave velocities are different corresponding to the different wavelengths. Here the pulse-to-pulse length

*l*

_{pp}is defined as

*l*

_{pp}(

*λ*

_{m}) =

*c*/(

*n*) =

_{g}f_{rep}*cT*/

_{rep}*n*where

_{g}*n*is the group refractive index. ${n}_{g}={n}_{m}-{\lambda}_{c}\frac{\partial {n}_{m}}{\partial {\lambda}_{m}}$. It is shown that

_{g}*n*is a function of the wavelength

_{g}*λ*

_{m}.

*λ*

_{c}is the center wavelength [3].

*T*

_{rep}is the time interval between the pulses.

*f*Δ

_{ceo}=*φ*

_{ce}f_{rep}/2π.In the Michelson interferometer, as shown in Fig. 1, the part of the pulse which goes into the measurement arm and is reflected by the target mirror can be expressed as:

*L*is the distance to be determined.

The part of the pulse which goes into the reference arm and is reflected by the reference mirror can be expressed as:

*floor*(2

*L*/

*l*

_{pp}),

*floor*rounds the element of 2

*L*/

*l*

_{pp}to the nearest integer less than or equal to 2

*L*/

*l*

_{pp}.

When the measurement part and the reference part overlap in space, the total field at BS is

A photodetector with a responding period *T*_{d} is used to detect the light intensity, and the intensity can be expressed as:

*g*is a positive integer,

*g*=

*floor*(

*T*

_{d}/

*T*

_{rep}).

Equation (5) can be calculated as:

Since the light from the source is a pulse train with a distance interval of *l*_{pp}, Eq. (6) can be rewritten as

_{I}. The AC part is simply an oscillation cosine function, and the envelop is determine by the factor ${P}_{m}\mathrm{exp}(-\frac{2{n}_{I}({\omega}_{m}){\omega}_{m}L}{c})$, ${P}_{m}=\frac{1}{{T}_{d}}{\displaystyle \underset{{T}_{d}}{\int}{E}_{0,m}{}^{2}(t,0)dt}$. Here,

*P*

_{m}is defined as the power spectral density. This shows that the temporal coherence function requires the knowledge of the optical source spectrum, and the absorption cannot be neglected. The AC part (arbitrary unit) of the waveform observed on the oscilloscope is determined by n

_{R}(

*ω*

_{m}),

*ω*

_{m},

*L*,

*N*, and Δ

*φ*

_{ce}, and is important for the distance measurement method proposed in this work, which we will discuss in next section. In our experiment,

*f*

_{rep}= 199.817 MHz, the center wavelength is 1548.2 nm, and the spectrum bandwidth is 58.8 nm.

## 3. Distance measurement principle and simulations

In the present section, we discuss a method to determine the distance by the intensity measurement. Since the shape of the pulse emitted from the optical frequency comb is not ideal and straightforward like Gaussian pulse and sech^{2} pulse, we compare the simulated interference
fringes based on the Gaussian pulse model, the sech^{2} pulse model, the asymmetric Gaussian pulse model, and the asymmetric sech^{2} pulse model, respectively. The comparison results can provide a reference to the distance measurement.

A distance measured can be calculated as:

where*N*is a positive integer, and

*d*is a small length, $d=2L-N\times {l}_{pp}$. We will determine the two parameters N and d to measure the distance.

#### 3.1 Determination of N and simulations

Traditionally, N can be determined roughly by the incoherence time-of-flight method [12], which is limited in measurement precision and resolution by the electronic instruments. Hence over the years researchers proposed combined approaches [12,19,24], however the system becomes complex when multiple detection instruments are involved. Actually when two pulses with different phases overlap completely in space, which means the envelops of the two pulses coincide completely, N can be measured through the light intensity generated by the two pulses directly due to the stable pulse-to-pulse phase relationship of the light from the optical frequency comb.

First let us consider the AC component of Eq. (7) with a given distance L. Assuming that the two pulses overlap completely (d = 0) in space, then the AC component of Eq. (7) can be rewritten as:

We have done simulations to verify the relation between the intensity and the parameter N without scanning of the reference arm. In the simulations, the center wavelength of pulse laser is 1550 nm, *f*_{rep} = 200 MHz, *f*_{ceo} = 2 MHz, d =
0, and the intensity is the average value. Figure 2 shows the
simulated results in a period. We can observe that the curve is a standard cosine function, and the intensity varies obviously when N is adjusted. It is necessary to indicate that the results shown in Fig. 2 are only theoretical values with no
consideration of the environment conditions.

When the pulse laser propagates in extremely absorptive medium, we can focus on the DC component of Eq. (7) in this case, and the DC part of the Eq. (7) can be rewritten roughly as:

*ω*

_{c}is the center frequency of the optical frequency comb. We can find that the intensity decreases when N increases, and the decreasing trend is square of a negative exponential function. The method requires the precision measurement of n

_{R}and n

_{I}and a very stable environment conditions. This can be a subject for future research.

#### 3.2 Determination of d and simulations

The accuracy of this method depends on the measurement of d. d is a small length in a range of tens of micrometers. The range is determined by the width of the pulse from the light source. We can observe the interference fringe when d is scanned continuously. Actually the AC part of Eq. (7) denotes the interference fringe theoretically. From Eq. (7), we can find that the interference fringe is an attenuate cosine function of d for a given parameter N. The angular frequency is about
2n_{R}(*ω*_{c})*ω*_{c}/*c*, and the non-ambiguity range of d is πc/(2n_{R}(*ω*_{c})*ω*_{c}). *ω*_{c}
is the center angular frequency of the pulse laser.

The shaping of the femtosecond pulse in the time domain is a subject which researchers have studied extensively for a long time [25–27]. In this section, we use the classical models, including Gaussian
pulse model, sech^{2} pulse model, the asymmetric Gaussian pulse model, and the asymmetric sech^{2} pulse model, to analysis the interference fringes theoretically, which can bridge the distance and the intensity smoothly.

The Gaussian pulse can be expressed as:

The sech^{2} pulse can be expressed as:

The asymmetric Gaussian pulse can be expressed as:

The asymmetric sech^{2} pulse can be expressed as:

*A*

_{1},

*A*

_{2},

*A*

_{3},

*A*

_{4}are the electric field amplitudes, and

*a*

_{1},

*a*

_{2},

*a*

_{3},

*a*

_{4},

*a*

_{5},

*a*

_{6}are the attenuation factors, respectively. When

*a*

_{3}>

*a*

_{4}, the pulse is left asymmetric;

*a*

_{3}<

*a*

_{4}, the pulse is right asymmetric. When

*a*

_{5}>

*a*

_{6}, the pulse is left asymmetric;

*a*

_{5}<

*a*

_{6}, the pulse is right asymmetric. Figure 3 shows the comparison between shapes of different pulses. The pulse width Δ

*t*and the spectral bandwidth Δ

*ν*maintain the relationship of Δ

*t*·Δ

*ν*= K where the constant K equals 0.32 and 0.44 for sech

^{2}and Gaussian pulses [28], respectively. The spectrum bandwidth is 58.8 nm, and the pulse width in our experiment is about 50 fs.

Based on Eqs. (11), (12), (13) and (14), we have done simulations of the interference fringes of
different pulses. In the simulations, the center wavelength of pulse laser is 1550 nm, *f*_{rep} = 200 MHz, *f*_{ceo} = 2 MHz, the scanning step size of d is 100 nm, and the scanning range is 100 µm. Figure 4 shows the interference fringes based on different pulse models. In the cases it is straightforward that all the fringes are symmetric. The Gaussian fringe and asymmetric Gaussian fringe attenuate a
little faster than the sech^{2} fringe and the asymmetric sech^{2} fringe, while all the pulse widths in the simulations are about 50 fs. While propagating in dispersive medium, the pulse laser suffers from the shape broadening and deformation in the long distance
measurement, known as the dispersion and the chirp. For the case of short distance, or even a tiny displacement measurement, the dispersion is not very obvious, and can be neglected. In this work, the fringes in Fig. 4 are generated based on an
interferometer at equal arms, which means the distance is not very long, and the patterns are simulated with no consideration of the dispersion. We will use the intensity of these fringes to measure distances in Sec. V.

## 4. Analysis of the position of the brightest fringe

M. G. Zeitouny reported in 2010 that the position of the brightest fringe is influenced by the dispersion, temperature, pressure, and humidity [22]. In this section, we derive an intuitive expression without dispersion to denote the relation between the shift of the position of the brightest fringe and the distance itself. Our analysis starts from Eq. (7). We can get that the brightest fringe emerges when the phase of cosine function equals to zero, that is

where N is a positive integer which is determined by the distance L. From Eq. (15), we can derive roughly:We set the point when N = 0 (d = 0) as the reference. Then the position shift of the brightest fringe can be expressed as:

Equation (17) shows that the position shift of the brightest fringe Δd increases linearly when N increases. We simulate the relation between Δd and N. The simulations are performed under a fixed environmental condition (20.5
°C, 1026 hPa, 20% humidity). The refractive index of air is 0.9982071 according to the Ciddor formula. The center wavelength of pulse laser is 1550 nm, *f*_{rep} = 200 MHz, *f*_{ceo} = 2 MHz, the scanning step size of d is 10 nm, and the
scanning range is 6 µm. The pulse-to-pulse length *l*_{pp} is 1.5016546 m. Figure 5 shows the position of the brightest fringe corresponding to
different N, and the shift displacement is shown in the figures significantly. The increasing step of the shifted displacement of the brightest fringe can be calculated as:

*λ*

_{c}is the center wavelength.

As shown in Fig. 5, the shifted displacements are 0, −0.01, −0.08, −0.16, −0.24, −0.32, −0.39, 0.32, 0.24, 0.16, 0.08, 0 µm corresponding to N = 0, 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 respectively. Due to the limitation of the resolution of the software, the simulation increasing step of the shifted displacement of the brightest fringe is not equal to 0.00776 µm, but about 0.008 µm. Figure 6 shows the relation between the shifted displacement and N, which is linear in two separate periods exactly. The period is 100. The positive displacement denotes that the position of the brightest fringe is shifted to the right side, and the negative displacement represents that the position is moved to the left side.

In this section we have studied the position of the brightest fringe without consideration of dispersion. We focus on the relationship between the shifted displacement of the brightest fringe and the pulse-to-pulse phase relation of the optical frequency comb essentially.

## 5. Experiment and deviation analysis

In this section, we do experiments to use the method mentioned in Sec. III to measure a distance. In our experiments, the light source is a mode-locked femtosecond pulse laser. We do experiments based on a Michelson interferometer at equal arms, as shown in Fig. 1, so the reference part and the measurement part at the BS are split from one pulse emitted from the light source. That means two things, one is the parameter N equals to zero, and the distance measured is in a range of tens of micrometers corresponding to the width of the pulse, the other is we do not need to consider the pulse-to-pulse phase relation of the femtosecond pulse laser which should be taken into account in long distance measurement.

In the experiments, the light source is a mode-locked femtosecond pulse laser designed by National Institute of Metrology, the repetition frequency is 199.817 MHz, the center wavelength is 1548.2 nm, the spectrum width is 58.8 nm, the oscilloscope is LeCroy waverunner 104Xi, the photodetector is Thorlabs PDB150 Balanced Amplified Photodetector, and the PZT nano-positioning platform is PI P-621.1. The environmental conditions are 20.5 °C, 1026 hPa, 20% humidity.

Figure 7 shows the interference fringe based on a Michelson interferometer at equal arms. The blue line (CH3) denotes the interference fringe, and the red line (CH2) is the PZT driving signal. The scanning range of the PZT nano-positioning platform is 55 µm, the scanning period is 7.82 s, and the center wavelength of the light source is 1548.2 nm. The frequency of the interference fringe can be calculated as:

where D is the scanning range of PZT nano-positioning platform, T is the scanning period,*λ*

_{c}is the center wavelength. As shown in Fig. 7, we can observe that there are three peaks in the fringe envelop. The reason is the pulses emitted from the light source are not ideal. The pulse itself has more than one peak in the time domain, and multiple peaks emerge when two pulses overlap in space. In this work, we focus on the intensity around the position of the brightest fringe.

We take the displacement of the PZT nano-positioning platform with a precision of 1 nm as a reference to measure a distance using the method of intensity detection. The maximum intensity of the brightest fringe shown in Fig. 7 is recorded as 1 (arbitrary unit), and the position of maximum intensity is regarded as the reference location, which means d = 0. We shift the nano-positioning platform by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 µm, respectively, and record the intensity correspondingly. In order to reduce the random error we measure each distance for 10 times, and take the average intensity to determine the distance. Figure 8 shows the intensity corresponding to 1, 2, 3, 4, 5 µm, respectively. Figure 9 shows the intensity corresponding to 6, 7, 8, 9, 10 µm, respectively.

We observe that there is slight difference between each measured intensity for each distance measured, as shown in Fig. 8 and Fig. 9, and the random error can be reduced effectively. The average intensities corresponding to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 µm are 0.05611, −0.8887, 0.32954, 0.85437, −0.34645, −0.5822, 0.63823, 0.4452, −0.42319, −0.11197. Figure 10 shows the positions of different intensities corresponding to different distances.

We measure the distances based on the Gaussian pulse model, sech^{2} pulse model, asymmetric Gaussian pulse model and asymmetric sech^{2} pulse model, respectively. We pick up the distances corresponding to the intensities of 0.05611, −0.8887, 0.32954, 0.85437,
−0.34645, −0.5822, 0.63823, 0.4452, −0.42319, −0.11197, respectively, and these distances are the measurement results based on different pulse models introduced in Sec. III. Table 1 shows the measurement results. Figure 11
shows the deviations of different pulse models.

As shown in Table 1 and Fig. 11, the difference of the measurement results based on different models is small. The maximum difference is 102 nm between asymmetric sech^{2} model and sech^{2}
model at 9 µm. We arrange the maximum difference of different models in Table 2. The maximum differences are all at the position of 9 µm, and the deviation of the asymmetric sech^{2} model is the smallest. The reasons of introducing deviations include the instability of the pulses, the variation of the environment
conditions, the resolution of the electric instruments, the vibration of the precision optical platform, the difference between the numerical models and the real one, and the resolution of the simulation software. We consider that the mean reason is the difference between the numerical
models and the real one. To build a more accurate model can reduce the deviation.

In this section, we have done experiments to realize absolute distance measurement in a range of 10 µm. The maximum deviation is 45 nm with the model of asymmetric sech^{2} pulse model. In ranging and manufacturing applications, there are two critical parameters: precision
and non-ambiguity range. We observe that the non-ambiguity range of this method is just 0.387 µm which can be calculated as λ_{c}/4 = 1548.2/4 = 0.387 μm theoretically. To solve this problem, we design an interference system which will be introduced in next
section.

## 6. A combined interference system using three Michelson interferometers

In this section, we design an interference system to extend the non-ambiguity range. This system combines the time-of-flight method and the intensity detection method, as shown in Fig. 12. The photodetector PD_{1} is Thorlabs PDB150 Balanced Amplified Photodetector, and the photodetector PD_{2} is EOT Amplified InGaAs Detector ET-3000A-FC.
S_{1} and S_{2} are two shutters. L is the distance to be measured. The interference system is a combination of three equal arms Michelson interferometer. As shown in Fig. 12, M_{R}, BS_{1}, BS_{2},
M_{T1} make up one Michelson interferometer named Mi_{α}, M_{R}, BS_{1}, BS_{2}, M_{T2} constitute another Michelson interferometer named Mi_{β}, and the last Michelson interferometer named Mi_{γ} is
composed of M_{T1}, BS_{2}, M_{T2}.

This system works as follows: firstly, S_{1} is open, S_{2} is open. The pulses reflected by M_{T1} and M_{T2} overlap in space, and PD_{2} can be illuminated. We record the intensity displayed on the oscilloscope. This intensity can be used to
measure the distance according to Sec. III and Sec. V. Secondly, S_{1} is open, S_{2} is closed. The pulses reflected by M_{T1} and M_{R} overlap and are detected by PD_{1}. The interference fringe observed on the oscilloscope like Fig. 7 is used to record the relative position of M_{T1}. Thirdly, S_{1} is closed, S_{2} is open. The pulses reflected by M_{T2} and M_{R} overlap, and the interference fringe is used to determine the relative position of
M_{T2}. We set the PZT driving signal as a reference of the position. Figure 13 shows the system photograph. The yellow line (CH1, upper line) indicates the interference fringe detected by PD_{1}, the red line (CH2, middle line) denotes the PZT driving signal, which is stable enough to be a position reference, and the blue line (CH3, lower line) is
the intensity detected by PD_{2}. We use this system to measure a distance, and the distances measured are 5, 10, 15 µm, respectively.

Let us take the distance of 5 µm as an example. Firstly, S_{1} is open, S_{2} is open. The intensity detected by PD_{2} is −0.34645. As shown in Fig. 14, there are several distances corresponding to the intensity of −0.34645, which means we cannot uniquely determine the distance
measured.

Secondly, S_{1} is open, S_{2} is closed. We can record the relative position of M_{T1} based on the interference fringe, as shown in Fig. 15(a) (upper yellow line, fringe generated by Mi_{β}). Thirdly, S_{1} is closed, S_{2} is open. We can determine
the relative position of M_{T2}, as shown in Fig. 15(a) (lower yellow line, fringe generated by Mi_{α}). The PZT driving signal (lower red line) is the position reference. As shown in Fig.
15(a), we can observe that the distance between the positions of the two brightest fringes is about 5 µm. Then we can uniquely pick up the distance around 5 µm in the red box, as shown in Fig. 14, and the measured value is 4.995
µm according to Sec. V. The deviation is −5 nm. We also measure the distances of 10 µm and 15 µm, as shown in Figs. 15(b) and 15(c).

We have described an interference system to uniquely determine a distance using the intensity detection method in this section. The experiment results show that this system can measure a distance with a higher accuracy and a larger range of non-ambiguity. The maximum non-ambiguity range for
the method of intensity detection can be expressed roughly as 2Δ*t·c*/n in air, where Δ*t* is the pulse width and *c* is the light velocity in vacuum. This system can extend the range of non-ambiguity to be largest. We can
find that the non-ambiguity range of the method of intensity detection is small, and this is a big limitation.

Essentially, this system takes the advantages of both the time-of-flight method and the intensity detection method. The three Michelson interferometers cannot work at the same time since the quality of the pulse from the light source is not very excellent. As shown in Fig. 15, there are several peaks in an interference fringe due to the characteristic of the pulse itself. The two shutters can prevent the mutual influence between the light reflected from M_{T1} and M_{T2}, and we can observe the pure interference
fringes. In fact, this system can satisfy the requirement of large scale distance measurement.

## 7. Conclusion and future plan

We propose an intensity detection method, to measure a distance using mode-locked femtosecond pulse laser. With consideration of dispersion, absorption, and environment conditions, we analyze the temporal coherence function of the pulse from the light source, and indicate that the function
requires the knowledge of the optical source spectrum. The absorption of the medium cannot be neglected, and actually it can be a method to determine a distance. The principle of the intensity detection method is analyzed. We can determine the two key parameters N and d by the intensity
measurement. Numerous simulations are developed based on the Gaussian pulse model, the sech^{2} pulse model, the asymmetric Gaussian pulse model, and the asymmetric sech^{2} pulse model. We investigate the relationship between the shifted displacement of the position of the
brightest fringe and the pulse-to-pulse phase relation of the optical frequency comb. We do experiments to verify the method of intensity detection under stable environment conditions. The displacement of piezo position platform has been taken as a distance reference. In a range of 10
µm, we measure each distance for 10 times to reduce the random error. The experimental results show that this method can realize absolute distance measurement. The maximum deviation of different pulse models all emerges at the position of 9 µm, which are 56, 57, 47, and 45 nm
corresponding to Gaussian, sech^{2}, asymmetric Gaussian, and asymmetric sech^{2} model, respectively. We observe that the deviation of the asymmetric models is smaller. There are two critical parameters for ranging system: precision/accuracy and non-ambiguity range. To
expand the non-ambiguity range, we design an interference system exploiting three Michelson interferometers. The working process of the system is introduced, and we measure distances of 5, 10, 15 µm using this system, respectively. The experimental results show that this system can
measure a distance with a higher accuracy and a larger range of non-ambiguity.

In future work, firstly we will measure a large distance in air with the intensity detection method using optical frequency comb. Secondly we will investigate the method to measure an arbitrary distance to expand the applications of the optical frequency comb.

## Acknowledgments

We would like to thank Division of Time and Frequency Metrology, National Institute of Metrology for great support. We gratefully acknowledge J. Ye and S. A. van den berg for their great encouragement. We also thank the peer reviewers for their very helpful comments on this manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 51105274), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120032130002), and the Fund of State key laboratory of precision measuring technology and instruments (Grant No. pil1201).

## References and links

**1. **K. N. Joo, Y. Kim, and S.
W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express **16**(24), 19799–19806 (2008). [CrossRef] [PubMed]

**2. **M. Cui, M.
G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long
distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express **19**(7), 6549–6562 (2011). [CrossRef] [PubMed]

**3. **P. Balling, P. Křen, P. Mašika, and S.
A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express **17**(11), 9300–9313 (2009). [CrossRef] [PubMed]

**4. **Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le
Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. **47**(14), 2715–2720 (2008). [CrossRef] [PubMed]

**5. **M. G. Zeitouny, M. Cui, A. J. E.
M. Janssen, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach,
“Time-frequency distribution of interferograms from a frequency comb in dispersive media,” Opt. Express **19**(4), 3406–3417 (2011). [CrossRef] [PubMed]

**6. **M. Cui, M.
G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H.
P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. **34**(13), 1982–1984 (2009). [CrossRef] [PubMed]

**7. **H. Matsumoto, X. Wang, K. Takamasu, and T. Aoto,
“Absolute measurement of baselines up to 403 m using heterodyne temporal coherence interferometer with optical frequency comb,” Appl. Phys. Express **5**(4), 046601 (2012). [CrossRef]

**8. **S. Hyun, Y. Kim, Y. Kim, J. Jin, and S. Kim,
“Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. **20**(9), 095302 (2009). [CrossRef]

**9. **J. Ye and S. T. Cundiff, *Femtosecond Optical Frequency Comb: Principle, Operation, and Applications* (New York, 2005), pp. 12–23.

**10. **J. Lee, S. Han, K. Lee, E. Bae, S. Kim, S. Lee, S. Kim, and Y. Kim,
“Absolute distance measurement by dual-comb interferometry with adjustable synthetic wavelength,” Meas. Sci. Technol. **24**(4), 045201 (2013). [CrossRef]

**11. **K. Minoshima and H. Matsumoto, “High-accuracy measurement of
240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. **39**(30), 5512–5517 (2000). [CrossRef] [PubMed]

**12. **J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical
fringe,” Opt. Lett. **29**(10), 1153–1155 (2004). [CrossRef] [PubMed]

**13. **K. N. Joo and S. W. Kim, “Absolute distance measurement by
dispersive interferometry using a femtosecond pulse laser,” Opt. Express **14**(13), 5954–5960 (2006). [CrossRef] [PubMed]

**14. **N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth,
“Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. **31**(21), 3101–3103 (2006). [CrossRef] [PubMed]

**15. **D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto,
“Time-of-flight method using multiple pulse train interference as a time recorder,” Opt. Express **19**(6), 4881–4889 (2011). [CrossRef] [PubMed]

**16. **X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto,
“Space position measurement using long-path heterodyne interferometer with optical frequency comb,” Opt. Express **20**(3), 2725–2732 (2012). [CrossRef] [PubMed]

**17. **P. Balling, P. Mašika, P. Křen, and M. Doležal,
“Length and refractive index measurement by Fourier transform interferometry and frequency comb spectroscopy,” Meas. Sci. Technol. **23**(9), 094001 (2012). [CrossRef]

**18. **S. A. van den Berg, S. T. Persijn, G. J.
P. Kok, M. G. Zeitouny, and N. Bhattacharya, “Many-wavelength interferometry with thousands of lasers for absolute
distance measurement,” Phys. Rev. Lett. **108**(18), 183901 (2012). [CrossRef] [PubMed]

**19. **I. Coddington, W.
C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long
range,” Nat. Photonics **3**(6), 351–356 (2009). [CrossRef]

**20. **J. Lee, Y. Kim, K. Lee, S. Lee, and S. Kim,
“Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics **4**(10), 716–720 (2010). [CrossRef]

**21. **D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto,
“Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express **17**(9), 7011–7018 (2009). [CrossRef] [PubMed]

**22. **M.
G. Zeitouny, M. Cui, N. Bhattacharya, H. P. Urbach, S.
A. van den Berg, and A. J. E. M. Janssen, “From a discrete to a continuous model for inter pulse interference with a frequency-comb laser,” Phys. Rev.
A **82**(2), 023808 (2010). [CrossRef]

**23. **P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,”
Appl. Opt. **35**(9), 1566–1573 (1996). [CrossRef] [PubMed]

**24. **K. N. Joo, Y. Kim, and S.
W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express **16**(24), 19799–19806 (2008). [CrossRef] [PubMed]

**25. **A. M. Weiner, D. E. Leaird, J.
S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. **15**(6), 326–328 (1990). [CrossRef] [PubMed]

**26. **S. H. Shim, D. B. Strasfeld, E.
C. Fulmer, and M. T. Zanni, “Femtosecond pulse shaping directly in the mid-IR using acousto-optic modulation,” Opt. Lett. **31**(6), 838–840 (2006). [CrossRef] [PubMed]

**27. **M. Bitter, E.
A. Shapiro, and V. Milner, “Enhancing strong-field-induced molecular vibration with femtosecond pulse shaping,” Phys. Rev. A **86**(4), 043421 (2012). [CrossRef]

**28. **S. Kim and Y. Kim, “Advanced optical metrology using
ultrashort pulse lasers,” Rev. Laser Eng. **36**(suppl), 1254–1257 (2008). [CrossRef]