## Abstract

The temporal coherence function of the femtosecond pulse train from femtosecond optical frequency comb (FOFC) has been studied. The theoretical derivation, which is based on the electric field equations of a pulse train, has been used to model the temporal coherence function of the FOFC and shows good agreement with experimental measurements which are taken with a modified Michelson interferometer. The theoretical and experimental points of view provide useful information for applications of FOFC in imaging and metrology.

© 2009 Optical Society of America

## 1. Introduction

The increased frequency stability and the very broad frequency band of FOFC[1] has led to the application of this device in several precision metrology application areas such as precision optical frequency metrology, high-precision spectroscopy, and distance measurements. In 1996, the time-averaged carrier envelope shift between different pulses of a pulse train was first measured, with the pulse train from an FOFC being introduced into an unbalanced optical-path Michelson interferometer and a delayed one-pulse train relative to the other pulse train [2]. The result provides insight into the high temporal coherence between one pair of pulse trains. In 1998, Chekhovsky et al. suggested a possible scheme for a pulse distance-measuring interferometer [3]. As a pioneering work, Minoshima and Matsumoto reported in 2000 that high temporal coherence of an unbalanced optical-path Michelson interferometer produced an adjacent pair of pulse trains from an FOFC for length measurement [4]. After that, various experiments were proposed using high temporal coherence between a pair of pulse trains for measurement of the group refractive index of air [5], absolute measurement of a long distance [6–8], and precision surface-profile metrology [9]. Interferometric measurements using FOFC are in progress at present, all of which put stringent demands on the high temporal coherence of the FOFC source. To the best of our knowledge, few studies have been carried out regarding the temporal coherence function (TCF) of a pulse train from an FOFC, though such an FOFC is likely to be useful for metrology applications.

In the present study, we investigated the TCF of a pulse train from FOFC. Note that the characteristic of showing high temporal coherence between a pair of pulse trains from an FOFC is essential for the fields mentioned above. We propose a simple analytical model for the TCF of a pulse train from a FOFC and simultaneously demonstrate high temporal coherence between different pairs of pulse trains using a modified Michelson interferometer of optical path differences up to 6m.

## 2. Principles

In this section we focus on a model of the TCF for the emitted pulse train from an FOFC. For convenience of explanation, let us first consider the electric field of a pulse train from an FOFC. A very stable pulse train is generated with the FOFC and is generally expressed as [10]:

where *E*
_{train}(*t*) and *E*̃_{train}(*f*) are the electric fields of a pulse train in the time domain and the frequency domain as shown in Fig. 1(a), (b), respectively, and *A*(*t*) is the pulse envelope, *φ*
_{0} is an arbitrary initial phase of the “carrier” pulse. In the time domain, the “carrier” pulse moves with the (angular) carrier frequency *ω _{c}*. When the electric field packet repeats at the pulse repetition period

*T*, the “carrier” phase slips by Δ

_{R}*φ*

_{ce}to the carrier-envelope phase because of the difference between the group and phase velocities. In the frequency domain, a mode-locked laser generates equidistant frequency comb lines with the pulse repetition frequency

*f*

_{rep}∝1/

*T*, and due to phase slip Δ

_{R}*φ*

_{ce}, the whole equidistant frequency comb is shifted by

*f*

_{CEO}.

In this experiment, we are interested in the TCF of an FOFC. Based on the Wiener-Khintchine theorem, the interferometric signal of the autocorrelation function is given by the inverse Fourier transform of the spectrum of the source *S*(*f*), which is proportional to the squared modulus of the Fourier spectrum ∣*E*̃_{train} (*f*)∣^{2}. The power spectrum of an FOFC light source can be expressed as

and the interferometric signal Γ(*τ*) is given by the inverse Fourier transform of Eq. (3):

$$\phantom{\rule{1.5em}{0ex}}\propto {F}^{-1}\left[{\mid \tilde{A}\left(f-{f}_{c}\right)\mid}^{2}\right]\otimes \sum _{m=-\infty}^{+\infty}\delta \left(\tau -m{T}_{R}\right).$$

The following two expressions were used here to obtain Eq. (4), as expressed above.

We may assume Γ(0) like Eq. (7)

where * denotes a complex conjugate, 〈 〉 denotes time integration over the pulse envelope and carrier period, and we have

From Eq. (8), the TCF of the FOFC periodically displays a high temporal coherence peak where the pulse train signal of the FOFC displays a high-intensity peak as shown in Fig. 1(c).

Let us consider the interference fringes formed by an unbalanced optical-path Michelson interferometer, as shown in Fig. 2. First, an incoming pulse train is split into two identical parts, *E*
_{train 1} (*t*) and *E*
_{train 2} (*t*) at the beam splitter BS, *E*
_{train 2} (*t*) delays relatively to *E*
_{train 1} (*t*) and they finally are recombined at the BS. When the pulse train *E*
_{train 1} (*t*) and the relatively delayed pulse train *E*
_{train 2} (*t*) overlap in space, one would expect that interference fringes can be observed.

The intensity due to the linear superposition of N pulse pairs from *E*
_{train 1} (*t*) and *E*
_{train 2} (*t*) with parallel polarization is just

and

$$\phantom{\rule{2em}{0ex}}\propto \frac{1}{2}{\mid {E}_{\mathrm{train}1}\left(t\right)\mid}^{2}\mid \gamma \left(\tau \right)\mid \mathrm{cos}\left[\mathrm{mod}\left(h\times \Delta {\phi}_{\mathrm{ce}},2\pi \right)\right]$$

where mod(*h* × Δ*φ*
_{ce}, 2*π*) returns *h* × Δ*φ*
_{ce} - *n* × 2*π* where *n* = floor(*h* × Δ*φ*
_{ce}/2*π*) ( floor(*h* × Δ*φ*
_{ce}/2*π*) rounds the elements of *h* × Δ*φ*
_{ce}/2*π* to the nearest integers less than or equal to *h* × Δ*φ*
_{ce}/2*π*), *hT _{R}* is the relative delay between

*E*

_{train 1}(

*t*) and

*E*

_{train 2}(

*t*), and Am is the “carrier” phase slip defined by Eq. (1). After performing the time integration we obtain

In the case of a pulse train from an FOFC, since N is typically a large number, the mode-lock technique results in interference fringes reappearing at delays equal to *hT _{R}* an integer multiple

*h*of the pulse spacing

*T*. And the information about the time-averaged value first provides$\sum _{i=1}^{N}\u3008\mathrm{cos}\left[\mathrm{mod}\left(h\times \Delta {\phi}_{\mathrm{ce}},2\pi \right)\right]\u3009=\mathrm{cos}\left[\mathrm{mod}(h\times \Delta {\phi}_{\mathrm{ce}},2\pi )\right]$by Xu et al [2].

_{R}Next, let us consider how to reconstruct TCF from the obtained interference fringes. Figure 3 (a) shows an example of typical interference fringes *I*(*t*) actually measured experimentally. First, Fourier-transforming Eq. (11) leads to the result of

$$\phantom{\rule{1.5em}{0ex}}\propto F\left[\mid \gamma \left(\tau \right)\mid \mathrm{cos}\left(\mathrm{mod}\left(n\times \Delta {\phi}_{\mathrm{ce}},2\pi \right)\right)\right]$$

$$\phantom{\rule{1.5em}{0ex}}\propto \mid \tilde{\gamma}\left(f\right)\mid \otimes \left[\frac{1}{2}\delta \left(f+{f}_{\mathrm{CEO}}\right)+\frac{1}{2}\delta \left(f-{f}_{\mathrm{CEO}}\right)\right]$$

where *δ*(*f*) is the Dirac delta function and *f* is the frequency. *G*(*f*) is the Fourier transform of *I*(*t*). The interference fringe *I*(*t*) is a real function, so its Fourier spectra become symmetrical at about *f*
_{CEO} = 0 and are separated by the frequency *f*
_{COE}, as shown in Fig. 3(b). The unwanted noise has been filtered out by a band pass filter, and the peak at *f* =+*f*
_{COE} is inverse Fourier-transformed into the time domain, whose result is derived as

Finally, the TCF ∣*γ*(*t*)∣ can be obtained as,

## 3. Experiment

The experimental setup is simple, and its optical schematic is illustrated in Fig. 4. The experiment is carried out with a system consisting of a polarization-mode-locked femtosecond fiber laser (FC1500, MenloSystems), a modified Michelson interferometer, and system control. The pulse duration, repetition rate, and total output power of the fiber laser are 180 femtosecond, 100 MHz, and 20 mW, respectively. The output wavelength of the pulse is centered at 1550 nm with a bandwidth of 20 nm.

The pulse train from the FOFC is expanded and collimated by a collimator C and introduced into a modified Michelson interferometer. The modified Michelson interferometer is a combination of an ordinary Michelson interferometer and two unbalanced optical-path Michelson interferometers as introduced in the principles. The ordinary Michelson interferometer is composed of a beam splitter BS, a reference mirror M_{1}, and an object mirror (half-reflection mirror) HM_{1}. One unbalanced optical-path Michelson interferometer is composed of the common BS and M_{1}, and a different object mirror (half-reflection mirror) HM_{2}. The other unbalanced optical-path Michelson interferometer is composed of the common BS and M_{1}, and a different object mirror M_{2}. The mirrors HM_{2} and M_{2} are arranged at space position far away from HM_{1} about 1.5 m and 3 m in space, respectively.

The pulse train is split into two identical parts at the beam splitter BS. One part of the pulse train goes into the common reference arm of the three interferometers with length LR and is reflected by mirror M_{1}. The other part of the pulse train goes into the other arm with lengths Lo, Lo+cT_{R}+cΔ_{P1}, and Lo+2cT_{R}+cΔ_{P1}+cΔ_{P2} (c is the light velocity in air.) and are sequentially reflected by mirrors HM_{1}, HM_{2}, and M_{2}, respectively. The displacement cΔ_{P1} and cΔ_{P2} are introduced to avoid overlap with each other between interference fringes in space. During the measurement, by moving the common reference arm of the interferometers by means of a computer-controlled and calibrated ultrasonic stepping motor (TULA-OP-03, Technohands, Inc), we could vary the relative delay between the two output pulse trains of the three pairs.

After traveling different path lengths, these three pairs of pulse trains sequentially overlap at the beam splitter. Lens L images the interference fringes onto a photo detector PD (Front-end optical receivers Model 2011, New Focus, Inc.). The intensity of the interference fringes signal through PD is measured with a digital oscilloscope (TDS1000B, Tektronix, Inc.) and is sent to a computer.

In Fig. 5 we show the acquired interference fringes recorded in three different situations. The 2nd and 3rd interference fringe peaks appear when the shutters S_{1} and S_{2}, respectively, are opened. As predicted, the interference fringe signals exhibit a high contrast between the two pairs of pulse trains by the relative displacements cT_{R}+cΔ_{P1} (about 1.5 m) and 2cT_{R}+cΔ_{P1}+cΔ_{P2} (about 3 m). To see more clearly, we analyzed the interference fringe signals in Fig. 5(c) by the Fourier transform method that is described in the principles. Because two half-reflection mirrors were used, the intensities of the interference signals were adjusted, respectively. In consequence, Fig. 6 shows the reconstructed TCF with the relative different delay times.

Theoretically, the three determined ∣*γ*(*τ*)∣ peaks should have the same height value according to Eq. (10). The experimental measurements are achieved by displaying three interference fringe signals to one screen of an oscilloscope in order to shorten the measurement time and suppress the influence due to air turbulence and mechanical vibration. The measurement results primarily suffer from the error arising from peak overlap caused by side lope noise on interference fringes, because three interference fringe signals are not sufficiently separated due to the restricted resolution of the oscilloscope. However, it should be stressed that these limitations are not related to the principle itself. The performance can be improved by the optical component with high accuracy and the data-acquisition equipment with high resolution.

## 4. Summary and future work

In closing, we have studied the TCF of FOFC. The results show that high temporal coherence peaks exist during the period equal to the repetitions interval in the traveling direction of the used FOFC. The theoretical derivation of the expected temporal coherence function and the interference fringes agrees with the results of a simple proof-of-the-principle experiment using a modified Michelson interferometer with optical path differences up to 6m. To our knowledge, this is the first report regarding the general principle and the experimental demonstrations of the TCF of FOFC. Based on this new understanding of the TCF of FOFC, new applications can be proposed fairly readily.

In future work, by making the best use of the unique temporal coherence characteristic of FOFC, we plan to propose a brand-new interferometer that can considerably expand the vertical measurement range of conventional white-light interferometry, which also can be applied to optical tomography and profilometry.

## Acknowledgments

Part of this research work was supported by the Global Center of Excellence (COE) Program on “Global Center of Excellence for Mechanical Systems Innovation” granted to The University of Tokyo, from the Japanese Government. We are also grateful to NEOARK Corporation for providing the femtosecond fiber laser. Dong Wei gratefully acknowledges the scholarship given from Takayama International Education Foundation, Heiwa Nakajima Foundation, Ministry of Education, Culture, Sports, Science, and Technology of Japan, respectively.

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