## Abstract

A new approach is described to compensate the variations induced by laser frequency instabilities in the recently demonstrated Fourier transform spectroscopy that is based on the RF beating spectra of two frequency combs generated by mode-locked lasers. The proposed method extracts the mutual fluctuations of the lasers by monitoring the beating signal for two known optical frequencies. From this information, a phase correction and a new time grid are determined that allow the full correction of the measured interferograms. A complete mathematical description of the new active spectroscopy method is provided. An implementation with fiber-based mode-locked lasers is also demonstrated and combined with the correction method a resolution of 0.067 cm^{-1} (2 GHz) is reported. The ability to use slightly varying and inexpensive frequency comb sources is a significant improvement compared to previous systems that were limited to controlled environment and showed reduced spectral resolution. The fast measurement rate inherent to the RF beating principle and the ease of use brought by the correction method opens the venue to many applications.

©2008 Optical Society of America

## 1. Introduction

Fourier Transform Spectroscopy (FTS) is a key diagnostic method long used for its spectral accuracy and energy efficiency in environmental monitoring and in forensic analysis among many applications. Some years ago, a new kind of FTS method has been proposed [1,2] that is based on the beating spectrum between two closely tuned frequency comb sources. The original idea emerged for THz spectroscopy [3], with some results reported where short pulse lasers and a photoconductive emitter are used to generate the THz combs [4]. Very recently high-resolution complex HCN spectroscopy has been reported using mode locked fiber lasers stabilized on two narrow continuous wave (CW) fiber lasers [5]. The main advantages compared to standard FTS are: very fast optical path difference scanning rates equivalent to several centimeters in milli- or micro-seconds, no moving parts and potential high repetition rate of the measurement. In this type of FTS system, named hereafter cFTS, the optical spectrum is directly mapped into the radio frequency region (RF) by the two-by-two beating of the optical components from both lasers. In published papers, the frequency comb sources were generated by two mode-locked Ti:Sapphire lasers in the 800 nm region. In order to keep the beating frequency constant between two components at a given optical frequency, the constraints are extremely high on the stability of the repetition rates and of the carrier envelope offset (CEO) frequencies of both mode-locked lasers. The solutions implemented in [1,2] are: CEO frequency cancellation by nonlinear frequency difference generation of two harmonic frequency comb sources in the 9–12 µm band, use of two identical Ti:Saphhire lasers using the same optical pump and passive stabilization in a controlled environment (temperature and vibration). The demonstrated measurement showed limited spectral resolution of 2 cm^{-1} for a measurement duration of 70 µs.

This paper proposes a new approach to overcome most of the constraints of the first reported cFTS systems: costly lasers, controlled environment and limited resolution. Based on relatively inexpensive fiber-based mode-locked lasers, a direct measurement of the beating spectra is performed in conditions where small variations of the lasers are allowed. In addition these frequency combs are periodic but modes are not necessarily on exact multiples of the repetition rates, as the CEO frequencies are not cancelled (combs are not harmonic). To correct for the inevitable variations in the RF beat note, we propose to record the beating fluctuations at two known optical frequencies and from this information, determine a correction phase and a new sampling grid for the measured interferogram. This approach is very similar to using a reference channel in traditional FTS where the interference signal of a laser is used to determine the real optical path length difference (OPD) at each measurement point, regardless of the moving carriage speed. A complete mathematical description of the cFTS method is first proposed for the stationary case. It is then demonstrated that for small time variations of the repetitions rates and CEO frequencies in a defined range, it is possible to adapt the equations found in the stationary case. The use of narrowband fiber Bragg gratings filters is then discussed for monitoring the source changes. The new correction method is applied on a cFTS system based on two all-fiber mode-locked lasers around 1550 nm and measurements with spectral resolution of 0.067 cm^{-1} (2 GHz) are demonstrated using the absorption bands of an hydrogen cyanide (HCN) gas cell.

The cFTS technique is at the junction of two very wide research fields: the frequency comb sources mostly studied and developed for applications in the absolute frequency referencing and infrared spectroscopy based on time domain measurements (standard FTS). Both the cFTS technique and the absolute frequency referencing exploit the discrete, stable and periodic nature of frequency comb sources but they differ in their stability requirements. A large part of the research in the referencing domain is made for increasing the long term stability of the sources whereas the cFTS technique only requires short term stability. In addition, the correction method proposed in this paper relaxes most of the stability issues. In the spectroscopy field, the cFTS technique proposes promising features in terms of measurement durations that are not easily if not at all reachable with standard FTS and this with good spectral resolution and signal to noise ration (SNR). On the other hand, the cFTS method is not well adapted for ultrahigh spectral resolution, due to the comb nature of the used light sources and also due to limited equivalent OPD range where the correction method is applicable.

## 2. Principles of cFTS

#### 2.1 Frequency comb light source

The frequency comb sources considered in this paper are generated by two fiber-based mode-locked lasers using the Erbium gain medium in the 1550 nm band (193.5 THz). The main parameters are presented in Fig. 1(b): the repetition rate f_{r} (16.9 MHz in our case), the CEO frequency f_{0} and the source envelope A(ν) bounded in the optical range around ν_{0}. In addition, the envelope A(ν) is considered slowly varying compared to f_{r}. The linear part of the phase component in the electrical field is neglected as it can be viewed as a simple delay in the time domain. By definition, the light source is said dispersion-free if A(ν) is purely real.

For a stationary comb source having a constant repetition rate and CEO, the electrical field E_{m} of the m^{th} component in the frequency domain is given by E_{m}=A(ν_{m})·δ(ν-ν_{m}), where δ is the Dirac function and ν_{m}=m·f_{r}+f_{0} is the frequency of the m^{th} component. The electrical field E_{ν}(ν) for the whole frequency comb source is simply the sum of all components E_{m} and it can be written as the product of the envelope function A(ν) with a frequency-shifted Dirac comb ^{ν}III(ν), E_{ν}(ν)=A(ν)·^{ν}III(ν) where ^{ν}III(ν) is defined as:

In the time domain, the stationary frequency comb light source is viewed as an infinite pulse train, see Fig. 1(c). The electrical field E_{τ}(τ) is the inverse Fourier transform (iFT) of E_{ν}(ν) and it corresponds to the convolution of a(τ) and ^{τ}III(τ) where a(τ) is the iFT of A(ν) and ^{τ}III(τ) is the iFT of ^{ν}III(ν). The convention used in this paper for the Fourier transformation is based on the signal processing formulation that is:

It is interesting to define the carrier-free envelope function A_{0}(ν) that corresponds to A(ν-ν_{0}). Using the shifting property of Fourier transforms, a(τ) can be written as a(τ)=a_{0}(τ)·exp(i2πν_{0}τ), where a_{0}(τ) is the iFT of A_{0}(ν) that provides the pulse envelope and ν_{0} brings the very fast modulation present in the pulse as observed in Fig. 1(d). For a dispersion-free light source, the amplitude of a_{0}(τ) is symmetric whereas its phase is anti-symmetric (it becomes a purely real function if A(ν) is symmetric around ν_{0}).

The inverse Fourier transform of the shifted Dirac comb corresponds to an harmonic Dirac comb multiplied by a scale factor and a phase shift function; alternatively, it can also be expressed as an infinite summation of exponentials:

The electrical field E_{τ}(τ) can then be rewritten as:

From this equation, it is clear that the infinite pulse train is periodic for the pulse envelope (period of 1/f_{r}) but not periodic for the pulse itself due to the constant phase shift of 2πf_{0}/f_{r} that exists between two successive pulses. It is interesting to note that any time shift of m_{0}/f_{r}, m_{0} integer, only affects E_{τ} by a constant phase offset: E_{τ}(τ-m_{0}/f_{r})=E_{τ}(τ)·e^{-iφ} where φ=2πf_{0}m_{0}/f_{r}. This property means that any pulse can be used to define the time axis and that the electrical field remains defined by Eq. (4) but with an additional phase constant φ that is omitted in most cases. Equation (4) also indicates that E_{τ} is close to a discrete iFT of the frequency domain envelope A(ν) but on a non-harmonic (shifted) frequency grid.

Assuming that the electrical field E_{τ}(τ) is measured with an infinite bandwidth detector, the resulting intensity signal I_{D}(τ) is given by:

It is appropriate to apply a variable change from n to p=m-n that can be viewed as a simple re-arrangement of the terms in the double summation and in this case the source intensity becomes:

where α_{p} are time independent coefficients. The frequency domain description of I_{τ}(τ) is provided by its Fourier transform I_{D}(ν) that takes a very simple expression I_{D}(ν)=Σα_{p}·δ(ν-p·f_{r}), thus discrete and periodic components at multiples of f_{r}.

#### 2.2 Principles of the cFTS method with stationary non-harmonic frequency combs

The two frequency comb sources are defined by the parameters [A_{1}(ν), f_{r1}, f_{01}] and [A_{2}(ν), f_{r2}, f_{02}], respectively. Both light sources are considered stationary and the repetition rate difference is Δf_{r}=f_{r2}-f_{r1}≪f_{r1}. The principle of the cFTS technique is illustrated in Fig. 2(a) in the frequency domain. The pairs of optical components are chosen deliberately such that the m-th line in the first comb spectrum is the closest to the line having the same m index in the second comb. Throughout the text, such lines will be called nearest neighbors and the shown RF spectrum only represents the RF beating between the components of those pairs. As two different frequency comb sources are used, their envelope A_{1} and A_{2} are different. The constant phase offsets φ discussed in the previous section are neglected for the moment. The beating components in the RF band are revealed by the intensity measurement of the interference of both light sources on a detector and it will be demonstrated that the amplitude of the beating components of interest S is proportional to a scaled and shifted version of the envelope product U=A_{1}·A_{2}*. The relation between the functions S and U is the cornerstone of the cFTS method.

When considering the beat note between components m and m+x from the two sources (x integer), it is clear that there will be a replica S_{x} of U in every [x·f_{r}, (x+1)·f_{r}] RF interval. In addition, the intern beating components for each source are observed in the RF band exactly at integer multiples of the repetition rate (nearly identical for both light sources). The intensities of the RF replicas S_{x} and of the intern RF beatings are shown in Fig. 2(b). As only positive RF frequencies are experimentally observed, the negative part of the RF band is aliased and is effectively seen flipped on the positive frequencies such that there are in fact two replicas in every [x·f_{r}, (x+1)·f_{r}] interval.

The width of the function S is determined from the width of the function U in the optical domain and the scale factor Δf_{r}/f_{r} that can be derived from the sampling grids. The position of the function S is also determined by the CEO frequencies. It is mandatory in the cFTS method to be able to isolate a single RF replica. In the presented example, it is possible to adjust one of the CEO frequency f_{0}, see Fig. 2(c) to remove the overlap because the width of the functions S_{x} is smaller than f_{r}/2. It is possible to reduce the width of S by decreasing the value of the repetition rate difference Δf_{r}. The ability to position the replicas S_{x} using f_{0} is a great advantage over the cFTS using purely harmonic frequency comb sources (as in [1, 2]).

In the time domain, the pulses from both light sources are observed at the detector location. It is possible to use the pulse centers from the first comb source as a trigger to observe the incoming pulses from the other source. From this point of view a relative slide is observed with a speed proportional to the difference of repetition rates, see Fig. 2(d). It is clear that the relative movement of the pulses is a continuous process and that pulse trains are perfectly superimposed at every 1/Δf_{r}. For a cFTS measurement, a single occurrence of the time domain response is considered and the time axis origin is chosen at the time when both pulses are superimposed.

The main application of the cFTS method is active determination of the transmission or reflection property of a sample probed by one or both frequency comb light sources. For instance with both sources probing the sample and if t(ν) is the transmission function for the electrical field of the sample (amplitude and phase), the cross-correlation becomes T(ν)·U(ν) where T(ν)=|t(ν)|^{2} is a real function. A measurement without the sample provides U(ν) and thus T(ν) after a normalization in the frequency domain. It is also possible to directly measure the complex transmission function t(ν) by probing the sample with only one of the frequency comb and to later mix both sources to generate the cross-correlation function as theoretically proposed in [6] and realized in [5] and in a microscopy context in [7].

#### 2.3 Stationary case

This section provides a mathematical description of the cFTS technique with stationary frequency comb sources and discusses the conditions to produce useable measurements. The sources are described by the parameters [A_{1}(ν), f_{r}, f_{0}] and [A_{2}(ν), f_{r}+Δf_{r}, f_{0}+Δf_{0}], respectively (Δf_{r}≪f_{r}). The constant phase shifts (φ) are ignored for the moment but will be taken into account properly in Eq. (17). The frequency grids ν_{1}(m) and ν_{2}(n) determines the components for both frequency combs and the grid ν_{1} is also used to define a generic optical grid ν_{opt}(m)=ν_{1}(m)=m·f_{r}+f_{0}. The electrical fields in the frequency domain are E_{ν1}(ν) and E_{ν2}(ν), respectively, and they are given by E_{ν1}(ν)=A_{1}(ν)·III(ν-f_{0},f_{r}) and E_{ν2}(ν)=A_{2}(ν)·III(ν-(f_{0}+Δf_{0}),(f_{r}+Δf_{r})). The time domain equivalent E_{τ1}(τ) and E_{τ2}(τ) are, due to the stationarity assumption, the iFT of E_{ν1}(ν) and E_{ν2}(ν), respectively. The iFT of the envelope functions are a_{1}(τ) and a_{2}(τ), respectively, and those for the frequency-shifted Dirac combs are given by Eq. (3):

The acquired signal in the cFTS method is the time domain intensity I_{D}(τ) of the interference corresponding to the sum of both frequency comb light source fields E_{τ1}+E_{τ2}. The description of I_{D}(τ) is made here for an infinite bandwidth detector:

The first two terms represent the inner interference between the components of each light source separately and the third term is the cross-correlation term of interest. It is preferred to define the cross-correlation term, labeled I_{τ}(τ), in its complex declination and to neglect the factor 2. Using again the variable change p=m-n, it is found that:

An important approximation is made at this stage to remove the dependency on p in the bracket term of Eq. (9):

This approximation is at its best for nearest neighbors as in the simplified example of Fig. 2(a) and it is very well adapted for experimental measurements that are performed in the lowest part of the RF band. Using components that are not nearest neighbors induces a resolution loss in the cFTS method. Depending on the value of m·Δf_{r} compared to f_{r} (m being the average value of the frequency component indices), the nearest neighbors can be observed for non-null values of p, meaning that the first RF replica near zero is not necessarily the replica obtained from the components of identical index m.

The envelope product U is defined as U(ν_{opt}(m))=A_{1}*(ν_{opt}(m))·A_{2}(ν_{opt}(m)). It is appropriate to define a frequency grid ν_{rf}(m) in the RF band that is defined as ν_{rf}(m)=mΔf_{r}+Δf_{0}. The frequency axis ν_{rf}(m) is adapted for the fundamental RF replica (p=0) generated by the beating between the components of identical index m. For the other RF replicas, a simple frequency shift of p·f_{r} is required. Using U and ν_{rf}, I_{τ}(τ) becomes:

It is possible to interpret the function g(τ) as the time domain response of the RF beating components from the optical components of identical index (mostly a scaled version of the iFT transform of the envelope U) and to see that the exponential term plays the role of frequency shifter in the frequency domain that explains the RF replicas every f_{r}+Δf_{r}≅f_{r}.

It is possible to define continuous versions of the optical and RF axis ν_{opt} and ν_{rf}, respectively. The relationship between the continuous axis ν_{opt} and ν_{rf} is:

The envelope functions A_{1} and A_{2} are assumed slowly varying compared to f_{r}, thus the beating envelope functions U can be defined in a continuous formulation and exploiting the relation between ν_{opt} and ν_{rf} it is possible to define an envelope function S directly in the RF band: S(ν_{rf})=U(ν_{opt}(ν_{rf}))=A_{1}*(ν*opt*(ν*rf*))·A_{2}(ν_{opt}(ν_{rf})). The fundamental intensity g(τ) takes the form of a discrete iFT on a frequency shifted and uniformly spaced grid in the RF band:

In the frequency domain, the Fourier transform G(ν) of g(τ) can be expressed as the product of the continuous envelope function S in the RF band and a shifted Dirac comb, also in the RF band:

Back in the time domain, the intensity g(τ) becomes:

where s(τ),the iFT of S, can be expressed using u(τ) the iFT of U(ν):

In this form g(τ) clearly shows the time domain replication of s(τ) every 1/Δf_{r} and the requirement of a constant phase correction for each time replica to take into account the frequency shift component Δf_{0}.

Some undesired overlapping occurs depending on the very nature of s(τ) as viewed in Fig. 3. The cropping process required to isolate a single time replica q in a cFTS measurement is performed by multiplying the function g(τ) with an adequate boxcar window function Rect((τ-q/Δf_{r})/Δτ_{crop}) where Rect(τ)=1 for |τ|<0.5 and 0 anywhere else. The crop interval Δτ_{crop} is chosen to balance the amount of acceptable overlap and the desired frequency resolution. The dispersion properties of the envelopes A_{1} and A_{2} and thus of U or S become a potential problem in this context as the temporal spreading of the pulses leads to a significant increase of the overlapping region between two adjacent replicas g_{q} and g_{q+1}. In consequence, the crop interval length is reduced proportionally to the amount of dispersion of S and thus the spectral resolution is reduced accordingly.

It is straightforward to demonstrate that all time copies are identical at a constant phase shift: g_{q}(τ)=g_{0}(τ+q/Δf_{r})·exp(i2πΔf_{0}q/Δf_{r}). It is then possible to redefine the time axis at the center of any cropped time copy:

where φ is an unknown phase constant that contains all phase offset constants (the phase from both lasers not taken into account until here and the phase from the time shift used here). The frequency domain response I_{ν,crop}(ν) is given by:

where Sinc(x)=sin(π·x)/(π·x) is the well known normalized cardinal sine function that determines in the standard FTS the spectral resolution for a given time domain interval of measurement. Here, the spectral resolution in the RF band needs to be multiplied by the scale factor f_{r}/Δf_{r} to obtain the resolution in the optical band. The intensity in the frequency domain is essentially the sum of the frequency shifted versions of S(ν) with the potential overlapping problem discussed previously in Fig. 2(b). The number of available RF replicas is only limited by the detector bandwidth but for high RF replica index the validity of the approximation in Eq. (10) is questionable as much as the usage for the FTS applications. A cFTS measurement is exploitable if it is possible to isolate a single RF replica, electronically or during the data processing. For a chosen RF replia of index p (often unknown a priori), the frequency domain intensity I_{ν,FTS}(ν) is given by:

For a RF replica mirrored in the positive part of the RF band by the detection process, it is required to apply a time axis inversion (τ to -τ) to compensate the frequency inversion. Using the relation between s and u it is possible to express the cFTS time response I_{τ,cFTS}(τ) for a chosen RF replica and for a single time copy:

This is the fundamental equation of the cFTS method using shifted frequency comb light sources. The target function U(ν) can be recovered from I_{τ,cFTS}(τ) with a limited resolution (U_{lim}(ν)) after a phase correction, a scale compensation and a Fourier transform. If the RF replica index p is known, U_{lim}(ν) is directly obtained after a simple axis conversion from I_{ν,cFTS}(ν) using the Eq. (12).

#### 2.4 RF to optical band mapping

After selection of a single RF replica from a single time copy, it is possible to define the frequency axis of measurement ν_{meas}=ν_{rf}-p·f_{r} and using Eq. (12) it is possible to define an affine relation between ν_{meas} and ν_{opt}:

where G_{opt} and O_{opt} can be viewed as a gain and an offset factors, respectively, because those factors give ν_{meas}=G_{opt}·ν_{opt}+O_{opt}. It is also possible to define a RF gain G_{rf} and RF offset O_{rf} that provide the affine relation ν_{opt}=G_{rf}·ν_{meas}-O_{rf} and those factors are directly connected to G_{opt} and O_{opt}:

In the case where G_{rf} and O_{rf} are known, the relation of Eq. (20) allows to directly provide U_{lim}(ν) from the Fourier transform of I_{τ,cFTS}(τ) without requiring the a priori knowledge of the laser parameters (f_{r}, Δf_{r}, f_{0} and Δf_{0}) and the RF replica index p. An option to determine the parameters G_{rf} and O_{rf} is to measure the RF frequency of the beating coming from two different, very small and known optical bandwidths. This is the approach followed in this work that also leads to a very efficient monitoring tool for the mapping between the RF and the optical bands when small variations of the lasers are experienced during the measurement time window.

Fiber Bragg gratings with narrow bandwidths are chosen in our cFTS system to filter out the desired optical bands required to determine G_{rf} and O_{rf}. Uniform FBGs of several centimeters are ideal because they provide symmetric optical filtering bandwidth between 1 to 5 GHz with a dispersion that is negligible on most of the reflection bandwidth. The reflection properties of the FBG are contained in the complex reflectivity r(ν) defined for the electrical field. The electrical field from a frequency comb source reflected by the FBG is given by E(ν)·r(ν) and the envelope U_{fbg}(ν_{opt}) corresponding to the interference of both lasers filtered by the FBG is given by:

where the reflectivity in intensity R=|r|^{2} is a real function. Recalling the approximation of Eq. (10), it follows that the dispersion distribution of the filter is not as innocuous as one would think by analyzing Eq. (23), which explains why other type of filters with phase discontinuities in the filtering bandwidth (Fabry-Perot filters or phase-shift FBGs) are to be avoided. The function R can be viewed as a frequency shifted filter Q around zero R(ν_{opt})=Q(ν_{opt}-ν_{0}). The envelope function U is assumed slowly varying compared to the bandwidth of the FBG, thus the filtered envelope U_{fbg} is approximated by it value in ν_{0}: U_{fbg}(ν_{opt})≅Q(ν_{opt}-ν_{0})·U(ν_{0}). The iFT of U_{fbg}(ν_{opt}) is u_{fbg}(τ_{opt})=q(τ_{opt})·exp(i2πν_{0}τ_{opt})·U(ν_{0}) where q(τ_{opt}) is the iFT of Q(ν_{opt}). The measured signal I_{τ,fbg}(τ) is deduced from Eq. (20) using the definition of G_{rf} and O_{rf}:

It is advised to use a symmetric envelope Q(ν) because in this case the phase of q(τ) is zero everywhere (this is true for uniform FBGs). For non-symmetric functions Q(ν), it is possible to map the time domain envelope |q(τ)| and deduce the expected phase of q(τ) from a preliminary characterization of Q(ν) but this is not a trivial task and this method is more sensitive to measurement noise. For a symmetric FBG envelope, the phase φ_{fbg}(τ) is given by:

The phase function φ_{fbg} is only defined in the crop interval Δτ_{crop} and in addition, φ_{fbg} is limited on the time bandwidth on which the function q(G_{opt}·τ) is above the noise floor. The derivative of Eq. (25) provides a first equation for determining G_{opt} and O_{rf}:

Using two FBGs around two different optical frequencies ν_{01} and ν_{02}, it is straightforward to determine G_{rf} and O_{rf} by solving the system of two linear equations provided by Eq. (26). For a uniform FBG with a length of D, the potential measurement range of I_{τ,fbg} is estimated around 4·D/c_{0}·G_{opt} where c_{0} is the speed of light in vacuum. For stationary sources, the factors G_{rf} and O_{rf} are constants and determined in this interval.

#### 2.5 cFTS method with small variations of the frequency comb light sources

The question arises of what happens when one or several parameters of the frequency comb light sources varies during the measurement acquisition. In general, the RF signals is scattered on a large RF zone that, among other problems, breaks the fundamental assumption for the cFTS method, that is, to be able to isolate a single RF replica for a single time copy. It is fortunate that the dynamics of mode-locked lasers ensures a continuous and moderately slow time variations of the repetition rates and CEO frequencies (or the mode-locking is lost during the measurement). For the cFTS method, the dominant effect is the variation of Δf_{r} that modifies the gain factor G_{rf} dramatically even for very small variations. As an example, considering lasers with repetition rates of 17 MHz and optical components near 200 THz, a variation of 1 Hz for Δf_{r} during the measurement is viewed in the RF band as a shift of 11.8 MHz that is even larger than f_{r}/2=8.5 MHz (an overlap on the RF band of the nearest RF replica is thus unavoidable in this example).

It is demonstrated hereafter that the equations obtained for the stationary case can be adapted to describe a cFTS measurement using frequency comb sources with small variations of their parameters. As a matter of course several conditions applies: 1) the time variations are expected small enough to ensure that no overlapping of the RF replicas occurs, 2) the sources are considered nearly dispersion-free at the detector location in order to guarantee synchronous changes for all frequencies and minimal overlapping of the time copies, 3) the spectral density of the sources are considered constant for at least the measurement duration, only the repetition rates and CEO frequencies vary and 4) the time variations of the source parameters are known for all the time being of the measurement (the knowledge of G_{rf} and O_{rf} during the measurement is sufficient).

The first condition is evident and compared to what is observed in Fig. 2(b) for the stationary case, the fundamental RF replica is widened and highly distorted when small time variations are experienced. A reduction of the scanning speed Δf_{r} can be used to compress the RF replica if the widening becomes too large (and adjusting f_{0} in consequence to remain in the allowed frequency window). The second condition deals with the dispersion properties of U or S and as stated previously, the lesser dispersion the better for the control of the time domain spreading (time copy overlapping problem). The other aspect of dispersion on the synchronous changes is less obvious at first sight but appears when considering the dispersion as a frequency dependant time of travel. Any time variation of the light sources is scattered over the time offset bandwidth between the frequency components and this effect is very difficult, if not impossible, to compensate a posteriori. For this reason it is necessary to use any means to reduce the pulse dispersion as much as possible. The amount of acceptable dispersion is related to the changing rate of the laser parameters. The third condition is very important as it ensures that the function u(τ_{opt}) at the basis of the fundamental Eq. (20) is not perturbed during the measurement and usually this is not a problem as mode-locked lasers exhibit stable spectrum for periods much longer than a batch of measurements.

As regards of the fourth condition, it has been demonstrated in the previous section that two FBGs can be used to determine the value of G_{rf} and O_{rf}. The envelope drop of q(τ_{opt}) limits the time window Δτ_{opt} where their phase functions are defined and those functions are necessary to determine the coefficients G_{rf} and O_{rf} using Eq. (26). It follows that 1/Δτ_{opt} defines the ultimate resolution in the optical band for the cFTS method that requires the monitoring of the small time changes with two FBGs. The starting point for the perturbation development is Eq. (20) that is reformulated as:

The perturbations are small and continuous such that the constant phase factor φ remains constant as it represents the pulse number of each laser at τ_{opt}=0. The relative variations on f_{r} are so small that the factor 1/f_{r} is also considered constant. Only a coarse windowing is used, larger than the desired final boxcar window. The function τ_{opt} and θ are better defined as their derivative to the variable τ_{meas}:

With this formulation it is straightforward to reconstruct the scanning time τ_{opt} and the correction phase θ from the knowledge of G_{rf}(τ_{meas}) and O_{rf}(τ_{meas}). The fundamental equation becomes:

Using the phase φ_{1} and φ_{2} of the two FBGs centered at optical frequencies ν_{1} and ν_{2}, respectively, the factors G_{rf} and O_{rf} are determined from Eq. (26):

The fundamental equation can be expressed in terms of the phase function φ_{1} and φ_{2} and the optical references ν_{1} and ν_{2}:

The distributions τ_{opt}(τ_{meas}) and θ(τ_{meas}) are thus given by:

Equation (32) is not well suited for numerical processing as it directly provides the time domain response of the original optical band signal U(ν) whereas it would be preferred to work without the optical frequency carrier, that is U(ν-ν_{0}). It is straightforward to demonstrate that a phase correction with the phase of one of the reference is the solution:

In this form, the Fourier transform of I_{num} gives U(ν-ν_{1}), that is with a frequency carrier offset of ν_{1}. The optical delay axis τ_{opt} is not on a uniformly sampled grid due to the Δf_{r} variations. In order to use FFT algorithms, another uniformly spaced optical delay axis τ_{lin} is defined on the same range as τ_{opt}. It is appropriate to define the boxcar window at this moment, symmetrically around the position τ_{lin}=0 that corresponds to the center of the interferogram (the point of symmetry in dispersion-free conditions). The re-interpolation between I_{num}(τ_{opt}) and I_{num}(τ_{lin}) is better performed separately on the amplitude and the unwrapped phase functions.

## 3. Experimental setup

The experimental setup is presented in Fig. 4. This configuration is designed to measure the transmission properties in intensity of a sample under test. Other designs for reflection or complex measurements are possible without limitations. Two frequency comb light sources (FC_{1,2}) are amplified (A) and their polarization matched with a polarization controller (PS) prior to be combined with a 50/50 coupler (X_{3}). There are six optical paths that provide six signals measured with six filtered detectors (low-pass under f_{r}/2 for detectors 1 to 4 and band-pass at f_{r} for detectors 5 and 6). Path 1 is used to probe the sample with the combined sources. Path 2 is used as reference and this signal is very useful to normalize the sample response. Paths 3 and 4 provide the optical signals of the combined sources filtered by the FBGs that are required for monitoring G_{rf} and O_{rf} during the measurement. The signals from paths 1 to 4 are acquired at 50 MS/s with a data acquisition card (DAQ). The paths 5 and 6 provide a direct measurement of the repetition rate for both lasers separately and this information is used by a feed-back system to maintain the repetition rate difference Δf_{r} around the desired value. The variations of Δf_{r} are estimated from the time interval between the time copies of the interferograms and variations under ±0.05 Hz are observed. The feed-back effect is based on optical length tuning of the ring cavities of the mode-locked lasers that combines two complementary methods: 1) fiber stretching with a piezoelectric stack for fast and small range effects and 2) temperature change on 5 m of the erbium fiber for slower effects but on a much larger range. The positioning of the RF replica in the [0,f_{r}/2] band is done manually for the time being with a small tuning of the laser pumping power that changes f_{0}. Measurements presented in this paper were acquired at values of Δf_{r} around 0.5 Hz.

The two frequency comb light sources (FC_{1,2}) are mode-locked, all-fiber, solitonic ring lasers using erbium-doped media for amplification [8]. The output power is 1 mW, the repetition rate 16.9 MHz and the pulses are linearly polarized and nearly dispersion-free. The spectrum at the laser output is smooth with a bell shape and with some expected and characteristic Kelly sidebands [9]. The spectral bandwidth is close to 15 nm at the half maximum (25 nm of useful bandwidth) and the central frequency can be tuned between 1550 to 1580 nm. The pulses in the time domain are 350 fs in the near dispersion-free condition. In order to increase the average power and the useful spectral bandwidth, each laser is amplified separately with an erbium-based optical amplifier (A). The output spectrum bandwidth is extended to cover approximately 70 nm but its shape shows important inhomogeneities that are very sensitive to the tuning of the mode-locking conditions. The amplification process also induces nonlinear and frequency dependant polarization rotation that becomes a problem for the cFTS measurements as it reduces or even cancels the interference process for some optical bands. It is thus important to tune the lasers and the polarization states in order to obtain the smoothest output spectrum after the cFTS process and to acquire a reference spectrum without the sample for normalization. For high energy pulses, non-linear rotation can also occur in the optical fiber after the amplifiers and thus discrepancies are possible in the normalization process depending on the splitting ratios of the couplers and the losses between the sample and reference paths.

It is important that the measurement time axis are identical for the four signals acquired by the DAQ card in order to apply the correction method (known and constant delays are acceptable at the price of an appropriate increase of the acquisition duration and post-processing data shifting). For this reason, the optical and electrical path lengths are matched to meet the synchronous condition. The dispersion of both laser pulses at the detector locations also needs to be minimized to avoid the undesired frequency dependent time shifts. Fortunately, the erbium amplifiers add a large dispersion component of inverse sign compared to the dispersion of standard SMF-28 fiber. The pulse width is a good indicator of the amount of dispersion (the smaller the better) and for this reason the pulse width is measured with an autocorrelator at the location of detector 2, the path for the normalization signal. The width of the pulses for each laser separately is minimized by adding an adapted length of SMF-28 fiber anywhere before the mixing coupler X_{3} (compensation of the first order of dispersion). In most cases, the dispersion added by the sample compared to the amount of dispersion for the same optical path length in standard fiber is small and can be neglected. For very dispersive samples, special dispersion compensation could be necessary. A final note on dispersion: a moderate amount of dispersion is not necessarily detrimental and even proved to be beneficial. This is a little counterintuitive as the dispersion in the correction method adds uncertainty in the frequency domain response. Nonetheless, the noise figure is well advantaged for broadband optical signals as the dispersion spreads the large and sharp peak at the center of the interferogram and thus allows a better detection of the smallest signals (the dynamic range of the time signal is compressed). Two sets of measurements are presented in the next section, in the first case with nearly dispersion-free condition and in another case with a moderate amount of dispersion.

The two fiber Bragg gratings used in the setup (FBG_{1,2}) are 5 cm-long with expected uniform amplitude and period (no chirp, no apodization). The first grating at 1549.88 nm is close to the ideal case, see Fig. 5(a) but the second grating at 1542.18 nm is less than perfect due to writing inhomogeneities that cause a small chirp in the grating. The autocorrelation functions of both gratings are presented in Fig. 5(b). As expected the first grating is very close to the ideal case compared to the second grating. For the ideal grating, the envelope is nearly triangular with an equivalent OPD window of 18 cm (close to four times the grating length). The phase is linear in the window and corresponds to the phase of the carrier frequency. Removing this carrier phase, the phase is zero everywhere except at the zeros of amplitude where a π-shift is observed. The phase deviation of the first grating is in a ±0.1 rad for the first 10 cm and of ±0.2 rad for the useful window. For the second grating, a discrepancy of ±0.5 rad is observed in the first 10 cm and get much worse outside this window. These values remain very small compared to the pertinent range of phase to consider that is given by the phase from the frequency difference 2π·(f_{2}-f_{1})·τ whose range is 2026 and 3648 rad for the 10 and 18 cm-width window, respectively. The narrower envelope shape for the second grating is a concern as it means more sensitivity to noise. Experimentally, it is demonstrated that the amplitude and phase signals for the first grating are easily obtained up to a 15 cm-width window where the chosen criteria is that the unwrapped phase deviation from the linear fit is continuous without 2π shift on a short scale (this is primordial as the τ_{opt} axis is defined from the phase distributions). For the second grating, a direct determination of the unwrapped phase is limited to a 10 cm-width window. It is possible to extend the useful window for the second grating using a prior filtering based on the phase of the first grating followed by a spectral domain filtering (this aspect is explained in details in the next section).

## 4. Results

The sample chosen to demonstrate the capabilities of the cFTS is a fiber-pigtailed HCN cell with several narrow absorption bands in the optical range of the frequency comb light sources. The data treatment is explained using the measurement made in the condition of very low dispersion. The comparison with the measurements performed with moderately dispersed pulses is then proposed. The uncertainty, resolution and accuracy are discussed and compared to a reference measurement of the gas cell made with a commercial instrument.

#### 4.1 Example of data treatment

From the acquired interferograms, several manipulations are required to retrieve the transmission response of the sample. At first, it is necessary to isolate the signal from the desired RF replica for the four signals (the [0,f_{r}/2] range in our case) and to determine the phase distributions for the first FBG. Based on Eq. (34), the four signals are multiplied by exp(-iφ_{1}) to provide lower frequency signals. This operation is surprisingly efficient to clean up the spectral responses, such that the phase distribution for the second FBG can be retrieved on the full interval after a second frequency domain filtering. Finally a re-interpolation is performed to be able to use FFT’s algorithms.

The duration of the acquired interferograms is 20 ms and they are centered on one of the time domain copies. A first filtering is applied in the frequency domain to isolate the RF single sided replica of interest. The resulting filtered interferograms are thus complex. The acquired interferograms for the two FBGs are presented in Fig. 6(a) directly after the acquisition (oscillating real part function) and the amplitude of the complex interferograms are also presented after the first filtered transform where a significant noise amplitude is discernable (the figure also shows less noisy amplitudes after the second spectral filtering discussed hereafter). In Fig. 6(b) are presented the frequency domain responses in the RF band. The central frequency is around 4.8 MHz and the selected bandwidth is 1.2 MHz. The signals from the two FBGs are scattered over a large zone due to the small variations of the comb sources. The signal from the sample is smoother but shows none of the expected absorption peaks.

The phase distribution φ_{1} is extracted from the filtered interferogram of FBG_{1} and the phase correction is applied to all interferograms. The new interferogram for FBG_{2} is filtered in the frequency domain to improve the SNR in the time domain and the retrieved phase distribution corresponds to φ_{2}-φ_{1}. Fig. 7(a) shows the amplitude of the frequency response of both FBGs and the signal from the sample after the phase correction. Apart for filtering the signal of FBG_{2}, it is not necessary to calculate those frequency responses. The grey curves on Fig. 6(a) present the amplitude of the time domain response for both FBGs after this second spectral filtering. In Fig. 7(a), the amplitude of the first FBG is centered at zero due to the frequency shift (ν_{rf}(ν_{1})) and its shape mostly corresponds to the convolution of the grating spectrum with a Sinc function due to the time window limitation. For large variations of Δf_{r} that pinch and stretch the time domain interferogram, the shape of this amplitude could be more distorted. The amplitude of the second FBG is located around -29 kHz and as expected, it differs notably from the one of the first grating because the scale variations are not yet compensated. Compared to the original response observed in Fig. 6(b), the signal is now very narrow, especially when looking at the sample response bandwidth. This explains why the second frequency filtering on FBG_{2} is so efficient to clean up the phase of the time domain response. The frequency response of the sample with the phase correction is also presented in Fig. 7(a) and several absorption peaks are clearly visible. A detailed analysis of each peak indicates that the correction is not yet complete, in the same fashion as observed for FBG_{2} (broadening and parasitic peaks).

The instantaneous frequency of the RF beating for FBG_{1} is calculated from the derivative of φ_{1} and presented in Fig. 7(b). This frequency oscillates rapidly with a 250 kHz deviation around 4.8 MHz. The unwrapped phase difference φ_{2}-φ_{1} is viewed in two manners: the phase deviation from the best linear fit and the instantaneous RF interval corresponding to ν_{2}-ν_{1} (proportional to G_{rf}). The phase difference is directly connected to the optical time axis (Eq. (33)) but the required unwrapping of the phase difference can introduce inopportune and erroneous 2π shifts. The linear deviation of the phase difference is very useful in that matter to determine the maximum time interval where no such 2π shift occurs. For the example presented here, the useful time window is 17 ms, slightly smaller than the 20 ms of acquisition duration. The frequency of the RF interval between FBGs 1 and 2 corresponds to the optical bandwidth of ν_{2}-ν_{1} and it is nearly constant around 29 kHz with small oscillations that are partly real and partly related to the noise in the measurement (noise becomes the dominant contribution outside the ±5 ms window). The averaged Δf_{r} is deduced from the gain factor and corresponds to 0.5 Hz (Eq. (22) and Eq. (26)).

The optical time delay axis τ_{opt} is determined from the phase difference (Eq. (33)) and a linearly spaced time delay axis τ_{lin} is defined on the same time range but with a coarser step corresponding to the desired frequency bandwidth around ν_{1}. A final time cropping can be necessary to position the center of the interferogram in the center of the time interval. The signals defined on the τ_{opt} grid are interpolated on the linear grid τ_{lin}, separately on their amplitude and unwrapped phase distributions for better performances. The amplitudes of the four corrected interferograms are presented in Fig. 8(a) using a distance axis τ_{lin}·c_{0} commonly used in standard FTS. The distance range of 15 cm determines the ultimate achievable frequency resolution that corresponds to 0.067 cm^{-1} in wavenumber or 2 GHz in frequency (18 pm at 1550 nm). The amplitude of FBG_{1} is clean on the entire interval whereas the amplitude of FBG_{2} shows increasing noise on the side (the noise floor is not yet reached and side lobes are expected from the spectral density function of the grating). The dynamic range of the sample and source signals is 30 dB. The interferograms of the sample and the sources are nearly symmetrical due to the very low level of dispersion.

Finally, the frequency responses are calculated using a FFT algorithm where zero padding is used to increase the viewing resolution (Fourier interpolation). The optical frequency grid is calculated from the range of the kept part of τ_{lin} and the shift frequency ν_{1}. The amplitudes of the frequency domain results are presented in Fig. 8(b) on a wavelength scale. The spectra of the FBGs are very sharp and the spectrum of the sample channel presents very clean and defined absorption peaks. The matching with the response of the unperturbed path is good, indicating low non-linear polarization rotation in the fibers after the amplifiers. The signal to noise ratio of a single measurement is acceptable and reasonable for a 17 ms acquisition time.

#### 4.2 Dispersion

Several measurements with a moderate amount of dispersion were performed to study the robustness of the correction method. The amplitude of the sample response in the time domain after the whole correction process is presented in Fig. 9(a) alongside with the results obtained in the previous section for the nearly dispersion-less case. The measurement window is 10 cm in distance and an average on the amplitude of 10 measurements is performed. As expected, the symmetry of the response is broken by the dispersion and a broadening of the central part is clearly observed. It seems that the dynamic range of the dispersion case is reduced by 5 dB, indicating a potential advantage for the small signals compared to the low dispersion case. The corresponding spectra are shown in Fig. 9(b) for the sample responses alone and for the normalized responses (transmittances). The curves are averaged on the amplitude of the frequency domain response (10 measurements for each case). The measurements were conducted on different days with different tuning of the mode-locked lasers and that explains the dissimilar shape of the envelope (function U(ν_{opt})). Both measurements clearly identify the absorption peaks of the HCN gas and both spectra are very similar after normalization. The measurements with low dispersion appear a little noisier than in the case with moderate dispersion. This is probably explained by the pulse widening that reduces the maximum level of the optical signal arriving on the detectors. On the other hand, it is observed that the minimum values of the absorption peaks are less uniform for the case with dispersion, indicating the potential limitation in the correction process. For the resolution of 0.1 cm^{-1} of these measurements, no clear widening of the absorption peaks is observed.

These results on the dispersion are preliminary and further work is necessary to determine the expected resolution loss and added uncertainty. The good news is that a moderate amount of dispersion (from the sample for instance) is not a fundamental barrier to the cFTS method proposed in this paper.

#### 4.3 Accuracy and resolution

The uncertainty of the retrieved spectra is estimated from the normalized results of the 10 measurements in the case with moderate dispersion around one of the absorption peak. The Fig. 10(a) presents the 10 curves and the averaged one and a value of ±5 % for the uncertainty is derived. This value includes errors from the correction process, the noise in the measurements and the normalization (in this example the spectra for the unperturbed path show similar uncertainty behavior).

The efficiency to improve the resolution with increasing length of the measurement is shown in Fig. 10(b) where the final Fourier transform from the corrected interferograms is performed on several window widths (averaged in the frequency domain for the 10 measurements with moderate dispersion). The expected behavior is observed where the peak gets narrower and deeper with increasing resolution. The enhancement between 5 and 10 cm is a good sign of the potential of the cFTS method with correction for higher resolutions. The bottom part of Fig. 10(b) provides a comparison between the absorption peak measured with the cFTS system at 0.1 cm^{-1} resolution and the corresponding spectrum at the same resolution measured with a tunable laser instrument that uses an internal gas cell for wavelength calibration (Luna Technologies OVA). The position and the width of the main peak and the two smaller peaks are identical but the minimum value is a little different.

This behavior is also observed for the other peaks (same position and width but variations of the depth) as observed in Fig. 11 for 14 peaks around 1542 nm where the curves from the cFTS system and from the reference instrument are presented in a mirror fashion to better show the good agreement between them. In order to present the curves in such a way, it is necessary to remove the slowly varying loss background of the gas cell (determined from the external measurement) and then to also remove for the cFTS spectra the slowly varying part observed in the center part of Fig. 11 that is due to the non-ideal normalization process. The difference after removal of both slowly varying components is presented in the third part of Fig. 11 and an absolute error of ±5 % is obtained. This is an excellent result for the first tests on a cFTS system with small variations of the sources, a moderate amount of dispersion and less than perfect second reference FBG.

## 5. Conclusions

In this paper, the monitoring and correction method proposed for the cFTS radically improve the performances and usability of such systems. The first results are very promising with a good spectral resolution of 0.067 cm^{-1} and a good noise/uncertainty behavior for such time limited acquisition durations. This technology is at its infancy and much more research is still required to fully understand all aspects as for instance the nonlinear effects induced by the pulses or the impact of larger dispersion. Many projects are planed to improve the implemented system and many new applications are expected in a near future.

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