## Active Fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method

Optics Express, Vol. 16, Issue 6, pp. 4347-4365 (2008)

http://dx.doi.org/10.1364/OE.16.004347

Acrobat PDF (1313 KB)

### 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

1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

2. A. Schliesser, M. Brehm, F. Keilmann, and D.W. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing,” Opt. Express **13**, 9029–9038 (2005). [CrossRef] [PubMed]

4. T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. **88**, 241104 (2006). [CrossRef]

5. I. Coddington, W. Swann, and N. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

2. A. Schliesser, M. Brehm, F. Keilmann, and D.W. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing,” Opt. Express **13**, 9029–9038 (2005). [CrossRef] [PubMed]

^{-1}for a measurement duration of 70 µs.

^{-1}(2 GHz) are demonstrated using the absorption bands of an hydrogen cyanide (HCN) gas cell.

## 2. Principles of cFTS

### 2.1 Frequency comb light source

_{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.

_{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:

_{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:

_{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}).

_{τ}(τ) can then be rewritten as:

_{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.

_{τ}(τ) is measured with an infinite bandwidth detector, the resulting intensity signal I

_{D}(τ) is given by:

_{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

_{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.

_{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.

_{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

1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

2. A. Schliesser, M. Brehm, F. Keilmann, and D.W. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing,” Opt. Express **13**, 9029–9038 (2005). [CrossRef] [PubMed]

_{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.

^{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

6. S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. **27**, 766–768 (2002). [CrossRef]

5. I. Coddington, W. Swann, and N. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

7. M. Brehm, A. Schliesser, and F. Keilmann, “Spectroscopic near-field microscopy using frequency combs in the mid-infrared,” Opt. Express **14**, 11222–11233 (2006). [CrossRef] [PubMed]

### 2.3 Stationary case

_{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):

_{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:

_{τ}(τ), in its complex declination and to neglect the factor 2. Using again the variable change p=m-n, it is found that:

_{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.

_{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:

_{r}+Δf

_{r}≅f

_{r}.

_{opt}and ν

_{rf}, respectively. The relationship between the continuous axis ν

_{opt}and ν

_{rf}is:

_{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:

_{r}and the requirement of a constant phase correction for each time replica to take into account the frequency shift component Δf

_{0}.

_{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.

_{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:

_{ν,crop}(ν) is given by:

_{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:

_{τ,cFTS}(τ) for a chosen RF replica and for a single time copy:

_{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

_{meas}=ν

_{rf}-p·f

_{r}and using Eq. (12) it is possible to define an affine relation between ν

_{meas}and ν

_{opt}:

_{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}:

_{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.

_{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:

^{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}:

_{fbg}(τ) is given by:

_{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}:

_{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

_{rf}and O

_{rf}during the measurement is sufficient).

_{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:

_{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}:

_{opt}and the correction phase θ from the knowledge of G

_{rf}(τ

_{meas}) and O

_{rf}(τ

_{meas}). The fundamental equation becomes:

_{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):

_{1}and φ

_{2}and the optical references ν

_{1}and ν

_{2}:

_{opt}(τ

_{meas}) and θ(τ

_{meas}) are thus given by:

_{0}). It is straightforward to demonstrate that a phase correction with the phase of one of the reference is the solution:

_{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

_{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.

_{1,2}) are mode-locked, all-fiber, solitonic ring lasers using erbium-doped media for amplification [8

8. K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. **28**, 2226–2227 (1992). [CrossRef]

9. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. **28**, 806–807 (1992). [CrossRef]

_{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.

_{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

### 4.1 Example of data treatment

_{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.

_{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).

_{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.

_{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

_{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.

### 4.3 Accuracy and resolution

^{-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.

## 5. Conclusions

^{-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.

## References and links

1. | F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. |

2. | A. Schliesser, M. Brehm, F. Keilmann, and D.W. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing,” Opt. Express |

3. | D. van der Weide and F. Keilmann, “Coherent periodically pulsed radiation spectrometer,” US patent 5748309 (1998). |

4. | T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. |

5. | I. Coddington, W. Swann, and N. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. |

6. | S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. |

7. | M. Brehm, A. Schliesser, and F. Keilmann, “Spectroscopic near-field microscopy using frequency combs in the mid-infrared,” Opt. Express |

8. | K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. |

9. | S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(300.6190) Spectroscopy : Spectrometers

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

(300.6310) Spectroscopy : Spectroscopy, heterodyne

(300.6340) Spectroscopy : Spectroscopy, infrared

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: January 24, 2008

Revised Manuscript: March 9, 2008

Manuscript Accepted: March 13, 2008

Published: March 14, 2008

**Citation**

Philippe Giaccari, Jean-Daniel Deschênes, Philippe Saucier, Jerome Genest, and Pierre Tremblay, "Active Fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method," Opt. Express **16**, 4347-4365 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4347

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### References

- F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-4 (2004). [CrossRef] [PubMed]
- A. Schliesser, M. Brehm, F. Keilmann, and D.W. van der Weide, "Frequency-comb infrared spectrometer for rapid, remote chemical sensing," Opt. Express 13, 9029-38 (2005). [CrossRef] [PubMed]
- D. van der Weide and F. Keilmann, "Coherent periodically pulsed radiation spectrometer," US patent 5748309 (1998).
- T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006). [CrossRef]
- I. Coddington, W. Swann, and N. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008). [CrossRef] [PubMed]
- S. Schiller, "Spectrometry with frequency combs," Opt. Lett. 27, 766-768 (2002). [CrossRef]
- M. Brehm, A. Schliesser, and F. Keilmann, "Spectroscopic near-field microscopy using frequency combs in the mid-infrared," Opt. Express 14, 11222-11233 (2006). [CrossRef] [PubMed]
- K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-7 (1992). [CrossRef]
- S. M. J. Kelly, "Characteristic sideband instability of periodically amplified average soliton," Electron. Lett. 28, 806-807 (1992). [CrossRef]

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