Abstract:
There is provided a method for referencing and correcting the beating spectrum generated by the interference of the components of a frequency comb source. The proposed method allows monitoring of variations of a mapping between the source and the beating replica. This can then be used to compensate small variations of the source in Fourier transform spectroscopy or in any other interferometry application in order to overcome the accuracy and measurement time limitations of the prior art. Constraints on source stability are consequently reduced.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a divisional of U.S. application Ser. No. 12/664,887, filed on May 28, 2010, which claimed priority of U.S. provisional patent application No. 60/946,239 filed on Jun. 26, 2007, the specification of which is hereby incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to the referencing of frequency comb sources. More specifically, the present invention relates to the referencing of the beating spectrum generated by the interference of one or several frequency comb sources. 
     BACKGROUND 
     Spectroscopy uses the interaction of a radiation with a material sample to perform a spectral characterization. Fourier Transform Spectroscopy is an analysis method long used for its spectral accuracy and energy efficiency in environmental monitoring and in forensic analysis among many applications. Frequency comb spectroscopy uses electromagnetic pulses which provide a wide frequency range for the analysis. Frequency comb spectroscopy generally requires a pulsed light source generating the frequency comb. 
     In U.S. Pat. No. 5,748,309, a Fourier Transform Spectroscopy (FTS) method combining two frequency combs is proposed. The proposed method combines in the optical domain two frequency combs, namely mode-locked lasers, having slightly detuned repetition rates f r , and measures the beating spectra in the Radio Frequency (RF) domain, i.e. the RF beating replica of the optical domain spectrum to be characterized. The beating spectra are obtained by performing a Fourier transformation of the measured time response of the source interference signal (interferogram) in the considered RF band. The stability of the source combs is vital to ensure that the RF beating replica is consistent during a whole measurement interval. Even minor variations of f r  and of the carrier envelope offset (CEO) f 0  of any of the mode-locked lasers changes the mapping between the RF beating replica and the optical domain spectrum, thus severely limiting the accuracy of the Fourier transform of the interferogram. In order to maintain the mapping constant between the RF beating replica and the optical domain spectrum, the constraints are extremely high on the stability of the repetition rates and of the carrier envelope offset frequencies of both mode-locked lasers. 
     I. Coddington, W. C. Newbury and N. R. Newbury, “ Coherent multiheterodyne spectroscopy using stabilized optical frequency combs ”, Physical Review Letters 100, 013902 (2008) reports high-resolution complex spectroscopy using mode locked fiber lasers stabilized on two narrow continuous wave fiber lasers. This method transfers the continuous wave laser stability, on the frequency comb beating frequency, but is also sensitive to any remaining instabilities of the continuous wave laser. 
     The frequency of any line of a mode-locked laser is fully described by the laser repetition rate f r  and the carrier envelope offset frequency f 0 . The optical referencing is achieved for stabilized values of f r  and f 0 . In D. J. Jones, S. A. Diddams, J. K. Randa, A. Stentz, R. S. Windeler, J. L. Hall and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis”, Science Vol. 288, pp. 635-639 (2000), an auto-referencing of mode-locked frequency comb is proposed using the so-called 1f-2f technique for sources with a spectrum spanning of at least an octave. Using the 1f-2f technique, the carrier envelope offset frequency is canceled and in the case where the repetition rate is also actively locked, long-term stability and referencing of the optical comb is achieved with state of the art accuracy. Some limitations of the 1f-2f techniques are the short-term instabilities induced or left by the active feedback of the mode-locked lasers, and the overall complexity of construction. 
     Regarding the dual comb Fourier transform technique, international patent application bearing publication no. WO 2007/045461 A1 provides a way to improve the measurement duty cycle by periodically changing the repetition rate of one of the lasers in order to scan only the desired portion of the cross-correlation function, while still relying on the stability of the sources for the measurement duration. U.S. patent application bearing publication no. 2007/0182966 A1 provides an apparatus and method to conduct frequency comb heterodyne spectroscopy with continuous wave lasers. With this method, more than two parameters are likely to be needed to define the mapping between the RF beating replica and the optical domain spectrum. 
     Therefore, there is a need to address the stability issues of sources such as frequency comb sources when analyzing samples. 
     SUMMARY 
     There is provided a method for referencing and correcting the beating spectrum generated by the interference of the components of a frequency comb source. The proposed method allows monitoring of variations of a mapping between the source and the beating replica. This can then be used to compensate small variations of the source in Fourier transform spectroscopy or in any other interferometry application in order to overcome the accuracy and measurement time limitations of the prior art. Constraints on source stability are consequently reduced. 
     The referencing and correction method described herein is based on the determination of the mean beating frequency for narrowband parts of the source around different constant frequencies and the use of this knowledge in a mapping model that relates the beating spectrum to the source spectra. 
     It is noted that, throughout the specification, the expression “beating frequency” is intended to include a beating phase temporal distribution such that, when the beating phase temporal distribution is measured or otherwise determined, the beating frequency is thereby also retrieved and is considered to be measured or otherwise determined. 
     According to a first aspect, there is provided a method for correcting a beating interference signal. The method comprises: 1) providing a frequency comb having frequency components non-uniformly distributed in a source spectrum; 2) interfering the frequency components to produce the beating interference signal having a beating spectrum with beating components, the frequency comb having at least one variable frequency comb parameter; 3) isolating at least one reference beating component subset among said beating components; 4) monitoring at least one of a phase and a frequency of the reference beating component subset, the at least one of the phase and the frequency being related to the at least one variable frequency comb parameter; 5) determining from the at least one of the phase and the frequency a correction function to be applied to the beating interference signal, the correction function to correct for a variation of the variable frequency comb parameter; 6) recording the beating interference signal; and 7) correcting a recorded beating interference signal using the correction function. 
     According to a second aspect, there is provided an interferometric method for analyzing a sample having a sample signature. The method comprises: 1) providing a frequency comb having frequency components non-uniformly distributed in a source spectrum; 2) interfering the frequency components to produce a beating interference signal having a beating spectrum with beating components, at least a portion of the beating spectrum having a relation with at least a portion of the source spectrum, the frequency comb having at least one variable frequency comb parameter, a variation of the at least one frequency comb parameter causing a variation of the relation; 3) filtering at least part of the frequency comb with the sample to be analyzed to provide a sample interference signal holding the sample signature; 4) recording the sample interference signal, the sample interference signal showing an error due to the variation of the at least one frequency comb parameter; 5) selecting at least one reference beating component subset among said beating components; 6) monitoring at least one of a phase and a frequency of the reference beating component subset, the at least one of the phase and the frequency being related to the at least one frequency comb parameter; 7) determining from the at least one of a phase and a frequency a correction function to be applied to the sample interference signal, the correction function to correct for the variation of the at least one frequency comb parameter; and 8) correcting the recorded sample interference signal using the correction function. 
     According to a third aspect, there is provided a system for correcting a beating interference signal. The system comprises: 1) a source system for producing a frequency comb having frequency components non-uniformly distributed in a source spectrum, the frequency components to be interfered to produce the beating interference signal having a beating spectrum with beating components, a portion of the beating spectrum having a relation with a portion of the source spectrum, the frequency comb having at least one variable frequency comb parameter; 2) at least one narrowband filter receiving the frequency comb and having a known frequency passband in the source spectrum for isolating a subset of frequency components among the frequency components, the subset of frequency components interfering to provide a reference beating component subset; 3) a phase/frequency analyzer for monitoring at least one of a phase and a frequency of the reference beating component subset; 4) a model calculator receiving the at least one of the phase and the frequency for determining a correction function to be applied to the beating interference signal, the correction function to correct for the at least one variable frequency comb parameter; 5) an acquisition unit for recording the beating interference signal; and 6) a correction unit receiving the recorded beating interference signal and the correction function for correcting the recorded beating interference signal using the correction function. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  comprises  FIG. 1(A)  and  FIG. 1(B)  which are graphs respectively showing an example optical frequency spectrum and the corresponding beating spectrum of an example dual frequency comb source; 
         FIG. 2  is a block diagram illustrating an example optical frequency Comb Fourier Transform Spectroscopy (cFTS) system for analyzing spectroscopic properties of a sample and using a referencing technique to reference a frequency comb source; 
         FIG. 3  is a graph illustrating the offset and the scale effect in accordance with the referencing technique, when used with two mode-locked lasers; 
         FIG. 4  is a block diagram illustrating a processing unit for correcting an interferogram for a variation in the scaling factor and the offset parameters of the frequency comb source; 
         FIG. 5  comprises  FIG. 5A  and  FIG. 5B ;  FIG. 5A  includes three graphs showing a sample beating spectrum (i) and two reference beating spectra (ii) and (iii) before correction;  FIG. 5B  includes three graphs showing a sample optical spectrum (i) and two reference optical spectra (ii) and (iii) after correction; 
         FIG. 6  comprises two graphs showing the corrected sample optical spectrum of  FIG. 5B  superposed with two other similar measurements presented on a linear scale (top graph) and on a logarithmic scale (bottom graph); 
         FIG. 7  is a block diagram illustrating an interferometric system using the referencing method, according to a generalized example of the system of  FIG. 2 ; 
         FIG. 8  is a block diagram illustrating a cFTS system similar to the system of  FIG. 2 , but wherein the sample is probed in reflection; 
         FIG. 9  is a block diagram illustrating a cFTS system similar to the system of  FIG. 2 , but wherein a differential spectroscopy scheme is used with a sample probed in reflection; 
         FIG. 10  is a block diagram illustrating an interferometric system similar to the system of  FIG. 2 , but adapted for use in complex spectroscopy and wherein the sample is probed in transmission; 
         FIG. 11  is a block diagram illustrating an interferometric system similar to the system of  FIG. 10 , but wherein the sample is probed in reflection; and 
         FIG. 12  is a block diagram illustrating a cFTS system similar to the system of  FIG. 2 , but using active filtering instead of passive optical filtering. 
     
    
    
     DETAILED DESCRIPTION 
     The herein presented referencing technique is best understood considering the illustrative example of frequency comb Fourier Transform Spectroscopy (cFTS), which is first described in reference to  FIGS. 1 to 6 , but it is noted that the referencing technique can also be used beyond the herein described applications. For example, the herein described referencing technique can be adapted to be used in Optical Coherence Tomography (OCT). For example,  FIG. 11  shows a configuration more adapted to OCT. Other interferometric applications not specifically described herein may also benefit from the referencing technique. 
     The technique described herein is used to relax the strict requirements on the sources used in cFTS systems. A direct measurement of the beating spectra is performed in conditions where small variations of the sources are allowed. To correct for the resulting variations in the RF beating frequencies, the present technique records the beating fluctuations at two known optical frequencies and, from this information, it determines a correction phase and a new sampling grid for the measured interferogram. 
       FIG. 1  illustrates a cFTS beating spectrum obtained by interfering two slightly detuned frequency comb sources, and the instantaneous relation between the optical frequency components in the source spectrum and the beating components in the beating spectrum. The frequency components of each frequency comb are determined with two parameters: the repetition rate f r  and the carrier envelope offset frequency f 0 . The frequency of the k th  comb line is thus given by f k,1 =f 0,1 +k·f r,1  for the first comb source and f k,2 =f 0,2 +k·f r,2  for the second comb source. The complete source is an interleaving of both combs as shown in  FIG. 1(A)  where the solid lines represent the components from the first comb and the dotted lines, the components from the second comb. In this illustrative example, f r,1 =1, f r,2 =1.02, f 0,1 =1000 and f 0,2 =1000.1 and a linear amplitude variation is chosen. The beating components b m  of consideration are presented in  FIG. 1(B)  and are those corresponding to the beating of the source frequency components f k,1  and f k,2 . Due to the multiplication in the beating process, if the amplitude of the source frequency components increases linearly, the amplitude of the beating components is quadratic. The frequency of the beating component m is given by b m =f 0 +m·Δf r  where f 0 =f 0,2 −f 0,1  (=0.1) and Δf r =f r,2 −f r,1  (=0.02). Using the fact that the frequency f m1  is simply the m th  line of the first comb source (f m1 =f 0,1 +m·f r,1 ) the mapping function between b m  and f m1  can be determined and this mapping is a simple affine function, i.e. b m =G f ·f m1 +O f , where G f  is a scaling factor and O f  is an offset. Other beating subsets may be used as for example the beating components corresponding to the source components f k,2  and f k-1,1  or with f k,1  and f k-1,2 . Using small enough values of the repetition rate difference (|f r,1 −f r,2 |) and adjusting the offset frequencies (f 0,1  and f 0,2 ) ensures a valid mapping function between the subset of beating components and the source domain for the whole source range. In selecting the appropriate repetition rate difference Δf r  and offset frequencies f 0 , one should consider the folding points reached when the beating frequency modulo f r /2 is close to zero. 
     Now,  FIG. 2  shows an example of an optical frequency cFTS system  200  for analyzing spectroscopic properties of a sample  216 . The cFTS system  200  makes use of the herein described technique for correcting a beating interference signal, measured at detector D s , for frequency drifts between the first comb source  210  and the second comb source  212 . As described below, the beating fluctuations at two known optical reference frequencies f 1 , f 2  are recorded and, from this information, a correction phase and a new sampling grid for the measured interferogram are determined using the algorithm illustrated in  FIG. 4  and described hereinbelow. In the system  200  of  FIG. 2 , the comb sources  210  and  212  are based on two all-fiber mode-locked lasers having slightly different repetition rates f r . The central wavelengths of both lasers are around 1550 nm. The lasers are based on an all-fiber solitonic ring laser configuration using an erbium-doped media for amplification. Even though only thermal stabilization of the lasers is typically used, the remaining slow fluctuations of the repetition rates of both lasers are compensated using the provided referencing technique. 
     In the cFTS system  200  of  FIG. 2 , the two comb sources  210 ,  212  are combined to interfere using an optical coupler  213 . The two comb sources  210  and  212  are two nearly identical frequency comb sources adjusted in such a way that the beating spectrum between the modes of the both combs in the radio frequency (RF) beating spectrum is a replica of the optical domain spectra in the source spectrum (the RF spectrum is in fact the multiplication of both comb spectra). The main parameters considered here for each laser are the repetition rate f r  and the carrier envelope offset (CEO) frequency f 0 . 
     The combined sources is split onto two optical paths  214  and  215  at the output of the optical coupler  213 . In the optical path  214 , the optical signal probes a given sample  216  under test and thus experiences a spectral filtering prior to arriving on the detector D s . The resulting interferogram is measured at detector D s  and acquired using the first channel of a three-channel, 50 MS/s acquisition system  250  to provide a beating signal D s (t) in the RF domain. 
     The second part of the optical signal which propagates in the optical path  215  is split a second time using an optical coupler  220  in two optical referencing paths  222  and  224  each comprising a narrow band-pass filter  226 ,  232  used to isolate specific beating components in the optical signal, corresponding to known optical reference frequencies f 1 , f 2 . Each band-pass filter  226 ,  232  consists of a narrow band reflective Fiber Bragg Grating (FBG)  230 ,  236  centered at a known optical reference frequency, respectively f 1  and f 2 , and an optical circulator  228 ,  234  to convert the reflection FBG filter  230 ,  236  into a narrow band-pass transmission filter. The detectors D 1  and D 2  are used to measure the two reference signals D 1 (t), D 2 (t) acquired using the two remaining channels of the acquisition system  250 . The beating signal D s (t) is corrected using the two reference signals D 1 (t), D 2 (t) in the processing unit  400  further described below with reference to  FIG. 4 . 
     According to a specific example embodiment, the optical frequency region is centered at 193.5 THz, f 0  is around 2.5 THz, f r  is close to 17.58 MHz, the f r  detuning Δf r  between the two frequency combs is about 0.3 Hz and the beating bandwidth in the RF band that is equivalent to the laser optical bandwidth is 0.5 MHz, which allows a large working range between 0.5 and 8 MHz. The first FBG filter  230  is centered at λ 1 =1555.55 nm for a bandwidth of 50 GHz whereas the second FBG filter  236  is centered at λ 2 =1549.88 nm with a bandwidth of 3.5 GHz. In this illustrative example, the sample  216  is arbitrarily a FBG used in transmission and having a center wavelength of 1555.55 nm. The optical path lengths of the three optical paths are substantially equal such that negligible degradation of the correction method is observed. It was shown that an optical distance mismatch under tens of centimeters is suitable. The total dispersion of the pulses is estimated at the detector positions using an autocorrelator and the resulting time spreading of the pulses is very small compared to the acquisition sampling interval and thus negligible in this case. 
     A cFTS measurement using the herein described referencing and correction technique consists of three interferogram signals D s (t), D 1 (t) and D 2 (t), one for each path, and measured at detectors D s , D 1  and D 2 . The spectra obtained after a Fast Fourrier Transform (FFT) is applied to these raw interferograms present very narrow and strong components at the repetition rate f r  of the comb sources  210 ,  212 , i.e. around 17 MHz, and at its harmonics (folded in the 25 MHz of available band). It is noted that in order to reduce those unnecessary components that would mask the beating components of interest between the two frequency comb sources  210 ,  212 , 6 dB-slope low-pass electrical filters at 11 MHz are placed before the data acquisition system  250 . Two mirrored replicas of the source spectrum are expected in the RF domain under f r  and due to the electrical filtering, only the one under f r /2 is considered. 
     When dual mode-locked lasers are used as the comb source, the frequency variations of the source can be expressed in terms of the time variations of the parameters G f (t) and O f (t) where G f  (t) is a scaling factor and O f (t) is an offset. G f (t) and O f (t) are related to f r  and f 0  of the laser comb source. As illustrated in  FIG. 3 , taking into account small temporal changes of the lasers (i.e. of f r  and f 0 ), variations in the mapping between the frequency components f and the beating components b are expected, which can be seen as a bandwidth spreading and shifting for the beating replica in the RF domain. A variation ΔO f  of the parameter O f  induces an offset effect (also called translation) in the mapping function and this effect is mostly due to changes in the parameters f 0,1  and f 0,2  of the mode-locked laser comb sources  210 ,  212 . A variation ΔG f  of the parameter G f  induces a scale effect (also called homothety) in the mapping function and this effect is mostly due to changes in the parameters f r,1  and f r,2  of the mode-locked laser comb sources  210 ,  212 . In the point of view of the Fourier transformation, the translation induces a time dependent phase contribution whereas the homothety modifies the time delay spacing of the acquired interferograms. For a given optical frequency range where no folding problems are expected, the beating spectrum b(f,t) can be expressed as a translated homothety of the optical source spectrum:
 
 b ( f,t )= f·G   f ( t )+ O   f ( t ).
 
For the specific case of the cFTS, the optical referencing technique allows to determine G f  and O f . According to the described technique, two fixed, distinct and predetermined optical source reference frequencies f 1  and f 2  in the source domain are considered. The optical source reference frequencies f 1  and f 2  are defined by the selected optical filters  226  and  232  in the system  200 . The beating component subsets, i.e. beating frequencies b 1 (t) and b 2 (t) in this case in the RF domain are measured for the two fixed source reference frequencies f 1  and f 2 , where t is the measurement time and the beating frequencies are used as a reference to correct the interferograms measured at detector D s . The functions G f (t) and O f (t) can be determined with the knowledge of the two reference beating frequencies b 1 (t) and b 2 (t) as follows:
 
 O   f ( t )=[ f   2   ·b   1 ( t )− f   1   ·b   2 ( t )]/( f   2   −f   1 ), and
 
 G   f ( t )=[ b   2 ( t )− b   1 ( t )]/( f   2   −f   1 ).
 
Using G f (t) and O f (t), the referencing of the whole subset of beating components, is accomplished by the inversion of the mapping function:
 
 f =( b−O   f ( t ))/ G   f ( t ).
 
For the cFTS case, it is however more convenient to work with beating components expressed in beating phases rather than in beating frequencies. Integrating the relation linking optical and beating frequencies over time yields:
 
φ( f,t )= f·G   φ ( t )+ O   φ ( t ),
 
with G φ (t) and O φ (t) being respectively defined as:
 
 G   φ ( t )=2π∫ 0   t   G   f (τ) d τ, and
 
 O   φ ( t )=2π∫ 0   t   O   f (τ) dτ.  
 
Accordingly, the phase gain and offset are calculated as their frequency counterparts:
 
 O   φ ( t )=[ f   2 ·φ 1 ( t )− f   1 ·φ 2 ( t )]/( f   2   −f   1 ), and
 
 G   φ ( t )=[φ 2 ( t )−φ 1 ( t )]/( f   2   −f   1 ),
 
where φ 1 (t) and φ 2 (t) are respectively the beating phases corresponding to source reference frequencies f 1  and f 2 .
 
     This shows that the referencing can as well be performed using the inverse phase relation:
 
 f =(φ− O   φ ( t ))/ G   φ ( t ).
 
     For this application of the provided technique, p 1 (t)=O φ (t) and p 2 (t)=G φ (t) are the two correction parameters required to track and correct the impact of the comb source fluctuations on the beating spectra. 
     The resulting product of the cFTS is a corrected interferogram. It is therefore not required to strictly invert the model in this case. G φ (t) and O φ (t) are simply used to directly compute a corrected interferogram. 
     The variations of the function G φ (t) highlights the fact that the original signal was sampled on an equidistant time grid but not on an equidistant phase grid. A constant phase grid is usually desired in interferometry and especially in Fourier transform spectroscopy. Consequently, for correcting the beating interference signal D s (t), a new phase equidistant grid is defined on the range of G φ (t). Before being allowed to apply a reinterpolation of the cFTS interferogram on the new grid, the contribution of O φ (t) is removed by the multiplication with the correction phase function e −jOφ(t) . This is valid as long as the mapping function exists. 
       FIG. 4  illustrates this referencing technique typically performed in the processing unit  400 . The processing unit  400  receives the beating interference signal D s (t) of the interferogram to be corrected and the reference signals D 1 (t), D 2 (t) acquired at D 1  and D 2  in the system  200  of  FIG. 2 . As described previously, the two reference signals D 1 (t), D 2 (t) correspond to the optical signal filtered in narrow bands around the known reference frequencies f 1  and f 2  to isolate specific beating components. The beating signal D s (t), the reference signal D 1 (t) and the reference signal D 2 (t) are respectively received at a signal conditioner  410 , a first phase analyzer  412  and a second phase analyzer  414 . A model calculator  416  is connected to the output of the two phase analyzers  412 ,  414  to compute the gain G φ  and offset O φ  parameters which are provided to a correction unit  420  for correcting the beating interference signal. 
     In the signal conditioner  410 , an analytic interferogram signal S(t) arising from the beating interference signal D s (t) can be computed. This is done using the Hilbert transform  417 . Some filtering may be performed using band-pass filter  411  before the demodulation. 
     In the phase analyzers  412  and  414 , the beating phases φ 1 (t), φ 2 (t) (or alternately beating frequencies b 1 (t), b 2 (t)) corresponding to reference optical frequencies f 1  and f 2  are determined from the reference signals D 1 (t), D 2 (t). This involves basically a phase (or frequency) demodulation that can be implemented efficiently using a Hilbert transform in  418  and  419 . Some filtering may be performed using band-pass filters  413  and  415  to better isolate the relevant signal before the demodulation. 
     In the model calculator  416 , the gain G φ  and offset O φ  parameters are computed from the determined beating phases φ 1 (t), φ 2 (t), using the relations provided by the physical model given above. 
     The computed parameters G φ (t) and O φ (t) are provided to the correction unit  420  for correcting the interferogram signal S(t). The correction comprises an offset correction  422 , followed by a gain correction  424 . The offset correction  422  is performed by subtracting the offset phase O φ (t) from the analytic interferogram signal S(t) through a multiplication by e −jOφ(t) . The gain correction  424  uses the phase relation G φ (t) to resample the analytic interferogram signal S(t) on an equidistant phase grid (or equivalently on an equidistant optical time delay grid). The result is a corrected complex interferogram S c (φ s ). 
     The corrected complex interferogram S c (φ s ) is output and can be Fourier transformed to obtain a corrected beating spectrum. The computed parameters G φ (t) and O φ (t) are also output. 
     It is pointed out that for cFTS systems where the mapping between the optical and RF domain is a pure homothety (no offset frequency), such as when the CEO of the lasers are cancelled, a single reference beating frequency b 1 (f 1 ,t) is required, which provides the gain function G φ (t). More than two parameters may also be corrected for in applications where it is required, using more than two reference beating components (see  FIG. 7 ). 
     It is noted that in applications where a Fourier transform of the interference pattern is not needed and where only its local phase (or frequency) is of interest, the correction processing can be somewhat different from that illustrated in  FIG. 4 . The correction may then be performed without any resampling if the derivative of the sample phase is extracted. The correction is then a direct inversion of the model and the corrected phase (or frequency) may be reconstructed by integrating the inverted frequency estimate. Monitoring passive fiber optic sensors in distributed networks can be achieved at great speed and sensitivity using such an approach. 
       FIG. 5A  shows the uncorrected spectral response of the beating interference signal S(t) (i) corresponding to the characterized sample and both reference signals D 1 (t) (ii) and D 2 (t) (iii) according to the example embodiment of  FIG. 2  described hereinabove. In this measurement, the central beating frequency is about 2 MHz and the bandwidth is about 0.5 MHz. The significant spectral spreading of the reference signals D 1 (t), D 2 (t) is due to the fluctuations of the comb sources  210 ,  212 . The spectrum response of beating interference signal S(t) and corresponding to the probed sample is also very different from a smooth laser spectrum as would be expected and no clear filtered band due to the sample is discernable. 
     The offset O φ (t) and the gain G φ (t) phases required to correct the interferogram from the sample path are then calculated as described above and the beating interference signal S(t) is corrected.  FIG. 5B  shows the corrected spectral response corresponding to the beating interference signal S(t) (i) and to reference signals D 1 (t) (ii) and D 2 (t) (iii). The corrected spectral response of the reference signals are narrowband and properly located around f 1  and f 2  as expected, and the corrected spectral response of the beating interference signal S(t) is clean, the band filtered by the sample is clearly discernable. 
       FIG. 6  shows the superposition of three corrected spectral responses corresponding to the beating interference signal S(t), presented on a linear scale (top) and on a logarithmic scale (bottom). As expected, the grating notch filtering at 1555.55 nm and corresponding to the sample is observed in the corrected spectra. The optical path length difference range equivalent in a standard FTS for these measurements is larger than 70 mm, which provides a spectral resolution under 0.15 cm −1 . 
     Now referring to  FIG. 7 , the referencing technique described herein can be generalized for use in various applications and with various types of frequency comb sources, not necessarily based on a combination of two mode locked-lasers. The system  700  can be understood as a generalization of the system  200  of  FIG. 2 . 
     The interferometric system  700  of  FIG. 7  is used to analyze a sample  716 . As in system  200  of  FIG. 2 , the system  700  has a comb source  710  which may be, as in the system  200 , a combination of two frequency comb sources each with uniformly distributed frequency components, resulting in a comb with frequency components non-uniformly distributed. More generally, any comb source  710  producing a frequency comb with non-uniformly distributed frequency components may also be used. The comb source  710  is split on multiple paths  712 ,  713 ,  714 ,  715 , one of which, i.e. paths  712 , to probe the sample  716  and the others  713 ,  714 ,  715  reaching respectively the narrowband filters  726 ,  732 ,  736  with central frequencies f 1 , f 2 , f Q  respectively. The filters  726 ,  732  and  736  are used to isolate specific beating components of the frequency comb by each isolating a subset of frequency components corresponding to known reference frequencies f 1 , f 2 , f Q . Detector D s  detects the beating interference signal at the output of the sample  716  while D 1 , D 2 , D Q  are used to measure the reference signals at the output of filters  726 ,  732 ,  736 . The detected beating interference signal and reference signals are acquired using the acquisition unit  750 . 
     The acquired beating interference signal and reference signals are respectively received at a signal conditioner  760  and model calculator  766 . The model calculator  766  generally computes parameters p 1 , p 2 , p P  which are provided to a correction unit  770  for correcting the beating interference signal, to output a corrected beating interference signal. 
     The frequency comb is defined as consisting of N distinct frequency components f n  (n from 1 to N and f n &lt;f n +1). The interference of two different comb components m 1  and m 2  (m 1 &lt;m 2 ) generates a beating signal b m  at their frequency difference (b m =f m2 −f m1 ). The referencing is applicable for a subset of M contiguous beating frequency components b m  (m from 1 to M) that fulfills the following conditions: 
     1) b m &lt;b m+1 ; 
     2) The mapping function that relates the beating component b m  to the frequency f m1  in the interval [f 1 ,f N ] can be expressed using a set of P frequency independent parameters p x  (p 1  to p P ), possibly with the knowledge of the mapping function at a given initial time t 0 ; It is noted that in the specific example described herein in reference to  FIG. 2 , P is equal to 2, the two parameters being the gain p 2 =G f  and the offset p 1 =O f ;
 
3) The parameters p x (t) at a given time t can be determined from at least Q beating frequencies B q  (B 1  to B Q ) that correspond to the mean beating frequency in the interval [b 1 ,b M ] for narrowband filtered versions of the source around the known and distinct frequencies f q , respectively; It is noted that in the specific example described herein in reference to  FIG. 2 , Q is equal to 2.
 
     The referencing is obtained by measuring the beating frequencies B q , calculating the parameters p x  (possibly measuring the initial mapping at t 0 ) and finally inverting the mapping function to translate the beating frequencies into source frequencies. 
     Starting form the specific embodiment described above, several alternative embodiments can be described for a variety of applications or for different comb sources. 
       FIG. 8  illustrates a cFTS application system similar to the system  200  of  FIG. 2 , but where the sample  216  is probed in reflection instead of transmission. An optical circulator  840  is used to probe the sample  216 . For simplicity of illustration only, the acquisition system  250  and the processing unit  400  of  FIG. 2  are combined in a single acquisition and procession box  850  in  FIG. 8 . Other components remain unchanged or equivalent from the system  200  of  FIG. 2  and are therefore not repetitively described. It is noted that a retro-reflecting scheme can also be used for free-space reflecting samples. 
       FIG. 9  illustrates a differential spectroscopy scheme used with a reflective sample  216 . Again, the scheme of  FIG. 9  is similar to the one of  FIG. 2  and similar or equivalent components and arrangements are not repetitively described. In the system of  FIG. 9 , the interference beating signal is split on two optical paths  917  and  918 . The optical path  917  probes the sample  216 , while the optical path  918  remains unfiltered. A balanced photodetector D s  performs the subtraction of the optical signals at the end of the two optical paths  917  and  918 . This allows for a direct measurement of the sample response, provided that the two paths  917 ,  918  to the differential detector are as identical as possible. The same scheme can also be used with a sample  216  in transmission. 
     All the discussion about Fourier transform spectroscopy with frequency combs can be extended to complex spectroscopy where one of the comb source probes a sample prior to the combination with the second comb source. The obtained interference pattern is a convolution between the combined source and the complex impulse response of the sample.  FIG. 10  illustrates a scheme that can be used for complex spectroscopy with a sample  216  probed in transmission. In such a configuration the sample  216  is probed only by the frequency comb generated by the comb source  212 . The other frequency comb, generated by the source  210 , remains unfiltered before being combined with the frequency comb at the output of the sample  216 . It is the resulting combined frequency combs that are detected at detector D s . The remaining of the scheme remains unchanged or equivalent compared to the system of  FIG. 2  and such similar components and arrangements are not repetitively described. Using this scheme the amplitude as well as the phase spectrum of the sample  216  is measured. This can be useful to determine the complex transmittance (absorbance and dispersion) of the sample  216 . 
       FIG. 11  shows an approach similar to the one illustrated in  FIG. 10  but probing the sample  216  in reflection using an optical circulator  840 . Besides the complex spectroscopy application, this embodiment can be used to retrieve the spatial phase and amplitude profile along the depth of the sample  216 . Optical low coherence reflectometry and optical coherence tomography are techniques based upon a similar principle embodied with a classical Michelson or Mach-Zehnder interferometer. 
       FIG. 12  shows another interferometric system that makes use of the herein described referencing technique. The system of  FIG. 12  uses an active filtering technique instead of the passive filters  226  and  232  of the system of  FIG. 12 . Such an active technique involves continuous wave sources and electrical filtering, where continuous wave sources  1222 ,  1224  are used to down-convert a limited number of lines from each frequency comb in the band-pass of an electrical filter  1230 ,  1232 . 
     Similarly to the system  200  of  FIG. 1 , the two comb sources  210  and  212  are combined to interfere using an optical coupler  1213 , resulting in an optical signal probing the sample  216  before being detected at detector D s . 
     In parallel, for referencing, the comb sources  210  and  212  are each split using respectively an optical coupler  1218  and  1220  to be each combined with a first continuous wave sources  1222  and with a second continuous wave sources  1224 . The combination of the first comb source  210  with the first continuous wave source  1222  is detected at detector D 1     —     1 ; the combination of the second comb source  212  with the first continuous wave source  1222  is detected at detector D 1     —     2 ; the combination of the first comb source  210  with the second continuous wave source  1224  is detected at detector D 2     —     1 ; and the combination of the second comb source  212  with the second continuous wave source  1224  is detected at detector D 2     13      2 . The signals detected at D 1     —     1  and D 1     —     2  are then mixed to interfere using a multiplier  1226  and filtered with a low-pass electric filter  1230  in order to isolate one first reference beating component subset. Similarly, the signals detected at D 2     —     1  and D 2     —     2  are mixed to interfere using a multiplier  1228  and filtered with a low-pass electric filter  1232  in order to isolate one second reference beating component subset. This active filtering which uses a continuous wave source is equivalent to a narrow band-pass filtering with a central frequency corresponding to the frequency of continuous wave source  1222  or  1224 . 
     It is noted that in the active filtering scheme of  FIG. 12 , the low-pass electric filters  1230 ,  1232  may also be placed before the multipliers  1226 ,  1228 , i.e. one after each detector D 1     —     1 , D 1     —     2 , D 2     —     1 , D 2     —     2 , instead of after. 
     In another embodiment, the comb source(s) may be replaced by one or several continuous wave lasers in which the several spectral modes have a non-equidistant spacing in such a way as to produce a non-degenerate beating spectrum, such a source is provided for example in U.S. patent application bearing publication no. 2007/0182966 A1. This is achieved, for example, with significant dispersion in the laser cavity. The referencing technique then allows the characterization of the dispersion and its temporal evolution. 
     A non-uniform frequency comb or multiple combs may also be generated with a single mode-locked laser source, using polarization multiplexing or pulse interaction within the laser cavity for example. The referencing technique then still yields a reliable optical referencing. 
     Embodiments involving frequency comb sources of a different nature than optical, such as Terahertz or acoustical combs can also benefit from the referencing technique. In all cases the down-conversion occurring in the beating spectrum process provides a measurement gain proportional to the down-conversion factor. The referencing technique references the beating signal rather than the frequency comb sources and the characterization of the down-conversion relation is accordingly more precise in the beating spectrum domain than in the original source domain. 
     While illustrated in the block diagrams as groups of discrete components communicating with each other via distinct data signal connections, it will be understood by those skilled in the art that the preferred embodiments may be provided by a combination of hardware and software components, with some components being implemented by a given function or operation of a hardware or software system, and many of the data paths illustrated being implemented by data communication within a computer application or operating system. The structure illustrated is thus provided for efficiency of teaching the present preferred embodiment. 
     The embodiments of the invention described above are intended to be exemplary only. The scope of the invention is therefore intended to be limited solely by the appended claims.