OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 4347–4365
« Show journal navigation

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

Philippe Giaccari, Jean-Daniel Deschênes, Philippe Saucier, Jérôme Genest, and Pierre Tremblay  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 4347-4365 (2008)
http://dx.doi.org/10.1364/OE.16.004347


View Full Text Article

Acrobat PDF (1313 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

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

,2

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]

] that is based on the beating spectrum between two closely tuned frequency comb sources. The original idea emerged for THz spectroscopy [3

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

], with some results reported where short pulse lasers and a photoconductive emitter are used to generate the THz combs [4

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]

]. 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

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

]. 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

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

,2

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]

] 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 fr (16.9 MHz in our case), the CEO frequency f0 and the source envelope A(ν) bounded in the optical range around ν0. In addition, the envelope A(ν) is considered slowly varying compared to fr. 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.

Fig. 1. Electrical field for a dispersion-free and stationary frequency comb in the frequency domain (a and b) and in the time domain (c and d); A(ν) is the envelope distribution, νIII(ν) is a shifted Dirac comb, fr is the repetition rate and f0 is the CEO frequency; a0(τ) is the pulse envelope, a(τ) is the fundamental pulse for p=0, τIII(τ) is the iFT of νIII(ν) (only the amplitude is shown here) and ν0 is the mean frequency of the light source.

For a stationary comb source having a constant repetition rate and CEO, the electrical field Em of the mth component in the frequency domain is given by Em=A(νm)·δ(ν-νm), where δ is the Dirac function and νm=m·fr+f0 is the frequency of the mth component. The electrical field Eν(ν) for the whole frequency comb source is simply the sum of all components Em and it can be written as the product of the envelope function A(ν) with a frequency-shifted Dirac comb νIII(ν), Eν(ν)=A(ν)·νIII(ν) where νIII(ν) is defined as:

ν(III)(ν)=III(νf0,fr)=m=δ((νmf0)m·fr).
(1)

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

Eτ(τ)=iFT{Eν(ν)}(τ)=Eν(ν).ei2πντ.
(2)

It is interesting to define the carrier-free envelope function A0(ν) that corresponds to A(ν-ν0). Using the shifting property of Fourier transforms, a(τ) can be written as a(τ)=a0(τ)·exp(i2πν0τ), where a0(τ) is the iFT of A0(ν) that provides the pulse envelope and ν0 brings the very fast modulation present in the pulse as observed in Fig. 1(d). For a dispersion-free light source, the amplitude of a0(τ) is symmetric whereas its phase is anti-symmetric (it becomes a purely real function if A(ν) is symmetric around ν0).

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:

τIII(τ)=1fr·III(τ,1fr)·ei2πf0τ=ei2πf0τfr·mδ(τmfr)=mei2π(mfr+f0)τ.
(3)

The electrical field Eτ(τ) can then be rewritten as:

Eτ(τ)=1frma(τmfr)·ei2πf0mfr=mA(m·fr+f0)·ei2π(mfr+f0)τ.
(4)

From this equation, it is clear that the infinite pulse train is periodic for the pulse envelope (period of 1/fr) but not periodic for the pulse itself due to the constant phase shift of 2πf0/fr that exists between two successive pulses. It is interesting to note that any time shift of m0/fr, m0 integer, only affects Eτ by a constant phase offset: Eτ(τ-m0/fr)=Eτ(τ)·e-iφ where φ=2πf0m0/fr. 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 Eτ 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 Eτ(τ) is measured with an infinite bandwidth detector, the resulting intensity signal ID(τ) is given by:

ID(τ)=Eτ(τ)2=Eτ(τ)·Eτ*(τ)=mnA(m·fr+f0)A*(n·fr+f0)·ei2π(mn)frτ.
(5)

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:

ID(τ)=pei2πpfrτ·[mA(m·fr+f0)A*((mp)·fr+f0)]=pei2πpfrτ·αp,
(6)

where αp are time independent coefficients. The frequency domain description of Iτ(τ) is provided by its Fourier transform ID(ν) that takes a very simple expression ID(ν)=Σαp·δ(ν-p·fr), thus discrete and periodic components at multiples of fr.

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

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 Sx of U in every [x·fr, (x+1)·fr] 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 Sx 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·fr, (x+1)·fr] interval.

The width of the function S is determined from the width of the function U in the optical domain and the scale factor Δfr/fr 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 f0, see Fig. 2(c) to remove the overlap because the width of the functions Sx is smaller than fr/2. It is possible to reduce the width of S by decreasing the value of the repetition rate difference Δfr. The ability to position the replicas Sx using f0 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

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]

]).

Fig. 2. Principle of cFTS; (a) Simplified cFTS example: in the optical band with both the electrical fields of the two frequency comb light sources and in the RF band with the beating components from the interference of neighboring lines from both combs; (b) the RF band is occupied by several replicas of the envelope product S and the internal beatings located exactly at multiple of fr; (c) an adequate change of f0 eliminates the overlap between two replicas if their widths are not larger than fr/2; (d) time domain representation with the relative sliding of the second source pulses (dotted) compared to the center of the pulses of the first source (solid line) that can be viewed as triggers.

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/Δfr. 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

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

] and realized in [5

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

] and in a microscopy context in [7

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

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 [A1(ν), fr, f0] and [A2(ν), fr+Δfr, f0+Δf0], respectively (Δfr≪fr). The constant phase shifts (φ) are ignored for the moment but will be taken into account properly in Eq. (17). The frequency grids ν1(m) and ν2(n) determines the components for both frequency combs and the grid ν1 is also used to define a generic optical grid νopt(m)=ν1(m)=m·fr+f0. The electrical fields in the frequency domain are Eν1(ν) and Eν2(ν), respectively, and they are given by Eν1(ν)=A1(ν)·III(ν-f0,fr) and Eν2(ν)=A2(ν)·III(ν-(f0+Δf0),(fr+Δfr)). The time domain equivalent Eτ1(τ) and Eτ2(τ) are, due to the stationarity assumption, the iFT of Eν1(ν) and Eν2(ν), respectively. The iFT of the envelope functions are a1(τ) and a2(τ), respectively, and those for the frequency-shifted Dirac combs are given by Eq. (3):

Eτ1(τ)=mA1(ν1(m))·ei2πν1(m)τandEτ2(τ)=nA2(ν2(n))·ei2πν2(n)τ.
(7)

The acquired signal in the cFTS method is the time domain intensity ID(τ) of the interference corresponding to the sum of both frequency comb light source fields Eτ1+Eτ2. The description of ID(τ) is made here for an infinite bandwidth detector:

ID(τ)=Eτ1(τ)+Eτ2(τ)2=Iτ1(τ)+Iτ2(τ)+2·Re{Eτ1*(τ)·Eτ2(τ)}.
(8)

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

Iτ(τ)=Eτ1*(τ)·Eτ2(τ)=pei2πp(fr+Δfr)τ·[mA1*(ν1(m))·A2(ν2(mp))·ei2π(mΔfr+Δf0)τ].
(9)

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

A2(ν2(mp))A2(ν1(m)).
(10)

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·Δfr compared to fr (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(νopt(m))=A1*(νopt(m))·A2opt(m)). It is appropriate to define a frequency grid νrf(m) in the RF band that is defined as νrf(m)=mΔfr+Δf0. The frequency axis νrf(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·fr is required. Using U and νrf, Iτ(τ) becomes:

Iτ(τ)pei2πp(fr+Δfr)τ.[mU(vopt(m)).ei2πvrf(m)τ]=pei2πp(fr+Δfr)τ.g(τ).
(11)

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 fr+Δfr≅fr.

It is possible to define continuous versions of the optical and RF axis νopt and νrf, respectively. The relationship between the continuous axis νopt and νrf is:

vopt=(vrfΔf0)·frΔfr+f0.
(12)

The envelope functions A1 and A2 are assumed slowly varying compared to fr, thus the beating envelope functions U can be defined in a continuous formulation and exploiting the relation between νopt and νrf it is possible to define an envelope function S directly in the RF band: S(νrf)=U(νoptrf))=A1*(νoptrf))·A2optrf)). The fundamental intensity g(τ) takes the form of a discrete iFT on a frequency shifted and uniformly spaced grid in the RF band:

g(τ)=mS(vrf(m))·ei2πvrf(m)τ.
(13)

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:

G(v)=mS(vrf(m))·δ(vvrf(m))=S(v)·III(vΔf0,Δfr).
(14)

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

g(τ)=s(τ)*[ei2πΔf0τΔfr·III(τ,1Δfr)]=q[ei2πΔf0qΔfrΔfr]·s(τqΔfr)=qgq(τ),
(15)

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

s(τ)=Δfrfr·u(Δfrfrτ)·ei2π(Δf0Δfrfrf0)τ.
(16)

In this form g(τ) clearly shows the time domain replication of s(τ) every 1/Δfr and the requirement of a constant phase correction for each time replica to take into account the frequency shift component Δf0.

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/Δfr)/Δτcrop) where Rect(τ)=1 for |τ|<0.5 and 0 anywhere else. The crop interval Δτcrop is chosen to balance the amount of acceptable overlap and the desired frequency resolution. The dispersion properties of the envelopes A1 and A2 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 gq and gq+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.

Fig. 3. Time replicas q and q+1 from the fundamental RF replica g(τ).

It is straightforward to demonstrate that all time copies are identical at a constant phase shift: gq(τ)=g0(τ+q/Δfr)·exp(i2πΔf0q/Δfr). It is then possible to redefine the time axis at the center of any cropped time copy:

Iτ,crop(τ)eΔfr·Rect(τΔτcrop)·pei2πp(fr+Δfr)τ·s(τ),
(17)

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 Iν,crop(ν) is given by:

Iv,crop(ν)=eΔfr·sinc(νΔτcrop)*{ps(νp·(fr+Δfr))},
(18)

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 fr/Δfr 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 Iν,FTS(ν) is given by:

Iν,cFTS(ν)=eΔfr·sinc(νΔτcrop)*S(νp·(fr+Δfr)).
(19)

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 Iτ,cFTS(τ) for a chosen RF replica and for a single time copy:

Iτ,cFTS(τ)=u(Δfrfrτ)·ei2π(Δf0Δfrfrf0+p(fr+Δfr))τ·{eiϕfr·Rect(τΔτcrop)}.
(20)

This is the fundamental equation of the cFTS method using shifted frequency comb light sources. The target function U(ν) can be recovered from Iτ,cFTS(τ) with a limited resolution (Ulim(ν)) after a phase correction, a scale compensation and a Fourier transform. If the RF replica index p is known, Ulim(ν) is directly obtained after a simple axis conversion from Iν,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 νmeasrf-p·fr and using Eq. (12) it is possible to define an affine relation between νmeas and νopt:

vopt=frΔfr(vmeas(Δf0Δfrfr·f0+p·(fr+Δfr)))=1Gopt·(vmeasOopt),
(21)

where Gopt and Oopt can be viewed as a gain and an offset factors, respectively, because those factors give νmeas=Gopt·νopt+Oopt. It is also possible to define a RF gain Grf and RF offset Orf that provide the affine relation νopt=Grf·νmeas-Orf and those factors are directly connected to Gopt and Oopt:

Grf=1Gopt=ΔfrfrandOrf=OoptGopt=Δf0Δfrfr·f0+p·(fr+Δfr).
(22)

In the case where Grf and Orf are known, the relation of Eq. (20) allows to directly provide Ulim(ν) from the Fourier transform of Iτ,cFTS(τ) without requiring the a priori knowledge of the laser parameters (fr, Δfr, f0 and Δf0) and the RF replica index p. An option to determine the parameters Grf and Orf 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 Grf and Orf. 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 Ufbgopt) corresponding to the interference of both lasers filtered by the FBG is given by:

Ufbg(vopt)=[A1*(vopt)·r*(vopt)]·[A2(vopt)·r(vopt)]=U(vopt)·R(vopt),
(23)

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(νopt)=Q(νopt0). The envelope function U is assumed slowly varying compared to the bandwidth of the FBG, thus the filtered envelope Ufbg is approximated by it value in ν0: Ufbgopt)≅Q(νopt0)·U(ν0). The iFT of Ufbgopt) is ufbgopt)=q(τopt)·exp(i2πν0τopt)·U(ν0) where q(τopt) is the iFT of Q(νopt). The measured signal Iτ,fbg(τ) is deduced from Eq. (20) using the definition of Grf and Orf:

Iτ,fbg(τ)=q(Grf·τ)·ei2πv0Grfτ·ei2πOrfτ·{efr·Rect(τΔτcrop)}.
(24)

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 φfbg(τ) is given by:

φfbg(τ)=2π(Grf·ν0+Orf)·τ+ϕ·
(25)

The phase function φfbg is only defined in the crop interval Δτcrop and in addition, φfbg is limited on the time bandwidth on which the function q(Gopt·τ) is above the noise floor. The derivative of Eq. (25) provides a first equation for determining Gopt and Orf:

dφfbg(τ)dτ=2π(Grf·ν0+Orf).
(26)

Using two FBGs around two different optical frequencies ν01 and ν02, it is straightforward to determine Grf and Orf 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 Iτ,fbg is estimated around 4·D/c0·Gopt where c0 is the speed of light in vacuum. For stationary sources, the factors Grf and Orf 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 Δfr that modifies the gain factor Grf 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 Δfr during the measurement is viewed in the RF band as a shift of 11.8 MHz that is even larger than fr/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 Grf and Orf during the measurement is sufficient).

Iτ,meas(τmeas)=u(τopt(τmeas))·e(τmeas)·{efr·Rect}.
(27)

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 τopt=0. The relative variations on fr are so small that the factor 1/fr is also considered constant. Only a coarse windowing is used, larger than the desired final boxcar window. The function τopt and θ are better defined as their derivative to the variable τmeas:

dτoptdτmeas=frΔfr=Grf(τmeas),
(28)
dθdτmeas=2π·(Δf0f0·Δfrfr+p·(fr+Δfr))=2π·Orf(τmeas).
(29)

With this formulation it is straightforward to reconstruct the scanning time τopt and the correction phase θ from the knowledge of Grfmeas) and Orfmeas). The fundamental equation becomes:

Iτ,meas(τmeas)=u(0τmeas·Grf(τ))·exp(i2π·0τmeas·Orf(τ))·{efr·Rect}.
(30)

Using the phase φ1 and φ2 of the two FBGs centered at optical frequencies ν1 and ν2, respectively, the factors Grf and Orf are determined from Eq. (26):

1,2(τmeas)meas=2π(Grf(τmeas)·ν1,2+Orf(τmeas)).
(31)

The fundamental equation can be expressed in terms of the phase function φ1 and φ2 and the optical references ν1 and ν2:

Iτ,meas(τmeas)=u(φ2(τmeas)φ1(τmeas)2π(ν2ν1))·ei·ν2·φ1(τmeas)ν1·φ2(τmeas)ν2ν1·{efr·Rect}·
(32)

The distributions τoptmeas) and θ(τmeas) are thus given by:

τopt(τmeas)=φ2(τmeas)φ1(τmeas)2π(ν2ν1)andθ(τmeas)=ν2·φ1(τmeas)ν1·φ2(τmeas)ν2ν1.
(33)

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-ν0). It is straightforward to demonstrate that a phase correction with the phase of one of the reference is the solution:

Inum(τopt)=Iτ,meas(τmeas)·eiφ1(τmeas)=u(τopt(τmeas))·ei2π·ν1·τopt(τmeas)·{efr·Rect}·
(34)

In this form, the Fourier transform of Inum gives U(ν-ν1), that is with a frequency carrier offset of ν1. The optical delay axis τopt is not on a uniformly sampled grid due to the Δfr variations. In order to use FFT algorithms, another uniformly spaced optical delay axis τlin is defined on the same range as τopt. It is appropriate to define the boxcar window at this moment, symmetrically around the position τlin=0 that corresponds to the center of the interferogram (the point of symmetry in dispersion-free conditions). The re-interpolation between Inumopt) and Inumlin) 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 (FC1,2) are amplified (A) and their polarization matched with a polarization controller (PS) prior to be combined with a 50/50 coupler (X3). There are six optical paths that provide six signals measured with six filtered detectors (low-pass under fr/2 for detectors 1 to 4 and band-pass at fr 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 Grf and Orf 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 Δfr around the desired value. The variations of Δfr 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,fr/2] band is done manually for the time being with a small tuning of the laser pumping power that changes f0. Measurements presented in this paper were acquired at values of Δfr around 0.5 Hz.

Fig. 4. Experimental setup: solid lines represent optical fiber paths and dashed lines electrical wires; FC1,2 are the two frequency comb light sources, A’s are erbium amplifiers, PS is a polarization controller, FBG1,2 are the two fiber Bragg gratings, X’s are optical couplers, D’s are detectors, C’s are circulators, F’s are electrical low-pass filters (2nd order at 10 MHz) and P’s are band-pass filters around fr.

The two frequency comb light sources (FC1,2) 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]

]. 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

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

]. 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.

Fig. 5. (a) Reflection intensity (solid lines) of the FBGs used as metrology for the cFTS system; the dashed lines represent the theoretical spectrum for 5 cm-long uniform gratings; (b) autocorrelation function for both FBGs (solid lines) in amplitude and the phase deviation compared to the phase of the shift term exp(i2πfτ) where f is the Bragg frequency of the grating; the dashed lines represent the theoretical autocorrelation curves.

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,fr/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(-iφ1) 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.

Fig. 6. (a) Black lines: acquired interferograms for the two reference FBGs; grey: amplitude of the frequency filtered interferograms and white: amplitude of the interferograms after the second frequency filtering; (b) frequency response of FBG1,2 and of the HCN cell in transmission corresponding to the Fourier transform of the acquired interferograms.

The phase distribution φ1 is extracted from the filtered interferogram of FBG1 and the phase correction is applied to all interferograms. The new interferogram for FBG2 is filtered in the frequency domain to improve the SNR in the time domain and the retrieved phase distribution corresponds to φ21. 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 FBG2, 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 (νrf1)) 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 Δfr 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 FBG2 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 FBG2 (broadening and parasitic peaks).

Fig. 7. (a) Frequency domain response of FBG1,2 and of the HCN cell in transmission after the phase correction; (b) RF frequency for FBG1, RF frequency difference between the two gratings and linear phase deviation from φ21.

The optical time delay axis τopt is determined from the phase difference (Eq. (33)) and a linearly spaced time delay axis τlin is defined on the same time range but with a coarser step corresponding to the desired frequency bandwidth around ν1. 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 τopt grid are interpolated on the linear grid τlin, 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 τlin·c0 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 FBG1 is clean on the entire interval whereas the amplitude of FBG2 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.

Fig. 8. (a) Amplitude in logarithmic scale for the corrected interferograms of the four channels of the cFTS method and presented on a distance scale equivalent the optical path length difference (OPD) used in standard FTS; (b) amplitude response of all four channels after the whole correction process of the cFTS measurements.

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 τlin and the shift frequency ν1. 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

Fig. 9. (a) Amplitude of the corrected interferograms (average on 10 measurements) for the case with nearly no dispersion (top) and with a moderate amount of dispersion (bottom); (b) direct (top) and normalized (bottom) spectra of the HCN sample in transmission in the case of low dispersion (light grey) and in the case of moderate dispersion (black).

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).

Fig. 10. (a) Normalized spectra for 10 cFTS measurements with a moderate amount of dispersion (thin lines) and average curve (thick line); (b) zoom on the same frequency range around an absorption peak for different window width (and thus resolution) and comparison between the result for the 10 cm width window (solid line) and an external measurement for the same spectral resolution (dashed line).

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.

Fig. 11. Top: transmission spectra measured with the cFTS system and with another calibrated instrument (in mirring 200%-transmission); center: slowly varying error from the normalization process; bottom: difference between the cFTS and the external measurements.

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.

References and links

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]

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. 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]

6.

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

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-4 (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, 9029-38 (2005). [CrossRef] [PubMed]
  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. 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]
  6. S. Schiller, "Spectrometry with frequency combs," Opt. Lett. 27, 766-768 (2002). [CrossRef]
  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]
  8. 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]
  9. S. M. J. Kelly, "Characteristic sideband instability of periodically amplified average soliton," Electron. Lett. 28, 806-807 (1992). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited