Abstract

We report a detailed theoretical and experimental study of fiber-optic cavities under broadband excitation by mode-locked laser combs. We calculate the effects of fiber dispersion on the cavity transmission. For any integer ratio between the comb repetition rate and cavity free spectral range, the theoretical resonant output spectrum exhibits a narrow group of resonant teeth, surrounded by minor, unevenly spaced resonances. Also, the central resonance can be rapidly and precisely tuned over the entire comb span by only acting on its repetition rate. Experimental observations are provided by a single-mode fiber ring and a telecom-wavelength comb laser. The resulting spectral pattern agrees very well with our theoretical prediction, allowing a thorough characterization of the cavity dispersion and opening new perspectives for comb spectroscopy in dielectric resonators.

© 2013 OSA

1. Introduction

Optical frequency combs (OFCs), originally developed for metrology and fundamental physics experiments [14], have recently shown a surprising potential for sensing and spectroscopy applications [58]. In particular, the use of OFCs combined with high-finesse resonators led to impressive advances in broadband molecular gas spectroscopy, thus becoming competitive with more traditional Fourier-transform spectrometers [912]. So far comb-based high sensitivity spectroscopic applications have been implemented mostly with conventional mirror cavities. In these systems, the comb spectrum is matched to a multiple of the cavity free spectral range and the small wavelength dispersion introduced by the cavity mirrors is compensated [9] in order to maximize the optical bandwidth injected in the resonator. Then, simple or more sophisticated devices that spatially disperse the cavity output radiation are necessary for spectral reconstruction of absorption lines or bands. The dispersive element at the cavity output often complicates the spectrometer set-up and poses limitations to the resolution and the minimum detectable signal, although molecular fingerprinting over broad spectral regions has been demonstrated with several detection schemes [913]. In principle, broadband comb spectroscopy could be combined also to optical-fiber cavities, which are particularly suitable for a number of applications, such as remote sensing, liquid-phase spectroscopy or in situ chemical analysis in harsh environments. Unfortunately, in these resonators, it is hard to achieve efficient spectral coupling because the effects of intra-cavity fiber dispersion are much stronger.

In this paper, we theoretically and experimentally investigate the resonance spectrum of an optical fiber resonator under coherent, multiple-wavelength excitation of a mode-locked laser comb. We start from a simple linearly-dispersive model of the cavity material and consider effects of the fiber group-velocity dispersion (GVD) on the comb pulses. Applying dispersion relations to the standard coupling conditions, corresponding to the magic points defined in [13], we obtain analytical formulas that describe the comb transmitted spectrum, which exhibits a heterogeneous resonant structure. In addition, we point out a peculiar ‘leverage effect’ due to the fiber GVD, which allows to tune the magic point over an extremely-broad wavelength interval by only acting on the comb repetition frequency. In this way, intra-cavity dispersion, an unavoidable shortcoming of previous systems, can be used for direct, reliable spectral analysis of the comb light without resorting to any additional dispersive element at the cavity output. All predictions were experimentally verified with a 20-m single-mode fiber ring cavity that was frequency locked to an Er-fiber mode-locked laser comb. The very good agreement between the theory and experimental findings also allowed a precise characterization of the intracavity dispersion.

2. Theoretical model

2.1 Magic resonance conditions

Let us consider a comb of discrete optical frequencies, equally spaced by the quantity ωr, that excite a closed fiber-loop cavity with total length L. The comb frequency components can always be written as ωN=ω0±Nωr, where ω0 is an arbitrary comb tooth. At the same time, the resonant frequencies of the loop are instead defined by the general condition

k(ω)L=2Aπ
with A any positive integer number. A linear expansion of this dispersion relation around ω0 yields
[k(ω0)+1vg(ω0)(ωω0)]L=2Aπ
where vg=ωk(ω)|ω0is the group velocity in the loop. Assuming that the comb tooth at ω0 is frequency locked to a cavity mode, k(ω0)L in Eq. (2) is a multiple of 2π and can be included in the right-hand side term. The resonance condition for any comb tooth N can then be found by replacing ω with ωN in Eq. (2), obtaining
NLvg(ω0)fr=B
where fris the comb repetition rate in natural frequency units and B is still a positive integer. The above equation defines the so-called magic point [13] between the comb and the cavity that is satisfied when
Lvg(ω0)fr=G
where G is a positive integer that will be referred to as magic number in the following. Equation (4) states that if the comb repetition rate is an entire multiple G of the cavity free-spectral-range (FSR) vg(ω0)/L, the resonant condition of Eq. (1) is fulfilled by any comb tooth N. In this condition, a virtually infinitely-broad comb can be coupled to the cavity. Equation (4) defines all the magic numbers when fr ≥ FSR.

2.2 Primary resonance and secondary resonances

In practical situations, the dispersion relation k(ω) can be considered linear only in a small interval around ω0 . To investigate the comb resonances over a wavelength span larger than a few nanometers, the dispersion of the group velocity (GVD) must be taken into account by including a quadratic term in the spectral phase expansion. Equation (2) then becomes

k(ω0)L+1vg(ω0)(ωω0)L+12D(ωω0)2L=2Cπ
with C again a positive integer number. A non-zero group velocity dispersion D = 2k(ω)ω2|ω0has several important implications on the broad resonance of a comb in a fiber loop cavity.

The main effect of GVD is the limitation of the comb resonant bandwidth. It is straightforward to see that if ω0 is a resonant tooth while L and fr are tuned to a magic point, for any comb frequency the first and the second terms in Eq. (5) are multiples of 2π. The GVD term then represents an excess round-trip phase delay φrt(ω) that increases quadratically with the distance from ω0, and causes the comb teeth to gradually mismatch from the cavity resonances. Such a “phase walk-off” effect can be either positive or negative, depending on the sign of the GVD, in both cases resulting in a finite resonant bandwidth around ω0 in the transmission spectrum that we will refer to as the “primary comb resonance” (PCR).

The PCR central wavelength strongly depends on the comb repetition rate. In the presence of GVD, the cavity FSR is in fact chirped, and it can be matched to the comb teeth spacing only in a particular wavelength region. Therefore, the coupling condition corresponding to a magic point depends on the precise value of the repetition rate fr. Quantitatively, this can be seen by generalizing the magic condition to the case with GVD. A dispersion of the group velocity means that 1/vg can be no longer considered constant but a linear function of ω, that is

1vg(ω)=1vg(ω0)+D(ωω0).
By using this expression in Eq. (4), it is clear that when changing the repetition rate value, a magic condition with the same number G is now satisfied by comb teeth at different ω . Let us start from the main resonance atω0 and consider a small variation of the repetition rate δfr. The frequency-dependent magic condition (4) changes to
L[1vg(ω0)+D(ωω0)](fr+δfr)=G
and it is then satisfied by a frequency ω01 such that
ω01ω0=1vg(ω0)Dδfrfr+δfr1vg(ω0)Dδfrfr
Α small repetition rate adjustment (in the order of few kHz) then allows a large, linear wavelength tuning of the PCR, thanks to the huge leverage generated by the D (GVD) term in the denominator of Eq. (8).

The PCR is not the only possible resonance condition for a given magic number G. Indeed, the GVD-induced excess phaseφrt(ω) increases monotonically with the distance from ω0. It is then expected to periodically match a multiple of 2π at fixed distances from the center of the PCR, thus producing an additional set of resonances. We refer to these ones as “secondary comb resonances” (SCRs). SCRs occur for optical frequencies ωM such that

12|D|(ωMω0)2L=2Mπ
In Eq. (9), M is again a natural number and the absolute value of D accounts for the fact that only the amount of walk-off (not its sign) is important. Solving Eq. (9) yields the positions of SCRs with respect to ω0:
ωMω0=±4πM|D|L
Equation (10) gives a full description of the positions of the comb resonances. M = 0 corresponds to the primary comb resonance at ω0, while for any natural number M > 0, a couple of SCRs at distance ±ωM from ω0 are found. According to Eq. (10), the spacing between secondary resonances is not constant but scales as M.

The width of the comb resonances can be calculated by differentiating the GVD phase walk-off term with respect to ω:

Δφrt=φrtωδω=LD(ωω0)δω
This relation indicates that the variation of the round trip phase associated with a frequency variation δωis larger for optical frequencies far from ω0. In the limit where the width of the comb tooth is much smaller than the cavity resonance, the optical bandwidth coupled to the PCR can be readily calculated from Eq. (11). Since a fiber ring is analogous to a flat Fabry-Pérot cavity [14], from the Airy formulas, the transmission of a resonant frequency by an optical cavity is half the peak output level when its roundtrip phase mismatch is ∆φrt = 2arcsin(π/2F) (F is the cavity finesse). To calculate the Half Width Half Maximum (HWHM) of any comb resonance ωM is then sufficient to find from Eq. (11) the δωcorresponding to this value of ∆φrt:
δωHWHM=2arcsin(π2F)LD(ωMω0).
Substituting (ωMω0)with the positions of the secondary resonances given by Eq. (10), we have:
δωHWHMM=2arcsin(π2F)4πM|D|L
In a similar way, the width of the primary comb resonance can be calculated by considering that, when the starting frequency is ω0, (ωω0)δω. In this case Eq. (12) gives:
δωHWHMPCR=2arcsin(π2F)|D|L
Equations (13) and (14), show that the widths of the PCR and SCRs are related to the fiber dispersion and the cavity length, as well as to the cavity finesse. The optical bandwidth coupled to the resonances is largest for the PCR, and it scales inversely with the square root of the resonance order M in the SCRs.

Equations (10), (13) and (14) completely define the position, width and relative amplitude of all the comb resonances in the fiber loop. A simulation of the entire comb transmitted spectrum based on this model is shown in Fig. 1(a). In the theoretical spectrum, the relative amplitude of each resonance is calculated from the number of teeth contained within its FWHM in order to simulate the response of a low-resolution optical analyzer. In this case, the PCR appears much more intense while the intensity of the other resonances scales inversely to √M.

 

Fig. 1 a) Theoretical spectrum of the comb resonances for L ~20 m (λ0=2πc/ω0, c is the speed of light in vacuum). b) Wavelength position of the comb secondary resonances for 3 different fiber cavity lengths using the manufacturer specified D value (0.0196738 ps2/m).

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Finally, it is worth noting the dependence of the secondary comb resonances position on the cavity length L (or equivalently on the magic number G). From Eq. (10), in a very long cavity (large G) all the secondary resonances collapse within a few nm from ω0. On the other hand, in a short cavity (small G) the resonances are pushed tens of nm away from the PCR. This behavior is illustrated in Fig. 1(b), where the distance of the various resonances from ω0 is plotted for three different cavity lengths: L1 = 0.88 m (corresponding to G = 1), L2 = 20.4 m (G = 26) and L3 = 444 m (G = 500).

3. Experimental system

The experimental arrangement is schematically shown in Fig. 2. The resonator element consists of a 20-m single-mode optical-fiber (SMF28 Corning) loop cavity, whose length is finely controlled by a piezoelectric fiber stretcher. Light is coupled into the fiber-loop through low loss evanescent fiber couplers. The system is all-fiber made without any free-space gap. The cavity finesse is about 200, corresponding to a line-width of 50 kHz with a FSR of 10 MHz. The interrogating source is a 80-fs pulsed modelocked erbium-fiber laser (Menlo Systems FC1500) which provides an OFC with an average output power of 30 mW in the region from 1520 nm to 1580 nm and a fr of 250 MHz. The fr is frequency locked to an external synthesizer.

 

Fig. 2 Experimental set-up. EOM: electro-optic modulator; PM: phase modulation; RR: repetition rate; PZT: piezo electric transducer.

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To make the OFC resonating with the fiber loop the spacing between its teeth must be an integer multiple of the cavity FSR. In order to match our cavity, the fr is tuned to be around 26 (i.e. the magic number) times the FSR. In a first approximation, in this condition the whole OFC should resonate with the cavity. Actually, because of fiber dispersion only a limited wavelength interval can be effectively coupled. Hence the fiber cavity acts as a natural dispersive element (FDS). As shown in the previous section, a fixed relation exists between the fr radiofrequency and the FDS center: the operating wavelength can be tuned acting on the fr synthesizer.

The OFC is phase modulated before entering the fiber-loop. Radio frequency (RF) sidebands are simultaneously superimposed to all comb’s teeth. Thus, a Pound-Drever-Hall (PDH) scheme [15] allows locking of the cavity modes to the OFC teeth by the fiber stretcher. At the same time, the fiber cavity is designed to have a magic point within the repetition rate tuning range. Then the cavity and the comb find a natural match at about the center of its emission spectrum (~1560 nm). Resonances centered at different wavelengths are seen when the fr is slightly changed.

4. Results and discussion

In Fig. 3, a typical spectral response of the FDS to comb excitation is recorded with an optical spectrum analyzer (OSA). The central peak on the cavity transmission is due to the primary resonance. i.e. the broadest resonant group of teeth corresponding to the spectral interval selected by the magic condition.

 

Fig. 3 Resonance spectrum of the fiber loop as observed by an OSA (resolution 0.05 nm) when the comb teeth are frequency locked to the cavity modes. The original comb spectrum is also shown (blue line, right vertical axis).

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Besides the PCR at 1558 nm, a large population of narrower and unevenly spaced resonances similar to the secondary resonances of Fig 1(a) can be observed. As discussed above, these extra resonances can be likely attributed to GVD effects on the comb pulses. Their existence and distribution were predicted by Eqs. (4)-(10). The asymmetric power envelope of the corresponding transmission peaks is due to the non-flat comb emission spectrum, as can be seen from the overlapped comb spectrum. As shown in the inset of Fig. 3, the FWHM of the PCR is about 0.7 nm, in good agreement with the 0.9 nm predicted by Eq. (14) for a fiber loop of finesse 200. The amplitude noise on the transmission peak is likely due to acoustic noise on the long fiber loop leading to relative comb-cavity frequency fluctuations that are not completely suppressed by the locking servo. Secondary resonances, as expected, become narrower and narrower when moving farther from the center. Also, a couple of sidebands around each resonance can be noted. These are due to phase modulation of the EOM which is necessary to implement the PDH locking scheme described above.

Equation (10) can be fit to the position of the resonances in the experimental spectrum of Fig. 3 leaving the D term as a free parameter. The very good agreement between the resonance distances and the theoretical values of Eq. (10) can be appreciated from the curve of Fig. 4. A non-linear least-square routine yields the best value of the intra-cavity Dbest = 0.02179 ± 0.00002 ps2/m (relative error = 0.09%). This value is slightly larger than the value given by manufacturer for the fiber alone (Dfiber = 0.0196738 ps2/m). The slight discrepancy may be attributed to the presence of the evanescent couplers as additional dispersive elements of the cavity. The experimental validation of the expression that describes the SCRs position also confirms the possibility to design a cavity where these extra resonances are pushed out of the comb emission span so that only the narrow PCR is transmitted.

 

Fig. 4 Distance of the secondary comb resonances from the central condition (λ0) – Full circles: experimental data. Solid line: best fit with Eq. (10) using L = 20.43 m, resulting in a Dbest = 0.02179 ± 0.00002 ps2/m.

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The possibility of fine tuning the PCR with small variations of the comb repetition rate, predicted by Eq. (8), is also demonstrated. Experimental results are shown in Fig. 5(a). The entire set of automodes excited in the cavity by the broadband comb shifts towards longer wavelengths (or shorter, depending on the sign of D) as fr is increased. Changes of the order of few kHz cause the primary resonance to move by tens of nm, in agreement with the optical leverage effect predicted by Eq. (8). Figure 5(b) shows a comparison between these experimental data and the theoretical prediction confirming a perfect linear dependence on fr change (δfr) as expected from the theory. A linear fit with Eq. (8) using the Dbest value from the previous measurement yields a vg0)best = (1.95 ± 0.03)·108 m/s.

 

Fig. 5 a) Tuning of the comb resonances in the optical fiber loop with repetition rate steps of about 10 kHz. b) Linear fit of the tuning response with Eq. (8), using fr = 249.9184 MHz, Dbest = 0.02179 ± 0.00002 ps2/m, resulting in a vg0)best = (1.95 ± 0.03)·108 m/s.

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Based on the arguments that have been experimentally verified above, the fiber-optic loop resonator behaves as a transmission spectrometer for the optical comb radiation, whose central pass-band wavelength can be finely and precisely tuned over the entire comb span in a few ms by acting on its repetition rate. This aspect makes the presented setup promising for broadband cavity-enhanced spectroscopy applications. In fact, measurements of intra-cavity loss due to chemical species that absorb the comb light (e.g. by evanescent-wave interaction) are feasible with our system. The spectrum of the SCRs can be made very sparse using a relatively-short fiber cavity (L < 1 m) and thus substantially reduce their effect for detection purposes. Thanks to the intrinsically-wide tunability of the primary coupling condition, whole absorption bands can be recovered using the fiber as a spectrometer and miniature probe at the same time. According to Eq. (14), the resolution of the spectrometer can be easily tailored by changing the cavity finesse and the cavity length, respectively. Spectroscopic detection using this scheme is currently under investigation and will be the subject of a future work.

5. Conclusions

We characterize the resonance spectrum of a fiber-optic cavity under broadband excitation by optical frequency combs. An analytical model of the cavity-comb coupling is developed, which predicts an intrinsic spectral filtering action over the comb beam. When a magic coupling condition between the comb and the cavity is met, the transmitted spectrum shows a central resonant region, surrounded by several extremely narrow and unevenly spaced secondary resonances. The width of the central region only depends on the GVD and cavity length, while its position can be finely tuned over the entire comb span by small changes of the repetition rate. Experimental observations are provided by a 20-m single-mode fiber ring and a telecom-wavelength comb laser spanning a 60-nm interval, showing a spectral pattern that agrees very well with the theory. A thorough characterization of the cavity dispersion is possible by fitting the presented analytical expressions to the experimental spectra. We point out that the tunability and resolution of this fiber spectrometer can be tailored by a careful design of the fiber ring material and geometry, thereby enabling cavity-enhanced spectroscopic detection over a wavelength region of hundreds nanometers.

Acknowledgments

The authors acknowledge partial support from Italian Ministry for Education, University and Research (MIUR) in the framework of FIRB RBFR10Q0PT_001 “DROPS” and PON PON01_01209 “BACKOP” projects.

References and links

1. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef]   [PubMed]  

2. J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003). [CrossRef]  

3. N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006). [CrossRef]   [PubMed]  

4. P. Maddaloni, P. Cancio, and P. De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009). [CrossRef]  

5. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007). [CrossRef]   [PubMed]  

6. J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009). [CrossRef]  

7. Y. Bitou, T. R. Schibli, and K. Minoshima, “Accurate wide-range displacement measurement using tunable diode laser and optical frequency comb generator,” Opt. Express 14(2), 644–654 (2006). [CrossRef]   [PubMed]  

8. G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010). [CrossRef]   [PubMed]  

9. M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006). [CrossRef]   [PubMed]  

10. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010). [CrossRef]  

11. R. Grilli, G. Mèjean, C. Abd Alrahman, I. Ventrillard, S. Kassi, and D. Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012). [CrossRef]  

12. C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007). [CrossRef]   [PubMed]  

13. T. Gherman and D. Romanini, “Modelocked cavity--Enhanced absorption spectroscopy,” Opt. Express 10(19), 1033–1042 (2002). [CrossRef]   [PubMed]  

14. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7(6), 288–290 (1982). [CrossRef]   [PubMed]  

15. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]  

References

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  1. T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
    [CrossRef] [PubMed]
  2. J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
    [CrossRef]
  3. N.  Schuhler, Y.  Salvadé, S.  Lévêque, R.  Dändliker, R.  Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006).
    [CrossRef] [PubMed]
  4. P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
    [CrossRef]
  5. S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
    [CrossRef] [PubMed]
  6. J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
    [CrossRef]
  7. Y.  Bitou, T. R.  Schibli, K.  Minoshima, “Accurate wide-range displacement measurement using tunable diode laser and optical frequency comb generator,” Opt. Express 14(2), 644–654 (2006).
    [CrossRef] [PubMed]
  8. G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
    [CrossRef] [PubMed]
  9. M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
    [CrossRef] [PubMed]
  10. B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
    [CrossRef]
  11. R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
    [CrossRef]
  12. C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
    [CrossRef] [PubMed]
  13. T.  Gherman, D.  Romanini, “Modelocked cavity--Enhanced absorption spectroscopy,” Opt. Express 10(19), 1033–1042 (2002).
    [CrossRef] [PubMed]
  14. L. F.  Stokes, M.  Chodorow, H. J.  Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7(6), 288–290 (1982).
    [CrossRef] [PubMed]
  15. R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
    [CrossRef]

2012 (1)

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

2010 (2)

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

2009 (2)

P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
[CrossRef]

J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[CrossRef]

2007 (2)

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

2006 (3)

2003 (1)

J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
[CrossRef]

2002 (2)

T.  Gherman, D.  Romanini, “Modelocked cavity--Enhanced absorption spectroscopy,” Opt. Express 10(19), 1033–1042 (2002).
[CrossRef] [PubMed]

T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

1983 (1)

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

1982 (1)

Abd Alrahman, C.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Avino, S.

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

Bernhardt, B.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Bitou, Y.

Cancio, P.

P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
[CrossRef]

Chodorow, M.

Dändliker, R.

De Natale, P.

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
[CrossRef]

Diddams, S. A.

S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Ferraro, P.

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

Ford, G. M.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Gagliardi, G.

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

Gherman, T.

Gohle, C.

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

Grilli, R.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Guelachvili, G.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[CrossRef]

Hall, J. L.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Hänsch, T. W.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Hollberg, L.

S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Hollberg, L. W.

J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
[CrossRef]

Holzwarth, R.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

N.  Schuhler, Y.  Salvadé, S.  Lévêque, R.  Dändliker, R.  Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006).
[CrossRef] [PubMed]

T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Hough, J.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Jacquet, P.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Jacquey, M.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Jones, R. J.

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

Kassi, S.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Kobayashi, Y.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Kowalski, F. V.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Lévêque, S.

Maddaloni, P.

P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
[CrossRef]

Mandon, J.

J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[CrossRef]

Mbele, V.

S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Mèjean, G.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Minoshima, K.

Moll, K. D.

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

Munley, A. J.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Ozawa, A.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Picqué, N.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[CrossRef]

Romanini, D.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

T.  Gherman, D.  Romanini, “Modelocked cavity--Enhanced absorption spectroscopy,” Opt. Express 10(19), 1033–1042 (2002).
[CrossRef] [PubMed]

Safdi, B.

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

Salvadé, Y.

Salza, M.

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

Schibli, T. R.

Schliesser, A.

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

Schnatz, H.

J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
[CrossRef]

Schuhler, N.

Shaw, H. J.

Stein, B.

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

Stokes, L. F.

Thorpe, M. J.

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

Udem, T.

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Ventrillard, I.

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Ward, H.

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

Ye, J.

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
[CrossRef]

Appl. Phys. B (1)

R. W. P.  Drever, J. L.  Hall, F. V.  Kowalski, J.  Hough, G. M.  Ford, A. J.  Munley, H.  Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J.  Ye, H.  Schnatz, L. W.  Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9(4), 1041–1058 (2003).
[CrossRef]

Meas. Sci. Technol. (1)

P.  Maddaloni, P.  Cancio, P.  De Natale, “Optical comb generators for laser frequency measurement,” Meas. Sci. Technol. 20(5), 052001 (2009).
[CrossRef]

Nat. Photonics (2)

J.  Mandon, G.  Guelachvili, N.  Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[CrossRef]

B.  Bernhardt, A.  Ozawa, P.  Jacquet, M.  Jacquey, Y.  Kobayashi, T.  Udem, R.  Holzwarth, G.  Guelachvili, T. W.  Hänsch, N.  Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4(1), 55–57 (2010).
[CrossRef]

Nature (2)

T.  Udem, R.  Holzwarth, T. W.  Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

S. A.  Diddams, L.  Hollberg, V.  Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

R.  Grilli, G.  Mèjean, C.  Abd Alrahman, I.  Ventrillard, S.  Kassi, D.  Romanini, “Cavity-enhanced multiplexed comb spectroscopy down to the photon shot noise,” Phys. Rev. A 85(5), 051804(R) (2012).
[CrossRef]

Phys. Rev. Lett. (1)

C.  Gohle, B.  Stein, A.  Schliesser, T.  Udem, T. W.  Hänsch, “Frequency comb Vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99(26), 263902 (2007).
[CrossRef] [PubMed]

Science (2)

G.  Gagliardi, M.  Salza, S.  Avino, P.  Ferraro, P.  De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[CrossRef] [PubMed]

M. J.  Thorpe, K. D.  Moll, R. J.  Jones, B.  Safdi, J.  Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311(5767), 1595–1599 (2006).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

a) Theoretical spectrum of the comb resonances for L ~20 m ( λ 0 =2πc/ ω 0 , c is the speed of light in vacuum). b) Wavelength position of the comb secondary resonances for 3 different fiber cavity lengths using the manufacturer specified D value (0.0196738 ps2/m).

Fig. 2
Fig. 2

Experimental set-up. EOM: electro-optic modulator; PM: phase modulation; RR: repetition rate; PZT: piezo electric transducer.

Fig. 3
Fig. 3

Resonance spectrum of the fiber loop as observed by an OSA (resolution 0.05 nm) when the comb teeth are frequency locked to the cavity modes. The original comb spectrum is also shown (blue line, right vertical axis).

Fig. 4
Fig. 4

Distance of the secondary comb resonances from the central condition (λ0) – Full circles: experimental data. Solid line: best fit with Eq. (10) using L = 20.43 m, resulting in a D best = 0.02179 ± 0.00002 ps2/m.

Fig. 5
Fig. 5

a) Tuning of the comb resonances in the optical fiber loop with repetition rate steps of about 10 kHz. b) Linear fit of the tuning response with Eq. (8), using fr = 249.9184 MHz, D best = 0.02179 ± 0.00002 ps2/m, resulting in a vg0)best = (1.95 ± 0.03)·108 m/s.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

k(ω)L=2Aπ
[ k( ω 0 )+ 1 v g ( ω 0 ) (ω ω 0 ) ]L=2Aπ
N L v g ( ω 0 ) f r =B
L v g ( ω 0 ) f r =G
k( ω 0 )L+ 1 v g ( ω 0 ) (ω ω 0 )L+ 1 2 D (ω ω 0 ) 2 L=2Cπ
1 v g (ω) = 1 v g ( ω 0 ) +D(ω ω 0 ).
L[ 1 v g ( ω 0 ) +D(ω ω 0 ) ]( f r +δ f r )=G
ω 0 1 ω 0 = 1 v g ( ω 0 )D δ f r f r +δ f r 1 v g ( ω 0 )D δ f r f r
1 2 | D | ( ω M ω 0 ) 2 L=2Mπ
ω M ω 0 =± 4πM | D |L
Δ φ rt = φ rt ω δω=LD(ω ω 0 )δω
δ ω HWHM = 2arcsin( π 2F ) LD( ω M ω 0 ) .
δ ω HWHM M = 2arcsin( π 2F ) 4πM| D |L
δ ω HWHM PCR = 2arcsin( π 2F ) | D |L

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