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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 4845–4858
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High-speed off-axis Cavity Ring-Down Spectroscopy with a re-entrant configuration for spectral resolution enhancement

Jérémie Courtois, Ajmal Khan Mohamed, and Daniele Romanini  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 4845-4858 (2010)
http://dx.doi.org/10.1364/OE.18.004845


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Abstract

Monitoring of changing samples by Cavity Ring-Down Spectroscopy (CRDS) is possible using fast frequency scans of the laser and/or the cavity resonance. Mode-matched cavity excitation improves performance of fast CRDS but data-points result separated by the cavity Free Spectral Range (FSR): low pressure samples demand long cavities. We demonstrate fast CRDS with off-axis injection of a “re-entrant” resonator yielding FSR/N data-points separation. Our N = 4 short-cavity setup is found to perform well compared with other fast-CRDS implementations. Interestingly, the intrinsic chirped ringing affecting ring-down signals in mode-matched fast-CRDS disappear with off-axis injection. This is due to a fine splitting of the re-entrant-cavity degenerate groups of modes by astigmatism.

© 2010 OSA

1. Introduction

Cavity Ring-Down Spectroscopy is among the most sensitive trace gas analysis tool as it allows effective absorption path lengths of several kilometres, depending on the cavity finesse F (L eff = 2F/П). CRDS was originally developed to characterize high-reflectivity (HR) mirrors [1

1. J. M. Herbelin, J. A. MacKay, M. A. Kwok, R. H. Ueunten, D. S. Urevig, D. J. Spencer, and D. J. Benard, “Sensitive measurement of photon lifetime and true reflectance in an optical cavity by a phase-shift method,” Appl. Opt. 19(1), 144–147 (1980). [CrossRef] [PubMed]

,2

2. D. Z. Anderson, J. C. Frisch, and C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23(8), 1238–1245 (1984). [CrossRef] [PubMed]

], but O’Keefe and Deacon [3

3. A. O’Keefe and D. A. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59(12), 2544–2551 (1988). [CrossRef]

] first exploited it with a pulsed laser source as a spectroscopy tool. In their implementation, each laser pulse is coupled by direct transmission through one of the mirrors of a cavity containing the sample of interest. The small amount of stored light propagates back and forth within the resonator while gradually leaking out through the mirrors. The transmitted intensity I(t,ν) exhibits an exponential decay of the initial intensity I0 given by the equation:
I(t,ν)=I0exp(tτ(ν)),
(1)
where τ(ν) is the characteristic ring-down time of the cavity or more simply the trapped photon lifetime. The loss rate per unit distance is then given by:
1cτ(ν)=1R+α(ν).LL,
(2)
with c the speed of light, L the cavity length, R the reflectivity of the mirrors and α(ν) the absorption coefficient (supposed <<1/L). The term (1-R)/L on the right-hand side of Eq. (2) is usually slowly frequency dependent and appears as a baseline offset that can be measured in the absence of the intracavity absorbing medium. The loss rate α as a function of the laser frequency is the CRDS spectrum.

These schemes allow recording absorption line profiles over seconds or minutes by step-scanning the laser frequency while measuring τ at each step. The resulting acquisition times are incompatible with applications dealing with transient phenomena occurring in chemically reacting flows. In our blow down hypersonic wind tunnels case, reacting gas flows are generated during short gusts, typically 200 ms long, with aerodynamic conditions changing by 1% ms−1. For such applications, a high speed acquisition scheme involving fast wavelength tuning is therefore needed.

As Y. He, B. J. Orr [8

8. Y. He and B. J. Orr, “Ring-down and cavity–enhanced absorption spectroscopy using a continuous-wave tuneable diode laser and a rapidly-swept optical cavity,” Chem. Phys. Lett. 319(1-2), 131–137 (2000). [CrossRef]

] and J. W. Hahn et. al. [9

9. J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, and H. Lee, “Cavity ring-down spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38(9), 1859–1866 (1999). [CrossRef]

] have shown, a ring-down signal may be obtained during a rapid and continuous sweep of the cavity length. Cavity injection is obtained when a passage through resonance occurs, without any optical switch. More recently, the first team adopted an equivalent cw-CRDS scheme to record an absorption spectrum during a unique and rapid laser frequency sweep across the comb of the cavity modes during few milliseconds [10

10. Y. He and B. J. Orr, “Rapid measurement of cavity ring-down absorption spectra with a swept frequency laser,” Appl. Phys. B 79(8), 941–945 (2004). [CrossRef]

]. In this approach, it has been shown that as the optical frequency is rapidly swept through a cavity resonance, optical power builds up inside the cavity and then undergoes a ring-down decay. However, the generated exponential-like temporal profile exhibits a chirped beating note superposed to it, due to interference between the stored decaying intracavity field and the incoming field [11

11. Z. Y. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86(1), 51–57 (1991). [CrossRef]

,12

12. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perot interferometer,” J. Opt. Soc. Am. B 16(4), 523–532 (1999). [CrossRef]

]. Theoretical modelling, matching laboratory observations, on high-finesse cavity injection during a sweep through resonance has been carried on accounting for a realistic laser linewidth [13

13. J. Morville, D. Romanini, M. Chenevier, and A. A. Kachanov, “Effects of laser phase noise on the injection of a high-finesse cavity,” Appl. Opt. 41(33), 6980–6990 (2002). [CrossRef] [PubMed]

], i.e. laser field phase fluctuation inherent to spontaneous emission. These highlight the necessity of using a sufficiently spectrally narrow laser conjugated with high speed scanning in order to avoid excess amplitude noise.

Apart from these considerations, in the case of a TEM00 on-axis mode matched injection, fast laser scanning gives a maximum frequency definition limited by the cavity’s Free Spectral Range (FSR = c/2L). Indeed, the cavity FSR imposes a sampling grid: absorbance-dependent ring-down times are measured only at the successive etalon-resonance transmission frequencies. While the spectral resolution of each data point is extreme, and basically given by the cavity mode width (tens of kHz), the definition of the resulting spectrum is inherently limited by the interval between successive resonances. This limitation must be overcome if one intends to perform CRDS measurements using small cavity lengths associated with low sample pressures. As an example, the A band of O2 exhibits, at 10 mbar, Gaussian absorption features with a Full Width at Half Maximum (FWHM) near 3 × 10−2 cm−1 while a 30-cm cavity FSR is 1.66 × 10−2 cm−1. Thus only about 3 data points inside the Gaussian FWHM appear to be not sufficient to determine the line width and intensity in order to derive intrinsic gas properties.

This inherent lack of spectral resolution was circumvented [14

14. Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44(31), 6752–6761 (2005). [CrossRef] [PubMed]

] by the use of an additional piezoelectric transducer (PZT) to vary the cavity length in small stepwise increments (~0.27 μm for a 0.453 m-long cavity thereby shifting the cavity-resonance grid ~0.35 × FSR). Reducing the gaps between data points was then possible by merging a few interlaced spectra. The corresponding recording-time penalty was then addressed by I. Debecker et al. [15

15. I. Debecker, A. K. Mohamed, and D. Romanini, “High-speed cavity ringdown spectroscopy with increased spectral resolution by simultaneous laser and cavity tuning,” Opt. Express 13(8), 2906–2915 (2005). [CrossRef] [PubMed]

] who presented an alternative cw-CRDS design based on fast tuning of the laser frequency combined with a simultaneous sweep of the cavity length. By scanning in opposite directions both the laser mode and the comb of TEM00 cavity modes with the help of a PZT, spectral coincidences occur more closely spaced than for a fixed cavity. They experimentally demonstrated a reduction of 17% for the FSR of a 50 cm long cavity with the help of a 5 μm cavity sweep range and a conjugated ~0.6 cm−1 laser frequency sweep in less than 2 ms.

Here, we investigate an alternative approach allowing much larger spectral resolution enhancement. This is based on the off-axis injection using a fast frequency sweep of a cavity with fractionally degenerate modes (we will refer loosely to a “fractionally degenerate cavity” in the following) where we take advantage of its transverse mode structure. In such a cavity configuration, groups of degenerate transverse cavity modes are separated by an integer fraction of the cavity FSR, allowing for a substantial refinement of the spectral sampling grid.

2. The re-entrant ring-down cavity

To understand the fractionally degenerate case of a resonator, we start from the well known expression for eigenfrequencies νq,mn [19

19. A. E. Siegman, “Lasers” (University Science Books, Mill Valley, CA, 1986).

] of a spherical two mirror paraxial resonator which is given by:
νq,mnSpherical=c2L(q+(m+n+1)θ2π),
(3)
where θ=2arccos(g1g2) represents the Gouy phase shift accumulated by the TEM00 mode over one cavity round-trip and g1/2=1L/R1/2 is the geometric cavity parameter. The resulting spectrum consists, for every transverse mode of indices m and n, of combs of longitudinal modes (as a function of q) with the same basic spacing given by νq+1,mnνq,mn=ΔνL=c2L, called the Free Spectral Range (FSR). On the other hand the transverse mode spacing is ΔνT=ΔνL.θ2π. In principle, the sum of all mode combs may lead to a high density mode structure (limited by the transverse size of the mirror), if no special assumption is made about the Gouy phase, i.e. about the cavity length for a given mirror curvature. In practice, however, only a limited number of cavity modes will receive an appreciable amount of injected intensity as a scanning laser frequency passes through resonance with them, and that depends on the value of the overlap integral between each transverse mode profile and the laser beam profile [20

20. K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105(23), 10263–10277 (1996). [CrossRef]

]. This provides a transverse mode selection mechanism. Since external-cavity diode laser (ECDL) output beams are often close to a fundamental Gaussian mode, it is convenient to optimise injection to the TEM00 cavity modes, which also have a simple Gaussian profile. This ensures the best signal-to-noise-ratio (S/N) on cavity transmitted signals.

If we consider, now, a cavity length (depending on the mirror curvature) leading to a rational value of the TEM00 round-trip Gouy phase shift, that is, θ = 2πK/N where K and N are two integers with no common factor verifying 0<K/N<1 so as to satisfy the stability condition, the eigenfrequencies present the characteristic form:
νq,mn=c2LN(Nq+K(m+n+1)).
(4)
For such a re-entrant or “magic-length” resonator, the cavity exhibits degeneracy involving different transverse and longitudinal mode families. Indeed, as previously shown [21

21. I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

], incrementing ‖decrementing| the longitudinal-mode index q by K while simultaneously decrementing (incrementing| the sum of the transverse-mode indices (m + n) by N leaves the frequency unchanged. Consequently, by considering, e.g., an on-axis multimode mismatched input beam or equivalently, as seen below, an off-axis Gaussian beam, we find that there will exist N degenerate groups of modes being excited between two subsequent longitudinal resonances. This resonator, involving a high degree of frequency degeneracy, therefore allows the spectral sampling grid of a CRDS absorption spectrum to be increased N times.

Of course, in the general case, no special assumptions are needed on the cavity length to observe excitation of cavity transverse modes but, in the magic-length resonator case, their excitation will produce a transmitted pattern of equally frequency-spaced peaks rather than an irregular sequence of peaks whose structure depends on the incident beam profile and alignment. This is illustrated by the experimental cavity transmission spectra of Fig. 1
Fig. 1 Experimental non mode-matched multimode cavity transmission signals for a laser scan spanning one cavity FSR, for cavity lengths of (a) ~23.8 cm and (b) ~29.3 and for 1-m mirror curvature. For this last case of a magic-length resonator, with (N = 4, K = 1), transverse modes are degenerate in frequency at 4 positions equally dividing the cavity FSR. That is, only 4 distinct equally spaced resonances are observed, independently of the incident laser beam geometry
.

Disordered excitation of transverse modes as shown in Fig. 1(a) makes it difficult to extract the decay time of each mode in contrast to when the cavity is mode matched to selectively excite only the TEM00 modes. In the case of a magic-length resonator, as shown in Fig. 1(b), the TEMmn cavity mode structure is fractionally degenerate and perfectly regular as in the mode matched case, only the apparent mode spacing is reduced (by a factor 4 here).

It is interesting and relevant to connect here the fractionally degenerate resonator case, described above by wave-optics, to the case of an off-axis re-entrant spherical multipass resonator described by ray-optics [18

18. D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-Axis Paths in Spherical Mirror Interferometers,” Appl. Opt. 3(4), 523–526 (1964). [CrossRef]

]. In this latter representation, the off-axis paraxial ray follows a trajectory that will close on itself after N cavity round-trips, irrespective of the input beam slope injection. This is a direct result of the fact that in the ABCD-matrix formalism (ruling Gaussian mode propagation and ray propagation as well) the magic-length resonator is satisfying the identity rule after N cavity round-trips. If the re-entrant condition is satisfied and the injection is sufficiently off-axis, when the laser frequency is tuned across the cavity FSR one obtains N resonances equally strongly excited (neglecting both cavity and laser source jittering). Each of these resonances corresponds to an intracavity Gaussian-beam trajectory closing onto itself after N round-trips, following the trace of a re-entrant ray path. In the frequency domain, the situation looks as if one had stretched the cavity by a factor N, except that each resonance transmits only 1/N of the input intensity injected. In fact, in the case of a lossless cavity and a monochromatic laser beam, the N-folded linear trajectory transmits N light beams possessing 1/N 2 of the incoming intensity, as required to conserve energy (note that at the input mirror we also have N-1 output beams of field amplitude 1/N, plus the direct reflection beam carrying a field amplitude (1-1/N)...). As a compensation for this degraded impedance matching leading to an N-fold reduction of the signal at cavity output compared with the mode matched case, the off-axis injection leads to collect N times as many data points for an identical frequency scanning interval. Indeed for an ideally optimal shot-noise limited setup, while the spectral resolution is greatly improved, the signal reduction of N leads to a S/N reduced by N which is exactly compensated by the acquisition and averaging of N times more data points in the same time frame. In the present case, as the fluctuations of spectral data points are larger (1% in the best case of a complete off-axis injection, as discussed below for Fig. 4
Fig. 4 Experimental absorption profiles obtained from three distinct multimode cavity injections schemes (more or less multimode on-axis injection). For a better understanding, two of the profiles are shifted from the zero baseline.
) than what expected from the low noise level of ring-down events (0.2% fit error is found on the ring-down time parameter for the exponential fit of data such as those of Fig. 6(a)
Fig. 6 The experimental cavity ring-down signals on the left, obtained with off-axis (top) and on-axis(bottom) excitation are compared with simulated signals (right curves) obtained respectively by the transient excitation of several transverse modes slightly split (by a weak astigmatism), which are thus excited at slightly different times, and by the transient excitation of a single mode (equivalent to several perfectly degenerate modes). Insets: Logarithm of the signals showing single exponential decay.
), the performance of our injection scheme is limited by other factors than the reduced signal level.

3. Experimental procedures and results

Our experimental scheme is presented in Fig. 2
Fig. 2 Experimental setup for high speed CRDS measurements. ECDL: external cavity diode laser. Fd: Fibredock. The scheme distinguishes two different experimental procedures in order to take advantage of the cavity transverse mode structure. The first setup concerns the on-axis multimode excitation by using a multimode optical fibre permitting to easily transfer incident radiation to a few low-order transverse modes, while the second scheme involves an off-axis excitation. In both cases the cavity length is taken at the N = 4 re-entrant order.
. We use an ECDL (Toptica DL-100) emitting around 766 nm as a narrow-linewidth cw tuneable source. The system includes a Faraday optical isolator to prevent undesired optical feed-back and a fibre coupling system. The fibre core diameter is specified to be 9 μm and allows propagation of several transverse modes at our working wavelength, however it can be injected to obtain a rather clean Gaussian mode output. As explained below, we exploited this multimode fibre also to produce a non TEM00 beam for testing multimode on-axis cavity injection.

The laser frequency is rapidly swept over almost 0.5 cm−1 with a 200-Hz triangular modulation (4.2 THz s−1) to cover one O2 absorption line profile at atmospheric pressure. For wavelength calibration, we use a 23.5 cm vacuum-spaced Fabry-Perot low finesse cavity. As the laser frequency is widely swept over the O2 absorption line, a sequence of ring-down events, corresponding to successive passes through the cavity resonances, is recorded. Each ring-down decay is then numerically processed under Labview by a nonlinear Levenberg-Marquardt exponential fit to extract the linear absorption coefficient α(ν). At this stage, we exclude (depending on the multimode excitation scheme employed as discussed below) the first ~1 μs of the exponential decay where a chirped fast oscillation may be present (see discussion on this point later below).

The empty cavity decay time, τ0 = 3.78 μs, is deduced from the baseline on the wings of the absorption line. As presented on the experimental scheme, we tested two different cavity injection schemes yielding multimode ring-down spectra, using the same ¼-fractionally degenerate cavity.

The first set up concerns cavity injection with an on-axis spatial multimode beam obtained by stressing a multimode optical fibre. Given its Numerical Aperture value (N.A.) 0.11 and a core diameter of 9 μm, one can estimate that the net output resulting field will be a weighted sum of at most 8 propagating transverse modes (depending on the physical stress). While an aligned on-axis axisymmetric beam (say a mismatched Gaussian TEM00 beam) with input values for the spot size and radius of curvature that do not match those of the cavity modes will excite (to a varying extend) only even cavity eigenmodes [20

20. K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105(23), 10263–10277 (1996). [CrossRef]

], on-axis injection with a “slightly” multimode beam will ensure excitation of both even- and odd-order cavity modes.

The second procedure, involving off-axis injection, allows exciting superpositions of degenerate groups of high order transverse modes as the laser come into resonance with them (Fig. 3(b)
Fig. 3 Experimental CRDS transmission for a ~29.3-cm long 4-times re-entrant cavity with (a) a TEM00 injection and (b) an off-axis multimode injection during identical laser scans (~4.6 THz.s−1) and time interval (3.1 ms). Taking advantage of the cavity transverse modes structure leads to increase the spectral resolution by a factor corresponding to the re-entrant order. In (c) is displayed an absorption profile of atmospheric oxygen at 1 bar obtained in 2.3 ms thanks to this ¼-fractionally degenerate cavity in the particular off-axis injection case.
). As an example, reasonable off-axis injection conditions (given our 12.7 mm mirror diameter) involve input beam slopes of ~2.5 × 10−3 rad and ~-6 × 10−3 rad for the x and y transverse direction, respectively, and input coordinates (x0,y0) = (−2.5 mm,0 mm) in order to obtain at cavity output Lissajous patterns contained in an approximately squared region of size x0.

Independently of the experimental multimode scheme employed, the resulting spectral definition is improved by a factor which equals the re-entrant order N employed. To illustrate this, the monomode TEM00 excitation of a (N,K) = (4,1) magic-length resonator, depicted in Fig. 3(a) (transverse modes are here coupled at less than 1%), is compared with the recording of Fig. 3(b) which corresponds to a multimode injection (identical laser scans). Figure 3(c) displays an example of an off-axis multimode absorption line profile generated within a single fast sweep (2.3 ms) of the laser frequency. Absorbance values are sampled at intervals corresponding to a quarter of the cavity FSR. The (rms) scatter of data points relative to the Lorentzian curve of best fit, i.e. the residual for the spectrum, indicates that the minimum detectable absorption for a single laser sweep is ~3 × 10−6 cm−1. This yields a data-rate normalized minimum detectable absorption loss of 1.6 × 10−8 cm−1.Hz-1/2 given a 36 kHz repetition rate for the ring-down events.

4. Discussions

4.1 Multimode fractionally degenerate cavity excitation: two injection schemes

As illustrated above, multimode excitation of a fractionally degenerate cavity may either be performed by a mismatched and even multimode beam aligned along its axis or by a strongly off-axis Gaussian beam.

The comparison of the spectral noise level resulting from these two configurations, which will determine the minimum possible detectable absorption, makes us prefer the off-axis injection. Figure 4 presents three profiles of an absorption line belonging to the (0,0) vibrational transition of the weak b1Σg +←X3Σg - electronic “A” band of molecular oxygen. These profiles of the PQ(25) line at 13023.079 cm−1 are measured during a single fast (~2.5 ms) laser scan without any averaging. At atmospheric pressure, the Lorentzian pressure broadened profiles have a FWHM around 0.1 cm−1, while the effective “mode spacing” for the magic-length resonator corresponding to a ¼-fractionally degenerate cavity is 4.267 × 10−3 cm−1 (for different configurations of cavity injection).

We see from Fig. 4 that, depending on the injection scheme, structured noise on the CRDS spectra vary significantly (up to Δτ~0.6 μs, i.e. ΔR ~5.5×10-5 for the on-axis multimode injection). From the first 10 points of the spectra, we obtain relative ring-down time fluctuations Δτ/τ of ~1.3 %, ~3.5 % and ~7 %, respectively from off- to on-axis multimode injection. These last large variations may be understood by the fact that, due to a mirror surface inhomogeneity, different transverse modes do not experience the same losses, which is a well known property of high finesse cavities. These mode-to-mode ring-down time variations present a periodicity over one cavity FSR, since the same superposition of transverse modes is attained after the laser tunes one FSR, if its beam profile is unchanged. Indeed, after one FSR the laser goes through resonance with the same group of degenerate transverse modes (but with a longitudinal quantum order changed by +/- 1), and it presents the same projection coefficient on each mode of the group. The same injection and decay profiles are then produced, if the effects of laser phase noise are negligible (as in the present case).

Particular attention is focused in Fig. 5
Fig. 5 On-axis multimode cavity injection for the same cavity configuration as in Fig. 3. (a): Ring-down time measurements without the absorber (green round points) reveal the periodic reflectivity dependence, whose period is identical to the cavity re-entrant order. In this same graph is illustrated, with the blue square points, the improvement in sensitivity when the 4 interlaced traces are translated to minimize their relative offset. (b): Due to coating inhomogeneity, each of the interlaced groups of modes [(m + n) modulo N] = 0..N-1 does not see the same reflectivity, thus absorption line profile appear very noisy if no correction is applied. In (c) we show the noise reduction on absorption spectra after minimising the loss difference between the N groups of modes.
on the on-axis multimode excitation. We can see that it is possible to separate the interlaced spectra corresponding to the N = 4 groups of degenerate modes. Each of these interlaced spectra has its own baseline cavity losses which are given by the losses experienced by its specific distribution of the intracavity field over the reflective surfaces of the mirrors. It is important to underline that the additional losses produced by the intracavity sample absorption are the same for each of these spectra. Since losses simply add, once the recorder ring-down values are converted into losses by the basic CRDS relation in Eq. (2), it is perfectly acceptable to try equalizing the offset losses of the N interlaced spectra in order to obtain a compound high-resolution spectrum with reduced periodic noise. As Figs. 5(a) and 5(c) show, this correction leads to an appreciable sensitivity improvement. Figure 5(c) presents two absorption spectra resulting from the recording of 136 ring-down events over ~2.7 ms respectively from raw and corrected data. The detection limit (corresponding to the (rms) noise-equivalent absorption) is found to be ~9 × 10−5 cm−1 and ~3 × 10−5cm−1 before and after the mode-to-mode calibration, or 4 × 10−7 cm−1 Hz-1/2 and 1.3 × 10−7 cm−1 Hz-1/2 respectively given the 50 kHz acquisition rate of ring-down events.

Even if correcting for the systematic and periodic ring-down variations permits an improvement in the detection limit, still better results are achieved with the off-axis multimode injection scheme, needing no special data processing. An order of magnitude improvement in the detection limit is observed, down to 3 × 10−6 cm−1, according to the line profile presented in Fig. 3(c), i.e. a bandwidth-normalized detectivity of almost 1.6 × 10−8 cm−1 Hz-1/2 for the 36 kHz data rate. To put our sensitivity in perspective, we compare our spectrometer with the similar high speed CRDS setup developed by Y. He and B. J. Orr [14

14. Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44(31), 6752–6761 (2005). [CrossRef] [PubMed]

], even if one important difference is that they employed an optical heterodyne detection scheme (OHD). For the detection of CO2 at ~1.54 μm, they used mirrors with reflectivity specified ~0.9998% at that working wavelength and placed at 45.3 cm from each other giving an empty-cell ring-down time of about 10 μs. Using a cw tuneable diode laser source, they obtained recordings of 21 ring-down events over about 3 ms. Their detection limit is estimated at ~9 × 10−8 cm−1 or a data-rate-normalized minimum detectable absorption of 1.1 × 10−9 cm−1 Hz-1/2. Although our detection limit is more than one order of magnitude worse, our ring-down time (proportional to the effective path length) is about 3 times shorter. Thus, we are within a factor 3 or 4 from the performance demonstrated by using a more complex and in principle more sensitive OHD detection scheme for fast-CRDS. However, given the high S/N of our ring-down recordings visible in Fig. 6 below, it is clear that we are not limited by low cavity output signal but by other noise sources, which in the future we should be able to reduce.

It is important to underline the relaxed requirements with regard to adjustment of both the cavity length and the optical alignment when using off-axis cavity injection. Indeed, the fact that cavity modes cannot be perfectly degenerate due to astigmatism, to which an error in cavity length may just add a contribution, does not appear to be critical in fast CRDS. It is sufficient that the time of passage through a group of quasi-degenerate modes be shorter than the ringdown time, which is granted using a fast laser sweep. The excitation of a group of quasi-degenerate modes should then be considered in the impulsive limit. It appears, however, that the transverse size of mirrors is important since periodic patterns in the spectra are reduced when using cavity injection further from the cavity axis.

4.2 Cavity transient response profile (ring-down) behaviour

Another important feature relative to the off-axis injection concerns the unexpected disappearance of the ringing normally observed for on-axis injection as presented in Fig. 6. As noted above, in the case of a sufficiently narrowband laser, the output transient response presents a decay with chirped oscillations originating from the beating of the built-up intracavity field with the progressively detuned laser field. Some authors exploited this property, as in the case where a swift mirror displacement is used to set the laser out of resonance, in order to measure the cavity finesse and/or the mirror speed [11

11. Z. Y. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86(1), 51–57 (1991). [CrossRef]

,25

25. K. An, C. Yang, R. R. Dasari, and M. S. Feld, “Cavity ring-down technique and its application to the measurement of ultraslow velocities,” Opt. Lett. 20(9), 1068–1070 (1995). [CrossRef] [PubMed]

27

27. J. Poirson, F. Bretenaker, M. Vallet, and A. Le Floch, “Analytical and experimental study of ringing effects in a Fabry–Perot cavity. Application to the measurement of high finesses,” J. Opt. Soc. Am. B 14(11), 20811–22817 (1997). [CrossRef]

].

In CRDS with on-axis injection, this chirped beating note is a limiting factor for accurate exponential fitting. A larger ringing-free ring-down portion can be obtained by increasing the laser tuning speed, but at the price of reduced cavity injection efficiency.

In the off-axis scheme the exponential decay turns out to be smooth without the need for increasing the frequency tuning speed. Our explanation is that this originates from the averaging of the transient excitation of not completely degenerate transverse modes due to slight cavity astigmatism, unavoidable in normal experimental condition. We note in particular that the specified surface quality of our mirrors is λ/10 (taken at 630 nm usually), which gives the max deviation of the mirror surface from a perfect sphere. If we take this error over the mirror diameter (half inch) we estimate a maximum allowable error on the 1 m mirror curvature radius of as much as 2 mm, enough to produce a splitting of the re-entrant cavity modes larger than their spectral width. On the right side of Fig. 6 we show simulations in good agreement with observed recordings (left side in Fig. 6). The weak astigmatism can be taken into account by distinguishing two radii of curvature rX and rY corresponding to the curvatures along the two axes. Eigenfrequencies are then different from those of Eq. (3) and are given as:
νq,mnAstigmat=c2L(q+(m+12)θrX2π+(n+12)θrY2π),
(8)
where it appears that each dimension x, y has its own Gouy phase expressed as θrX,Y=2arccos(gX,YgX,Y). This picture which supposes that the axes of the two astigmatic mirrors are parallel is perfectly suitable here. In the general case where the small astigmatisms of the two cavity mirrors do not have the same orientations, one can show that the problem is equivalent to the first order in the curvature differences to that of a cavity with only one defective mirror whose astigmatism is oriented at some intermediate angle. Apart from such considerations, according to Eq. (8) the modes sharing an identical transverse order S = m + n can no longer be degenerate when astigmatism is present. The different rate of change of the two Gouy phases with L implies that for some values of L, each group of initially degenerate modes split by astigmatism may become re-organized into several closely spaced degenerate subgroups, forming a “quasi-resonance”. Thus, for each of the N resonances dividing the re-entrant cavity FSR, the astigmatic cavity displays a quasi-resonance, i.e. a packet of much more finely spaced sub-resonances (the width of the packet is limited since in practice one cannot consider arbitrarily large m, n values). If we additionally consider the injection of the transverse modes by a Gaussian laser beam, we find that the sub-groups of a quasi-resonance possess a bell-shaped excitation envelope (in frequency domain). By the way, this envelope corresponds to the energy distribution for a “coherent” state. For an injection close to the optical axis, the quasi-resonance envelope collapses to only one degenerate sub-group, but the more off-axis is the injection, the more sub-groups are excited and the bell shaped spectral envelope of the quasi-resonances becomes wider and wider.

If we go back to the observation of sharp Lissajous-like patterns as L is tuned, our simulation shows that these occur just right when the split modes composing a quasi-resonance become degenerate in sub-groups. For intermediate cavity lengths the excitation of a quasi-resonance still gives a bell shaped envelope but with a disorganized fine structure. As the split modes composing the quasi-resonance are not regrouped, the peak intensity is strongly reduced for these intermediate cavity lengths. However the cavity transmitted signal for a tuned laser would always display a peak of the same intensity until the bell shaped excitation envelope becomes broader than the laser (short-term) linewidth. This may occur when L is far enough from the magic-length or when the injection is sufficiently off-axis. Thus the mechanical precision on the adjustment of L needed to observe sharp resonances like in a simple re-entrant case is also function of the laser linewidth. And the laser linewidth should be considered over the timescale of the duration of the passage through resonance. For fast laser tuning the linewidth may even be Fourier Transform limited over this short time.

Our model calculation accounting for the observed behaviour of the ringing effect is based on the numerical analysis developed in reference [13

13. J. Morville, D. Romanini, M. Chenevier, and A. A. Kachanov, “Effects of laser phase noise on the injection of a high-finesse cavity,” Appl. Opt. 41(33), 6980–6990 (2002). [CrossRef] [PubMed]

] (curves on the right in Fig. 6). It takes into account a weak astigmatism and supposes an intracavity field formed by the superposition of several transverse modes composing a quasi-resonance.

An experimental measure of the weak astigmatism has been extracted from the selective beating note of TEM01 and TEM10 modes. With a proper cavity alignment one can ensure the injection of mostly these two modes (none of the other [(m + n) modulo 4] = 1 modes sharing the degeneracy) and deduce the surface deviation from a perfect sphere thanks to the oscillation period appearing on the ring-down event. To create and observe the transverse beat frequencies of the two modes split by astigmatism, it is enough to block part of the resulting transverse pattern before the photodiode [22

22. J. B. Goldsborough, “Beat frequencies between modes of a concave-mirror optical resonator,” Appl. Opt. 3(2), 267–275 (1964). [CrossRef]

]. Given the measured 3.7 μs for the beating note period, we then find a |rY-rX| value of 2 mm relative to the specified curvature value r = 1 m. This weak astigmatism (only 0.2%) will produce a splitting of sub-resonances inside a quasi-resonance of about 270 kHz, which is hardly resolvable by an ECDL laser considering our typical laser tuning speed of 4.2 THz.s−1. Those sub-resonances will then be excited at different instants separated by ~65 ns. For a modest off-axis parallel injection at 1 mm distance from the cavity axis, the FWHM of the quasi-resonance spans about 6 sub-resonances, and the total time of passage through the quasi-resonance will be about 400 ns, which corresponds to an FT-limited (Gaussian) laser linewidth of about 2.5 MHz. Since the linewidth of ECDL lasers over much longer timescales is sub-MHz, we may be confident that this FT limit is the effective laser linewidth during the transient excitation. It is not surprise that the superposition of the transient responses of several shifted sub-resonances (each simulated as a single mode transient injection) washes out the ringing present in each individual transient response. Clearly, the mode splitting may have other causes such as the cross-coupling of all transverse modes by light scattering on mirror defects [28

28. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006). [CrossRef] [PubMed]

,29

29. T. Klaassen, J. de Jong, M. van Exter, and J. P. Woerdman, “Transverse mode coupling in an optical resonator,” Opt. Lett. 30(15), 1959–1961 (2005). [CrossRef] [PubMed]

]. However our observation of Lissajous figures nicely reproduced by our astigmatic cavity model is a compelling reason for believing this is the correct interpretation of the disappearing ringing effect for an off-axis excitation.

Moreover it is consistent with the fact that oscillations reappear when the laser is aligned close to the optical axis, even if with a multimode beam, as presented in the recording of Fig. 6(b). The small number of low-order modes co-excited is here not sufficient to induce the averaging out of the ringing.

5. Conclusion

We explored the use of multimode re-entrant high-finesse cavity configurations in order to decrease the spectral sampling interval in CRDS. This was coupled with fast laser scan cavity injection, allowing millisecond timescale acquisition of complete absorption line profiles as needed for monitoring fast environment changes in molecular gas flows. We have shown that using this type of cavity configuration leads to periodic noise structures on the CRDS spectra if the injection is performed close to the cavity axis, even if multimode. Processing independently each structure allows removing most of this periodic noise. However better performance, without numerical post-processing, is obtained by off-axis injection of superpositions of high-order transverse cavity modes. This leads to a detection limit of about 3.10−6 cm−1 over a single 2.5 ms laser scan and for 3.7 μs ring-down-time cavity, which translates into a bandwidth normalized baseline spectral noise level of 1.6 × 10−8 cm−1 Hz-1/2.

We have probably achieved the highest ring-down repetition rates so far reported, up to 50 kHz, practically limited by the tuning rate of our ECDL. Still higher repetition rates together with smaller spectral sampling intervals would be feasible by using a cavity with a re-entrant order larger than 4, which would require changing the mirror curvature in order to maintain the same cavity length.

A strong feature of the scheme presented is robustness with respect to alignment (multimode strongly off-axis injection) and also with respect to cavity length variations around the chosen degeneracy point. Indeed for variations of up to 1 mm in our case, mode splitting does not produce a sizeable distortion of the ring-down events which remain perfectly mono-exponential (mode beatings occur over timescales longer than the ring-down time). In this regime of injection, we also observed an unexpected absence of the ringing oscillations which we explain (as modelling confirms) by the averaging of modes slightly split from the degenerate case. Mirror astigmatism appears to be the principal reason for a basic splitting which does not allow obtaining perfect mode degeneracy by adjusting the cavity length to an expected re-entrant configuration.

Acknowledgments

The authors would like to acknowledge the support of the French space agency “Centre National d’Etudes Spatiales”.

References and Links

1.

J. M. Herbelin, J. A. MacKay, M. A. Kwok, R. H. Ueunten, D. S. Urevig, D. J. Spencer, and D. J. Benard, “Sensitive measurement of photon lifetime and true reflectance in an optical cavity by a phase-shift method,” Appl. Opt. 19(1), 144–147 (1980). [CrossRef] [PubMed]

2.

D. Z. Anderson, J. C. Frisch, and C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23(8), 1238–1245 (1984). [CrossRef] [PubMed]

3.

A. O’Keefe and D. A. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59(12), 2544–2551 (1988). [CrossRef]

4.

K. K. Lehmann, U.S. Patent No., 5 528 040 (1996).

5.

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring-down spectroscopy,” Chem. Phys. Lett. 264(3-4), 316–322 (1997). [CrossRef]

6.

D. Romanini, A. A. Kachanov, and F. Stoeckel, “Diode laser cavity ring-down spectroscopy,” Chem. Phys. Lett. 270(5-6), 538–545 (1997). [CrossRef]

7.

D. Romanini, P. Dupré, and R. Jost, “Non-linear effects by continuous wave cavity ring-down spectroscopy in jet-cooled NO2,” Vib. Spect. 19, 99–106 (1999). [CrossRef]

8.

Y. He and B. J. Orr, “Ring-down and cavity–enhanced absorption spectroscopy using a continuous-wave tuneable diode laser and a rapidly-swept optical cavity,” Chem. Phys. Lett. 319(1-2), 131–137 (2000). [CrossRef]

9.

J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, and H. Lee, “Cavity ring-down spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38(9), 1859–1866 (1999). [CrossRef]

10.

Y. He and B. J. Orr, “Rapid measurement of cavity ring-down absorption spectra with a swept frequency laser,” Appl. Phys. B 79(8), 941–945 (2004). [CrossRef]

11.

Z. Y. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86(1), 51–57 (1991). [CrossRef]

12.

M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perot interferometer,” J. Opt. Soc. Am. B 16(4), 523–532 (1999). [CrossRef]

13.

J. Morville, D. Romanini, M. Chenevier, and A. A. Kachanov, “Effects of laser phase noise on the injection of a high-finesse cavity,” Appl. Opt. 41(33), 6980–6990 (2002). [CrossRef] [PubMed]

14.

Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44(31), 6752–6761 (2005). [CrossRef] [PubMed]

15.

I. Debecker, A. K. Mohamed, and D. Romanini, “High-speed cavity ringdown spectroscopy with increased spectral resolution by simultaneous laser and cavity tuning,” Opt. Express 13(8), 2906–2915 (2005). [CrossRef] [PubMed]

16.

J. B. Paul, L. Lapson, and J. G. Anderson, “Ultrasensitive absorption spectroscopy with a high-finesse optical cavity and off-axis alignment,” Appl. Opt. 40(27), 4904–4910 (2001). [CrossRef]

17.

G. Meijer, M. G. H. Boogaarts, and A. M. Wodtke, “Coherent cavity ring-down spectroscopy,” Chem. Phys. Lett. 217(1-2), 112–116 (1994). [CrossRef]

18.

D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-Axis Paths in Spherical Mirror Interferometers,” Appl. Opt. 3(4), 523–526 (1964). [CrossRef]

19.

A. E. Siegman, “Lasers” (University Science Books, Mill Valley, CA, 1986).

20.

K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105(23), 10263–10277 (1996). [CrossRef]

21.

I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

22.

J. B. Goldsborough, “Beat frequencies between modes of a concave-mirror optical resonator,” Appl. Opt. 3(2), 267–275 (1964). [CrossRef]

23.

D. Romanini, Laboratoire de Spectrométrie Physique, Université Joseph Fourier de Grenoble, 140 Rue de la physique, Grenoble, cedex France, is preparing a manuscript to be called “The optical resonator and a Gaussian beam: Their perfect marriage by superposition of transverse modes”.

24.

Y. He and B. J. Orr, “Detection of trace gases by rapidly-swept continuous-wave cavity ringdown spectroscopy: pushing the limits of sensitivity,” Appl. Phys. B 85(2-3), 355–364 (2006). [CrossRef]

25.

K. An, C. Yang, R. R. Dasari, and M. S. Feld, “Cavity ring-down technique and its application to the measurement of ultraslow velocities,” Opt. Lett. 20(9), 1068–1070 (1995). [CrossRef] [PubMed]

26.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry–Perot cavity using the ringing effect,” Phys. Lett. A 271(5-6), 314–318 (2000). [CrossRef]

27.

J. Poirson, F. Bretenaker, M. Vallet, and A. Le Floch, “Analytical and experimental study of ringing effects in a Fabry–Perot cavity. Application to the measurement of high finesses,” J. Opt. Soc. Am. B 14(11), 20811–22817 (1997). [CrossRef]

28.

R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006). [CrossRef] [PubMed]

29.

T. Klaassen, J. de Jong, M. van Exter, and J. P. Woerdman, “Transverse mode coupling in an optical resonator,” Opt. Lett. 30(15), 1959–1961 (2005). [CrossRef] [PubMed]

OCIS Codes
(280.3420) Remote sensing and sensors : Laser sensors
(300.6190) Spectroscopy : Spectrometers

ToC Category:
Spectroscopy

History
Original Manuscript: September 21, 2009
Revised Manuscript: December 1, 2009
Manuscript Accepted: January 5, 2010
Published: February 24, 2010

Citation
Jérémie Courtois, Ajmal Khan Mohamed, and Daniele Romanini, "High-speed off-axis Cavity Ring-Down Spectroscopy with a re-entrant configuration for spectral resolution enhancement," Opt. Express 18, 4845-4858 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4845


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References

  1. J. M. Herbelin, J. A. MacKay, M. A. Kwok, R. H. Ueunten, D. S. Urevig, D. J. Spencer, and D. J. Benard, “Sensitive measurement of photon lifetime and true reflectance in an optical cavity by a phase-shift method,” Appl. Opt. 19(1), 144–147 (1980). [CrossRef] [PubMed]
  2. D. Z. Anderson, J. C. Frisch, and C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23(8), 1238–1245 (1984). [CrossRef] [PubMed]
  3. A. O’Keefe and D. A. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59(12), 2544–2551 (1988). [CrossRef]
  4. K. K. Lehmann, U.S. Patent No., 5 528 040 (1996).
  5. D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring-down spectroscopy,” Chem. Phys. Lett. 264(3-4), 316–322 (1997). [CrossRef]
  6. D. Romanini, A. A. Kachanov, and F. Stoeckel, “Diode laser cavity ring-down spectroscopy,” Chem. Phys. Lett. 270(5-6), 538–545 (1997). [CrossRef]
  7. D. Romanini, P. Dupré, and R. Jost, “Non-linear effects by continuous wave cavity ring-down spectroscopy in jet-cooled NO2,” Vib. Spect. 19, 99–106 (1999). [CrossRef]
  8. Y. He and B. J. Orr, “Ring-down and cavity–enhanced absorption spectroscopy using a continuous-wave tuneable diode laser and a rapidly-swept optical cavity,” Chem. Phys. Lett. 319(1-2), 131–137 (2000). [CrossRef]
  9. J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, and H. Lee, “Cavity ring-down spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38(9), 1859–1866 (1999). [CrossRef]
  10. Y. He and B. J. Orr, “Rapid measurement of cavity ring-down absorption spectra with a swept frequency laser,” Appl. Phys. B 79(8), 941–945 (2004). [CrossRef]
  11. Z. Y. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86(1), 51–57 (1991). [CrossRef]
  12. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perot interferometer,” J. Opt. Soc. Am. B 16(4), 523–532 (1999). [CrossRef]
  13. J. Morville, D. Romanini, M. Chenevier, and A. A. Kachanov, “Effects of laser phase noise on the injection of a high-finesse cavity,” Appl. Opt. 41(33), 6980–6990 (2002). [CrossRef] [PubMed]
  14. Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44(31), 6752–6761 (2005). [CrossRef] [PubMed]
  15. I. Debecker, A. K. Mohamed, and D. Romanini, “High-speed cavity ringdown spectroscopy with increased spectral resolution by simultaneous laser and cavity tuning,” Opt. Express 13(8), 2906–2915 (2005). [CrossRef] [PubMed]
  16. J. B. Paul, L. Lapson, and J. G. Anderson, “Ultrasensitive absorption spectroscopy with a high-finesse optical cavity and off-axis alignment,” Appl. Opt. 40(27), 4904–4910 (2001). [CrossRef]
  17. G. Meijer, M. G. H. Boogaarts, and A. M. Wodtke, “Coherent cavity ring-down spectroscopy,” Chem. Phys. Lett. 217(1-2), 112–116 (1994). [CrossRef]
  18. D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-Axis Paths in Spherical Mirror Interferometers,” Appl. Opt. 3(4), 523–526 (1964). [CrossRef]
  19. A. E. Siegman, “Lasers” (University Science Books, Mill Valley, CA, 1986).
  20. K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105(23), 10263–10277 (1996). [CrossRef]
  21. I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]
  22. J. B. Goldsborough, “Beat frequencies between modes of a concave-mirror optical resonator,” Appl. Opt. 3(2), 267–275 (1964). [CrossRef]
  23. D. Romanini, Laboratoire de Spectrométrie Physique, Université Joseph Fourier de Grenoble, 140 Rue de la physique, Grenoble, cedex France, is preparing a manuscript to be called “The optical resonator and a Gaussian beam: Their perfect marriage by superposition of transverse modes”.
  24. Y. He and B. J. Orr, “Detection of trace gases by rapidly-swept continuous-wave cavity ringdown spectroscopy: pushing the limits of sensitivity,” Appl. Phys. B 85(2-3), 355–364 (2006). [CrossRef]
  25. K. An, C. Yang, R. R. Dasari, and M. S. Feld, “Cavity ring-down technique and its application to the measurement of ultraslow velocities,” Opt. Lett. 20(9), 1068–1070 (1995). [CrossRef] [PubMed]
  26. L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry–Perot cavity using the ringing effect,” Phys. Lett. A 271(5-6), 314–318 (2000). [CrossRef]
  27. J. Poirson, F. Bretenaker, M. Vallet, and A. Le Floch, “Analytical and experimental study of ringing effects in a Fabry–Perot cavity. Application to the measurement of high finesses,” J. Opt. Soc. Am. B 14(11), 20811–22817 (1997). [CrossRef]
  28. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006). [CrossRef] [PubMed]
  29. T. Klaassen, J. de Jong, M. van Exter, and J. P. Woerdman, “Transverse mode coupling in an optical resonator,” Opt. Lett. 30(15), 1959–1961 (2005). [CrossRef] [PubMed]

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