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
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 [1,2], but O’Keefe and Deacon  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: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.
As first suggested by the demonstration of Anderson et.al , CRDS could be performed with continuous wave lasers (cw-CRDS). In this case, to obtain a ring-down event, it is necessary to first couple the continuous light beam at a cavity resonance and then abruptly switch it off (by means of an active optical switch, e.g. an acousto-optic deflector or an electro-optic modulator). While K. K. Lehmann patented  the use of narrow bandwidth continuous wave lasers for the excitation of ring-down cavities, D. Romanini et.al. first demonstrated in 1997 a simple and now popular implementation exploiting a single-frequency ring-dye laser near 570 nm  and an external-cavity diode laser (ECDL) near 785 nm . Besides the strong appeal of such continuous light sources (relatively low cost, ease of use, small size, and low power consumption), cw-CRDS offers important advantages relative to pulsed CRDS in terms, for instance , of sensitivity and spectral resolution.
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  and J. W. Hahn et. al.  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 . 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 [11,12]. 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 , 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  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.  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.
The use of off-axis injection instead of TEM00 selective excitation by careful mode-matching appeared as a solution for Integrated Cavity Output Spectroscopy (ICOS) to create an extremely dense mode pattern at cavity output . Since the concept introduced in 1994 by Meijer et.al . of using small cavities (10 cm) having dense modal structure for improved resolution, off-axis-ICOS was exploited to average the cavity transmission spectrum more effectively, resulting in an important improvement in sensitivity. This off-axis cavity enhanced scheme was already demonstrated for resonators of moderate finesse at the time when the Herriott multi-pass cell was developed . The use of a re-entrant cavity configuration is demonstrated here with a high finesse cavity. We believe this is the first application of such an injection scheme to CRDS, where it presents a particular interest when combined with fast laser scans.
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  of a spherical two mirror paraxial resonator which is given by:20]. 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:21], 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 .
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 . 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 ) 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) ), the performance of our injection scheme is limited by other factors than the reduced signal level.
In our experimental implementation of a re-entrant cavity, we succeeded in observing the N spots at cavity output only for a moderate finesse (F~2000). When we tried with the higher reflectivity mirrors used in this work, the cavity output would rather consist of Lissajous-like sharp figures appearing at several specific cavity lengths closely spaced around the magic-length. The “degeneracy” interval over which such sharp patterns could be observed was about ± 2 mm in our case. According to a complete simulation of the transverse modal structure of an astigmatic cavity, which we are developing, we achieved a good understanding of the situation which will demand a separate paper to be exposed in detail. Here we will give a brief preliminary account, needed for explaining only one of the experimental results presented below. In fact most of our observations are qualitatively the same as expected from a perfectly re-entrant non-astigmatic cavity, notably the appearance of 4 sharp resonances over the cavity FSR, which display the same excitation amplitude when using an off-axis injection. The only surprising observation, which can be nicely explained in terms of the broken cylindrical symmetry, is the disappearance of the ringing effect on the transient ring-down events observed with off-axis excitation. We will thus postpone our discussion of the astigmatism case until we will consider this point further below.
3. Experimental procedures and results
Our experimental scheme is presented in Fig. 2 . 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.
Our cavity is formed by two concave mirrors (by Layertec GMBH) with a specified reflectivity of 99.97% at 765 nm and a 1-m radius of curvature. Their diameter is 12.7 mm. The results presented here are obtained with a cavity length of ~29.3-cm ± 2 mm corresponding to a K/N = 1/4-fractionally degenerate resonator allowing a 4 times spectral sampling grid improvement. The specified ± 2 mm cavity length uncertainty arises from experimental observation. Actually, for a perfectly spherical symmetrical cavity, the magic-length resonator is rigorously given by L = r(1-Cos(ПK/N), i.e. 29.2893 cm for mirrors with radius of curvature r = 1 m exactly. The cavity output is collected on a 1.5 mm diameter amplified silicon avalanche photodiode (Hamamatsu) coupled to a 12 bit, 20 MS/s digitizer board (ADLink PCI-9812). One of the cavity mirrors is mounted on a 3 axes manual translation stage permitting a fine 3-D positioning. In our N = 4 re-entrant order, the off-axis light leaking out of the cavity is focused by a 8-cm lens on the active photodetector surface in order to prevent undesired mode beating responsible for non-exponential ring-down relaxations .
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 , 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) ). 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.
4.1 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).
It is more complicated to account for the reduction of ring-down time variations in the case of complete off-axis excitation. In fact, the intracavity transverse field distribution can be simulated for an off-axis excitation of a fractionally degenerate cavity . One may show that the N groups of degenerate modes giving N resonances dividing the FSR correspond to very similar field distributions. Given the distance of the incident beam from the axis used here (~2.5 mm), the distribution of excited transverse modes is peaked around m = n = 70 (FWHM of this distribution is about 25 along m or n). The transverse field distributions obtained are very similar from a resonance to the next (only fine details do change), in particular they all present the same N Gaussian spots on each mirror, thus the resulting loss variations are small. On the contrary, multimode on-axis excitation involves an intracavity field reconstructed from the superposition of a small number of degenerate low-order modes (m,n<10 here) which do not form N Gaussian spots on the mirrors but more complex patterns very different from a resonance to the next. In this case, photons trapped at each of the N resonances will suffer more pronounced loss differences.
Particular attention is focused in Fig. 5 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 , 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.
We should also mention that Y. He and B. J. Orr  recently presented instrumental improvements yielding a 10 times lower normalized detection limit of 5 × 10−10 cm−1 Hz-1/2. They used higher reflectivity mirrors (99.993%, with an empty-cavity ring-down time of 25 μs) together with a more efficient data-acquisition procedure allowing higher CRDS data rates. These remarkable performances are based on a 4 kHz rate for ring-down events, almost half that in their previous work and about 10 times below what we obtain thanks to the more closely spaced passages through resonance, as the scanning rate is basically imposed by the maximum for a grating-tuned ECDL laser. We note that our application imposes a strong demand for millisecond timescale measurements, so that data averaging is barely possible. A more in-depth investigation of the noise sources should allow us to achieve even higher performance in the future.
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 [11,25–27].
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: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  (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 . 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 [28,29]. 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.
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.
The authors would like to acknowledge the support of the French space agency “Centre National d’Etudes Spatiales”.
References and Links
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