1 Introduction

Short-pulse lasers are becoming indispensable tools in many research areas and in industrial or medical applications. Several commercial systems are available to produce sub-picosecond pulses in different spectral ranges. Perhaps the best known is the titanium-doped sapphire laser (Ti:Sa), where the Kerr effect in the laser crystal itself can sustain a stable self-modelocked regime of operation (passive locking). A single ultrashort pulse is then formed in the laser cavity and produces a train of self-similar pulses through the output coupling mirror. The repetition rate of this pulse train is given by the round–trip time T of the pulse in the cavity (twice the cavity length divided by the pulse group velocity). Dispersion compensation of the laser cavity is usually implemented in order to further stabilize and shorten the pulse. By the indetermination relation between time and energy (Fourier Transform – FT principle), the shorter is the pulse the larger must be its spectrum. Ti:Sa gain is sufficiently large for supporting pulses shorter than 100 fs.

Passive mode locking is thus a simple way for producing ultra-short pulses, whose high peak power renders nonlinear optical techniques very efficient. For example, frequency doubling easily attains better than 10% energy conversion. Use of photonic fibers allows stretching out the laser spectrum to more than one frequency octave by self–phase–modulation: The high frequencies at the blue end of the generated spectrum are more than twice those at its red end.

The extreme stability of the modelocked pulse train is well known, but the perfect regularity of the spacing of the modes composing its spectrum have been demonstrated only very recently[1,2]. The uniformity of an octave–stretched frequency comb was shown to be better than a few parts in 10¹⁷. The term ‘frequency comb’ nicely describes the fine structure characterizing the emission spectrum of a modelocked laser when one looks at it at very high spectral resolution. This comb is associated with the Fourier Transform (FT) of the train of pulses. Each single pulse, if extracted from the train and independently analyzed, would simply give a broad spectrum with no mode structure. On the other hand, if we consider the pulse train as a perfectly periodic function of time, it is clear that its frequency representation is a Fourier series, a sum of sinusoidal functions whose frequencies are integer multiples (harmonics) of the repetition rate: In reality, small fluctuations of the repetition rate or of the pulse profile or carrier phase (e.g. originating from spontaneous emission) give a finite width to these harmonics, which are indeed the modes of the laser comb (the continuous FT must be then applied). The fact that the comb spacing (Free Spectral Range – FSR) is exactly given by the pulse repetition rate, which might seem evident, has been experimentally confirmed very recently[1,2].

Highly accurate frequency measurements have been performed using modelocked frequency combs[3,4,5,6,7]. As a general metrological tool, an octave–stretched self-stabilized optical comb was developed [2]: By doubling part of its low frequency modes and by measuring the beat note against the high frequency modes, a handle was obtained for controlling the carrier-envelope frequency slip f _ceo. In fact a light pulse can be written as a monochromatic carrier wave at the central frequency of the pulse spectrum, multiplied by a profile (for example ‘bell shaped’) containing the phase chirp. The frequency f _ceo slip is then defined as the phase change between the envelope and the carrier occurring from a pulse to the next in the train, divided by the time interval. This change is produced by a difference between the round–trip phase and group propagation delays. By FT it is easy to verify that the frequency slip also gives the offset of the comb with respect to zero frequency. We will see that this might have implications also for the observations reported in the present work. Since the comb FSR can be directly measured by pulse counting and locked to a stable radiofrequency reference, controlling f _ceo gives a frequency ruler with known frequencies fn=n × FSR + f _ceo. Precise and broad optical frequency rulers based on such a scheme, packaged as compact devices, will likely become commercially available.

An optical cavity (or Fabry-Perot resonator) also possesses a comb of transmission resonances which are equally spaced if the cavity is empty. This uniformity is rather well maintained if inside the cavity we place a gas possessing weak absorption lines, since its index of refraction will be just weakly dispersive. Light resonating inside the cavity will interact with the intracavity medium, before being transmitted, during a time which is on the order of the single-pass delay ℓ/c, where ℓ is the cavity length, multiplied by (1 – R)^-1, where R is the reflectivity of cavity mirrors. This response-time effect, or multi-pass effect, has already been employed for cavity-enhanced absorption spectroscopy (CEAS)[8] but only with single-frequency CW laser sources [9,10, 11,12]. The theory of optical resonators shows that a small intracavity absorption per pass α(ν)ℓ (where α(ν) is the sample absorption coefficient) will induce a fractional loss of the peak transmission of a cavity mode at frequency ν by α(ν)ℓ × 2F/π (where F = π√R/ (1 – R) is the cavity finesse, which is defined as the ratio of the cavity FSR and the Full Width at Half Maximum-FWHM of its resonances). As it will become clear, it is appropriate here to consider the cavity transmission integrated over the whole mode, that is, the mode area. As intracavity sample absorption induces an increase of the mode width, the mode area fractional change will be smaller by a factor 2, as it is shown easily.

If one succeeds injecting light with a broad spectrum into a cavity, the output spectrum will thus present absorption lines of the intracavity gas enhanced by F/π. This principle should work with any broad band light source, such as a lamp or a LED. However, due to their low spatial coherence (large emitting surface and beam divergence) and to their lack of mode structure (smooth spectrum) the cavity throughput is poor, and by decreasing as 1/F it becomes rapidly useless for cavities of finesse larger than 100. With broad-band nanosecond pulsed lasers (e.g. dyes), cavity injection is more effective due to their good spatial coherence and high peak power, but the laser spectrum is highly structured, giving noisy CEAS spectra.

On the other hand, modelocked lasers, when operating close to the FT limit, possess a spectral envelope which is smooth and structureless, an ideal background for observing cavity enhanced absorption spectra. Besides, the mode structure of a modelocked laser allows obtaining a large throughput even for a high finesse cavity when the laser modes are made to coincide with cavity transmission resonances. This occurs when the laser repetition rate and the cavity FSR are in an integer ratio. More precisely, if the cavity FSR is equal to N times the laser repetition rate, then 1 every N laser modes will be transmitted by the cavity (the rest being reflected, which implies a significant loss of power). In theory, a lossless cavity would then achieve 1/N peak transmission as long as the laser modes are narrower than the cavity modes. Otherwise, the peak transmission is reduced by approximately the ratio of laser to cavity mode linewidth. In the time domain, each laser pulse entering the cavity (by partial transmission of input mirror) will add up to the next laser pulse after N cavity round–trips, during which it will be partially transmitted by both cavity mirrors and produce N replicas at the output (of slightly decreasing amplitude). This fact has already been exploited for multiplying the laser repetition rate by a factor 20[3]. Simultaneous multi-mode resonance was also used in order to stabilize a modelocked laser comb to a passive confocal cavity[13].

For the first time to our knowledge, we report here a high–sensitivity cavity–enhanced absorption measurement exploiting the spectral properties of a modelocked laser for injection of a high finesse cavity. For our demonstration of this Mode–Locked Cavity Enhanced Absorption Spectroscopy (MLCEAS) technique, we used a cavity of finesse F ~ 420 with length finely adjustable to achieve the mode combs matching condition with N = 2. As for all ideas which appeal for their simplicity, an experimental proof of principle was demanded, notably to demonstrate that unexpected (e.g. nonlinear) effects do not interfere with the cavity enhanced principle. After all this principle is but a consequence of the linear response theory of optical resonators.

In the following we strive to go beyond a simple demonstration by giving a detailed accounting of the physics lying behind the technique. We introduce here what is perhaps the main point. The presence of a (generally) non–zero phase–envelope frequency offset f _ceo implies that exact matching of mode combs cannot be achieved in principle (since the two combs do not have a common origin). However, this may have little bearing over the limited spectral range of the laser spectrum, especially if we consider that the matching of modes should not be infinitely precise but just better than the largest between the cavity and the laser modes. More important may be dispersion effects in the cavity, in particular in the presence of absorption lines. Even a small local mismatch of mode combs, when using high finesse cavities, may thus induce distortion of the cavity enhanced spectrum. Therefore, our scheme of injection is not based upon a static comb matching but rather upon dynamic injection during a passage through resonance. All laser modes are thus equally injected (proportionally to their area) even if at slightly different instants, during the cavity output observation time. This is the multi-mode analogous of the injection scheme used in the CW–CRDS technique (Cavity Ring Down Spectroscopy with CW lasers)[14]. On the basis of the presented results, it will be clear that a much higher finesse cavity can be used to attain higher sensitivity, with little additional effort (involving a ‘modulated–locking’ scheme…).

2 Experimental

The experimental setup is shown in Fig. 1. We use a passively modelocked Ti:Sa laser source (Tsunami Spectra-Physics, model 3941) pumped by an argon laser, which gives a train of 100 fs pulses with approximately 81 MHz repetition rate. During our measurements the laser spectrum is Gaussian in shape with a FWHM of about 46 cm^-1. As expected, the spectrum is smooth, with only slight broadband modulations which are stable in time. The central laser wavelength is tuned to 860 nm. In front of the laser source we mount an optical isolator (I) which eliminates feedback from the cavity. Using a beamsplitter we direct part of the laser beam into a fast photodiode associated with a 1 GHz oscilloscope for monitoring the pulse train. Two lenses (L1, L2) and a pinhole (PH) are used for mode–matching purposes. The cavity, of length ℓ = 92 cm, is composed by a glass tube and two mirrors (M1, M2) with 500mm radius of curvature and 0.75% transmittivity (measured at 860nm using the modelocked laser as a source). From this and neglecting mirror losses, we estimate our cavity finesse F =420. The output mirror sits on a cylindrical piezoelectric actuator (PZT) which allows modulating the cavity length. The signal transmitted through the cavity is send to a photodiode (response time a few μs) and to a spectrograph equipped with a CCD (Charge Coupled Detector) linear detector array (integration time 40 ms).

 

Fig. 1. Schematic diagram of the experimental setup: I is an optical isolator, PD are photodiodes, PH is a pinhole, L1 and L2 are lenses, M1 and M2 are cavity mirrors, and PZT is a piezoelectric actuator.

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The position of pinhole, lenses and cavity is chosen in order to obtain transverse mode matching. A preliminary alignment is done by looking at the cavity output with a camera. We use a translation stage for the output mirror in order to adjust the cavity length and approach the “magic point”. This is the point where the comb of cavity modes and the comb of laser modes are in perfect resonance. By modulating the cavity length close to this point (by the piezoelectric actuator) we observe narrow transmission resonances as on the bottom trace in Fig. 2. The intense resonances correspond to fundamental transverse cavity modes (TEM_0,0), while weak resonances correspond to higher order transverse cavity modes. This is easily confirmed by looking with a camera at the transmitted profile when scanning the cavity at a slow rate. Strong Gaussian profiles are seen flashing by corresponding to fundamental cavity modes, followed by much weaker profiles with a number of lobes disposed as in higher order transverse modes. The horizontal scale in the figure represents the time necessary for cavity length to scan in one direction (approximately 4 ms) and pass over 4 resonances. The cavity length change between two resonances is in our case λ/4 because the ratio of cavity and laser FSR’s is 2. By looking at these signals and by minimizing intensity of higher order transverse modes we could finely adjust the cavity length and also improve the laser beam alignment for better transverse mode–matching.

3 Results and Discussion

The bottom trace in Fig. 2, reveals that one of the resonances (close to 2 ms) is more intense. This is actually the magic point for the cavity length where mode combs are as closely matched as possible. With respect to other resonances, the cavity length mismatch is sufficient for not allowing simultaneous transmission of all laser modes, which explains their lower intensity. However, it is puzzling that the resonances on the two sides of the magic point are not of equal intensity. Modeling of the intensity of these resonances confirms that this asymmetry could well be a manifestation of f _ceo. This effect could actually be explored as an alternative method to control the laser comb origin without need for frequency doubling[2]. Other traces of Fig. 2 make it evident that the resonances not only loose intensity but become broader as we move away from the magic point. The width increase is due to modes in different parts of the laser spectrum passing through resonance with cavity modes at different times as the cavity length scans. It is to be noted that, as long as resonances are well resolved, their area is constant, as is confirmed by a detailed analysis of the two bottom traces in Fig. 2.

 

Fig. 2. Cavity transmission as a function of displacement from “magic point”. These oscilloscope traces correspond to a piezoelectric scan of about 1 λ.

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With respect to the magic point resonance, its shape cannot be fit by a Lorentzian, as it would be expected if it was dominated by the profile of the cavity modes. It can instead be well fit by a Voigt profile, indicating the presence of ‘inhomogeneous’ broadening (Gaussian FWHM close to the Lorentzian FWHM). This is likely a manifestation of frequency dependent local dispersion effects (by air, mirror coatings, or absorption lines) not allowing perfect coincidence of mode combs. In addition the Lorentzian mode width appears about 25% larger than estimated from the mirror transmittivity, which could be due to non–negligible (about 0.2%) mirror reflection losses.

The spectra of the cavity transmission for different displacements from the magic point are shown in Fig. 3. These are taken without cavity modulation except close to the magic point (0 μm), where cavity length fluctuations give bursts of transmitted light only when an occasional passage through resonance occurs. In order to avoid these fluctuations, we modulate the cavity length so that during the CCD integration time (40 ms) the transmission over several passages through resonance is averaged (1 per ms, as in Fig. 2). Indeed the same cavity transmission spectrum can be obtained by modulating the cavity around any of the isolated resonances shown in the 2 bottom traces of Fig. 2. Structured cavity transmission spectra only begin to appear for sufficiently large length mismatch, such as for the 150μm trace in Fig. 3. These beating patterns on the transmitted spectrum correspond to modes going in and out of resonance periodically. The beating period ∆ν_b, can be used to measure the displacement δ from the magic point, since ∆ν_b, = c/2Nδ = c/4δ. Finally it is to be noted that the intensity of the spectrum at the bottom of Fig. 3 is corrected by a duty-cycle factor accounting for the fraction of time spent in resonance during cavity length modulation. This gives the same intensity scale as other traces in the same figure and allows appreciating the cavity transmission increase close to the magic point.

 

Fig. 3. Spectra transmitted by the cavity for different displacements from magic point. Peaks correspond to groups of transmitted modes, unresolved by the spectrograph. Smaller peaks are due to excitation of transverse cavity modes.

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These observations show that MLCEAS is more robust and simple than it could be expected. Even if MLCEAS measurements should be done close to the magic point, an exact coincidence to the central resonance is not necessary nor even especially convenient (average transmission over a passage through resonance is the same). Cavity transmission spectra shown in the following are all taken by modulating the cavity length around the magic point.

In Fig. 4a we present the transmission spectrum after filling the cavity with 1 atm of acetylene. As expected we obtain a cavity-enhanced spectrum, with an effective pathlength given by

The resolution of our spectrograph is about 0.2cm^-1 (resolving power λ/∆λ =50000). The acetylene absorption lines visible in this spectrum belong to a rovibrational overtone band previously studied at higher resolution by FT spectroscopy coupled with a multipass absorption cell with a path between 32 and 64 m, an acetylene pressure from 300 to 700 torr, and 0.05 cm^-1 spectral resolution[17].

In the same figure we also display (conveniently rescaled) the laser spectrum taken by removing the output cavity mirror, with the input mirror acting as an attenuator. By comparing with the empty cavity transmission it appears that indeed the cavity transmits the laser spectrum without distortion. The small oscillating structure on the laser spectrum is stable in time. This allows dividing out the laser spectrum from the cavity enhanced spectrum, giving the cavity transmission T_c(ν) in Fig. 4b. In order to obtain the linear molecular absorption from the cavity transmission (α(ν) ← T_c(ν)), a nonlinear transformation should be further applied, which is the inverse of the cavity integrated-transmission function:

With respect to the transverse mode matching, we have seen that a simplified cavity transmission structure with length modulation is an aid to cavity alignment and also to the localization of the magic point. In addition, clean mode-beating figures as in Fig. 3 are observed only if very few transverse modes are excited. It was thus necessary at a demonstration stage for a good physical understanding of the system. However, mode matching is not strictly necessary for adjusting the cavity close to the magic point, since this point would be anyway characterized by narrow transmission peaks visible by cavity length modulation. Even for the accurate determination of spectral intensities, TEM_0,0 cavity enhanced transmission is not strictly needed. The integrated cavity output over several transverse modes is the weighted sum of the same spectrum multiplied by slightly different cavity enhancement factors, since the mirror transmission and reflection coefficients are actually surface-averaged effective parameters which depend on the transverse mode. The weights in the sum correspond to the injection efficiency of each transverse mode. It would then be enough to determine the average cavity enhancement factor, the only difficulty being that it is slightly frequency dependent according to the mirror frequency response. However, as the emission spectrum of a modelocked laser is usually narrower than the mirrors working range, close to its center the frequency dependence of the enhancement factor should give negligible inaccuracies in most applications. Under these conditions, it is possible to measure the mirror reflectivity and thus deduce an accurate value of the finesse and of the enhancement factor by a simple (multispectral) ringdown measurement done by interrupting the laser beam at the passage through resonance, exactly as in CW-CRDS[14].

It should be remarked that, due to the presence of intracavity field buildup, even the low spectral density corresponding to the broad spectrum of a femtosecond laser might be sufficient to induce saturation of molecular transitions or multiphoton absorption. For weak transitions, and in presence of collisional broadening such as in the present case, such eventuality is to be discarded.

4 Conclusions

In conclusion, MLCEAS promises to be a robust high-sensitivity spectroscopic tool exploiting femtosecond lasers. These are becoming fairly common in research laboratories and allow accessing any wavelength from the near infrared to the UV by nonlinear frequency conversion. Their spectral stability and width should even allow liquid-phase broad-band MLCEAS measurements. Advantages with respect to direct multipass methods include higher sensitivity with a much smaller sample volume (a few cm³). Time response is limited by the need to integrate enough light on the CCD at the spectrograph output. In our demonstration, we simply modulate the cavity length to insure periodic passages through resonance of laser and cavity mode combs. Even with the small injection duty cycle (α 1/ F) of this simplified setup,40 ms are sufficient to ‘fill-up’ the CCD. During this integration time we obtain a transmission spectrum of about 800 points (Fig. 4) with 1% noise over 120 m of effective path length. The smallest detectable absorption (detection limit) is thus α _min = 1 % √/40 ms/120 m≃ 2 × 10^-7/cm/√Hz, while the figure of merit including the number of data points M simultaneously recorded is α _min/√M ≃ 6 × 10^-9/cm/√Hz. The principal noise source in our MLCEAS demonstration is to be attributed to laser spectral noise or structure. However the ultimate limit for an ideal FT-limited laser is the shot noise associated to the photons collected onto the CCD. This should become dominant when using higher finesse cavities from which we expect a smaller throughput.

 

Fig. 4. Cavity transmission with length modulation around magic point. a) Cavity filled with acetylene, empty cavity, and laser spectrum. Intensities have been arbitrarily adjusted for clarity. b) Ratio of cavity transmission with/without acetylene over laser spectrum.

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Finally, a simple ‘modulated-locking’ scheme where the cavity is tightly modulated around a resonance near the magic point, would reduce the integration time possibly down to the fundamental limit of the cavity response time (a few microseconds), and also allow using higher finesse cavities (~10⁴ or 10⁵) by compensating for their smaller throughput. Such an increase in finesse is expected to improve the detection limit of MLCEAS not far from the best obtained by CEAS with CW lasers[12] (without using locking…). Multispectral measurements could then be used to follow real–time variations of species concentrations, with high sensitivity. Using a Fourier Transform spectrometer instead of a grating spectrograph is possible, and scan–step recording of the interferogram should allow time–resolved (repetitive) operation. On the other hand, a dedicated high–diffraction–order spectrograph as routinely used in ICLAS (Intra Cavity Laser Absorption Spectroscopy)[15,16] will allow higher spectral resolution down to the molecular Doppler width.

This work was sponsored by EU (SPHERS contract HPRN-CT-2000-00022). We wish to thank P. Baldeck for kindly letting his modelocked laser system at our disposal.