1. Introduction

Recently, we are witnessing new developments which further broaden the already wide application scope of femtosecond laser frequency combs in frequency metrology [1], extreme non-linear optics [2] and other areas of fundamental and applied science [3]. Very notable among these is the successful implementation of high-harmonic generation of near-infrared femtosecond pulses at unprecedented repetition rates, providing coherent light sources for high-precision spectroscopy at previously inaccessible extreme ultraviolet wavelengths [4, 5]. An indispensable prerequisite for these experiments are passive, high-finesse optical resonators, in which the energy of the circulating pulse can be enhanced by orders of magnitude with respect to the driving pulse train from the oscillator [6].

In order to couple the largest possible fraction of the driving laser’s power into the enhancement cavity, a set of longitudinal modes of the passive cavity must match the frequencies of the femtosecond comb. Therefore, means to accurately measure and tailor the net intracavity dispersion are required. Traditionally, white-light interferometric techniques (see for example in [7]) or CW laser based techniques that determine the free spectral range of an optical cavity [8, 9] have been used to determine the dispersion of optical elements. Here we demonstrate a method based on frequency combs unifying the advantages of a time-efficient parallel approach, the precision provided by laser-based measurements, and the enhanced sensitivity due to the high finesse of the cavity under study. Beyond the results that have already been achieved with a similar method [10], we are able to extract also the round-trip damping or, correspondingly, the finesse of the cavity in a spectrally resolved manner from our data. Additionally, by monitoring all degrees of freedom of the frequency comb, our technique is insensitive to drifts or fluctuations in the passive cavity length.

2. Theoretical concept and experimental realization

In the pulse train derived from a mode-locked laser, the pulses bear a certain mutual phase relation since they all derive from the same solitary pulse that is stored in the laser cavity. With a precision that is far beyond what is of concern here, the spectrum of such a pulse train has a regular comb-like spectrum where the optical frequencies ω _n =nω _r+ω _CE may be numbered by an integer n typically on the order of 10⁵ to 10⁶. Here ω _r/2π is the pulse repetition rate and w _CE is the angular carrier envelope offset frequency that is related to the pulse-to-pulse phase shift ϕ _CE of the carrier with respect to the envelope via ω _CE=ϕ _CE/T with the pulse repetition time T=2π/ω _r (see for example [1] for details).

The resonance locations of the passive cavity, in contrast, are determined by the phase delay φ(ω), which a wave oscillating at frequency ω experiences during a round-trip through it, by the condition φ(ω)=2πm (m integer). For a cavity in vacuum of length L with ideal mirrors, the round-trip phase can be written as a linear function in frequency, φ(ω)=Lω/c,with the vacuum light speed c. In this case, the resonant frequencies are evenly spaced, allowing us to drive all of them simultaneously with an appropriate frequency comb. In the general case however, we have to account also for a number of higher-order contributions (in ω) to the round-trip phase due to the dispersion of intracavity elements and mirror coatings. They can distort the equidistancy of the resonator modes, rendering a full-bandwidth match to the laser frequency comb impossible. In the context of laser pulse propagation, it is common to expand the round-trip phase delay in a power series $\phi \left(\omega \right)=\phi \left({\omega }_c\right)+\phi \prime \left({\omega }_c\right)\left(\omega -{\omega }_c\right)+\frac{1}{2}\phi ″\left({\omega }_c\right){\left(\omega -{\omega }_c\right)}^2+\dots$ in terms of the detuning from ω _c, usually the center frequency of the comb. To provide an intuitive meaning of the constant, linear, and remaining terms we express this series as follows:

Very similar to the frequency comb this expression includes carrier envelope phase shift per round-trip ${\varphi }_{\mathit{\text{CE}}}^{\prime }$ . The linear term contains the large dispersion free term L/c and a term τ that accounts for any additional group delay of the pulse at ω _c . According to these definitions the cavity round-trip time of a pulse centered at ω _c is given by T ^′=L/c+τ. We will not further discuss the constant and linear term here, but simply assume that they exist and focus on the determination of the nonlinear term ψ(ω) that distinguishes the modes of a passive cavity from a regular spaced comb and limits the pulse enhancement. This term contains all higher-order contributions to the round-trip phase that are of interest here. These contributions originate from the dispersion of intracavity media and cavity mirrors. This non-linear phase term is related to the cavity’s group delay dispersion (GDD) via GDD(ω)=∂² ψ(ω)/∂ω ².

For an understanding of our approach to measure the GDD and the round-trip losses with high accuracy it is instructive to recall the steady-state expression for the electric field inside a resonator (e.g. at the inner surface of the input coupler) when it is driven by an external electric field [11]

where E _i(ω _n ) denotes the complex driving field amplitude at frequency ω _n , t(ω _n ) the coupler (amplitude) transmittance and r(ω _n )<1 the amplitude loss factor per round-trip. The latter entity describes the reduction of electric field amplitude after one round-trip. It equals the product all mirrors’ reflectance amplitudes and intracavity media (additional optical elements, gases) transmittance amplitudes and is close to unity. The resonator finesse F is related to the loss factor via F=π√r/(1-r).

For the description of the response of the cavity subjected to a frequency comb, it is convenient to introduce the phase detuning function

Evaluated at the comb frequencies ω _n , it equals φ(ω _n ) apart from an irrelevant phase offset -ω _n T+ϕ _CE=-2πn and can therefore replace it in (2) without modifying the result. Then it is directly seen that in the case of a small round-trip time difference T-T ^′, the phase function (3), and consequently, when individual comb modes are not resolved, also the intracavity power spectrum (∝|E _c(ω,ϕ _CE)|²) varies only slowly with frequency ω, governed mainly by the nonlinear dispersion term ψ(ω).

The basic idea to attain quantitative information about the cavity’s properties is to measure the intracavity power spectrum |E _c(ω,ϕ _CE)|² as a function of ϕ _CE and ω. This is accomplished by measuring ϕ _CE=Tω _CE with the standard f-2f -interferometer [1] and the power spectrum with a grating spectrometer that is placed behind one of the high reflecting cavity mirrors. The circulating power peaks on resonance, corresponding to the condition Δφ(ω,ϕ _CE)=2πm with an integer valued m. Instead of tracking the position of the odd-shaped (governed by ψ(ω)) peak in the spectrum when the frequency comb is tuned as in ref. [10], we analyze the response of the cavity as a function of ϕ _CE separately for every frequency ω. In this way, we obtain a perfectly symmetric Airy-shaped (2) response for every frequency ω, which we can fit over the entire measured ϕ _CE range. This has a two-fold advantage: First, as will be detailed below, we can not only derive the GDD, but also the round-trip losses in a spectrally resolved manner. Second, by using all measured data points for the fit, we fully exploit the available measurement time for an optimum signal-to-noise ratio (SNR).

While it is safe to assume that ${\varphi }_{\text{CE}}^{\prime }$ and ψ(ω), like all material parameters, are constant during the experiment, T ^′ may vary due to mechanical vibrations or thermal drifts of the mirror positions. Due to the high mode numbers (ωT ^′≈n≈10⁶) and finesse (~10²) of the resonator, fractional length changes on the order of 10^-8 already lead to a substantial change of the circulating power. Therefore, an active feedback is implemented on T-T ^′, forcing the cavity and the laser on resonance at a certain frequency ω _lock, such that Δφ(ω _lock,ϕ _CE)=2πl (l integer) during the entire experiment. As a consequence, ϕ _CE and T-T ^′ have a fixed algebraic relation that can be used to eliminate T ^′-T from (3).

Under this constraint the phase function φ(ω) in the response model (2) is modified once more and replaced by

As this function still depends only linearly on ϕ _CE, it is straightforward to fit this model (with variable ϕ _CE) to the data measured at a frequency ω. To perform the fit, all unknown terms in the right side of (4) are summarized to the single fit parameter ${\varphi }_{\text{CE}}^{\text{res}}$(ω) by setting

with

At frequency ω, the cavity thus becomes resonant in the case Δφ(ω,${\varphi }_{\text{CE}}^{\text{res}}$(ω))=0, where we have set m=0. This is possible because it removes a constant that will drop out in the final result. By performing fits for a large set of frequencies with Δφ(ω;,ϕ _CE) of (5) replacing φ(ω _n) in the model (2), the function ${\varphi }_{\text{CE}}^{\text{res}}$(ω) is determined. Multiplying this function with ω/ω _lock-1 and computing the second derivative yields the group delay dispersion ∂² ψ(ω)/∂ω ² without ambiguity.

With the same fitting procedure we obtain the cavity loss factor r(ω) with high accuracy by virtue of the linear dependence of the phase function on the pulse-to-pulse carrier envelope phase shift. Of particular interest for the detection of weak absorbers like dilute gases within the cavity is the width of the Airy function since it is a measure of the cavity ring down time in inverse units. The spectrometer data allow to measure the cavity losses as function of ω so that this method also provides a sensitive tool for trace gas analysis.

It should be noted that the intracavity intensity is observed through one of the cavity mirrors that in general will modify the spectrum. Even though this modification might be compensated for by knowing the spectral transmittance of that particular mirror, such a correction does not seem to be necessary since only the resonance peak position and width are being evaluated. In addition, because of the finite resolution of the spectrometer it has to be assumed that the round-trip phase and losses vary only slowly within the latter.

The experimental setup (Fig. 1) is based on the setup detailed in the Methods section of ref. [4] and will only be described briefly. We derive the frequency comb from a chirped-mirror dispersion-compensated Ti:sapphire oscillator (repetition rate 112MHz, pulse duration 20fs, average power 850mW). The comb parameters ω _r and ω _CE can be adjusted by a piezo-mounted folding mirror and an intracavity fused silica wedge, respectively. Both parameters are continuously computer-monitored using a photodiode and a frequency counter (HP 53131A) for ω _r and an f-to-2 f -interferometer and an electronic spectrum analyzer (HP 8591A) to determine ω _CE. The laser light is coupled into an eight-mirror, bow-tie ring cavity (some of the mirrors used in [4] have been replaced, and the vacuum chamber has been removed). Of the light reflected off the cavity coupler, a small wavelength region at ω _lock is selected by a grating and a slit and used for the generation of an error signal [12] for a high-bandwidth (>100kHz open loop unity gain) feedback loop locking the laser and the cavity on resonance in this region. A small fraction of the light circulating in the cavity leaks through one of the mirrors and is analyzed with a grating spectrometer equipped with a CCD array detector (Ocean Optics USB2000) with a resolution of about 0.5 THz. Its output is fed to the computer recording also ω _CE and ω _r. When the wedge is stepped into the laser cavity, a changing transmission pattern is observed with the spectrometer, reflecting the varying matching conditions of the equidistant frequency comb and the cavity modes.

 

Fig. 1. Experimental setup. The Ti:Sapphire and the passive cavity (schematically represented by input coupler IC, and mirrors S, M, and OC) are locked to resonance at a frequency ω _lock selected by a grating G. The resonant spectrum |E _c(ω,ϕ _CE)|², and the comb parameters ω _r and ω _CE are simultaneously recorded with a computer.

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3. Dispersion and finesse measurement of a passive cavity

A typical raw transmission data set is shown in Fig. 2(A). Its acquisition with a ϕ _CE step size of 12×10^-3 rad and a spectrometer integration time of 60 ms per step takes about 5 minutes. After the acquisition, the data of each pixel of the CCD is separately fit by the square modulus of (2) with phase (4) as a model, an example being shown in Fig. 2(C). To determine the lock frequency ω _lock, the position of the peak in the spectrum at fixed ϕ _CE is used (Fig. 2(B)).

From such data the resonator round-trip loss factor r(ω), i.e. the finesse and the round-trip phase (up to a linear function) may be extracted as described in the previous section. As differentiation enhances fast (in frequency) fluctuations in the round-trip phase it can be useful to smooth the GDD data calculated from the obtained round-trip phase, as these pronounced high frequency fluctuations can obscure other (slower) features in the result. If not indicated otherwise, we used a Gaussian low pass filter of about 0.6THz width (4 Pixel) on all GDD data shown. This filter reduces noise in the result considerably while leaving the resolution essentially unchanged because most of the noise power in the GDD is in the roll-off of the spectrometer resolution.

 

Fig. 2. Raw data obtained using the described method. A: The cavity transmission (color coded) as a function of carrier envelope phase slippage ϕ _CE and optical wavelength. The lock frequency ω _lock of the cavity is close to 800 nm. B: section of the shown data at a fixed ϕ _CE. It can be used to determine the lock point ω _lock. C: section at a fixed wavelength (green) and a fit to it (black) using the square modulus of (2) with phase (4) as the model. Excellent agreement between data and model is observed.

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Figure 3 displays the results obtained from data as shown in Fig. 2. To assess the reproducibility of the measurement, we compare results from different runs with the same system. In the measured range between 750 and 840 nm, we find typical reproducibilities of 4×10^-4 rad, 4×10^-4 (no smoothing) and 1fs² (1.6THz BW) for ψ(ω), r(ω) and GDD(ω) respectively. Obtained phases were compared by taking the difference between two runs and subtracting a linear function of frequency from the result. The quality of the data is reduced in the wings of the 150 nm-wide laser spectrum, where both SNR and laser power stability decrease. We assume that the reproducibility in phase is dominated by changing atmospheric conditions. Round-trip loss changes are likely to be caused by a small aperture (1mm²-small piezo-mounted mirror) in the cavity that makes the loss sensitive to alignment. Once these sources of instability are eliminated by placing the cavity into vacuum and moving the piezo actuators entirely into the laser, a significant improvement in stability and reproducibility can be expected. The substantially larger discrepancy in the range between 790 and 800 nm in the round-trip loss is due to an imperfect determination of the lock frequency ω _lock. It can be argued that due to this uncertainty Δω _lock the measured round-trip loss factor is modified as Δr/(1-r ²)≈Δω _lock/(ω-ω _lock). This error becomes small as the distance to the locked point becomes much larger than Δω _lock. The dispersion measurement remains unaffected to lowest order in Δω _lock. Regions close to the locked frequency ω _lock therefore have been excluded in the round-trip loss factor reproducibility analysis. In principle uncertainties of this kind can be entirely eliminated by evaluating several measurements with different lock frequencies as demonstrated in Fig. 3.

 

Fig. 3. GDD and round-trip loss factor data of an 8-mirror passive cavity, obtained from three measurements with lock frequencies 780.5 (red) and 801.0nm (green and black), respectively. The dashed vertical lines mark the locked points. The GDD data have been smoothed with a 0.6THz BW filter. The modified cavity studied in this work (in contrast to ref. [4]) has not been dispersion engineered and thus displays significant GDD over the entire 150nm measurement bandwidth. The strong resonances around 760 and 822 nm will be discussed in section 5.

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4. Group-delay dispersion of single optical elements

The presented method may be used to accurately determine the dispersion properties of low loss optical elements. To this end, we first inserted a sapphire window at Brewster’s angle into the resonator and determined the change in GDD. A comparison to a Sellmeir formula [13] shows (Fig. 4) an agreement within about 2.5 fs² rms in a resolution bandwidth of 2 THz across the measured 725–850 nm range. In a second experiment, a pair of dielectric mirrors was inserted into the resonator in a Z-configuration to minimize the impact on the resonator geometry. The incidence angle on the newly inserted mirrors was determined to 3.5°. The other mirrors in the resonator were moved slightly in order to keep the cavity length (and thus air dispersion contributions) unchanged while minimizing angle changes on the mirrors. Assuming identical mirrors, the measured GDD of a single mirror agrees with the design goal at the given incidence angle to about 2 fs² rms in a 2 THz resolution bandwidth. The measured GDD is shifted by about -1 fs² to lower values compared to the design, indicating a slight deviation of the actual dielectric stack from the design goal.

5. Observation of atmospheric gas species

In all measurements of the dispersion of the entire cavity, we consistently observed a pronounced feature in both round-trip losses and GDD around 760nm resembling a double resonance (Fig. 5). From the observation of a reduction of the feature when we purged the cavity with argon, we inferred that it originates from air within the cavity light path. This is in contrast to previous high-precision measurements of air refractive index [14] or dispersion [10] with frequency combs, which both assume and confirm a smooth polynomial functional form. To corroborate our explanation, we calculated the absorbtion of the oxygen A-band at 296 K and 1 atm air pressure (20.95% O₂) over a 266 cm cavity length using data from the HITRAN database [15]. After convoluting the resulting transmission spectrum with the spectrometer’s instrumental line shape (approximated by a 0.83 THz-FWHM Gaussian) and multiplying it with an estimated baseline at 0.98 we find a quantitative agreement with the measured drop in round-trip loss factor. The absorption dip coincides with a very strong signature in the GDD data.

 

Fig. 4. Comparison between measured and expected change of resonator dispersion. The measured GDD of a 0.525 mm sapphire window at Brewster’s angle vs. Sellmeir formula [13] and a quarter wave stack dielectric mirror vs. coating design data both show agreement with deviations on the order of 2.5 fs² rms (resolution bandwidth 2 THz). The mirror data is systematically shifted by -1 fs² compared to the design indicating a wavelength shift of the actual stack due to production tolerances.

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The high sensitivity in phase helps identifying another feature in the vicinity of 822 nm (Fig. 3). Although limited SNR prevented us from clearly observing a reduction of this feature when purging with argon, we presume it is caused by water vapor absorption. Our assumption is again supported by comparison of our data with the absorption line data of HITRAN: A collection of water vapor lines in the vicinity of 822 nm constitutes an absorption feature similar to the measured drop in the squared round-trip loss factor, while no other species in the database (including the most common atmospheric gases) exhibits strong enough absorption in this region. The absorbtion depth of this feature is about 1×10^-4 per round-trip (in amplitude) and shows the sensitivity of the device. Although the absorption peak is close to the noise floor of the round-trip loss data, it is still clearly seen in the GDD, demonstrating also the advantage of a phase sensitive method.

We note here that high-finesse cavities have been employed for decades to enhance the sensitivity in the detection of molecular species. Most commonly, time-domain ring-down techniques are used. An elegant combination of the ring-down technique with frequency combs has been published during the preparation of this manuscript [16]. However, these methods lack the ability to observe spectral phases.

6. Conclusion

We have introduced and demonstrated a new method to characterize high-finesse passive optical cavities. For the first time, it enables a simultaneous, high-precision measurement of a cavity’s round-trip losses (finesse) and phase over the entire bandwidth of the frequency comb used as a light probe. Similar to well-established cavity ring-down techniques, our approach offers a highly enhanced sensitivity to changes in the round-trip losses and, in our case, phase. The method is intrinsically well suited for applications requiring the enhancement of ultrashort laser pulses in external cavities. All devices required for this method are already present in such an experiment. While challenging only a few years ago, the measurement and control of the carrier-envelope offset frequency ω _CE has become a standard task for which a number of techniques have emerged, partly commercially available today. Furthermore, the demonstrated pronounced signatures from atmospheric gas species suggest a potential application of the method for cavity-enhanced, high-sensitivity amplitude and phase spectroscopy of intracavity gas samples. With an increased resolution of the analyzing spectrometer and a high-repetition-rate laser, high-resolution spectroscopy only limited by the linewidth of the femtosecond laser frequency comb components is conceivable.

 

Fig. 5. The oxygen A-band, as observed in the power loss factor (squared amplitude loss factor r², upper panel, red) and the GDD (lower panel). The absorption of molecular oxygen, as calculated from HITRAN [15] data (blue spikes, right ordinates), can explain the increased losses very well. The expected reduction in the round-trip power loss factor for our experimental conditions is indicated with a dotted blue line (left ordinate).

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