Cavity buildup dispersion spectroscopy

Measurements of ultrahigh-fidelity absorption spectra can help validate quantum theory, engineer ultracold chemistry, and remotely sense atmospheres. Recent achievements in cavity-enhanced spectroscopy using either frequency-based dispersion or time-based absorption approaches have set new records for accuracy with uncertainties at the sub-per-mil level. However, laser scanning5 or susceptibility to nonlinearities limits their ultimate performance. Here we present cavity buildup dispersion spectroscopy (CBDS) in which the dispersive frequency shift of a cavity resonance is encoded in the cavity's transient response to a phase-locked non-resonant laser excitation. Beating between optical frequencies during buildup exactly localizes detuning from mode center, and thus enables single-shot dispersion measurements. CBDS yields an accuracy limited by the chosen frequency standard, a speed limited by the cavity round-trip time, and is currently 50 times less susceptible to detection nonlinearity compared to intensity-based methods. The universality of CBDS shows promise for improving fundamental research into a variety of light-matter interactions.

Measurements of ultrahigh-fidelity absorption spectra can help validate quantum theory, engineer ultracold chemistry, and remotely sense atmospheres [1][2][3][4] . Recent achievements in cavity-enhanced spectroscopy using either frequency-based dispersion 5 or time-based absorption 6 approaches have set new records for accuracy with uncertainties at the sub-permil level. However, laser scanning 5 or susceptibility to nonlinearities 6

limits their ultimate performance. Here we present cavity buildup dispersion spectroscopy (CBDS) in which the dispersive frequency shift of a cavity resonance is encoded in the cavity's transient response to a phase-locked non-resonant laser excitation. Beating between optical frequencies during buildup exactly localizes detuning from mode center, and thus enables single-shot dispersion measurements. CBDS yields an accuracy limited by the chosen frequency standard, a speed limited by the cavity round-trip time, and is currently 50 times less susceptible to detection nonlinearity compared to intensity-based methods. The universality of CBDS shows promise for improving fundamental research into a variety of light-matter interactions.
Highly accurate models of light-matter interactions are important for fundamental studies of molecular hydrogen 1,7 , tests of molecular structure calculations 8,9 , modelling of planetary atmospheres 3,10 , and the development of advanced spectroscopic databases 11 . The latter application is crucial for improvements in remote sensing and measurements of variations in greenhouse gas concentrations at 0.1% uncertainty levels required to better predict changes in Earth's climate 4 . To develop and test these models using first-principles approaches, accurate experimental techniques are required. Cavity mode-dispersion spectroscopy 12 (CMDS) is one such technique recently developed to meet these challenges 5 . While CMDS yields absorption spectra entirely in terms of measured optical frequency shifts 12 , the shifts are not read out instantaneously. Consequently, the CMDS technique is susceptible to drifts. Although frequency-agile rapid scanning (FARS) spectroscopy 13 provides single-shot acquisition of local absorption limited only by the cavity response time, it is an intensity-based technique and inherently susceptible to nonlinearities in the detection system 6 . Therefore, each of these established techniques in ultrasensitive absorption spectroscopy has a critical weakness.
Here, we present cavity buildup dispersion spectroscopy (CBDS): an accurate, phase-sensitive measurement of cavity mode frequency that can be implemented on time scales that are substantially shorter than the cavity buildup time. In CBDS, a phase-locked laser beam is instantaneously injected into a high-finesse cavity followed by observation of the transient transmitted signal. The net response involves optical interference between the excitation and the transient cavity fields, with the latter field always occurring at the local cavity resonance frequency 14 and in opposition to the former field. Thus, measurement of the resulting heterodyne beat frequency provides the local cavity mode position. In practice, absorptioninduced dispersion within the resonator leads to changes in the measured mode-by-mode beat frequencies, which yield dispersion spectra. We demonstrate that CBDS measurements can be made on timescales equal to or significantly less than the buildup duration and show the method to be relatively immune to nonlinear response in the detection system.
Recently reported techniques in condensed-phase sensing use micro-resonators with optical quality factors of Q ≤ 10 8 to readout dispersive signals on nanosecond timescales 15,16 . As described by Yang et al. 17 , those approaches clearly leverage heterodyne 18 and rapidly swept laser 19,20 cavity ring-down readout concepts developed decades ago to reduce technical noise and improve sensitivity. Collectively, the prior works in dispersive sensing utilize decaying signals which occur after the cavity driving field is effectively extinguished. Consequently, and unlike in the present work, dispersive micro-resonator sensors do not probe the temporal region associated with cavity buildup-a regime which includes two synchronous fields of interest here: a non-resonant laser-driven field with rapidly changing amplitude and the cavity's transient response to this field which always occurs at the cavity resonance frequency.
Uniquely, we demonstrate here dispersion spectroscopy performed in the transient buildup regime. The CBDS approach requires the sequential injection of discrete and arbitrarily detuned laser fields which are optically phase-coherent with the resonant cavity field 21 . Using high-Q (~10 11 ), macroscopic-length (~1 m) resonators we demonstrate acquisition times as short as 3 μs (Fig. 1b)-far from the fundamental lower limit set by the cavity round-trip time and Nyquist-Shannon sampling criteria (e.g., 2trt = 4nL/c ~ 1 ns for an effective geometric length of 0.1 m)-and therefore achieve a new measurement paradigm without sacrificing the ability to study a wide range of dynamic phenomenon. In addition, we establish the immunity of CBDS to common nonlinearities and biases which may occur with conventional intensity-based cavity-enhanced spectroscopy methods 6,22 . We note that the interrogation of consecutive cavity modes via buildup signals to measure broadband phenomena such as molecular absorption spectra has not been previously considered.
The general concept and experimental realization of CBDS are illustrated in Fig. 1. The CBDS method utilizes a single-frequency light source that is phase-locked to the optical resonator. We use a double-polarization phase-locked laser scheme 5,13 where one linear polarization of laser light is phase-locked to a cavity mode while the orthogonal polarization (having a wellcontrolled, constant frequency detuning MW from the locking point) is used for non-resonant excitation of the measured cavity mode (Fig. 1a). Additionally, the cavity length is stabilized to prevent thermal drift of the comb of modes over time scales >1 s. The buildup signal is initiated after rapidly switching on the frequency-detuned probe beam at the measurement mode, and the locking beam remains on during the entire cavity pumping process (but does not contribute to the CBDS signal thanks to a polarization-dependent optical filter and/or offset locking). Coherent averaging in the time-domain of repeated events is readily achieved because of the tight phase-locking scheme. The Fourier spectrum of the beating signal appearing in the transmitted light allows determination of the frequency detuning meas of the cavity resonance with respect to the probe beam frequency . Finally, frequency agile rapid scanning for fast spectral acquisition is achieved by adjusting the detuning frequency MW using a highbandwidth (~20 GHz) electro-optic modulator 5,13 .

Figure 1 Cavity mode localization and schematic of CBDS apparatus. (a)
The laser frequency is phaselocked to a TEM00 cavity mode, which is a local resonance of the cavity transmission spectrum indicated by the thick black line. An orthogonally polarized beam with frequency , is detuned from by frequency MW and excites another TEM00 mode shifted by dispersion. For non-resonant excitation, an oscillation on the transmitted buildup signal with a frequency meas corresponds to heterodyne beating between the non-resonant driving field and the resonant transient response of the cavity. For a given mode k relative to the locking point and cavity free spectral range FSR , the dispersive shift of the cavity mode can be retrieved from meas . (b) Schematic of the CBDS experiment. A broadband electro-optic modulator EOMP, driven by the microwave source MW, rapidly detunes the ECDL beam with frequency from the locking point. An acousto-optic modulator AOM prevents cavity excitation by the carrier frequency. A photodiode PDP records the buildup signal. Here the experimental signal for meas = 5 MHz and time interval 3 μs is shown. EOML, Circ. (circulator), PDL: elements in the Pound-Drever-Hall phase-locking loop.

(b)
The FFT spectrum of the buildup signal from the panel (a) and residuals from fits of Lorentzian (orange) and our asymmetric (blue) model.
In Fig. 2a a simulated buildup signal is shown. We developed a physically justifiable model for the transient cavity response to non-resonant single-mode excitation which is based on summation of delayed replicas of the driving field with a phase shift growing during each round-trip time 23 . Fourier transformation of our model reproduces the fast-Fourier transform (FFT) of the measured transmitted light intensity with 10 −7 accuracy and works 10 5 times better than a single Lorentzian function, which does not capture the predicted asymmetries (Fig. 2b). Even greater improvement is observed for retrieved values of meas leading to 10 −9 accuracy. The relation FSR − ( MW + meas ) yields the dispersive shift of the cavity resonance relative to the locking point frequency (Fig. 1a). Here, FSR is the cavity free spectral range (FSR) corresponding to cavity conditions outside the molecular resonance with potential contribution from the broadband intracavity dispersion, and is the integer number of modes between the locking point and the measured cavity mode. For each cavity resonance, a single buildup signal was recorded at a new value of MW (Fig.  3a). Corresponding FFT spectra, presented in Fig. 3b, show absorptive and dispersive changes in the width and position of cavity modes within the frequency range of the measured molecular line. Measured dispersive shifts ( − 0 ), determined for each cavity mode, were used to reconstruct the purely frequency-based complex-valued line shape of a CO transition of central frequency 0 (Fig. 3c). Allan deviation plots of meas , Fig. 3d, demonstrate excellent stability of the frequency measurement in the CBDS experiment yielding an equivalent absorption coefficient detection limit less than 3 × 10 −11 cm −1 , corresponding to a detection limit of ~100 mHz. Moreover, consistent with the rapid phase-sensitive nature of the measurement, a shortterm 20-Hz sensitivity to cavity resonance shifts was obtained in 400 s. In Figs. 4a-b we demonstrate excellent agreement between CO spectra and peak areas obtained from CMDS and CBDS experiments. We found that that systematic differences between line areas determined from CBDS and CMDS methods are only 0.07 % on average, with a standard deviation of 0.17 %. These observations indicate that the accuracy of these first CBDS measurements are already similar to the most accurate techniques currently available 5,6 . We emphasize that this level of agreement requires the proper frequency-domain modelling (Fig.  4c) of the transmission signals.
To quantify the effect of detection system nonlinearity on the accuracy of CBDS, we simulated buildup signals and assumed nonlinear quadratic or power-law deviations from linearity of the signal amplitude. Conventional cavity ring-down spectroscopy (CRDS) absorption spectra also were analyzed in the same fashion. Given the same degree of assumed nonlinear response, the maximum relative errors in CBDS analyzed in the time-domain (TD) and frequency-domain (FD) were found to be independent of detuning meas and 6 and 50 times smaller, respectively than those acquired using CRDS, see Fig. 4d. For the frequency-domain CBDS case, maximum errors were 0.02% for realistic non-linearities at the 2% level 6,22 .
CBDS achieves high accuracy through precise measurement of the cavity resonance frequencies, and accurate modeling and fitting of the buildup signals in the frequency domain. The generality of our field-based method enables applications to dynamic cavity-enhanced sensing throughout the electromagnetic spectrum, making the method amenable to the analysis of intermode 24 and multiplexed 25 buildup signals with detuned local excitation fields. When dynamic events are not of interest, the tightly locked optical scheme allows for coherent timedomain averaging of CBDS signals, and therefore ultrahigh precision with minimal data storage and no dead time. We see the potential of CBDS for improving the accuracy of fundamental and atmospheric absorption spectroscopy studies as well as metrological applications which to date have depended exclusively on intensity-based experiments. The rapidity and sensitivity of CBDS should also render it useful in fast biological processes and single-particle spectroscopy. Moreover, CBDS methods can be readily extended to broadband spectroscopic techniques using an optical frequency comb, which will open new possibilities for high-accuracy measurements in this field 26 .
The CBDS aligns well with general efforts to express physical quantities in terms of frequency 27 . Molecular spectra entirely measured in terms of cavity resonance frequencies can be easily referenced to the atomic frequency standard. CBDS will result in robust SI-referenced uncertainties and will greatly facilitate interlaboratory comparisons of data. In this context, we see clear applications of CBDS e.g. to Doppler thermometry 28 as well as to a new gas pressure standard currently being developed which is based on precisely measuring the dispersive shifts of optical cavity modes 29 . Also, recent nondestructive detection of Rydberg atoms based on cavity dispersion 30 indicates the potential of CBDS in terms of both speed and accuracy for determination of atomic population.

Transient cavity response to single-mode, non-resonant excitation
Consider a conventional, linear optical cavity formed by two mirrors having intensity reflectivity and separated by a distance . The cavity is filled with an intracavity gas medium described by an absorption coefficient . We define an effective mirror reflectivity and round-trip time of the empty cell as eff = − and = 2 ⁄ = 1 FSR ⁄ , respectively, where is the speed of light in vacuum, is the refractive index of absorptive medium and FSR is the cavity free spectral range. Let us consider excitation of the cavity by light electric field ( ) = (1 − − 0 ) characterized by an arbitrary angular frequency and amplitude . Here, we assume a finite switch-on time 0 = 0 −1 of the electric field. The time response of the cavity out ( ) can be calculated at a given time by summing the contribution of finite number of passes in the cavity realized up to this moment 23 We assumed that = 0 corresponds to the moment when the first transmitted field contribution leaves the cavity. Further expansion of Eq. (1) leads to the sum of two finite geometric series where the factor describes modification of the electric field amplitude after the first pass through the cavity. In transition from Eq. (1) to Eq. (2) we replaced the expression − by − , since the electric field angular frequency can be rewritten as = 2 ( FSR + ), where is the cavity mode number and = 2 is detuning of the light angular frequency from the cavity mode center. Small values of round-trip time allow us to replace the discrete time values by the continuous quantity and consequently use integrals instead of sums in Eq. (2) Here, for the convenience of calculations we expressed eff / as exp[ / ln( eff )]. Evaluation of Eq.
(4) leads to the final complex-valued expression for the electric field leaving the cavity where out 0 ( ) = (1 − − 0 ), and = − −1 ln( eff ) describes the width (HWHM) of q-th cavity mode. It can be easily shown that (2 ) −1 is the conventional intensity-based time constant of a light decay measured in cavity ring-down spectroscopy. The structure of Eq. (5) illustrates how the transient field tends to oppose the driving field and giving rise to interference between the two fields at frequencies and = − , respectively. Taking the real part of the field defined by Eq. (5) gives Re{ out ( )} = | out 0 ( )| cos( where In order to compute the intensity, we square the real-valued field (Eq. 10) and average all sinuosoidal terms over optical cycles to account for the finite detector bandwidth. Ignoring the sum frequency term occurring at optical frequencies, this operation yields the intensity exiting the cavity as that fully describes the shape of the buildup signal measured in the CBDS method. Here, the amplitude of the transmitted signal is defined as out 0 ( ) = | out 0 ( )| 2 /2. The intensity function given by Eq. (13) exponentially approaches a constant value for long times and exhibits damped oscillations at the beat angular frequency, between that of the excitation field, and the cavity resonance frequency, . Note the occurrence of two characteristic rates, one at 2 equal to the familiar ring-down intensity decay rate, and the other at half of this value, which corresponds to the characteristic decay rate of the field amplitude.
In general, the amplitude out 0 ( ), the factor | ( )| and the phase However, for small values of 0 all these functions can be treated as time independent. In practice, for 0 = 50 ns the expression − 0 is of the order of 10 -9 for > 1µs, so it can be neglected in out 0 ( ) and ( ) leading to timeindependent functions out 0 and . For the limiting case 0 → 0 ( 0 → ∞), which corresponds to an immediate switch-on of the incident light electric field, out 0 ( ) → | | 2 , ( ) → 1 and ( ) → 0 (because → 0).

Spectrum model of the buildup signal
For practical reasons mentioned above let us consider the buildup shape function Ф( ) approximating the ratio out / out 0 from eq. (13) in the form where This function, apart from Lorentzian components, also contains symmetric, dispersive terms. We checked numerically that out ( ) given by Eq. (16), scaled by an amplitude parameter, reproduces the fast Fourier transform (FFT) spectrum of the buildup signal simulated from Eq. (13) with 10 -7 accuracy.
Here, the limiting factor is the accuracy of the FFT spectrum calculation caused by the finite sampling density of the buildup signal. We also found that adding a constant background parameter to Eq. (16) further improves the agreement between our model and FFT spectrum by more than a factor 10 3 .

Effect of detection system nonlinearity on measurement accuracy
To quantify how system nonlinearity affects the accuracy of measured spectra, we performed simulations of normalized buildup signals, ( )/ max , with their amplitudes multiplied by the function 1 ( ) = 1 − [ ( )/ ], where a is a constant factor which scales the degree of nonlinearity and max is the maximum amplitude of ( ) for the whole spectrum (Fig. I). We also did analogous calculations to evaluate the sensitivity of conventional cavity ring-down spectroscopy (CRDS) absorption spectra to detector nonlinearity. We found a 0.1% -4% systematic bias in the y axis of CRDS spectra and a 1% -20% maximum systematic error for 2% < a < 30% (Fig. Ib). Here, detector nonlinearity of 2% corresponds to a realistic case 6,22 . For CBDS spectra retrieved from frequency-domain (FD) analyses of buildup signals, the maximum systematic error is up to 50 times smaller than that predicted for CRDS spectra and is independent of detuning meas . The systematic bias of the FD CBDS spectra averages close to zero within the entire spectrum (Fig. Ia). We estimate sub-per-mil accuracy in the FD CBDS even when the detector nonlinearity is as high as 8%. Moreover, we noticed that the choice of fitting the buildup signals in the time or frequency domain has a large impact on the sensitivity to detector nonlinearity. Dispersive spectra obtained through time-domain (TD) analyses of buildup signals have a nonlinear susceptibility intermediate between the CRDS and the FD CBDS cases, i.e. 6 times lower by comparison to the CRDS absorption case. In the case of the TD CBDS spectrum, we also calculate a systematic bias of the y axis ranging from 0.01% -0.2 % and we observed a slight asymmetry of TD CBDS spectra (Fig. Ia). As a further exploration of the effect of non-linearity, we assumed its power law response model has the form 2 ( ) = [ ( )/ ] , which in conventional CRD spectroscopy would lead to measured decay rates biased by the constant fractional amount, . Notably, this type of nonlinearity would not be evidenced in the fit residuals of individual decay signals because the decay signals remain exponential in form. As can be seen in Fig. Ib, this type of nonlinearity augments the bias for both the CRDS and TD CBDS spectra, by comparison to 1 ( ), but yields nearly identical results for the FD CBDS case.
Figure I Influence of detection system nonlinearity on the spectrum accuracy. (a) Relative systematic differences between absorptive/dispersive spectra obtained from ring-down/buildup signals simulated with and without nonlinearity of the amplitude for corresponding nonlinear factors a: 2%, 10% and 30%. The buildup signals were analyzed in the time and frequency (as a power spectrum) domains leading to spectra denoted as "TD" and "FD", respectively. For dispersion, we chose a detuning meas ≈ 100 kHz, corresponding to ≈30 cavity mode widths (HWHM). We modeled nonlinear distortion of the spectrum by multiplying the normalized time response of the cavity ( )/ max (buildup and ring-down signals) by 1 ( ) = 1 − [ ( ) max ⁄ ], where max is the maximum amplitude of ( ). (b) Solid lines -maximum relative systematic errors of absorptive and dispersive spectra from the panel (a) versus detection nonlinearity 1 for various detunings meas . Dashed purple linessimilar results when using the power-law model of detector nonlinearity given by 2 ( ) = [ ( ) max ⁄ ] and detuning meas ≈ 100 kHz.