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The Journal of Chemical Physics

The superposition principle and cavity ring-down spectroscopy

Kevin K. Lehmann, D. Romanini

Published Dec 15, 1996• 192 citations
Publisher

Abstract

Cavity ring-down is becoming a widely used technique in gas phase spectroscopy. It holds promise for further important extensions, which will lead to even more frequent use. However, we have found widespread confusion in the literature about the nature of coherence effects, especially when the optical cavity constituting the ring-down cell is excited with a short coherence length laser source. In this paper we use the superposition principle of optics to present a general and natural framework for describing the excitation of a ring-down cavity regardless of the relative values of the cavity ring-down time, the input pulse coherence time, or the dephasing time of absorption species inside the cavity. This analysis demonstrates that even in the impulsive limit the radiation inside a high finesse cavity can have frequency components only at the natural resonance frequencies of the cavity modes. As an immediate consequence, a sample absorption line can be detected only if it overlaps at least one of the cavity resonances. Since this point is of particular importance for high resolution applications of the technique, we have derived the same conclusion also in the time domain representation. Finally, we have predicted that it is possible to use this effect to do spectroscopy with a resolution much higher than that of the bandwidth of the excitation laser. In order to aid in the design of such experiments, expressions are derived for the temporal and spatial overlap of a Fourier transform limited input Gaussian beam with the TEMmn modes of the cavity. The expressions we derive for the spatial mode overlap coefficients are of general interest in applications where accurate mode matching to an optical cavity is required.

DOI
10.1063/1.472955

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Cavity Ring Down Spectroscopy has become an important method in gas phase spectroscopy and its uses promise to expand. While previous investigators have considered the decay of the stored energy in the cavity ring down cell following short pulsed excitation, this paper considers the effects of pulse reshaping during cavity ring down. It is found that in order to properly model these effects, one must consider not only the mode dependent loss of the ring down cavity, but also dispersion that shifts the resonant modes of the cavity. Only by considering the latter effect does one get a "causal" result where the free induction decay signal of the absorber occurs after the excitation pulse. It is demonstrated that in the limit that the cavity round trip time is long compared to both the T 2 time of the intracavity absorber and the temporal width of the excitation pulse, the reshaping of the n'th pulse out of the ring down cavity is the same as the reshaping produced by free space propagation over a distance of (2 n + 1) L where L is the separation of the mirrors in the cell. However, when the absorber T 2 becomes comparable to or longer than the cavity round trip time, then discrete modes of the ring down cavity become important. A general expression for the output is given for the important case of a weak absorber, and changes in the output are computed for a single absorption line as a function of the ratio of T 2 to the cavity round trip time for several values of the optical thickness of the sample. Lastly, it is suggested that monitoring changes in the pulse shape of the cavity ring down signal provides a way to characterize an absorption spectrum on a frequency scale of the laser bandwidth down to the cavity free spectral range.

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We present an ac heterodyne technique in cavity ringdown spectroscopy that permits $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ absorption sensitivity with microwatt-level light power. Two cavity modes, one probing the empty cavity and the other probing intracavity absorption, are excited simultaneously but with their intensities temporally out of phase, with one mode decaying and the other rising. Heterodyne detection between the two modes reveals the dynamic time constants associated with the empty cavity and the additional intracavity gas absorption. The method offers a quick comparison between the on- and off-resonance information, a prerequisite to reaching the fundamental shot-noise limit. This simple and yet important improvement of cavity ringdown spectroscopy should lead to the enhanced performance of a wide range of applications.

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Cavity ring-down is becoming a widely used technique in gas phase spectroscopy. It holds promise for further important extensions, which will lead to even more frequent use. However, we have found widespread confusion in the literature about the nature of coherence effects, especially when the optical cavity constituting the ring-down cell is excited with a short coherence length laser source. In this paper we use the superposition principle of optics to present a general and natural framework for describing the excitation of a ring-down cavity regardless of the relative values of the cavity ring-down time, the input pulse coherence time, or the dephasing time of absorption species inside the cavity. This analysis demonstrates that even in the impulsive limit the radiation inside a high finesse cavity can have frequency components only at the natural resonance frequencies of the cavity modes. As an immediate consequence, a sample absorption line can be detected only if it overlaps at least one of the cavity resonances. Since this point is of particular importance for high resolution applications of the technique, we have derived the same conclusion also in the time domain representation. Finally, we have predicted that it is possible to use this effect to do spectroscopy with a resolution much higher than that of the bandwidth of the excitation laser. In order to aid in the design of such experiments, expressions are derived for the temporal and spatial overlap of a Fourier transform limited input Gaussian beam with the TEMmn modes of the cavity. The expressions we derive for the spatial mode overlap coefficients are of general interest in applications where accurate mode matching to an optical cavity is required.
Cavity Ring Down Spectroscopy has become an important method in gas phase spectroscopy and its uses promise to expand. While previous investigators have considered the decay of the stored energy in the cavity ring down cell following short pulsed excitation, this paper considers the effects of pulse reshaping during cavity ring down. It is found that in order to properly model these effects, one must consider not only the mode dependent loss of the ring down cavity, but also dispersion that shifts the resonant modes of the cavity. Only by considering the latter effect does one get a "causal" result where the free induction decay signal of the absorber occurs after the excitation pulse. It is demonstrated that in the limit that the cavity round trip time is long compared to both the T 2 time of the intracavity absorber and the temporal width of the excitation pulse, the reshaping of the n'th pulse out of the ring down cavity is the same as the reshaping produced by free space propagation over a distance of (2 n + 1) L where L is the separation of the mirrors in the cell. However, when the absorber T 2 becomes comparable to or longer than the cavity round trip time, then discrete modes of the ring down cavity become important. A general expression for the output is given for the important case of a weak absorber, and changes in the output are computed for a single absorption line as a function of the ratio of T 2 to the cavity round trip time for several values of the optical thickness of the sample. Lastly, it is suggested that monitoring changes in the pulse shape of the cavity ring down signal provides a way to characterize an absorption spectrum on a frequency scale of the laser bandwidth down to the cavity free spectral range.
Optical cavities serve as powerful tools for sensing experiments, often relying on narrow-linewidth laser sources to minimize the impact of optical coherence on measurements. However, demands for affordable integrated and miniaturized sensing platforms in point-of-care diagnostics, environmental monitoring, and similar applications motivate switching to sources with broader linewidths suitable for both monolithic and heterogeneous integrations. Time-domain measurement techniques such as cavity ring-down spectroscopy (CRDS) are widely used in conjunction with optical cavities, but to date there has been no universal model that quantifies the impact of partial coherence on the cavity temporal transfer function. We apply a linear systems theory approach to develop a closed-form analytic model for cavity-based sensing that quantifies the influence of source bandwidth (i.e., temporal coherence) on spectroscopic measurements in the time domain. This approach can be applied to a variety of cavity-based spectroscopies. In this study, cavity-enhanced absorption spectroscopy (CEAS) and CRDS paradigms have been examined using standing- and traveling-wave resonator examples. Results show that although increased cavity loss is the primary factor reducing output power and photon lifetime in both CEAS and CRDS, broader source linewidths can also influence cavity buildup and transmission and must be accounted for when modeling the system response. The model is consistent with known results in the literature and provides a framework for evaluating source detuning and coherence effects on cavity dynamics.
The dynamic response of optical resonant cavity has been theoretically studied,by taking resonant cavity as a linear and time invariant system and through the approach of Signal Processing.The mathematical description obtained could fit any form of light waves.From the perspective of Signal Processing,the traditional pulse and continues wave cavity ring-down spectroscopy could be considered as observation of impulse response and step response respectively.Furthermore,the cavity response to the light-wave linearly frequency chirped has been studied,in the condition of slow chirp.The Fourier analysis of light power passed through cavity has been made and an analytic description has been given,which reveals the fact that the signal spectrum of light power is related to the absorption spectrum of material inside cavity.Based on this theoretic analysis,a new approach of cavity enhanced absorption spectroscopy has been put forward,called Frequency Domain Cavity Ring-Down Spectroscopy.The result would facilitates the research of cavity enhanced spectroscopy techniques,as well others using resonant cavity,such as P-D-H laser frequency stabilization.
The dynamic response of optical resonant cavity has been theoretically studied, by taking resonant cavity as a linear and time invariant system and through the approach of Signal Processing. The mathematical description obtained could fit any form of light waves. From the perspective of Signal Processing, the traditional pulse and continues wave cavity ring-down spectroscopy could be considered as observation of impulse response and step response respectively. Furthermore, the cavity response to the light-wave linearly frequency chirped has been studied, in the condition of slow chirp. The Fourier analysis of light power passed through cavity has been made and an analytic description has been given, which reveals the fact that the signal spectrum of light power is related to the absorption spectrum of material inside cavity. Based on this theoretic analysis, a new approach of cavity enhanced absorption spectroscopy has been put forward, called Frequency Domain Cavity Ring-Down Spectroscopy. The result would facilitates the research of cavity enhanced spectroscopy techniques, as well others using resonant cavity, such as P-D-H laser frequency stabilization. ©, 2015, Chinese Optical Society. All right reserved.
Due to the phase-coherent relationship of intra-cavity pulse reflections leading to the narrow frequency-selectivity of the cavity, Cavity Ring-Down Laser Absorption Spectroscopy (CRLAS) is a promising method lo study Doppler-free two-photon absorption at resolutions at least two orders of magnitude better than the pulse bandwidth of even Fourier Transform Limited nsec pulsed lasers. The nonlinear absorption will lead to a nouexponential cavity decay with can be used to discriminate it from both linear absorption and intrinsic cavity loss. In this study, several issues have been examined in order to access the feasibility of obtaining the high-resolution two-photon absorption spectra of the Rydberg states of alkali dimers by this technique. Using atomic transition dipoles as an approximation, the laser intensities needed to elicit reasonable excitation were estimated, and found to be well within presently available levels. The importance of the various linewidth- limiting mechanism was also examined. The relative magnitudes of pressure broadening, AC Stark broadening, cavity stability, and time-of-flight broadening lead us to expect a resolution of $\\sim$1 MHz or better.
We present an ac heterodyne technique in cavity ringdown spectroscopy that permits $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ absorption sensitivity with microwatt-level light power. Two cavity modes, one probing the empty cavity and the other probing intracavity absorption, are excited simultaneously but with their intensities temporally out of phase, with one mode decaying and the other rising. Heterodyne detection between the two modes reveals the dynamic time constants associated with the empty cavity and the additional intracavity gas absorption. The method offers a quick comparison between the on- and off-resonance information, a prerequisite to reaching the fundamental shot-noise limit. This simple and yet important improvement of cavity ringdown spectroscopy should lead to the enhanced performance of a wide range of applications.