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We describe an ultra-sensitive cavity ring-down spectrometer which operates in the mid-infrared spectral region near 4.5 µm. With this instrument a noise-equivalent absorption coefficient of 2.6×10−11 cm−1 Hz−1/2 was demonstrated with less than 150 nW of optical power incident on the photodetector. Quantum noise was observed in the individual ring-down decay events, leading to quantum-noise-limited short-time performance. We believe that this spectrometer’s combination of high sensitivity and robustness make it well suited for measurements of ultra-trace gas species as well as applications in optics and fundamental physics.
The mid-infrared spectral region is home to the fundamental vibrational transitions of a wide variety of molecular species. While these strong transitions should allow for measurements of ultra-low concentrations, only recently has the mid-infrared optical technology reached the point where highly sensitive spectroscopic techniques can be utilized. Of these techniques, continuous-wave cavity ring-down spectroscopy (cw-CRDS)  has shown great promise, however only a few groups have demonstrated CRDS noise-equivalent absorption coefficients below 1×10−10 cm−1 Hz−1/2 [2–4] in the mid-infrared. Here we describe a mid-infrared cavity ring-down spectrometer which exhibits an ultra-low detection limit and shows evidence of quantum noise [5–9] in the ring-down decay events.
CRDS is a cavity-enhanced technique in which a high finesse Fabry-Pérot cavity is used to achieve a long pathlength (10’s of km to 100’s of km) in the laboratory . During a CRDS measurement the cavity is first optically pumped by the laser source. This light is then extinguished and the corresponding passive decay of the intracavity optical power is monitored . The exponential decay time, the ring-down time constant (τ), is directly related to the absorption, α, within the cavity:
where c is the speed of light and τ0 is the empty-cavity ring-down time constant which accounts for mirror losses due to transmission, absorption, and scattering. Importantly, CRDS is insensitive to laser amplitude fluctuations.
While CRDS is routinely performed in the near-infrared, working in the mid-infrared spectral region can be far more challenging due to generally lower component performance and higher component costs. Importantly, supermirror coatings are more difficult to produce in the mid-infrared, leading to much higher absorption and scattering losses. These significant losses lead to far lower build-up of optical power within the cavity and correspondingly lower optical transmission. As a result, very low noise photodetectors are required to achieve high signal-to-noise ratios on the ring-down decay events. In addition, high efficiency electro-optic modulators are not available in the mid-infrared, which can make Pound-Drever-Hall locking  far more challenging  as well as precluding the use of recently developed frequency-agile techniques [6, 7, 13].
A schematic of our cavity ring-down spectrometer can be found in Fig. 1. Measurements were made with a distributed-feedback quantum cascade laser (DFB-QCL) having a maximum output power of 38 mW and a tuning range of 2205 cm−1 to 2213 cm−1. The resulting mid-infrared radiation was passed through an optical isolator and then a germanium acousto-optic modulator (AOM). This AOM was used as a fast optical switch to initiate the ring-down decays and to further prevent optical feedback to the laser. The optical extinction was improved by simultaneously chirping the laser frequency away from a given cavity resonance via a change in the DFB-QCL drive current. The 1st-order output of the AOM was sent through mode-matching optics, a half-wave plate, and then into the optical cavity, while the 0th-order output was launched into a wavelength meter. Typically, 12 mW of optical power was incident on the cavity input mirror. The half-wave plate was placed before the cavity to reduce the effects of mirror birefringence . This led to a roughly factor of two improvement in sensitivity.
The optical cavity had a nominal length of 1.5 m and a corresponding free spectral range of 100 MHz. The ring-down mirrors had a radius of curvature of 1 m and a power reflectivity of 99.99 % leading to a finesse of 31 000 and an effective pathlength of 15 km. We estimate that the cavity mode matching was >90%. The ring-down decays were recorded on a 0.1 mm diameter liquid-nitrogen-cooled InSb photodetector with a field of view of 60°, a responsivity of 3.7 A/W, a transimpedance gain of 106 V/A, and a measured noise-equivalent power of 70 fW Hz−1/2 and digitized at 1 MSamples s−1 by a 22-bit acquisition board. The 3-dB bandwidth of the acquisition board was measured to be B = 480 kHz. Given its high bit depth, the digitizer board did not contribute significantly to the overall technical noise. The laser was stabilized to the optical cavity via a low bandwidth (4 Hz) transmission lock which actuated the laser current . Spectral scanning was performed by stepping the laser temperature in increments of the cavity’s free spectral range.
We can calculate the expected root-mean-square (RMS) noise on a given ring-down decay event, σV(t), as a combination of two-sided quantum and technical sources as :
where e is the electron charge, G is the transimpedance gain, V(t) is the observed ring-down signal amplitude, and Δt is the sampling interval. The technical noise is given by , where NEP is the one-sided detector noise-equivalent power, R is the detector responsivity, and B is the electronic bandwidth. Importantly, the quantum noise component scales as the square-root of the signal amplitude while the technical noise component is constant.
The noise within each decay event can be measured by applying the Wiener-Khintchine theorem, in which we equate the noise power spectral density (V2/Hz) to the Fourier transform of the autocorrelation function of the fit residual r(t) = V(t) − Vfit(t). Here f is the Fourier frequency and Vfit(t) is an exponential fit to the decay signal. Symbolically,
from which the integrated noise power (variance) can be found as:
Equations (3) and (4) were evaluated using discrete summations for the autocorrelation and Fourier transform. In order to measure the time dependence of the noise power spectral density, was computed for averaging bins of width T <<τ and at times ti relative to the beginning of the fitting window. In the absence of correlation, Eqs. (3) and (4) yield the variance of the residual signal within an averaging bin.
The signal noise was also measured by analyzing the distribution of fit residuals for an ensemble of decays. For each time tj relative to the beginning of the fit window, the noise power was given by the variance in the ensemble of residuals. The time-dependent noise calculated in this manner was in good agreement with the single-shot analysis described above, indicating that the noise behaves ergodically. Both approaches yield quantum-noise-limited results which are consistent with Eq. (2), and which decay exponentially with a time constant of τ/2 until the technical noise floor is reached. The agreement between the autocorrelation of the fit residuals and Eq. (2) can be observed in Fig. 2. To the best of our knowledge this the first mid-infrared instrument, and only the third overall [7–9], which exhibited quantum noise in the ring-down decay events in a traditional (i.e. direct current) CRDS instrument. We note that when these measurements were repeated with a higher noise photodetector, the decay events did not show evidence of quantum noise (as would be expected).
In the quantum noise limit (QNL) the relative standard uncertainty in the determination of τ from an individual fit to a decay event is given by the inverse of the square root of the number of photoelectrons [9, 16]:
The quadrature sum of these two expressions can then be used to approximate the relative uncertainty in the regime where both quantum noise and technical noise are significant . From Eqs. (5) and (6) we can calculate the expected quantum-noise-limited and technical-noise-limited (i.e. detector limited) single-decay-event fit uncertainties to be 0.008 % and 0.015 %, respectively, where V(0) = 0.50(06) V (corresponding to 136(16) nW). Adding these uncertainties in quadrature yields an approximate total expected fit uncertainty of 0.017 %. Experimentally, we observe average fit uncertainties of 0.019 % which is in good agreement with the above calculations. This indicates that other noise sources do not significantly contribute to our observed decay curves. Furthermore, we observe that a typical 1 s ensemble of ring-down time constants (i.e. the first 30 time constants from Fig. 3) have a standard deviation of 0.023 %, thus, demonstrating that shot-to-shot fluctuations in the system are minimal.
The presence of quantum noise during a significant fraction of the cavity decays necessitates the use of a weighted least-squares fit with weighting factors w(t) = 1/σV (t)2 , where σV (t) is calculated from Eq. (2). An estimate of the proper weighting factors for any given ensemble requires V(t), which is approximated by the average fit of the first 10 cavity decay events using equal weights. When fitting the portion of each cavity decay that exhibits significant quantum noise (t<100 µs), equal weighting of the data yields an average fit uncertainty that is 20 % higher than that observed with proper weighting. Each cavity decay event has a relatively constant offset arising from dark current in the InSb photodetector. For the purpose of calculating w(t), this dark-current offset has been subtracted from V(t) to ensure that the quantum noise contribution to Eq. (2) is not overestimated.
An Allan variance plot , which is a measure of the system stability, can be found in Fig. 3. Based upon this plot we can determine after an optimal averaging time of 1.3 s (corresponding to 40 ring-down decay events) we can achieve a minimum detectable absorption of 2.3×10−11 cm−1 which corresponds to a noise-equivalent absorption coefficient of 2.6×10−11 cm−1 Hz−1/2. A representative spectrum of a low hydrocarbon air sample can be found in Fig. 4, with absorption noise of 1.6×10−10 cm−1 (which is within a factor of 4 of the value predicted by the Allan variance in Fig. 3 and likely limited by a combination of weak interfering molecular absorption and subtle coupled-cavity etalons). Based upon this spectrum we can estimate a detection sensitivity for N2O of 2 pmol/mol with only ten ring-down time constants averaged at each spectral point.
The highly sensitive nature of this spectrometer becomes clear upon inspection of Fig. 4. We observe, at a signal-to-noise ratio of 10:1, a weak 13CO2 rotational-vibrational transition originating with a rotational quantum number, J, of 76. This measurement of 13CO2 in natural abundance was performed on a low-hydrocarbon air sample with a manufacturer-specified total hydrocarbon concentration of <1 µmol/mol at room temperature (20 °C) and moderately low pressure (13 kPa). We are presently in the process of reducing the effects of etalons and long-term drifts to the observed absorption in order to produce spectra which are themselves quantum-noise limited. The use of an additional isolator between the mode-matching lenses and a more significant tilt on our detector reduced the influence of the observed etalons.
The relatively low cost instrument described herein is well suited for precise measurements of ultra-trace gas species including radiocarbon [20, 21] and atmospherically relevant free radicals. This instrument has been used to measure birefringence in supermirror coatings  and may be applied in addressing other challenges in fundamental physics including the search for symmetrization postulate violations in molecular physics  and measurements of absolute number densities for magnetically-trapped ultra-cold molecules using absorption-based techniques .
Funding. Support was provided by a NIST Innovation in Measurement Science grant and the NIST Greenhouse Gas Measurements and Climate Research Program.