## Abstract

This paper presents results from a pulsed-laser cavity ring-down spectrometer with novel field programable gate array real-time data collection. We show both theoretically and experimentally that the data extraction can be achieved from a single cavity ringdown event, and that the absorbance can be determined without the need to fit the ringdown time explicitly. This methodology could potentially provide data acquisition rate up to 1MHz, with the accuracy and precision comparable to nonlinear least squares fitting algorithms.

© 2012 OSA

## 1. Introduction

Cavity ringdown spectroscopy (CRDS) is a widely used ultra-sensitive direct absorption technique [1,2]. The cavity ringdown technique has been used to address a variety of research topics since its initial use to measure losses in mirror coatings by Anderson *et. al.* [3, 4] and first use with pulsed laser CRDS by O’Keefe *et. al.* [5], including the detection of species in harsh environments. CRDS is used in commercial instruments are designed, by-in-large, for species specific gas monitoring. The sensitivity of CRDS lies in the extraordinarily long path lengths attained and in the insensitivity of the absorption characteristic (the lifetime of photons trapped in an optical cavity) to fluctuations in the intensity of the light source. Like many spectroscopic methods which rely on scanning a light source through a range of frequencies of interest, traditional CRDS is becoming limited in its ability to rapidly acquire spectra. Interestingly, this limit is commonly imposed neither by the scanning rate nor the pulse rate of the laser system, but rather by the speed of the data acquisition and analysis process [6–9]: while a traditional experiment with a pulse rate less than 100 Hz may be able to keep up with the fitting of transients using non-linear least squares, new laser technologies (such as Quantum Cascade Lasers (QCLs)) are able to pulse at tens of kilohertz: far too fast for even the most well coded fitting regime to be able to analyse data in real time. Indeed, a recent publication provides an in-depth examination of the performance of several common techniques used to extract decay lifetimes from ring-down waveforms [10]. Furthermore, systems have been designed which can generate ring-down decay transients at rates in excess of 10 kHz. Such systems have limited scanning rates due to the speed of data acquisition and analysis systems [6–9].

Several cavity enhanced optical techniques are addressing the issue of data acquisition by attempting real-time data collection using field programmable gate arrays (FPGA) and similar technologies, see for example Sayres *et. al.* [11]. The approach presented here is an extension of the CW-CRDS work by Boyson *et. al.* [12], but for a pulsed dye laser system capable of acquiring data in real time using FPGA technology.

The signal generated by the photodetector in a standard pulsed or continuous wave-CRDS system, *I*(*t*), is a simple exponential decay which is usually digitized and then fit with a three parameter function,

*O*, and initial laser intensity

*I*

_{0}are required fit parameters, but the cavity decay lifetime,

*τ*, is the only fit parameter needed to measure absorbance,

*A*:

*n*is the index of refraction within the optical cavity,

*l*is the optical path length in the cavity,

*c*is the speed of light, and

*τ*

_{0}is the empty cavity decay lifetime. Several digitally based methods have been reported which can extract

*τ*from a digitized decay signal having 1000 points in as little as 200

*μ*s. It is worth noting that if the offset is stable over the course of an experiment, it may be treated as a constant and excluded from the fitting regime. With simultaneous fitting and acquisition, such systems could achieve data acquisition rates of 5 kHz. In previous work, Spence

*et. al.*[6] developed an analog system that could acquire and analyze decay transients at 50 kHz, however the system required the laser source to be locked to a cavity resonance. As laser pulse rates increase, the need for fast data processing is limiting the ability of CRDS to rapidly collect spectra.

Recent advances in digital technologies are making fast signal processing both possible and more accessible to scientists at large. In particular, FPGAs offer users a programmable chip capable of processor-speed data manipulation. FPGAs are a single chip comprised of logic blocks which, when “programmed,”are wired together to perform data manipulations at clock speeds of 0.1 to 1 GHz. This new kind of signal processing platform offers signal processing without the overhead of an operating system and programming environment. The methodology presented here is particularly amenable to implementation using an FPGA as it does not require the iterative refinement of non-linear least squares algorithms, but relies on simple multiplication and addition.

In this work we present a frequency component analysis (FCA) methodology for extracting decay lifetimes from digitally acquired signals using both traditional hardware and an FPGA. In what follows, this technique is first modeled using synthetic decay transients with added noise, and then applied using a traditional data acquisition system consisting of a fast digitizer and personal computer to provide fast acquisition of *τ*. The methodology then deployed on a combination digitizer/ FPGA evaluation board producing the first digitally based system capable of acquiring *τ* from fast exponentially decaying signals in real time at rates exceeding 1 MHz.

## 2. Theory: frequency component analysis

Presented here is an alternative approach to that of Boyson *et. al.* [12], Mazurenka *et. al.* [13] and Everest *et. al.* [8], and consists of a discrete sum method for the rapid extraction of decay lifetimes from exponentially decaying signals by measuring the relative intensity of two frequency components of an exponentially decaying signal. While this methodology can be applied to single-decay events from pulsed or continuous wave CRDS systems, for the purposes of this discussion it is convenient to consider CRDS waveforms like those shown in Fig. 1(a).

Figure 1(a) presents two ringdown waveforms, one with a long decay lifetime (blue) and another with a short decay lifetime (red), collected over the same time window *w*, with a sampling frequency of 100 MHz. These two signals could correspond to an empty cavity response (blue) and a cavity with a strongly absorbing species present (red). The waveform in Fig. 1(b) is constructed from five ringdowns by concatenating successive decay events after transposing every other event in time, and Fig. 1(c) is the power spectrum of this waveform. This power spectrum show a comb of characteristic frequency components whose magnitude vary with *τ*. With a strongly absorbing species present, the time-domain signal in altered in such a way that results in a comb of peaks in the frequency domain where the lowest frequency components have almost equal intensity. Without an absorber present, the intensity of the peaks in the frequency domain vary greatly, especially between the two lowest frequency components *I*_{1} and *I*_{2}. In principle, the intensity of any individual peak in the frequency domain could be used to determine *τ*, however, these individual peak intensities also vary with the initial intensity of the laser pulse, *I*_{0}. As shown below, the ratio of two frequency components, like
$\frac{{I}_{1}}{{I}_{2}}$, provides a light-intensity independent measure of *τ* which can be rapidly determined using basic digital processing techniques.

In the following analysis we will derive a relation for *τ* as a function of the powers of the fundamental frequency *I*_{1} and the first harmonic *I*_{2}. In this derivation we will show that the relation between *τ* and
$\frac{{I}_{1}}{{I}_{2}}$ is the same for 1 ringdown event or as it is for many events concatenated together. Figure 2 shows the schematic of the experimental setup used in the following sections and is the basis for the mathematical analysis.

As the Fourier transforms of even functions are real, extending the waveform in Fig. 1(b) to generate an even waveform greatly simplifies the mathematical analysis.The two lowest frequency components *I*_{1} and *I*_{2} obtained using *m* sampling windows can be evaluated by computing the integral of the product of the exponential decay described in Eq. (1) and a cosine function having the frequency of interest:

*R*, of

*I*

_{1}/

*I*

_{2}, is given by:

*τ*′ is the decay constant

*τ*normalized with respect to the sampling window time

*w*, that is

*τ*′ =

*τ/w.*

It is important to note that both *I*_{1} and *I*_{2} are independent of any systematic signal offset *O* and that the ratio *R* is independent of the initial signal intensity *I*_{0} or the number of windows, *m*. Indeed, *R* is only a function of *τ*′ and may be used to unambiguously determine *τ*.

For digitally acquired signals from a CRDS system, the integrals in Eq. (3) and Eq. (4) become discrete sums of the product of the recorded exponentially decaying signal, *S _{i}* =

*S*(

*t*),

_{i}*t*= 0,...,

_{i}*t*=

_{k}*w*, and the appropriate cosine function which, if the sampling window

*w*remains constant, is of fixed frequency:

*R*in Eq. (6) from a digitized signal is particularly suited for rapid acquisition using digital computing technology as it requires simple accumulation of two products of three discrete signals, two of which only change if the sampling window varies.

Unfortunately, obtaining *τ*′ from the *R* using Eq. (5) is problematic because the relationship needs to be solved numerically for each value of *τ*′. However, as shown in Fig. 1(d), *τ*′ is a near linear function of the ratio at *τ*′ values greater than about 0.1. At ratios less than 0.1, *τ*′ is small compared to *w* and the corresponding frequency-domain signal approaches a comb of evenly spaced peaks, with the two lowest frequency peaks having near equal magnitudes, as illustrated in the red spectrum in Fig. 1(c). As a result, *R* approaches 1 and *τ*′ ceases to be a stable function of *R*. Experimentally, this restriction is easily overcome by shortening the data acquisition window *w* when *R* falls below some threshold value.

Two approaches for extracting *τ*′ from experimentally determined *R* values are explored here. In the first method, referred to here as “theoretical,” Eq. (5) is used to generate a precise lookup table of pairs (*R*,*τ*′) that is then used, with linear interpolation, to determine *τ*′ for a particular observed *R*. Alternatively, a lookup table was also constructed using Eq. (6) by computing (*R*,*τ*′) with a set of noiseless ring-down waveforms having the same sampling frequency as the pulsed laser CRDS system used here. This second approach is referred to as the “empirical” method and has the same inherent digitization error as the experimental system.

A simulation study was performed to compare the performance of the theoretical and empirical methods to the traditional Levenberg-Marquard (LM) non-linear fitting methodology. Without loss of generality for the simulation study we set *w*, such that *τ*′ = *τ*. For selected values of *τ*, noisy signals were simulated using the model *S*(*t _{i}*) =

*e*

^{−ti/τ}+

*ε*, were

_{i}*ε*is white noise (normally distributed with mean 0 and standard deviation

_{i}*σ*∈ [0, .5]). The resulting signal

*S*(

*t*) was used to compute

_{i}*R*using Eq. (6) and to obtain

*τ*approximations for the theoretical and the empirical methods.

*S*(

*t*) was also fitted to an exponential decay using LM non-linear fitting. We repeated this simulation at least 100,000 times and for each of the three methods computed the relative error of the mean estimated

_{i}*τ*: that is,

The results of this simulation are given in Fig. 3 for *τ* = 0.3. At lower noise levels, both the empirical and the LM methods outperform the theoretical method. The differences in the theoretical and empirical methods reflect the error associated with using sums of discrete data points to approximate the integrals in Eq. (3) and Eq. (4). Interestingly enough, when the standard deviation of the noise is close to 20% of the initial intensity, *I*_{0}, the theoretical method dips below both the empirical and the LM methods, and remains roughly below the other two methods from then on. This suggests that if the signal to be analyzed is very clean one should use the empirical method, as its implementation is much faster than the LM method. If the signal is somewhat noisy, with noise levels 20% of initial intensity or larger the theoretical methodology is to be preferred. Simulation results using *τ* values between .1 and .6 show similar dips in relative error of the mean estimated *τ* at noise levels between 20% and 30% for the theoretical method.

In addition to assessing accuracy, the noisy waveforms described above were analyzed to assess the precision of the three methods. Simulations were performed as above. Figure 4 is a plot of the relative standard deviation in the determination of *τ* as a function of the amplitude of the normally distributed noise added to the synthetic decay transients. At all noise levels the traditional LM method provides slightly better precision, but again this must be weighed against the speed of data analysis which is two to three orders of magnitude faster using the frequency component analysis methodology presented here.

Finally, we have examined the effect of window size, *w*, on precision. Figure 5 shows a plot of the relative standard deviation in t as a function of window size. For these simulations, *τ* = 0.3, *δτ* = 0.001, and *w* is varied from 0.8 to 3.0. Gaussian distributed random noise was added to each trace with *σ* = 0.2. For both methods an optimal data acquisition window of 1.5 or 5 times the decay constant is observed. Performance degrades at shorter times due to truncation of the decay waveform. At longer times, performance degrades due to additional introduced noise.

## 3. Experimental

The CRDS system, shown schematically in Fig. 2, consisted of a pulsed nitrogen-laser pumped dye laser (Photon Technology International GL-3300 and GL-302, respectively). The output of the laser was mode matched to a 1.1 meter optical cavity consisting of a pair of high-reflectivity mirrors (Los Gatos Research, *R* = 0.9995). The optical cavity was evacuated using an oil-free diaphragm pump and *μ*L quantities of NO_{2}(g) mixed with room air were injected into the cavity through a septum using a gas-tight syringe. Light exiting the cavity was detected using a photomultiplier tube (Hamamatsu H8443). The resulting signal was digitized either by a PC interfaced LeCroy digital oscilloscope (LT3720, 500 MHz) or by an FPGA digital signal processing development kit (Altera EP3C120), which consists of a high-speed mezzanine data conversion daughter card interfaced with a Cyclone III FPGA development board. This relatively inexpensive development kit achieved data sample rates of 100 MHz.

Programming an FPGA is comparable to designing a circuit as once it is programmed the FPGA chip runs repetitive tasks autonomously. Figure 6 is a block diagram which describes the design of the real-time FCA system which calculated *R* by implementing Eq. (6). It is important to note that each stage of this circuit executes in a point-by-point fashion calculating *R* in real time. The circuit is enabled by the falling edge of a ring-down event from the photodetector. When enabled, on each clock cycle (at a rate 100 MHz) the daughter board digitizes the signal from the photodetector and provides the signal to the FPGA board. Simultaneously, and also at 100 MHz, the board generates corresponding cosine function values and multiplies the digitized waveform by the result. The results of this multiplication are accumulated in a sum and the ratio of the sums is calculated and provided to Latch 1. While enabled, the output of Latch 1 follows the input showing the value of *R* as the ringdown transient it obtained. Once data has been acquired, the sums accumulated, and *R* calculated over the pre-determined sampling window, *w*, the enable signal goes logic “low,” stops all calculations, resets the accumulators, disables Latch 1 freezing its output at the final value of *R*, and activates Latch 2 providing the most recent value of *R* to the digital-to-analog output. As a result, the output of Latch 2 changes at the pulse rate of the CRDS system potentially allowing a slower data acquisition system to monitor *R* in real time. Indeed, the purpose of the second latch is to provide constant output while the circuit is disabled or acquiring and analyzing a new decay trace.

## 4. Results and discussion

Figure 7 is a plot of four of the signals generated in real time and recorded by the FPGA. The signal from the photomultiplier tube, *S*, is monitored at all times at a rate of 100 MHz and shows noise typical of a pulsed laser based CRDS system. Prior to triggering, the value of *R* is latched at the result from the previous ring-down event. Upon detection of a laser pulse, the FPGA enables calculation of both cosine functions (*C*_{1} and *C*_{2}) and calculates *R* as the exponential decay is acquired. The choice of *C*_{1} =cos(*πt/w*) and *C*_{2} =cos(2*πt/w*) correspond to the numerical approximation given in Eq. (6) and the conditions described by Fig. 1(b) and Fig. 1(c). After acquiring and analyzing data over the entire decay window (*w* = 28*μ*s or 2800 data points), *R* is again latched at its final value until the next laser pulse is detected. One of the benefits of this type of analysis is immediately apparent; the high-frequency noise present on *S* is effectively filtered out when the ratio of the two sums is calculated.

It should be noted that the decay waveform was altered by the FPGA used in this study as the input to the analog-to-digital converter was coupled through a small transformer. As a result, the recorded decay was shorter than the actual cavity decay time of the optical cavity and, when connected to the FPGA acquisition board, *S* tended to overshoot the decay baseline. Attempts to bypass the transformer led to excessively noisy waveforms. Despite the perturbation to the decay waveform, accurate absorbance values were obtained as empty cavity and sample decay traces were impacted equally.

As the FCA/FPGA system demonstrated here is capable of monitoring *R* in real time, the only practical limitation to data acquisition speeds is the decay time of the cavity itself. Indeed, assuming a minimum of 100 points is needed to adequately characterize the noisy output of a CRDS system, the FCA/FPGA presented here could acquire and analyze decay transients at 1 MHz. Faster digitizer/FPGA systems could further increase the speed of data acquisition.

Currently, *R* is communicated by the FPGA as a DC signal through a digital-to-analog converter output. For slower pulsed systems, this method is problematic as it is sensitive to baseband noise in the analog transfer of data. In future systems, we will explore exporting *R* as either a frequency- or amplitude-modulated signal which should greatly reduce susceptibility to added noise. Depending on the application, we can also leave *R* digital.

Figure 8(a) is an absorption spectrum of NO_{2}(g) in room air obtained at the maximum laser firing frequency of 20 Hz. FCA analysis was used to obtain this spectrum in real time, and the ringdown waveforms were also recorded and later analyzed using Levenberg-Marquardt nonlinear least squares fitting (we used the value for *τ* obtained from the previous fit as our initial guess for the next fit). Both methods gave statistically equivalent results. A minimum detectable absorbance, calculated from Eq. (2) and Eq. (5), was determined to be 5.3×10^{−6}cm^{−1}, calculated with respect to the empty cavity ringdown time and standard deviation, *τ*_{0} = 16.5 ± 0.4*μs*. The molar absorption coefficient for NO_{2} at 539nm is 1.8 × 10^{−19}M^{−1}cm^{−1} [14] and hence the minimum detectable molecular concentration in air for this system is 10 × 10^{−9} or 10ppb.

Figure 8(a) shows a visible spectrum from a Cary 1E spectrophotometer (Varian) on NO_{2}(g) in room air. The NO_{2}(g) concentration was much greater than the CRDS system and is presented to show that the two methods qualitatively agree, and reveal the same spectral features.

We do note that our method does not give a figure of merit for the fit like *χ*^{2} given by least squares fitting regimes. We do, however, note that our fast analysis times allow us to analyze each transient in real time: this allows us to discard data points that appear to be well outside the mean.

## 5. Conclusions

A real-time frequency component analysis methodology has been demonstrated which is potentially capable of extracting decay constants from exponentially decaying signals at rates approaching 1 MHz. The methodology provides accuracy and precision comparable to nonlinear least squares fitting algorithms at significantly faster data acquisition rates. The methodology was used to acquire absorption spectra of NO_{2}(g) in room air using a pulsed-laser based CRDS system. This technology could greatly expand real time monitoring of gasses using this ultra-sensitive absorption technique.

As a note on future work, we are currently refining the mathematical basis of our technique to take account of the discrete signals. We are also working on a thorough noise analysis of our laser system so that we can more accurately characterise the accuracy and precision of our technique for a real laser source.

## Acknowledgments

We would like to thank the National Science Foundation (NSF Grant #: 0728285), The Louisiana Board of Regents and the Australian Research Council for their support of this research.

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