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In this paper we report on the development of a Fourier-transform based signal processing method for laser-locked Continuous Wave Cavity Ringdown Spectroscopy (CWCRDS). Rather than analysing single ringdowns, as is the norm in traditional methods, we amplitude modulate the incident light, and analyse the entire waveform output of the optical cavity; our method has more in common with Cavity Attenuated Phase Shift Spectroscopy than with traditional data analysis methods. We have compared our method to Levenburg-Marquardt non linear least squares fitting, and have found that, for signals with a noise level typical of that from a locked CWCRDS instrument, our method has a comparable accuracy and comparable or higher precision. Moreover, the analysis time is approximately 500 times faster (normalised to the same number of time domain points). Our method allows us to analyse any number of periods of the ringdown waveform at once: this allows the method to be optimised for speed and precision for a given spectrometer.
©2011 Optical Society of America
1. Introduction
Cavity ringdown spectroscopy (CRDS) is a sensitive spectroscopic technique that can be used to measure absorption due to weakly absorbing or dilute samples. In a CRDS measurement, light (generally this is from a laser, but broadband techniques have been demonstrated. e.g. [1]) is coupled into an optical cavity formed by two or more mirrors. Upon extinguishing the incident light, the field within the cavity, I(t), decays according to:
where α is the initial intensity, t is time, b is an offset, and τ is the decay length (the time taken for the field to decay to of its initial value). The decay length is a direct measure of the losses within the cavity, comprising of the reflectivity of the mirrors, a lumped term due to scattering and other small losses, and the loss due to an absorbing species in the cavity. τ may be thus quantified: where: Trt is the roundtrip time for light in the cavity, ε(λ) is the extinction coefficient (as a function of wavelength, λ) of an absorbing species with concentration c; n is the number of mirrors with reflectivity R; and A is a lumped term comprising of all other absorptions (such as scattering, and absorption at the surface of the mirrors). An absorption spectrum is generated by scanning the frequency of the incident light while recording the decay length. These measurements are scaled by a measurement of the empty cavity decay length, τ 0. Traditional data processing techniques rely on fitting the exponential decay of the cavity field with a least squares algorithm [2]. If the logarithm of the decay is taken, linear least squares may be used [3]; else, a non-linear least squares fitting algorithm (such as Levenburg-Marquardt, LM) must be used. Linear least squares has the advantage that it returns a closed form solution requiring only a single iteration to complete; however, it is susceptible to noise, and requires that the baseline be determined and subtracted [3, 4]. Non linear least squares must find the solution iteratively (each iteration requires a linear least squares fit). If it is required to fit all three parameters of the decay, it is slow (fitting an exponential in Matlab, for example, takes several ms to fit a 1000 point decay). Moreover, LM requires the operator (or computer) to make an initial guess of the fit parameters; a poor guess may result in the algorithm diverging from the solution, or taking an infinite amount of time to converge on a solution [4]. CRDS was originally developed with pulsed lasers [3]: at a pulse repetition rate of 20Hz, a least squares fitting regime may be fast enough to keep up with instrumental output; systems using CW lasers and a fast optical switch, however, may generate transients at rates exceeding 50kHz [5]. The problem of fitting an isolated exponential decay rapidly and accurately is not unique to CRDS. Indeed, the fitting of exponential decays is a fundamental problem of applied signal processing. Istratov and Vyvenko [4] comprehensively outline the available solutions to this problem. They conclude that, when considering speed, precision, and ease of implementation, a method based on the Fourier transform of the decay developed by Kirchner et al. [6] for deep transient level spectroscopy, and adapted by Mazurenka et al. [7] for CRDS was the best solution. Everest et al. [8] analysed the methodology of Mazurenka and found systematic errors in the derivation of equations resulting from the assumption that the data were continuous rather than discrete. Everest compared the correct Fourier transform method with an improved method based on that published by Halmer et al. [9] based upon corrected successive integration (CSI); both The FT and CSI methods were found to be significantly faster than LM, but with a comparable accuracy. All of these methods rely on building up light inside of the cavity, then capturing a single decay transient for analysis. Here, we propose a new methodology for the extraction of τ from the ringdown cavity. Our method has more in common with Phase-Shift CRDS (otherwise known as Cavity Attenuated Phase Shift Spectroscopy, or CAPS) [10] than it does with traditional fitting methods: rather than considering only the exponential decay of the cavity to either a pulse or a sudden shuttering of the input light and analysing a single decay, we consider the response of the cavity to an amplitude modulated field, and analyse the output as a whole. Not only is this fast, it makes acquisition electronics much easier to design. In this paper we will first outline the theory for our new technique. We will then give details of a comparison of our technique to Levenburg-Marquardt fitting with simulated data. We will then give details of the experimental setup that we have used for proof of concept test. We then give our conclusions, and details of where we hope to implement this new data processing technique.2. Theoretical description of the CRDS system
Consider the impulse response of the cavity:
This equation (and the following mathematics) may be extended to a multi-exponential decay, but here we only consider the case for a cavity modelocked to the incident laser light; such a setup is almost guaranteed to return a single decay (presuming that the detection electronics have been designed with enough bandwidth). By taking the Fourier transform of Eq. (3), we can generate the frequency response (F(ω)) of the cavity to an input at angular frequency ω: Thus: F(ω) may be broken into real and imaginary parts: These two equations, respectively, are the response of the cavity to a cosine and sine of frequency ω; equivalently, they are the real and imaginary components of the Fourier transform. Here, we will only consider the case of cosine, i.e. the real part of the Fourier transform, although an analogous formalism may be developed for sine. Consider measuring the response of the cavity to two frequencies, with one being equal to some frequency multiple, a, of the the other i.e. ω and aω), then we find: If we take the ratio of these, we find: and then by rearranging for τ, we obtain: where ; the ratio of the magnitudes of two peaks in frequency space. Thus, by calculating the ratio of the magnitudes of two peaks in frequency space, we can easily calculate τ. A measurement for τ can be obtained, for example, by modulating the light at ω, measuring the system response, then modulating at aω and taking the ratio. Alternatively, and more conveniently, we could choose to use a square-wave-modulated light source (i.e. rapidly switched on and off) with a 50% duty cycle incident on the cavity. The Fourier series for a square wave, s(t), is given by: It can thus be seen that a square-wave modulated field has frequency components at f, 3f, 5f, etc.. The response of a locked cavity to squarewave modulate light is shown in Fig. 1. Due to the orthogonality of the sinusoids, we can analyse each of these components separately; we can thus calculate τ from a single measurement. Practically, we use the fundamental and the first harmonic, as the signal-to-noise is highest. For the case outlined above, τ is given by:
Fig. 1 Simulated 25 kHz squarewave and ringdown waveforms used to analyse and extract τ. The squarewave is the light before the cavity; the ringdown waveform is the light after passing through a cavity with a decay time of 5μs .
In order to measure the response at each frequency of interest, we can of course calculate the discrete fourier transform (DFT) and measure the magnitude of the real peak, however, this is computationally wasteful as we are only interested in the value for two frequencies. Instead of evaluating the whole DFT:
where k is the frequency in samples, and f [n] is the sampled time domain waveform. We evaluate it only at the frequencies of interest, e.g. for k = 1: As we are only interested in the real part (the projection onto cosine), we do not need to evaluate the whole complex sum: we simply multiply the time-domain waveform by a cosine at the frequency of interest, and sum over n, for k = 1: The DFT maps a set of N time domain data points onto a set of N frequency domain points, running from 0,1,...,N – 1: these frequency domain points are evenly spaced from 0 to the sampling frequency. As such, there is no guarantee that a given point (say, the maximum of a sharp spectral feature) will be included in the output. The information, however is still present. We can thus, rather than calculating the DFT by definition (i.e. by choosing an integer value of k), we can choose our local oscillator (LO) frequency to sit precisely where the SNR is highest. Thus for a LO frequency ω, where ω may correspond to an integer value of k, but is not restricted to it: This is identical to a digital mixer, or, if we evaluated a continuous integral rather than the discrete sum, a lock-in amplifier. In CAPS, the value of ω is chosen such that [10]. This maximises the change in the phase shift as a function of τ. For our fourier method, the best modulation frequency would probably be where : this would result in the fundamental being essentially unattenuated by the cavity, while the first harmonic’s attenuation would vary strongly as a function of τ. For our system, with τ 0 = 5μs, this would correspond to a modulation frequency of approximately 10 kHz; however, instrumental limitations prevent us from modulating this slowly, so we have chosen to work as slowly as we can, at 25 kHz.3. Simulations
In order to test our frequency domain method, we have made a comparison to LM fitting in Matlab. Although there are other methodologies outlined in the literature [4, 6–8], most published papers, and, to our knowledge, all commercial instruments still use LM. To compare the precision of the two methods we have simulated a dataset consisting 10,000 ringups and ringdowns with a SNR of 40 dB. For our method, we broke the data up into sets consisting of 1-200 periods (and thus 500-10,000 individual datasets). Each waveform within a set was analysed, and the set of solutions used to construct the probability histograms in Figure 2. For the Levenburg-Marquardt fit, the ringups were discarded and each individual ringdown fit for initial amplitude, ringdown time, and offset; this set of 10,000 solutions was used to construct the probability histogram for the fitting method. The results in Fig. 2 show that our method has a comparable accuracy to LM fitting, and that the precision increases as the length of the analysed data increases. This is to be expected, as our method is based on taking a running average.
Fig. 2 Normalized histograms of Tau for various lengths of the ringdown waveform using our frequency domain method, and for Levenburg-Marquardt fitting. A sample waveform, consisting of 10,000 ringups and ringdowns, with a SNR of 40 dB was synthesised. For our method, the data were trimmed into various lengths, analysed, and the set of answers used to construct the probability histogram. For LM, the ringups were discarded, the ringdowns fit in Matlab, and the probability histogram generated from the set of answers. In reality, for our method, we would analyse the 10,000 period-long waveform as a whole, giving a single answer. This figure shows how the precision of our method increases with the length of the input data. All data lengths have a similar accuracy to LM, but for any length longer than 2 periods, our method has higher precision.
In order to further verify the performance of our fourier method, we have performed simulations that vary the number of periods analysed and the signal-to-noise of the waveform. For both of these simulations, we have simulated a ringdown waveform with 1000 periods (i.e. 1000 ringups and ringdowns); we have analysed single periods of the ringdown waveform with our fourier transform method, and discarded the ringups and analysed single ringdowns with LM fitting. The simulation for the data length vs. the ringdown time, Fig. 3, shows that our fourier method has a comparable precision and accuracy to LM fitting for all data lengths. The results in Fig. 4 show that, as per our simulations in Fig. 2, LM fitting gives a slightly higher, but comparable, precision to our FT method at all simulated noise levels. We note that the results are comparable for all of the simulated noise levels, even those well outside that expected from a locked CRDS spectrometer.
Fig. 3 Ringdown time vs. standard deviation for various numbers of periods analysed. This figure shows that our fourier method has a comparable precision and accuracy to LM fitting for all data lengths. The error bars represent one standard deviation.
4. Experimental data
To confirm our simulation results, we have analysed a sample ringdown waveform from a CWCRDS instrument that we have built. A schematic diagram of the system is shown in Fig. 5. Briefly, light from an external cavity tunable diode laser (New Focus 6330, 10mW tunable from 1540-1640 nm) is passed through a Faraday isolator in order to prevent unwanted optical feedback. The light is then passed through an electro-optic modulator (EOM) (Thorlabs). The EOM places FM sidebands (at ± 18 MHz) on the laser radiation; these are used to lock the cavity to the laser using the method of Drever et al. [11]. The phase modulated light is then passed through an Acousto-Optic Modulator (AOM) (Brimrose) that is used to rapidly switch the laser light on and off (for this work, at 25kHz) in order to generate the waveform shown in Fig. 1. The light then passes through a polarising cube beamsplitter (PCB) and a quarter wave plate (QWP) before falling onto the optical cavity. The reflected beam passes back through the optical circulator, and is tapped off to lock the cavity. The cavity is a stainless steel tube (Los Gatos Research) with a 99.96 % mirrors as both the input and output coupler (Advanced Thin Films, R > 99.96%, loss < 10ppm). These mirrors give a calculated empty cavity ringdown time of ≈ 5μs. Both reflected and transmitted photodetectors were designed and built in house, they have a 3 dB bandwidth > 20MHz. The cavity is locked with an in house designed analog PI controller with a unity gain bandwidth of 1kHz. Light exiting the cavity is acquired using a high-speed digitising oscilloscope (Cleverscope 3284A, 100MS/sec, 14 bits), and exported to Matlab for analysis. A sample ringdown waveform from our instrument is shown in Fig. 6.
Fig. 5 Schematic diagram of laser-locked CRDS system. Abbreviations are as follows: ISO is the Faraday isolator; MMO are mode matching optics; HWP are half wave plates; EOM is the electro-optic modulator; AOM is the acousto-optic modulator; MOD1 is the RF generator and amplifier for phase modulation; MOD2 is the signal generator and amplifier used to generate the chopping waveform; M1 and M2 are beam steering mirrors; PCB is a polarizing cube beamsplitter; PD are photodetectors; QWP is a quarter wave plate; SERVO is the controller; HV AMP is a ±200V amplifier to drive PZT, the piezoelectric actuator that controls the cavity length.
Fig. 6 25 kHz CW Ringdown waveform, taken from our spectrometer. These are the data that we have analysed using our method and LM fitting.
To test the theory above, 40 ms of data was taken and digitised on our Cleverscope 3284A with the full 14 bits, at a sampling frequency of 100 MHz. This resulted in the capture of 1000 ringups and ringdowns. For our FT method, the data were trimmed into various integer-number-of-period lengths, and then analysed as above. For the LM fits, the data were trimmed into individual ringdowns and analysed in Matlab. For our method, τ was found to be 5.521 μs, with an analysis time of 720 msec to analyse the entire waveform as a single entity, and (5.522 ± 0.151μs (± one standard deviation)) with an analysis time 86.0 milliseconds to analyse the waveform as individual periods. For LM τ was found to be (5.533 ± 0.137μs (± one standard deviation)), with an analysis time of 24.0 seconds. The analysis time for LM was found to be strongly dependant on the initial guesses for the fitting parameters; an initial guess of 5μs for τ led to the stated analysis time of 24.0 seconds, while an initial guess of 1μs led to an analysis time of almost 45 seconds. The results from our analysis are shown in Fig. 7.
Fig. 7 Normalized histograms of Tau from experimental data. 40 ms of data at a 25 kHz chopping frequency were captured from our spectrometer. For our method, the data were chopped into lengths of 2 and 5 periods of the ringdown waveform, and the solutions used to construct the probability histogram. For LM, the ringups were discarded, the ringdowns fit using the aforementioned procedure, and the histgram constructed. It can be seen that our method has a comparable (but slightly higher) precision to LM for a waveform of 2 periods, and a higher precision for 5 periods. This supports our findings in the simulations we have performed. We have only analysed up to 5 periods here, as any longer data length does not give enough solutions for τ to generate reliable statistics.
5. Discussion
In this paper, we have given a new method for analysing the output of a CWCRDS instrument. Rather than analysing single decay transients, we take advantage of the properties of our laser-locked spectrometer, and analyse the waveform at the output of the cavity as a whole. We find, in simulation, that our method has an accuracy comparable to Levenburg-Marquardt fitting (the defacto standard for CRDS) and a better precision for data similar to that from a CWCRDS instrument. We have built laser-locked CRDS instrument to perform proof of concept tests; experimentally, we find that the SNR at the frequencies we are analysing is comparable to that in our simulations: ≈ 40dB. Our method was found to be more than 250 times faster at analysing a dataset than LM; this increase in computational speed is in spite of the fact that our method analyses twice as much data as LM, as we analyse both the ringup and the ringdown. Because our method is based in the frequency domain, it is reasonably noise immune; the only noise that will affect the result is noise at the frequencies that we are analysing: as such we choose to work where the signal to noise ratio is highest. This may be compared to traditional fitting techniques, where noise at every frequency must be dealt with. Both our simulations and our experimental data were taken at a chopping frequency of 25 kHz and a sampling frequency of 100MHz; however, there is no reason not to work at a higher chopping frequency or a lower sampling rate. In theory, we could sample at the Nyquist frequency (for this work, at 150 kHz) without any ill effects. Moreover, chopping at a higher frequency has several potential benefits: a higher data throughput, and reduced contribution of 1/f noise. Fitting regimes work best when they have several ringdown lengths worth of data to analyse [8]; this limits the rate at which one can generate data. Our method does not have this restriction. Our method is very easy to code; the section of the matlab script that does the analysis is only four lines long. This simplicity should make implementation on an FPGA feasible; this could result in even greater speed gains, and the possibility for real time analysis. Because our method is based in the frequency domain, the acquisition electronics are much easier to design: we could AC couple the ringdown signal with no ill effects. This is convenient as many high speed digitisers only have AC coupled inputs. We have locked our cavity with a simple analog PI controller. We are presently investigating using modern control theory [12] to improve the quality of the lock between the laser and the cavity. The quality of the lock is directly related to the precision of the instrument. If we can increase the bandwidth of the controller, we can better counteract perturbations over a wider frequency range. This will reduce the amount of noise present on the ringdown waveform.
6. Conclusions
In this paper, we have proposed a new method for analysing the output of a CW Cavity ring-down spectrometer; our method analyses the output of the spectrometer as a whole, rather than just analysing individual ringdown transients. We have simulated the technique, comparing it to Levenburg-Marquardt non-linear least squares fitting, and used a modelocked CRDS instrument that we have built for proof-of-principle tests. Our method greatly simplifies the design of acquisition electronics. We have found that our method has a comparable accuracy, and comparable or higher precision, to LM, but analyses data 500 times faster.
Acknowledgments
We would like to thank the Australian Research Council, the Australian Federal Police, the Defence and Security Applications Research Centre, the National Science Foundation, and the Louisiana Board of Regents for their support of this research.
References and links
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