## Frequency domain analysis for laser-locked cavity ringdown spectroscopy |

Optics Express, Vol. 19, Issue 9, pp. 8092-8101 (2011)

http://dx.doi.org/10.1364/OE.19.008092

Acrobat PDF (902 KB)

### Abstract

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 OSA

## 1. Introduction

1. S. M Ball, I. M. Povey, E. G. Norton, and R. L. Jones, “Broadband cavity ringdown spectroscopy of the NO_{3} radical,” Chem. Phys. Lett. **342**, 113–120 (2001). [CrossRef]

*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

*τ*may be thus quantified: where:

*T*is the roundtrip time for light in the cavity,

_{r}t*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

2. J. Xie, B. A. Paldus, E. H. Wahl, J. Martin, T. G. Owano, C. H. Kruger, J. S. Harris, and R. N. Zare, “Near-infrared cavity ringdown spectroscopy of water vapor in an atmospheric flame,” Chem. Phys. Lett. **284**(5), 387–395 (1998). [CrossRef]

4. A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. **70**(2), 1233–1257 (1999). [CrossRef]

4. A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. **70**(2), 1233–1257 (1999). [CrossRef]

5. T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Wilke, and R. L. Byer, “A laser-locked cavity ring-down spectrometer employing an analog detection scheme,” Rev. Sci. Instrum. **71**(2), 347–353 (2000). [CrossRef]

4. A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. **70**(2), 1233–1257 (1999). [CrossRef]

*et al.*[6

6. P. D. Kirchner, W. J. Schaff, G. N. Maracas, L. F. Eastman, T. I. Chappell, and C. M. Ransom, “The analysis of exponential and nonexponential transients in deep level transient spectroscopy,” J. Appl. Phys. **52**, 6462–6470 (1981). [CrossRef]

*et al.*[7

7. M. Mazurenka, R. Wada, A. J. L. Shillings, T. J. A. Butler, J. M. Beames, and A. J. Orr-Ewing, “Fast Fourier transform analysis in cavity ring-down spectroscopy: application to an optical detector for atmospheric NO_{2},” Appl. Phys. B **81**, 135–141 (2005). [CrossRef]

*et al.*[8

8. M. A. Everest and D. B. Atkinson, “Discrete sums for the rapid determination of exponential decay constants,” Rev. Sci. Instrum. **79**, 023108 (2008). [CrossRef] [PubMed]

*et al.*[9

9. D. Halmer, G. von Basum, P. Hering, and M. Murtz, “Fast exponential fitting algorithm for real-time instrumental use,” Rev. Sci. Instrum. **75**, 2187 (2004). [CrossRef]

*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

10. R. Engeln, G. von Helden, G. Berden, and G. Meijer, “Phase shift cavity ring down absorption spectroscopy,” Chem. Phys. Lett. **262**, 105–109 (1996). [CrossRef]

## 2. Theoretical description of the CRDS system

*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

*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:

*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. R. Engeln, G. von Helden, G. Berden, and G. Meijer, “Phase shift cavity ring down absorption spectroscopy,” Chem. Phys. Lett. **262**, 105–109 (1996). [CrossRef]

*τ*. For our fourier method, the best modulation frequency would probably be where

*τ*. For our system, with

*τ*

_{0}= 5

*μs*, 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

**70**(2), 1233–1257 (1999). [CrossRef]

6. P. D. Kirchner, W. J. Schaff, G. N. Maracas, L. F. Eastman, T. I. Chappell, and C. M. Ransom, “The analysis of exponential and nonexponential transients in deep level transient spectroscopy,” J. Appl. Phys. **52**, 6462–6470 (1981). [CrossRef]

8. M. A. Everest and D. B. Atkinson, “Discrete sums for the rapid determination of exponential decay constants,” Rev. Sci. Instrum. **79**, 023108 (2008). [CrossRef] [PubMed]

*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.

## 4. Experimental data

*et al.*[11

11. R. W. P. Drever, J. L. Hall, F. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. **31**, 97–105 (1983). [CrossRef]

*μ*s. 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.

*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 ± 0.151

*μs*(± 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 ± 0.137

*μs*(± 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 5

*μ*s for

*τ*led to the stated analysis time of 24.0 seconds, while an initial guess of 1

*μ*s led to an analysis time of almost 45 seconds. The results from our analysis are shown in Fig. 7.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## References and links

1. | S. M Ball, I. M. Povey, E. G. Norton, and R. L. Jones, “Broadband cavity ringdown spectroscopy of the NO |

2. | J. Xie, B. A. Paldus, E. H. Wahl, J. Martin, T. G. Owano, C. H. Kruger, J. S. Harris, and R. N. Zare, “Near-infrared cavity ringdown spectroscopy of water vapor in an atmospheric flame,” Chem. Phys. Lett. |

3. | A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. |

4. | A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. |

5. | T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Wilke, and R. L. Byer, “A laser-locked cavity ring-down spectrometer employing an analog detection scheme,” Rev. Sci. Instrum. |

6. | P. D. Kirchner, W. J. Schaff, G. N. Maracas, L. F. Eastman, T. I. Chappell, and C. M. Ransom, “The analysis of exponential and nonexponential transients in deep level transient spectroscopy,” J. Appl. Phys. |

7. | M. Mazurenka, R. Wada, A. J. L. Shillings, T. J. A. Butler, J. M. Beames, and A. J. Orr-Ewing, “Fast Fourier transform analysis in cavity ring-down spectroscopy: application to an optical detector for atmospheric NO |

8. | M. A. Everest and D. B. Atkinson, “Discrete sums for the rapid determination of exponential decay constants,” Rev. Sci. Instrum. |

9. | D. Halmer, G. von Basum, P. Hering, and M. Murtz, “Fast exponential fitting algorithm for real-time instrumental use,” Rev. Sci. Instrum. |

10. | R. Engeln, G. von Helden, G. Berden, and G. Meijer, “Phase shift cavity ring down absorption spectroscopy,” Chem. Phys. Lett. |

11. | R. W. P. Drever, J. L. Hall, F. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. |

12. | S. Z. Sayed Hassen, M. Heurs, E. H. Huntington, I. R. Petersen, and M. R. James, “Frequency locking of an optical cavity using linear quadratic Gaussian integral control,” J. Phys. B |

13. | B. A. Paldus, C. C. Harb, T. G. Spence, B. Wilke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. |

14. | P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. |

**OCIS Codes**

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(280.3420) Remote sensing and sensors : Laser sensors

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: February 16, 2011

Revised Manuscript: April 1, 2011

Manuscript Accepted: April 1, 2011

Published: April 13, 2011

**Citation**

T. K. Boyson, T. G. Spence, M. E. Calzada, and C. C. Harb, "Frequency domain analysis for laser-locked cavity ringdown spectroscopy," Opt. Express **19**, 8092-8101 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8092

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### References

- S. M Ball, I. M. Povey, E. G. Norton, and R. L. Jones, “Broadband cavity ringdown spectroscopy of the NO3 radical,” Chem. Phys. Lett. 342, 113–120 (2001). [CrossRef]
- J. Xie, B. A. Paldus, E. H. Wahl, J. Martin, T. G. Owano, C. H. Kruger, J. S. Harris, and R. N. Zare, “Near-infrared cavity ringdown spectroscopy of water vapor in an atmospheric flame,” Chem. Phys. Lett. 284(5), 387–395 (1998). [CrossRef]
- A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 254-4-2551 (1988).
- A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. 70(2), 1233–1257 (1999). [CrossRef]
- T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Wilke, and R. L. Byer, “A laser-locked cavity ring-down spectrometer employing an analog detection scheme,” Rev. Sci. Instrum. 71(2), 347–353 (2000). [CrossRef]
- P. D. Kirchner, W. J. Schaff, G. N. Maracas, L. F. Eastman, T. I. Chappell, and C. M. Ransom, “The analysis of exponential and nonexponential transients in deep level transient spectroscopy,” J. Appl. Phys. 52, 6462–6470 (1981). [CrossRef]
- M. Mazurenka, R. Wada, A. J. L. Shillings, T. J. A. Butler, J. M. Beames, and A. J. Orr-Ewing, “Fast Fourier transform analysis in cavity ring-down spectroscopy: application to an optical detector for atmospheric NO2,” Appl. Phys. B 81, 135–141 (2005). [CrossRef]
- M. A. Everest and D. B. Atkinson, “Discrete sums for the rapid determination of exponential decay constants,” Rev. Sci. Instrum. 79, 023108 (2008). [CrossRef] [PubMed]
- D. Halmer, G. von Basum, P. Hering, and M. Murtz, “Fast exponential fitting algorithm for real-time instrumental use,” Rev. Sci. Instrum. 75, 2187 (2004). [CrossRef]
- R. Engeln, G. von Helden, G. Berden, and G. Meijer, “Phase shift cavity ring down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996). [CrossRef]
- R. W. P. Drever, J. L. Hall, F. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983). [CrossRef]
- S. Z. Sayed Hassen, M. Heurs, E. H. Huntington, I. R. Petersen, and M. R. James, “Frequency locking of an optical cavity using linear quadratic Gaussian integral control,” J. Phys. B 42(17), 175501 (2009). [CrossRef]
- B. A. Paldus, C. C. Harb, T. G. Spence, B. Wilke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83(8), 3991–3997 (1998) [CrossRef]
- P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708–2717 (1995). [CrossRef]

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