## Nonlinear estimation of ring-down time for a Fabry-Perot optical cavity |

Optics Express, Vol. 19, Issue 7, pp. 6377-6386 (2011)

http://dx.doi.org/10.1364/OE.19.006377

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### Abstract

This paper discusses the application of a discrete-time extended Kalman filter (EKF) to the problem of estimating the decay time constant for a Fabry-Perot optical cavity for cavity ring-down spectroscopy (CRDS). The data for the estimation process is obtained from a CRDS experimental setup in terms of the light intensity at the output of the cavity. The cavity is held in lock with the input laser frequency by controlling the distance between the mirrors within the cavity by means of a proportional-integral (PI) controller. The cavity is purged with nitrogen and placed under vacuum before chopping the incident light at 25KHz and recording the light intensity at its output. In spite of beginning the EKF estimation process with uncertainties in the initial value for the decay time constant, its estimates converge well within a small neighborhood of the expected value for the decay time constant of the cavity within a few ring-down cycles. Also, the EKF estimation results for the decay time constant are compared to those obtained using the Levenberg-Marquardt estimation scheme.

© 2011 OSA

## 1. Introduction

1. A. O’Keefe and D. A. G. Deacon, “Cavity Ring-Down Optical Spectrometer for Absorption Measurements using Pulsed Laser Sources,” Rev. Sci. Instrum. **59**, 2544–2551 (1988). [CrossRef]

4. T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Willke, and R. L. Byer, “A Laser-Locked Cavity Ring-DOWN Spectrometer Employing an Analog Detection Scheme,” Rev. Sci. Instrum. **71**, 347–353 (2000). [CrossRef]

*lock*with the cavity. Any deviation between these frequencies is characterized in terms of the

*detuning parameter*Δ and is an undesired effect.

*τ*if the logarithm of the decay of the cavity field is considered [1

1. A. O’Keefe and D. A. G. Deacon, “Cavity Ring-Down Optical Spectrometer for Absorption Measurements using Pulsed Laser Sources,” Rev. Sci. Instrum. **59**, 2544–2551 (1988). [CrossRef]

5. 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**, 387–395 (1998). [CrossRef]

*τ*in the case of isolated ring-downs and are susceptible to system noise characteristics and instrument offsets; e.g., see [6

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

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 NO2,” Appl. Phys. B: Lasers Opt. **81**, 135–141 (2005). [CrossRef]

8. M. A. Everest and D. B. Atkinson, “Discrete Sums for the Rapid Determination of Exponential Decay Constants,” Rev. Sci. Instrum. **79**, 023108–023108–9 (2008). [CrossRef] [PubMed]

*τ*and Δ) for a Fabry-Perot optical cavity. In this paper, we apply the EKF to estimate

*τ*for a set of experimentally obtained output intensity data for a Fabry-Perot optical cavity. During the course of the experiment, Δ was maintained near zero with the aid of a proportional-integral (PI) controller which was used to maintain the distance between the two mirrors inside the cavity so as to match the cavity’s resonant frequency with the input laser frequency. Also, the EKF estimation results for

*τ*are compared to those obtained by applying the LM technique to the same set of ring-down data.

*τ*). A detailed description of the experimental setup and the estimation results for

*τ*are presented in Section 5 with a comparison between EKF estimation results and LM estimation results. Finally, conclusions and a note on future work are outlined in Section 6.

## 2. Estimation and control for cavity ring-down spectroscopy

*ring-down*effect. Indeed, the ring-down time or decay time (

*τ*), which is the time taken for the light inside the cavity to decay to 1

*/e*of its original intensity, can be computed, which gives a good indication of the absorptive losses associated with the cavity.

*I*(

*t*) represents the decay amplitude at time

*t*,

*I*

_{0}is the initial intensity of the field within the cavity, and

*τ*is the decay time constant. Indeed, the decay time is a direct measure of losses within the cavity and can be described as, where

*t*is the round-trip time for light within the cavity,

_{rt}*ε*(

*λ*) is the extinction coefficient of an absorbing species with concentration

*c*computed as a function of the wavelength (

*λ*) of the incident light on the cavity. Also,

*n*is the number of mirrors with reflectivity

*R*and

*α*is a lumped term comprising of other absorptions.

*τ*for CRDS, they are slow [8

8. M. A. Everest and D. B. Atkinson, “Discrete Sums for the Rapid Determination of Exponential Decay Constants,” Rev. Sci. Instrum. **79**, 023108–023108–9 (2008). [CrossRef] [PubMed]

*τ*occur in real-time. Since the decaying light intensity as seen at the output of the cavity is a nonlinear function of the magnitude and phase quadratures, we need a suitable nonlinear estimator to estimate

*τ*. Hence, we propose the use of an extended Kalman filter (EKF), which is a suboptimal nonlinear estimator, in order to determine

*τ*.

*detuning parameter*Δ, needs to be maintained at zero. In other words, cavity lock should be maintained. This can be achieved by varying the length between the two mirrors, m1 and m2, in the Fabry-Perot cavity by means of a piezoelectric actuator (PZT), controlled using a suitable controller (see Fig. 1). As mentioned in Section 1, this process varies the length of the cavity, hence affecting its resonant frequency. As shown in the Fig. 1, one way to achieve this is by recording the light intensity at the reflected port of the cavity and using the Pound-Drever-Hall (PDH) method to obtain an analogue voltage (Δ

*) proportional to Δ. This information is then used by the controller to position mirror m2 via a PZT to maintain cavity lock.*

_{V}_{2}/H

_{∞}control methods can be used to improve locking in the presence of noise and uncertainties, they will not be considered in the scope of this paper. A description of the cavity dynamics in terms of a state-space representation follows.

## 3. Cavity dynamics

15. H. A. Bachor and T. C. Ralph, *A Guide to Experiments in Quantum Optics* (Wiley-VCH, Weinheim, Germany, 2004), 2nd ed. [CrossRef]

*a*denotes the annihilation operator for the cavity mode defined in an appropriate rotating frame, (·)

^{†}represents the operator adjoint operation,

*γ*=

*γ*+

_{m}*γ*is the total cavity coupling coefficient.

_{c}*γ*represents the cavity coupling coefficient at the mirrors in a vacuum cavity and

_{m}*γ*represents the cavity coupling coefficient corresponding to the absorbers within the cavity.

_{c}*ā*the laser input,

_{in}*y*is the measured output corresponding to the output light intensity, and

*w*and

*v*represent lumped process and measurement noise terms respectively.

*q*) and phase (

*p*) in terms of the annihilation operator (

*a*) as, which upon time-differentiation gives,

*w*=

_{q}*w*+

*w*

^{†}and

*iw*=

_{p}*w – w*

^{†}. Considering the time dependence of various terms and writing (7) and (8) in the state-space form, we get, which is in the state-space form, where

*x̄*(

*t*) = [

*q*(

*t*)

*, p*(

*t*)]

*is the state vector,*

^{T}*ū*(

*t*) = 2

*ā*is the input to the system, and

_{in}*w̄*(

*t*) = [

*w*(

_{p}*t*)

*, w*(

_{q}*t*)] is the lumped quadrature noise vector. Also,

*A*,

_{c}*B*, and

_{c}*D*are given matrices. In the sequel, the system (9) will be treated as a classical state-space system.

_{c}## 4. EKF recursion and design

*q*and

*p*. The estimation of states in this case requires the application of a nonlinear filter such as the EKF which is described in the rest of this section. Also, since the measurements are obtained at discrete intervals of time, we shall apply the recursion equations of a discrete-time EKF.

*x*

_{(·)}∈ℝ

*is the state,*

^{n}*u*

_{(·)}∈ ℝ

*is the known input,*

^{m}*w*

_{(·)}∈ ℝ

*and*

^{p}*v*

_{(·)}∈ ℝ

*are the process and measurement noise inputs respectively, and*

^{q}*y*

_{(·)}∈ ℝ

*is the measured output. Also,*

^{l}*f*(·) and

*h*(·) are given nonlinear functions. For the system described in (12) – (13), the EKF

*propagation*and

*update*recursion equations are given by,

**Update**Here, the

*propagation*step consists of estimating the value of the state

*x*and covariance

*P*(a matrix representing the approximate variance of the estimate of the state from its true value) one time-step ahead. These values are computed using available state(s) and input(s) at the current time-step and evaluating the state dynamics

*update*step. Also, in (14) – (18),

*y*

_{(·)}is the measured sensor output and

*F*

_{(·)},

*H*

_{(·)}are the linearized process and output matrices respectively, computed about the current operating point as, In addition,

*K*(·) represents the Kalman gain,

*P*is the covariance matrix,

*Q*and

*R*are the process and measurement noise matrices, and

*I*is the identity matrix of suitable dimensions. Also, (·)

*and (·)*

^{−}^{+}represent apriori and posteriori values respectively; and

*δ*is the sampling time constant.

*τ*(= 1

*/γ*), the continuous-time linear model in (9) is written in the following form,

*x*(

*t*) = [

*q*(

*t*)

*, p*(

*t*)

*, γ*(

*t*)]

*is the*

^{T}*augmented state vector*with the dynamics for the constant term

*γ*(= 1

*/τ*) added to the original state vector

*x̄*defined in (10). The nonlinear measurement equation, however, remains the same as in (11).

*δ*the sampling time. Also, the discrete-time measurement equation is computed as,

*γ*

_{m}*>> γ*, which meant that

_{c}*τ*was almost equal to the decay time constant for an empty cavity. Considering the reflectivity of mirrors used in the experiment (explained in Section 5), the value of

*τ*for the cavity was expected to be around 5.26

*μs*, corresponding to

*γ*= 1.9

*×*10

^{5}. This was used as an indication for the approximate true value for

*τ*during the estimation process. In accordance, the initial state vector [

*q*

_{0}

*, p*

_{0}

*, γ*

_{0}]

*representing initial values for the amplitude quadrature, the phase quadrature and the total cavity coupling coefficient were set to [0*

^{T}*,*0

*,*1.805

*×*10

^{5}]

*. Here,*

^{T}*γ*

_{0}was set with a 5% error from its expected true value of 1.9

*×*10

^{5}. Also, the corresponding covariance matrix

*P*reflecting error variance in initial conditions was set to,

*y*

_{(·)}defined in (24) and the sampling time constant

*δ*was set to 10

^{−}^{8}s.

## 5. Experimental setup and results

*±*18.5 MHz) on the laser radiation; these are used to lock the cavity to the laser using the method outlined in [16

16. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization using an Optical Resonator,” Appl. Phys. B: Lasers Opt. **31**, 97–105 (1983). [CrossRef]

*μ*s. Both reflected and transmitted photodetectors were designed and built in-house and 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 digitizing oscilloscope (Cleverscope 3284A, 100MS/sec, 14 bits), and exported to Matlab

^{®}for estimating the value for

*τ*. Twenty ring-down cycles of this data were used for the estimation process, with Fig. 3 depicting a sample ring-up and ring-down cycle.

*s*for one cycle of ring-up and ring-down data. Since the EKF is a suboptimal method, its estimates converge to a neighborhood of the expected true value for

*τ*at 5.26

*μs*and oscillate with a variation of ±0.012

*μs*. In addition, the LM technique was also applied to the recorded intensity data and the estimation results for the ring-down time were compared to that obtained from the EKF at the end of each ring-down cycle. This is depicted in Fig. 4. As seen from Fig. 4, EKF estimates converge to a small neighborhood of the expected true value for

*τ*whereas the LM estimates converge to within 0.1

*μs*of the expected true value for

*τ*at the end of 19 ring-down cycles.

*τ*. Since the EKF is a recursive method and relies on the states and parameters at the previous instant in time to estimate the corresponding states and parameters at the current time instant by propagating these values through the system dynamical Eqs. (20)–(23). The associated estimation error improves with time as successive measurements are obtained and the filter recursion is carried out until the estimate for the state(s) converges to a certain neighborhood of the expected true state. This is a direct consequence of the large deviation of the EKF estimated values for

*τ*during the initial few cycles as depicted in Fig. 4, after which the error in estimation gradually reduces and the estimate settles within a neighborhood of the expected true value for

*τ*. On the other hand, in the case of the curve-fitting LM method, every ring-down cycle is considered independently of the previous cycle and the value for

*τ*is estimated separately for each ring-down cycle. This is why the LM algorithm generally needs hundreds of ring-down cycles to obtain an acceptable value for

*τ*after averaging the statistics obtained at the end of each ring-down cycle, which is not the case with the EKF.

## 6. Conclusion

^{®}for the estimation process. Since the cavity was almost empty during the process of data accumulation, the losses in the cavity were mainly due to the mirrors, with very little or no effect due to other factors contributing to the absorption or scattering of light within the cavity. Hence, the approximate value for

*τ*in the estimation process was expected to be close to that of an empty cavity. Considering the reflectivity of mirrors used in the experiment, the value for

*τ*of the cavity was expected to be around 5.26

*μs*(corresponding to

*γ*= 1.9

*×*10

^{5}).

*τ*converged to the neighborhood of the expected true value of 5.26

*μs*within a few cycles of the output ring-down data. The Levenberg-Marquardt (LM) technique was also implemented and its estimation results were compared to that of the EKF at the end of each ring-down cycle. It was found that the LM estimate for

*τ*had a 0.1

*μs*deviation from the expected true value for

*τ*at the end of 19 ring-down cycles, whereas the EKF converged to a neighborhood of the expected true value for

*τ*oscillating with a variation of 0.012

*μs*after the same number of ring-down cycles.

*τ*in real-time using field programmable gate arrays (FPGA).

## Acknowledgments

## References and links

1. | A. O’Keefe and D. A. G. Deacon, “Cavity Ring-Down Optical Spectrometer for Absorption Measurements using Pulsed Laser Sources,” Rev. Sci. Instrum. |

2. | 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. |

3. | K. W. Busch and M. A. Busch, |

4. | T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Willke, and R. L. Byer, “A Laser-Locked Cavity Ring-DOWN Spectrometer Employing an Analog Detection Scheme,” Rev. Sci. Instrum. |

5. | 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. |

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

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 NO2,” Appl. Phys. B: Lasers Opt. |

8. | M. A. Everest and D. B. Atkinson, “Discrete Sums for the Rapid Determination of Exponential Decay Constants,” Rev. Sci. Instrum. |

9. | C. K. Chui and G. Chen, |

10. | A. G. Kallapur, I. R. Petersen, T. K. Boyson, and C. C. Harb, “Nonlinear Estimation of a Fabry-Perot Optical Cavity for Cavity Ring-Down Spectroscopy,” in “IEEE International Conference on Control Applications (CCA),” (Yokohama, Japan, 2010), pp. 298–303. |

11. | S. Z. Sayed Hassen, E. Huntington, I. R. Petersen, and M. R. James, “Frequency Locking of an Optical Cavity Using LQG Integral Control,” in “ |

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: At. Mol. Opt. Phys. |

13. | S. Z. Sayed Hassen and I. R. Petersen, “A time-varying Kalman filter approach to integral LQG frequency locking of an optical cavity,” in “ |

14. | C. W. Gardiner and P. Zoller, |

15. | H. A. Bachor and T. C. Ralph, |

16. | R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization using an Optical Resonator,” Appl. Phys. B: Lasers Opt. |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(200.3050) Optics in computing : Information processing

(300.6360) Spectroscopy : Spectroscopy, laser

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: February 2, 2011

Revised Manuscript: March 3, 2011

Manuscript Accepted: March 7, 2011

Published: March 21, 2011

**Citation**

Abhijit G. Kallapur, Toby K. Boyson, Ian R. Petersen, and Charles C. Harb, "Nonlinear estimation of ring-down time for a Fabry-Perot optical cavity," Opt. Express **19**, 6377-6386 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6377

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### References

- A. O’Keefe and D. A. G. Deacon, “Cavity Ring-Down Optical Spectrometer for Absorption Measurements using Pulsed Laser Sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988). [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, 3991–3997 (1998). [CrossRef]
- K. W. Busch and M. A. Busch, Cavity-Ringdown Spectroscopy. An Ultratrace-Absorption Measurement Technique, vol. 720 of ACS Symposium Series (American Chemical Society, Washington, DC, 1999). [CrossRef] [PubMed]
- T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Willke, and R. L. Byer, “A Laser-Locked Cavity Ring-DOWN Spectrometer Employing an Analog Detection Scheme,” Rev. Sci. Instrum. 71, 347–353 (2000). [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, 387–395 (1998). [CrossRef]
- A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena,” Rev. Sci. Instrum. 701233 (1999). [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: Lasers Opt. 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, 023108–023108–9 (2008). [CrossRef] [PubMed]
- C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applications , Theoretical, Mathematical & Computational Physics (Springer-Verlag, Berlin, Heidelberg, Germany, 2009), 4th ed.
- A. G. Kallapur, I. R. Petersen, T. K. Boyson, and C. C. Harb, “Nonlinear Estimation of a Fabry-Perot Optical Cavity for Cavity Ring-Down Spectroscopy,” in “IEEE International Conference on Control Applications (CCA) ,” (Yokohama, Japan, 2010), pp. 298–303.
- S. Z. Sayed Hassen, E. Huntington, I. R. Petersen, and M. R. James, “Frequency Locking of an Optical Cavity Using LQG Integral Control,” in “17th IFAC World Congress ,” (Seoul, South-Korea, 2008), pp. 1821–1826.
- 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: At. Mol. Opt. Phys. 42, 175501 (2009). [CrossRef]
- S. Z. Sayed Hassen and I. R. Petersen, “A time-varying Kalman filter approach to integral LQG frequency locking of an optical cavity,” in “American Control Conference ,” (Baltimore, MD, USA, 2010), pp. 2736–2741.
- C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, Germany, 2000).
- H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, Weinheim, Germany, 2004), 2nd ed. [CrossRef]
- R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization using an Optical Resonator,” Appl. Phys. B: Lasers Opt. 31, 97–105 (1983). [CrossRef]

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