## Power spectra for laser-extinction measurements

Optics Express, Vol. 14, Issue 13, pp. 6011-6019 (2006)

http://dx.doi.org/10.1364/OE.14.006011

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### Abstract

Recent laser technology provides accurate measures of the dynamics of fluids and embedded particles. For instance, the laser-extinction measurements (LEM) uses a laser beam passing across the fluid and measures the residual laser light intensity at the fluid output. The particle concentration is estimated from this measurement. However, the particle flow is submitted to random time-varying fluctuations. This study thus proposes to model the received intensity by an appropriate random process. This paper first models the particle flow by a queueing process. Second, the measured intensity power spectrum is derived according to this random model. Finally, the simple case of a constant particle velocity is developped. The proposed model allows to generalize results previously obtained in the litterature with simplified models. Moreover, the particle celerity estimate is provided.

© 2006 Optical Society of America

## 1. Introduction

7. K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. **16**, 519–528, (2005). [CrossRef]

7. K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. **16**, 519–528, (2005). [CrossRef]

9. M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame , **141**, 371–391, (2005). [CrossRef]

3. A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. **166**, 15–20, (1999). [CrossRef]

3. A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. **166**, 15–20, (1999). [CrossRef]

*𝒱*with constant cross-section

*𝓢*

_{1}(Fig. 1).

*I*in the emission direction provides an estimate of the particle concentration i.e. of the mean number of particles per volume unit (Fig. 1). From [3

3. A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. **166**, 15–20, (1999). [CrossRef]

*I*can be written as

*I*

_{0}denotes the incident laser beam intensity and N is the number of illuminated particles. For a given particle with index

*k*,

*B*

_{k}denotes the extinction cross-section. Uncertainties on the particle shape and orientation result in a random model for

*B*

_{k}. The factor (1-

*A*

_{k}) models the laser beam attenuation induced by this particle. Let

*E*[..] denote the mathematical expectation. Let define

*m*

_{1B}and

*m*

_{2B}characterize the particle mechanical properties whereas

*m*

_{1A}and

*m*

_{2A}characterize the interaction between the light and the particles. Moreover, the relative particle and laser beam motion leads to a random number of illuminated particles

*N*[3

**166**, 15–20, (1999). [CrossRef]

**J**

_{t}is the random index set of the illuminated particles at a given time

*t*. The proposed model considers two characteristics of each particle

*B*

_{k}the extinction cross-section and (1-

*A*

_{k}) the laser beam attenuation induced by this particle as well as a random number of illuminated particles. Note that the resulting optical medium model can be described by the classical scattering and absorption coefficients. These so-called optical characteristics are functions of the proposed model parameters according to appropriate physics considerations.

## 2. The general case

### 2.1. The queueing process model

*t*

_{n}and

*n*

^{th}particle in the illuminated volume

*𝒱*from a given time origin. Hence:

*ρ*is constant in the whole fluid flow. Under this hypothesis, the input time sequence

*t*={

*t*

_{n},

*n*∈Z} can be modeled as an Homogeneous Poisson Process (HPP) [10] with parameter

*λ*function of

*ρ*and of other physical parameters. The interarrival times are exponentially distributed with mean value given by:

*𝒱*can be described by a

*M*/

*G*/∞ queueing process [4], [6

6. B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. **232**, 83–90, (2004). [CrossRef]

*t*

_{n}are the “customers arrival times” whereas the particle lightening durations

*t*

_{n}are the “service duration” in the queueing denomination. According to queueing notations,

*M*is for an HPP.

*G*denotes the unspecified cumulative distribution of the independent “service durations”:

*G*(

*x*) characterizes the lightening duration with mean

*m*

^{′}=

*E*[

*t*

_{n}].

*G*(

*u*) depends on the particle celerity and on

*𝓢*

_{1}geometric properties. Finally, 𝒱 is for an unlimited number of stations i.e. “the service is instantaneous and equivalent for each customer” which means that the particle is not stopped when entering

*𝒱*. Under these assumptions, the number of elements of

**J**

_{t}i.e. the number of illuminated particles at time

*t*follows a Poisson distribution independently of the distribution

*G*. This last property justifies the results obtained in [2

2. A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. **25**, No. 10, 689–691, (2000). [CrossRef]

**166**, 15–20, (1999). [CrossRef]

*A*

_{n},

*n*∈Z} are independent identically distributed and are independent of the input times {

*t*

_{n},

*n*∈Z} and of the service times {

*t*

^{′}

*n*-

*t*

_{n},

*n*∈Z}. The mean value and correlation function of the received intensity

*I*(

*t*) can then be derived leading to the power spectrum. The proofs are given in the appendix. The following section provides these quantities and the relations with the physical parameters.

## 2.2. The intensity related measurements

### 2.2.1. The intensity mean value and correlation function

*m*

_{1I}and correlation function

*K*

_{I}(

*τ*) of the measured intensity in the queueing process stationary state:

*τ*>0 :

*K*

_{I}(

*τ*) depends on the medium, on the particle and laser beam shapes. [2

2. A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. **25**, No. 10, 689–691, (2000). [CrossRef]

**166**, 15–20, (1999). [CrossRef]

*K*

_{I}(0) is the received intensity power whereas

*λ*large enough, (8) leads to the following relation:

## 2.2.2. Laser scintillation measurement

*m*

_{2I}have been derived in the previous subsection 10, leading to:

*λm*

^{′}

*m*

_{2A}is small with respect to 1, a limited development leads to:

**166**, 15–20, (1999). [CrossRef]

*I*

_{0}and of

*m*

_{1A}. The scintillation relates to the particle physical properties through

*m*

_{2A}and to the particle celerity and illuminated volume geometry through

*m*

^{′}.

## 2.2.3. The intensity power spectrum

*I*(

*t*) is the Fourier transform of

*K*

_{I}(τ). Since

*m*

_{1I}>0, the power spectrum

*s*

_{I}(

*ω*) is composed of a mass at the origin point (equal to

*λ*large enough, according to (11), (16) can be approached by the following Lorentzian spectrum:

*s*

_{I}(

*ω*) given in (17) for

*I*

_{0}=1,

*m*′=1,

*m*

_{1A}=

*m*

_{2A}=1 and different values of

*λ*.

## 3. The constant celerity model

### 3.1. Particle concentration estimation

*𝒱*displayed in Fig. 1. Now, assume that the particles have a constant celerity

*ν*perpendicular to the cylinder axis. Let

*𝓢*

_{2}denote the projection of

*𝒱*perpendicularly to the celerity direction. Boundary effects are neglected leading to a rectangular approximation for

*𝓢*

_{2}. Let

*L*and

*l*denote

*𝓢*

_{2}length and width respectively (Fig. 3).

*λ*is the mean number of particles in a cylinder of basis

*𝓢*

_{2}and height

*ν*. Consequently,

*λ*is proportional to the particle concentration

*ρ*:

*X*denote the particle input abscissa in the laser beam (Fig. 3).

*X*is uniformly distributed on (0,

*l*). Consequently, conditional expectations allow to derive

*m*

^{′}the mean fluid crossing duration:

*m*

^{′}depends on the cross-section area

*𝓢*

_{1}and not on its shape. Consequently, in the weak extinction case (14) leads to:

*ν*. The measurement of Δ, together with

*m*

_{2B}estimation, provide the particle concentration

*ρ*.

## 3.2. Particle celerity estimation

*a*(τ)=

*v*

*G*(

*u*))

*du*,

*h*denotes the finite “height” of

*𝓢*

_{1}(Fig. 1). In the constant celerity case,

*h*/

*v*is the finite maximum fluid crossing duration. Now let consider Δ

_{τ}the generalization of Δ such as:

_{τ}is independent of the emitted laser intensity

*I*

_{0}. As a function of the product ρ

*m*

_{2B}, Δ

_{τ}does not provide additionnal information for

*ρ*estimation. Nevertheless, it provides an estimation of the celerity

*v*. Indeed, Δ

_{τ}decreases from Δ to 0 when τ goes from 0 to

*h*/

*v*which provides a simple estimation of the particle celerity. Moreover, the system can be calibrated, for a given value of

*ρ*, which allows to measure

*m*

_{2B}.

## 3.3. The intensity power spectrum

*α*(τ) and thus

*s*

_{I}(

*ω*) depend on the laser beam cross-section shape. As an example, for a unit area rectangle (with side lengths

*l*and 1/

*l*), the autocorrelation expresses as

*s*

_{I}(

*ω*) can be written as

*lv*, and flattens out when

*lv*increases. In the same time, the discrete part stays unchanged like

*E*[

*I*(

*t*)]. Fig. 4 displays the continuous part of the spectrum for,

*L*=1,

*l*=1,

*ρ*=0.7,

*m*

_{2B}=1 and different celerity values.

## 4. Conclusion

## 5. Appendix

**E**| denote the cardinal number of the set

**E**, and

**E**Ē its complementary set. Now, let define the sets

**B**,

**C**,

**D**, by

*N*(

*t*,τ) is the number of system input times in the interval [

*t*,

*t*+τ[. With these definitions, (3) yields

*N*(0,

*t*) and

*N*(

*t*,τ), which implies the independence of the sets

**J**

_{t}∩

**J**

_{t+τ}and

**J**

_{t}∩

**J̄**

_{t+τ}with

**J̄**

_{t}∩

**J**

_{t+τ}. Consequently, a conditional expectation leads to

*N*(0,

*t*)=

*n*”, the

*t*

_{k}in (0,

*t*) are independent uniformly distributed random variables. Consequently, the two-dimensional random variable (|

**B**|, |

**C**| |

*N*(0,

*t*)=

*n*) follows a trinomial law with parameters [5]

*G*(

*u*)=Pr[

*t*

_{k}<

*u*. In the same manner, (|

**D**| |

*N*(0,

*t*)=

*n*) is binomial with parameter

*N*(0,

*t*) and

*N*(

*t*,τ) are Poisson with parameters

*λt*and

*λt*, (27) leads to

*t*→∞. Finally, (27), (25), (28) lead to:

*I*(

*t*) cannot exist (despite what was written in [6

6. B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. **232**, 83–90, (2004). [CrossRef]

## References and links

1. | R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids , |

2. | A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. |

3. | A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. |

4. | D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998. |

5. | N. Johnson and S. Koltz, Discrete distributions, Houghton mifflin Co.1969. |

6. | B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. |

7. | K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. |

8. | E. Lukacs, Characteristic Functions, Griffin, London, 1970. |

9. | M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame , |

10. | A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991. |

**OCIS Codes**

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(120.1740) Instrumentation, measurement, and metrology : Combustion diagnostics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 17, 2006

Revised Manuscript: June 13, 2006

Manuscript Accepted: June 14, 2006

Published: June 26, 2006

**Citation**

B. Lacaze and M. Chabert, "Power spectra for laser-extinction measurements," Opt. Express **14**, 6011-6019 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6011

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### References

- R. J. Adrian, C. S. Yao, "Power spectra of fluid velocities measured by laser Doppler velocimetry," Exp. in Fluids, 5, 17-28, (1987).
- A. Chen, J. Hao, Z. Zhou, K. He, "Particle concentration measured from light fluctuations," Opt. Lett. 25, No. 10, 689-691, (2000). [CrossRef]
- A. Chen, J. Hao, Z. Zhou, J. Zu, "Theoretical solutions for particular scintillation monitors," Opt. Commun. 166, 15-20, (1999). [CrossRef]
- D. Gross, C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.
- N. Johnson, S. Koltz, Discrete distributions, Houghton mifflin Co. 1969.
- B. Lacaze, "Spectral properties of scattered light fluctuations," Opt. Commun. 232, 83-90, (2004). [CrossRef]
- K. Lee, Y. Han, W. Lee, J. Chung, C. Lee, "Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique," Meas. Sci. Technol. 16, 519-528, (2005). [CrossRef]
- E. Lukacs, Characteristic Functions, Griffin, London, 1970.
- M. Musculus, L. Pickett, "Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments," Combustion and Flame, 141, 371-391, (2005). [CrossRef]
- A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991.

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