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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 13 — Jun. 26, 2006
  • pp: 6011–6019
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Power spectra for laser-extinction measurements

B. Lacaze and M. Chabert  »View Author Affiliations


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

The dynamics of fluid and embedded particles is currently studied using laser techniques such as the laser-induced incandescence (LII) [7

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]

], the laser-induced scattering (LIS) [7

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]

], the laser doppler anemometry (LDA) [1

1. R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids , 5, 17–28, (1987).

] or the laser-extinction measurements (LEM) [9

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]

]. The aim is to determine the properties of emitted particles for in situ monitoring of combustion effluents [3

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

].

This paper focusses on the LEM technique. The LEM system is composed of a laser beam crossing the particle flow. In the opposite direction, an optical device measures the light beam intensity power for opacity monitors but also its temporal variation for more recent scintillation monitors [3

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

]. The intersection between the laser beam and the particle flow is a piece of cylinder 𝒱 with constant cross-section 𝓢 1 (Fig. 1).

When the laser beam crosses the fluid, the light is partially absorbed. The power spectrum of the residual light intensity 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]

] and others, the received intensity I can be written as

Fig. 1. Laser extinction measurement system
I=I0k=1N(1Ak)Ak=Bk𝓢1<1
(1)

where 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

m1B=E[Bk],m2B=E[Bk2]m1A=m1B𝓢1,m2A=m2B𝓢12

Then 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

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

]. This study proposes random and as well time-varying models for the number of illuminated particles and the received intensity. Indeed, the received intensity is appropriately modelled by the following random process:

I(t)=I0kJt(1Ak)
(3)

where 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

Let define t n and tn the input and output times of the n th particle in the illuminated volume 𝒱 from a given time origin. Hence:

Jt={n,tn<t,tnt}
(4)

Assume that the particle concentration 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

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

] with parameter λ function of ρ and of other physical parameters. The interarrival times are exponentially distributed with mean value given by:

E[tn+1tn]=1λ
(5)

The particle flow in the illuminated volume 𝒱 can be described by a M/G/∞ queueing process [4

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

], [6

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

]. The particle input times t n are the “customers arrival times” whereas the particle lightening durations tn -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)=Pr[tntn<x]
(6)

i.e. G(x) characterizes the lightening duration with mean m =E[tn -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]

], [3

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

].

Now, assume that the random variables {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

This section first provides the mean m 1I and correlation function K I (τ) of the measured intensity in the queueing process stationary state:

m1I=E[I(t)]KI(τ)=E[I(t)I(t+τ)]
(7)

The appendix proves the main result, for τ>0 :

KI(τ)=I02exp[λ(m(2m1Am2A)+m2A0τ(1G(u))du)]
(8)

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]

] and [3

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

] consider the particular cases τ=0 and ∞. Indeed K I (0) is the received intensity power whereas

limτKI(τ)=m1I2
(9)

Then, (8) leads to the intensity first and second order moments:

{m1I=I0exp[λmm1A]m2I=KI(0)=I02exp[λm(m2A2m1A)]
(10)

These relations explain the results obtained with scintillation monitors as shown in the following subsection.

Note that for λ large enough, (8) leads to the following relation:

KI(τλ)exp[λm(m2A2m1A)]I02exp[m2Aτ]
(11)

This correlation function leads to a Lorentzian spectrum as shown below.

2.2.2. Laser scintillation measurement

The scintillation measurement have been recently introduced because of its insensitivity to the optical receiver opacity. The system performance is thus independent of the lens contamination caused by dust accumulation. The scintillation is defined by:

Δ=var[I]m1I2=m2Im1I2m1I2
(12)

Δ=exp[λmm2A]1
(13)

For a weak extinction, i.e when λm m 2A is small with respect to 1, a limited development leads to:

Δλmm2A
(14)

This relation is in perfect agreement with the result in [3

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

]: the scintillation measurement is independent of the incident intensity 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 .

An appropriate system calibration and the a priori knowledge of the particle physical properties allow to estimate λ and thus ρ from D measurement. Indeed for a random celerity ν, the parameter λ (related to the particle input HPP) G(x) and m (related to the stay inside the laser beam) can generally be derived under the hypothesis that v is independent of the particle dimensions. In this case, the correlation function and the power spectrum of I (t) can be derived as functions of the physical parameters [6

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

]. Section 3 develops the simplified case of the constant celerity model.

The following subsection derives the received intensity power spectrum.

2.2.3. The intensity power spectrum

The power spectrum of 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 m1I2) added to a spectral density such as:

sI(ω)=m1I2δ(ω)+1π0[KI(τ)m1I2]cosωτdτ
(15)

The power spectrum is deduced from (8) and (15):

sI(ω)=I02e2λmm1A[δ(ω)+1π0(eλm2A(τ(1G(u)du))1)cosωτdτ].
(16)

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

sI(ω)I02eλ(m'(2m1Am2A))πλm2Aω2+(λm2A)2
(17)

Figure 2 displays s I (ω) given in (17) for I 0=1, m′=1, m 1A=m 2A=1 and different values of λ.

Fig. 2. Approached Lorentzian intensity power spectrum

The following section develops the case of a particle constant celerity.

3. The constant celerity model

3.1. Particle concentration estimation

Consider the illuminated volume 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).

For this simplified model, 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:

λ=ρvlL
(18)

Let 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=E[E[tntnX]]=E[1v(f2(X)f1(X))]
m=1v0l(f2(x)f1(x))dxl=𝓢1lv
(19)
Fig. 3. Constant celerity model notations

Note that m depends on the cross-section area 𝓢 1 and not on its shape. Consequently, in the weak extinction case (14) leads to:

Δ=m2Im1I2m1I2=exp[ρL𝓢1m2B]1ρL𝓢1m2B.
(20)

Therefore, Δ is independent of the celerity ν. The measurement of Δ, together with m 2B estimation, provide the particle concentration ρ.

3.2. Particle celerity estimation

In the case of constant particle celerity, the correlation function expresses as

KI(τ)={I02exp[ρL(2m1B+1𝓢12m2Bα(τ))],0<τ<hvI02exp[2ρLm1B],τ>hv
(21)

where a(τ)=vτh/v (1-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 Δτ the generalization of Δ such as:

Δτ=KI(τ)m1I2m1I2={exp[lL𝓢12ρm2Bα(τ)]10<τ<hv0τ>hv

Like Δ, Δτ is independent of the emitted laser intensity I 0. As a function of the product ρm 2B, Δτ does not provide additionnal information for ρ estimation. Nevertheless, it provides an estimation of the celerity v. Indeed, Δτ 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

(16) leads to the following intensity power spectrum:

sI(ω)=I02e2ρLm1B[δ(ω)+1π0hv(exp{ρLl𝓢12m2Bα(τ)}1)cosωτdτ]
(22)

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

KI(τ)={I02exp[ρL{2m1B+m2B(1lvτ)}]τ<1lvI02exp[2ρLm1B]τ>1lv
(23)

Then, s I (ω) can be written as

sI(ω)=I02e2d[δ(ω)+1π1lvg(ωlv)]with{g(ω)=sinωω+deddcosω+ωsinωω2+d2d=ρLm2B
(24)

The continuous part of the spectrum depends on the product 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.

Fig. 4. Continuous part of the intensity power spectrum-Constant celerity case

4. Conclusion

5. Appendix

Let |E| denote the cardinal number of the set E, and EĒ its complementary set. Now, let define the sets B, C, D, by

{Jt={n,tn<t,tnt}B=JtJt+τ={n,tn<t,tnt+τ}C=JtJ¯t+τ={n,tn<t,ttn<t+τ}D=J¯tJt+τ={n,ttn<t+τ,tnt+τ}N(t,τ)={n,ttn<t+τ}
(25)

N (t,τ) is the number of system input times in the interval [t,t+τ[. With these definitions, (3) yields

E[I(t)I(t+τ)]=E[kB(1Ak)2jC(1Aj)mD(1Am)]
(26)

A basic property of the HPP is the independence of N (0, t) and N (t,τ), which implies the independence of the sets J tJ t and J tt with tJ t. Consequently, a conditional expectation leads to

E[I(t)I(t+τ)]=E[(m2A2m1A+1)B(1m1A)C]E[(1m1A)D].
(27)

However, conditionally 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

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

]

{bt=1t0tPr[tktk>t+τu]du=1t0t[1G(u+τ)]duct=1t0tPr[tu<tktk<t+τu]du=1t0t[G(u+τ)G(u)]du
(28)

where G(u)=Pr[tk -t k <u. In the same manner, (|D| |N (0,t)=n) is binomial with parameter

d=1τtt+τPr[tktk>t+τu]du=1τ0τ[1G(u)]du
(29)

Expressions (10) are obtained from the multinomial distribution generating function [5

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

]:

{E[xByCN(0,t)=n]=(xbt+yct+1btct)nE[zDN(0,t)=n]=(zd+1d)n

Consequently:

KI(τ)=E[((m2A2m1A)btm1Act+1)N(0,t)]E[(1m1Ad)N(t,τ)]

Since N (0, t) and N (t,τ) are Poisson with parameters λt and λt, (27) leads to

KI(τ)=exp[λt((m2A2m1A)btm1Act)m1Aλτd]
(30)

The stationary limit is obtained when t→∞. Finally, (27), (25), (28) lead to:

{KI(τ)=exp[α+βτ(1G(u))du]m=E[tktk]G(u)=Pr[tktk<u]α=2λmm1Aβ=λm2A
(31)

using the equality

m=0(1G(u))du=0udG(u)

when this quantity is finite. From a mathematical point of view, the linked characteristic function belongs to the Polya class [8

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

]. Consequently, the mean-square derivative of 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]

]). This property is common for processes with discontinuous realizations.

References and links

1.

R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids , 5, 17–28, (1987).

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]

3.

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

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. 232, 83–90, (2004). [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]

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 , 141, 371–391, (2005). [CrossRef]

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

  1. R. J. Adrian, C. S. Yao, "Power spectra of fluid velocities measured by laser Doppler velocimetry," Exp. in Fluids, 5, 17-28, (1987).
  2. A. Chen, J. Hao, Z. Zhou, K. He, "Particle concentration measured from light fluctuations," Opt. Lett. 25, No. 10, 689-691, (2000). [CrossRef]
  3. A. Chen, J. Hao, Z. Zhou, J. Zu, "Theoretical solutions for particular scintillation monitors," Opt. Commun. 166, 15-20, (1999). [CrossRef]
  4. D. Gross, C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.
  5. N. Johnson, S. Koltz, Discrete distributions, Houghton mifflin Co. 1969.
  6. B. Lacaze, "Spectral properties of scattered light fluctuations," Opt. Commun. 232, 83-90, (2004). [CrossRef]
  7. 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]
  8. E. Lukacs, Characteristic Functions, Griffin, London, 1970.
  9. 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]
  10. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991.

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