## Fluctuations of scattered waves: going beyond the ensemble average

Optics Express, Vol. 17, Issue 13, pp. 10466-10471 (2009)

http://dx.doi.org/10.1364/OE.17.010466

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### Abstract

The interaction between coherent waves and random media is a complicated, deterministic process that is usually examined upon ensemble averaging. The result of one realization of the interaction process depends on the specific disorder present in an experimentally controllable interaction volume. We show that this randomness can be quantified and structural information not apparent in the ensemble average can be obtained. We use the information entropy as a viable measure of randomness and we demonstrate that its rate of change provides means for discriminating between media with identical mean characteristics.

© 2009 Optical Society of America

1. P. A. Lee and A. D. Stone, “Universal Conductance Fluctuations in Metals,” Phys. Rev. Lett. **55(15)**, 1622–1625 (1985).
[CrossRef]

_{2}, 3

3. S. Etemad, R. Thompson, and M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averagign,” Phys. Rev. Lett. **57(5)**, 575–578 (1986).
[CrossRef]

4. M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, “Weak Localization and Light Scattering from Disordered Solids,” Phys. Rev. Lett. **57(16)**, 2049–2052 (1986).
[CrossRef]

5. B. Shapiro, “Large Intensity Fluctuations for Wave Propagation in Random Media,” Phys. Rev. Lett. **57(17)**, 2168–2171 (1986).
[CrossRef]

6. M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. **59(3)**, 285–287 (1987).
[CrossRef]

*N*of scattering centers and by the scattering cross-section

_{V}*σ*describing the properties of a single scattering event. For each realization of disorder

*α*, the interaction will be defined by a specific distribution

*p*(

_{α}*s*) of available photon path-lengths s through the medium. When an ensemble average is taken over many such realizations, the photon interaction will be described by a probability distribution function

*p*(

*s*)=〈

*p*(

_{α}*s*)〉=

*f*(

*s,D*) that has a universal behavior depending only on the normalized diffusion coefficient

*D*∝1=(

*N*) [8]. Note that all of the experimentally observable properties of the stochastic interaction can be described in terms of the probability distribution

_{V}σ*p*(

*s*) whose exact functional form

*f*(

*s,D*) may also depend on the particular geometry of an experiment. Clearly, there could be many dissimilar media with different

*N*and

_{V}*σ*that nevertheless display the same characteristics upon ensemble averaging. In practice, this ensemble can be acquired in different ways for dynamic or stationary systems, but the final result is the same: the number density and the scattering cross-section are being coupled through the diffusion coefficient, and only their product is accessible.

*p*(

_{α}*s*) will deviate from the one corresponding to the ensemble average:

*δ*(

_{α}*s,ξα*) is specific to a particular realization of disorder, it depends on variables not present in the ensemble average. Specifically, this can be expressed through a configuration function

*ξα*describing the particular morphology of the given realization

*α*. In the example above, the function

*ξα*describes the locations of scattering centers available in the realization and depends only on the number density and not on the scattering cross-section. For media with continuously varying refractive indexes, one can still define a configuration function

*ξα*that is independent of the strength of the scattering. By examining the statistical properties of

*δα*(

*s,ξα*), one could infer information not available in the ensemble average.

*δ*(

_{α}*s,ξα*) are worth making. First, as the length

*s*of the path increases, more and more different trajectories of the same length are possible through the medium, and

*p*(

_{α}*s*) approaches the value corresponding to the weight of trajectories of length

*s*in the ensemble average. In other words, in terms of the variable

*s*, the function

*δ*(

_{α}*s,ξα*) represents a nonstationary random process. Second, because upon ensemble averaging a scattering region will exist at any position, this random function is of zero mean, 〈

*δ*(

_{α}*s,ξα*)〉=0. However, because of the implicit dependence on the density of scattering regions, it is expected that higher order statistics of

*δ*(

_{α}*s,ξα*) can be used to reveal characteristics of the wave-matter interaction not included in the value of

*D*.

*s*. The two media consist of scattering centers having different cross-sections but also different number densities such that, upon ensemble average, they are described by the same diffusion coefficient. Clearly, when compared to all potential trajectories, there are fewer available paths of given length

*s*through the medium with less scattering centers. Consequently, the path-length distribution

*p*(

_{α}*s*) deviates more significantly from the ensemble average

*p*(

*s*)=〈

*p*(

_{α}*s*)〉. A measure of the nonstationary fluctuations in

*p*(

_{α}*s*) should discriminate between the two media, as we will show in the following.

*p*(

_{α}*s*) displays not only fluctuations in s but also differences from one material realization

*α*to another. There are many ways in which the two-dimensional statistical characteristics of

*p*(

_{α}*s*) can be quantified. Of course, a simple averaging over a will provide a path-length distribution

*p*(

*s*)=

*f*(

*s,D*) which corresponds to the ensemble average. For a single realization

*α*on the other hand, higher order moments of the fluctuations in

*p*(

_{α}*s*) can be evaluated. Even though

*p*(

_{α}*s*) is nonstationary in s, one can still calculate simple estimators such as, for instance, the variance

*V*(

_{α}*ξα*)=

*∫*

_{δ}^{2}

*(*

_{α}*s,ξα*)

*ds*-|∫

*δ*(

_{α}*s,ξα*)

*ds*|

^{2}of the fluctuations along s. However, this simple estimate is inadequate because

*δ*(

_{α}*s,ξα*) is a zero-mean random function and, consequently, a unique and meaningful normalization is difficult to define.

*δ*(

_{α}*s,ξα*) from the ensemble average can be regarded as a form of disorder, we can choose to examine its variance in terms of the Shannon information entropy [9]:

*s*

_{1},

*s*

_{2}]. Furthermore, the finite scale entropy can be normalized to its maximum allowable value for the entire range

*S*=

*s*

_{2}-

*s*

_{1}as

*h*(

_{α}*S*

_{1},

*S*

_{2}) will still vary from realization to realization and one can further build its average ̄

*h*(

_{α}*S*

_{1},

*S*

_{2}) over the number of realizations available. Being constructed in terms of the specific fluctuations of each realization a, this average is a comprehensive measure of the overall fluctuations in

*δ*

*(*

_{α}*s,ξα*).

*p*(

_{α}*s*). The distribution of photon path-lengths through different multiply-scattering media was measured interferometrically using the procedure of optical path-length spectroscopy (OPS)[10

10. G. Popescu and A. Dogariu, “Scattering of low coherence radiation and applications,” Eur. Phys. J. Appl. Phys. **32(2)**, 73–93 (2005).
[CrossRef]

11. G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. **24(7)**, 442–444 (1999).
[CrossRef]

*p*(

*s*,Δ) corresponding to the ensemble average can be evaluated to be

*z*is the so-called extrapolation length [10

_{e}10. G. Popescu and A. Dogariu, “Scattering of low coherence radiation and applications,” Eur. Phys. J. Appl. Phys. **32(2)**, 73–93 (2005).
[CrossRef]

*µ*m and 1.2

*µ*m. Upon ensemble averaging, both are characterized by the same value of the transport mean free path of 10

*µ*m.

*p*(

*s*) is a clear indication that, on average, the two media are being described by the same diffusion coefficient. On the other hand, the fluctuations from the average are rather dissimilar as can be seen in Fig. 3 where we plot the typical mean square of the fluctuations

*δ*(

^{2}_{α}*s,ξα*) corresponding to the two media. In general, medium A exhibits smaller deviations from the average which can be interpreted as a larger number of scattering trajectories available for each

*s*. Note also that the fluctuations in

*δ*(

^{2}_{α}*s,ξα*) decrease for larger values of s because these random processes are nonstationary as discussed above.

*s*

_{1}=Δ and the upper one at

*s*

_{2}=Δ+

*S*. Here

*S*denotes the value of the total span of path-lengths available in the measurement;

*S*is constant in our experiments. Subsequent normalization and averaging over different realizations was performed following the procedure outlined by Eq. (3). In Fig. 4 we present the values of the normalized scale dependent entropy ̄

*h*(Δ) averaged over ten realizations of disorder for both media examined.

_{α}*α, δ*(

_{α}*s,ξα*) is a nonstationary process, and its fluctuations decrease at larger

*s*. The absolute values and the rate of increase for ̄

*h*(Δ) however are medium specific.

_{α}*s*. Therefore, there are smaller fluctuations in

*δ*(

^{2}_{α}*s,ξα*) as discussed before and, consequently, the entropy tends toward its value corresponding to an infinite number of possible trajectories of length

*s*.

*s*can be reached through a different number

*m*of scattering events. For independent scattering, the joint distribution

*p*(

*s,m*) of such a process is Poissonian and the cumulative probability of scattering orders up to

*M*that contribute to paths of length

*s*is described, in average, by a universal cumulative distribution function

*p*(

*s,M*) [12

12. A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” Prog. in Opt. **34**, 333–402 (1995).
[CrossRef]

*M*and tends to saturate for higher scattering orders. In one realization where the interaction volume is finite, the maximum scattering order

*M*contributing to a certain s is essentially determined by the number density of available scattering centers. Thus, processes involving different number densities will in fact experience different regions of the cumulative distribution function. For the sparser medium B, a change in

*M*results in a faster increase of the corresponding values of

*p*(

*s,M*) and, consequently, a faster decrease in the possible fluctuations. Because the entropy is a measure of magnitude of these fluctuations, it follows that the medium B should be characterized by a faster rate of entropy increase as can be seen in Fig. 4. As a result, in spite of being described in average by the same diffusion coefficient

*D*, the two media can be discriminated based on their corresponding densities of scattering regions. This information was not available in the ensemble average.

13. A. Apostol, D. Haefner, and A. Dogariu, “Near-field characterization of effective optical interfaces,” Phys. Rev. E **74(6)**, 066603–6 (2006).
[CrossRef]

14. E. Hartveit and M. L. Veruki, “Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches,” Nature Protocols **2(2)**, 434–448 (2007).
[CrossRef]

## References and links

1. | P. A. Lee and A. D. Stone, “Universal Conductance Fluctuations in Metals,” Phys. Rev. Lett. |

2. | S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and Fluctuations of Coherent Wave Transmission through Disordered Media,” Phys. Rev. Lett. |

3. | S. Etemad, R. Thompson, and M. J. Andrejco, “Weak localization of photons: universal fluctuations and ensemble averagign,” Phys. Rev. Lett. |

4. | M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, “Weak Localization and Light Scattering from Disordered Solids,” Phys. Rev. Lett. |

5. | B. Shapiro, “Large Intensity Fluctuations for Wave Propagation in Random Media,” Phys. Rev. Lett. |

6. | M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. |

7. | J. W. Goodman, |

8. | A. Ishimaru, |

9. | C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J. |

10. | G. Popescu and A. Dogariu, “Scattering of low coherence radiation and applications,” Eur. Phys. J. Appl. Phys. |

11. | G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. |

12. | A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” Prog. in Opt. |

13. | A. Apostol, D. Haefner, and A. Dogariu, “Near-field characterization of effective optical interfaces,” Phys. Rev. E |

14. | E. Hartveit and M. L. Veruki, “Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches,” Nature Protocols |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(290.1990) Scattering : Diffusion

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: March 26, 2009

Revised Manuscript: May 12, 2009

Manuscript Accepted: June 3, 2009

Published: June 8, 2009

**Citation**

J. Broky, K. M. Douglass, J. Ellis, and A. Dogariu, "Fluctuations of scattered waves: going beyond the ensemble average," Opt. Express **17**, 10466-10471 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10466

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### References

- P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985). [CrossRef]
- S. Feng, C. Kane, P. A. Lee, and A. D. Stone, "Correlations and Fluctuations of Coherent Wave Transmission through Disordered Media," Phys. Rev. Lett. 61(7), 834-837 (1988). [CrossRef]
- S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986). [CrossRef]
- M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986). [CrossRef]
- B. Shapiro, "Large Intensity Fluctuations for Wave Propagation in Random Media," Phys. Rev. Lett. 57(17), 2168-2171 (1986). [CrossRef]
- M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987). [CrossRef]
- J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, 2007).
- A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 1 (Academic, New York, 1971).
- C. E. Shannon, "A Mathematical Theory of Communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).
- G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005). [CrossRef]
- G. Popescu and A. Dogariu, "Optical path-length spectroscopy of wave propagation in random media," Opt. Lett. 24(7), 442-444 (1999). [CrossRef]
- A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995). [CrossRef]
- A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006). [CrossRef]
- E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007). [CrossRef]

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