## Transport properties of light in a disordered medium composed of two-layered dispersive spheres |

Optics Express, Vol. 19, Issue 4, pp. 2928-2940 (2011)

http://dx.doi.org/10.1364/OE.19.002928

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### Abstract

In this paper, we perform a coated coherent potential approximation method to investigate the transport properties of disordered media consisting of two-layered dielectric spheres whose constituent layer is dispersive. The admixture of quantum dots to polymers to a certain concentration is used as dispersive medium. We find that the dispersive inclusion of the two-layered spheres influences the transport velocities greatly and a resonant scattering taking place in a dilute disordered medium is smeared out in the corresponding densely disordered medium where the correlation effects of multiple scattering are taken into account.

© 2011 Optical Society of America

## 1. Introduction

1. P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B **52**, 505–509 (1985). [CrossRef]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

4. E. Akkermans and G. Montambaux, *Mesoscopic Physics of Electrons and Photons* (Cambridge University Press, 2007). [CrossRef]

*l*

^{*}. The behaviors of diffusive propagation are then characterized by

*l*

^{*}and a diffusion coefficient

*D*=

*v*

_{e}l^{*}/3, where

*v*is the velocity of electromagnetic energy. The value of

_{e}*D*can be measured from time resolved transmission [5

5. J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. **63**, 259–262 (1989). [CrossRef] [PubMed]

6. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. **96**, 063904 (2006). [CrossRef] [PubMed]

*D*, in the classical diffusion theory in which wave interference is ignored [7].

_{B}*kl*

^{*}, which is inversely proportional to the angular width of the albedo from the disordered sample [6

6. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. **96**, 063904 (2006). [CrossRef] [PubMed]

8. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. **56**, 1471–1474 (1986). [CrossRef] [PubMed]

*kl*

^{*}gives the corresponding value of

*l*

^{*}[6

6. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. **96**, 063904 (2006). [CrossRef] [PubMed]

*kl*

^{*}is much larger than unity, meaning low turbidity, constructive interference on counterpropagating paths of multiple scattering leads to enhanced backscattering [10

10. M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. **55**, 2692–2695 (1985). [CrossRef] [PubMed]

11. P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. **55**, 2696–2699 (1985). [CrossRef] [PubMed]

*D*decreases due to such an enhanced-backscattering effect. If the turbidity is higher, the probability of light returning to the origin of elastic multiple scatterings increases, and multiply-scattered waves interfere with each other along the time-reversed loops, which may lead to a significant downward renormalization of

*D*[4

4. E. Akkermans and G. Montambaux, *Mesoscopic Physics of Electrons and Photons* (Cambridge University Press, 2007). [CrossRef]

5. J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. **63**, 259–262 (1989). [CrossRef] [PubMed]

*kl*

^{*}approaches unity, strongly scatterings occur [9]. These closed loops caused by interference effects in multiple scatterings start to be macroscopically populated that finally leads to the absence of diffusions, known as the Anderson localization [12

12. P. W. Anderson, “Absence of diffusion in certain random lattices,” *Phys. Rev.*109, 1492–1505 (1958). [CrossRef]

*D*becomes zero at finite length scales [4

4. E. Akkermans and G. Montambaux, *Mesoscopic Physics of Electrons and Photons* (Cambridge University Press, 2007). [CrossRef]

*R*. If the wavelength

*λ*≫

*R*, the scattering is weak and

*l*

^{*}≫

*R*. If

*λ*is comparable to

*R*, light is strongly scattered by the spheres, and the behaviors for single scattering can be described rigorously by the Mie theory [13, 14

14. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

14. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*l*

^{*}due to the increased scattering cross section [3], as well as an increase of dwell time [15

15. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. **98**, 145–147 (1955). [CrossRef]

18. R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. **99**, 233902 (2007). [CrossRef]

*v*, thus decreases due to the increased dwell time [16

_{e}16. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. **66**, 3132–3135 (1991). [CrossRef] [PubMed]

17. M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E **73**, 065602 (2006). [CrossRef]

*l*

^{*}and

*v*decrease, leading to a reduction of light transport. It is necessary to separate the localization effects and resonant scattering, since only the reduction of

_{e}*l*

^{*}signifies the wave localization according to the aforementioned discussions.

*TiO*

_{2}samples [17

17. M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E **73**, 065602 (2006). [CrossRef]

18. R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. **99**, 233902 (2007). [CrossRef]

*v*due to the resonant scatterings. However, further investigations on the impact of the morphology and refractive indices of spheres upon transport properties of light are still needed.

_{e}20. C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B **49**, 3800–3810 (1994). [CrossRef]

22. K. Busch and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B **54**, 893–899 (1996). [CrossRef]

*v*for different packing fractions to study the influences on transport velocity by the dispersion, resonant scatterings and density of the spheres.

_{e}## 2. Configuration of the multilayered spheres with dispersive inclusions

*nm*[23

23. S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. **87**, 142107 (2005). [CrossRef]

*ɛ*

_{2}= 12.0. Since the wavelength of incident ligth is in the order of 100

*nm*which is much larger than the size of QDs, with the Maxwell-Garnett approach, the effective dielectric constant of the admixture of QDs with a concentration

*η*to the polymer (

*ɛ*= 2.56) can be written as [24

_{m}24. D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B **77**, 035112 (2008). [CrossRef]

*β*(

*ω*) = (

*ɛ*–

_{QD}*ɛ*)/(

_{m}*ɛ*+ 2

_{QD}*ɛ*), and

_{m}*ɛ*is the effective dielectric constant of the QDs which can be described by the single-resonance Lorentz model [23

_{QD}23. S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. **87**, 142107 (2005). [CrossRef]

24. D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B **77**, 035112 (2008). [CrossRef]

*ω*

_{0},

*ω*and

_{p}*γ*represent the resonance frequency, the oscillator strength and the damping constant, respectively. We use the parameters provided in Ref. [24

24. D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B **77**, 035112 (2008). [CrossRef]

*ω*

_{0}

*r*/2

_{mant}*πc*= 0.245,

*ω*= 0.8

_{p}*ω*

_{0}and

*γ*= 0.01

*ω*

_{0}, to calculate the effective

*ɛ*for different values of

_{core}*η*. The results are shown in Fig. 1. Since the absorption of the admixture is provided by the QDs, the imaginary part of

*ɛ*decreases when the concentration of QDs decreases. Hereof we only consider the lossless cases by neglecting the imaginary part of

_{core}*ɛ*, which is approximately valid when

_{core}*η*is very low.

## 3. Theory

20. C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B **49**, 3800–3810 (1994). [CrossRef]

21. K. Busch and C. M. Soukoulis, “Transport properties of random media: A new effective medium theory,” Phys. Rev. Lett. **75**, 3442–3445 (1995). [CrossRef] [PubMed]

*R*=

_{c}*r*

_{2}

*f*

^{−1/3}, where

*r*

_{2}is the radius of the actual two-layered spheres and

*f*is the volume fraction occupied by the two-layered spheres. The scheme of the coated CPA method is shown in Fig. 2.

*ɛ̄*in Fig. 2(b), respectively. The energy density for an electromagnetic vectorial field is given by,

## 4. Numerical results and discussions

*c*and

_{n}*d*for single Mie scattering, which are required for solving the self-consistent Eq. (3), one should treat the numerical stabilities of the Riccati-Bessel functions carefully. Fortunately, some stable algorithms have been proposed [25

_{n}25. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. **13**, 302–313 (1969). [CrossRef]

28. H. Du, “Mie-scattering calculation,” Appl. Opt. **43**, 1951–1956 (2004). [CrossRef] [PubMed]

*S*(

*θ*) is isotropic, i.e.,

*S*(

*θ*) = 1 for all angles. However, when the disordered medium is dense thus multiple scatterings play an important role, an angle-dependent

*S*(

*θ*) should be taken into account which can be described by the so-called Percus-Yevick structural factor [29

29. J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. **110**, 1–13 (1958). [CrossRef]

### 4.1. Scattering cross-section efficiencies of single scattering

*C*is defined as the fraction of the scattered electromagnetic energy in a certain time divided by the incident energy when a plan wave passes the scatterer. In order to compare with the geometrical cross section of the real sphere, a dimensionless constant called the efficiency factor for scattering is defined as

_{sca}*Q*=

_{sca}*C*/

_{sca}*πr*

^{2}, where

*r*is the radius of the real sphere. We use a standard Mie algorithm for a coated sphere described in Ref. [14

14. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*Q*to reveal the influence of dispersion relation of the material on single-scattering events.

_{sca}*r*

_{2}to 0.9

*r*

_{2}and incident wavelength from

*r*

_{2}/0.27 to

*r*

_{2}/0.24. The results are shown in Fig. 3. It is shown in Fig. 1 that near at

*r*

_{2}/

*λ*= 0.258, both the real and imaginary parts of

*ɛ*show the anomalous behavior of a pole resonance. When the concentration increases from

_{core}*η*= 0 to

*η*= 0.01, the calculated

*Q*shows the similar behavior of a pole resonance to that in Fig. 1, while the background beyond the vicinity is approximately unchanged.

_{sca}*Q*′

*s manifest themselves as the spectra consisting of rapid oscillations superimposed on slowly varying profiles in the region*

_{sca}*k*

_{0}

*r*≥ 1, which are the standard characteristics of the Mie scattering [14

*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

30. P. R. Conwell, P. W. Barber, and C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A **1**, 62–67 (1984). [CrossRef]

*Q*′

*s also display the behavior of the slowly varying profiles but only three branches of Mie resonance appear due to the fact that*

_{sca}*r*

_{2}is several times smaller than

*λ*. When the core size becomes smaller, e.g.,

*r*

_{1}< 0.3

*r*

_{2}, its contribution to the

*Q*is trivial.

_{sca}### 4.2. Effective refractive index in the long-wavelength limit

*λ*≫

*r*

_{2}, the scattering events by the boundary conditions can be neglected, and the average dielectric constant of the optical system composed of two-layered spheres is given by the Maxwell-Garnett theory [14

*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

31. C. Pecharroman, T. G. C. no, J. E. Iglesias, “Average dielectric constant of coated spheres: Application to the ir absorption spectra of nio and mgo,” Appl. Spectrosc. **47**, 1203–1208 (1993). [CrossRef]

*S*

_{1}and

*S*

_{2}are defined as follows, where

*ξ*= (

*r*

_{1}/

*r*

_{2})

^{3}is the volume fraction occupied by the core layer compared to the whole two-layered sphere.

*λ*≫

*r*

_{2}, so

*ɛ*

_{1}∼ 2.56 regardless of

*η*. We change the relative radius of the core layer

*r*

_{1}/

*r*

_{2}and the volume fraction

*f*for comparison. We find that the coated CPA method gives results in rather good agreement with that from the Maxwell-Garnett theory.

17. M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E **73**, 065602 (2006). [CrossRef]

20. C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B **49**, 3800–3810 (1994). [CrossRef]

### 4.3. Transport velocities of electromagnetic energy

16. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. **66**, 3132–3135 (1991). [CrossRef] [PubMed]

*v*by involving single scattering of scalar waves only, which gives results in good agreement with the experiments for dilute disordered samples [16

_{e}16. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. **66**, 3132–3135 (1991). [CrossRef] [PubMed]

18. R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. **99**, 233902 (2007). [CrossRef]

**73**, 065602 (2006). [CrossRef]

*v*, according to the Boltzmann expression of

_{e}*D*=

*v*

_{e}l^{*}/3. By measuring

*D*from the time-resolved transmission and

*l*

^{*}from the angular dependence of the albedo from

*TiO*

_{2}disordered samples, they found that the measured

*v*is in good agreement with that obtained by the coated CPA, i.e

_{e}*v*=

_{e}*c/n̄*.

*v*=

_{e}*c/n̄*to investigate the transport properties of the disordered medium composed of two-layered spheres with a high volume fraction approaching the close-packing limit,

*f*= 60%.

*ɛ*by the concentration

_{core}*η*does not result in obvious difference on

*Q*. It suggests that the contribution from the core layer is trivial if the core size is much smaller than the incident wavelength, similar as that shown in Fig. 3.

_{sca}*r*

_{2}= 0.262

*λ*, while the corresponding

*v*, shown in Fig. 5(c), manifests itself a smooth curve. It suggests that the correlation effects of multiple scattering in high packing situations may smear out the contribution from resonant single-scattering.

_{e}*f*= 1%, the curve of transport velocity shows a valley (Fig. 6(b)) which coincides with the resonant peak of the

*Q*curve in Fig. 6(a). We may attribute this phenomenon to the resonant effect based on the fact that the resonant scattering within the spheres leads to a sharply-increased dwell time of light [15

_{sca}15. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. **98**, 145–147 (1955). [CrossRef]

*v*. In general, the contribution of single scattering determines the behaviors of the transport velocity for a dilute disordered medium, which is in accordance with Ref. [16

_{e}**66**, 3132–3135 (1991). [CrossRef] [PubMed]

*v*curve will be smeared out, as indicated by Figs. 6(c–d). It can be attributed to the correlation effects of multiple scattering described by the higher-order terms of scattering matrix 〈

_{e}*T̄*〉

*[see Eq. (3.57) in Ref. [3]], which leads to the re-distribution of the electromagnetic fields. It suggests that in such a situation the two-layered sphere can not act as a microresonator to increase the dwell time or reduce*

_{c}*v*subsequently.

_{e}*f*, the renormalized

*D*as well as

*v*decrease due to the increasing of the correlation effects of multiple scattering, as indicated by Figs. 6(b–d), which are in accordance with Fig. 4. In other words,

_{e}*n̄*increases with an increase of

*f*or equivalently with a decrease of

*R*. Therefore, the effective indices calculated by a standard CPA method are larger than those obtained by the coated CPA method [20

_{c}**49**, 3800–3810 (1994). [CrossRef]

*R*=

_{c}*r*

_{2}. The larger values of

*c/v*obtained by van Albada et. al. compared with that from the coated CPA method may be understood on this mechanism as well. For a disordered medium at the extreme limit of

_{e}*f*→ 0, the standard CPA or van Albada’s method will give the same results of

*c/v*as that of the coated CPA method. Since in this situation their scattering configurations are essentially identical when

_{e}*R*→ ∞, that is why these three methods give approximately the same results for very dilute disordered media.

_{c}*r*

_{1}is large enough, e.g

*r*

_{1}= 0.9

*r*

_{2}in Fig. 5(b), the efficiency factor

*Q*for single scattering is sensitive to

_{sca}*ɛ*hence to the concentration

_{core}*η*. The pole-resonance of

*Q*is different from the Mie resonances, and it corresponds to the behavior of

_{sca}*ɛ*near

_{core}*r*

_{2}= 0.258

*λ*(the blue lines in Fig. 1). The corresponding pole-resonant pattern appears in the curve of transport velocities in Fig. 5(d), and one can find a similar pattern in the case of two-layered spheres with a smaller

*r*

_{1}in Fig. 5(c) as well.

**99**, 233902 (2007). [CrossRef]

## 5. Summary

## A. Mie theory for the field and its energy within constituent layers

*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*j*) and second (

_{n}*y*) kinds to expand (

_{n}**E⃗**

*,*

_{l}**H⃗**

*) in terms of vectorial harmonics [13, 14*

_{l}*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*y*will be infinite at the center, therefore we only use

_{n}*j*as the expansion base. Suppose the incident field is a plane wave,

_{n}**E⃗**

*=*

_{i}**E⃗**

_{0}

*exp*(

*ikz*). The electromagnetic field in the core layer can be expanded as, where

*k*

_{1}=

*n*

_{1}

*k*

_{0}, and

**k⃗**

_{0}(

*k*

_{0}= 2

*π*/

*λ*) is the wave vector in air.

*i*) means that the vectorial harmonic function has the radial dependence of the Bessel function of the

*i*-th order.

*l*-th (

*l*> 1) layer of the sphere, both

*j*and

_{n}*y*are finite, therefore the vectorial harmonics of the fields should include both of them,

_{n}*E*

^{(1)}contained within the multilayered spheres when the sphere is illuminated by a plane wave can be obtained in accordance to the left side of Eq. (3), while the density of the electromagnetic field

*E*

^{(1)}for a three-layered sphere under consideration, which is shown in Fig. 2(a), is the sum of the respective energy contained in the constitutent layers, so it can be written as follows,

*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

22. K. Busch and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B **54**, 893–899 (1996). [CrossRef]

*i*= 1,2, 3. The function

*W*is defined as follows [22

_{n}22. K. Busch and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B **54**, 893–899 (1996). [CrossRef]

**54**, 893–899 (1996). [CrossRef]

**E⃗**

_{2}and

**H⃗**

_{2}, i.e

**54**, 893–899 (1996). [CrossRef]

*x*=

*k*

_{0}

*r*

_{1},

*y*=

*k*

_{0}

*r*

_{2},

*z*=

*k*

_{0}

*R*, and

_{c}*m*=

_{i}*n*/

_{i}*n̄*.

*n*is the refractive index in the

_{i}*i*constituent layer of the multilayered spheres and

^{th}*n̄*is the effective refractive index of the disordered media,

*ψ*and

_{n}*χ*are the Riccati-Bessel functions, and the ratios of the Riccati-Bessel functions, i.e

_{n}*G*

_{1n},

*G*

_{2n}are defined as follows [14

*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*c*and

_{n}*d*, which can be derived by a typical Mie theory considering the boundary conditions [14

_{n}*Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc, 1998). [CrossRef]

*and 3*

^{nd}*layers respectively, i.e Eq. (23) can be easily extended to the case of n-layer spheres (*

^{rd}*n*≥ 3) by performing the Mie theory [27

27. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. **26**(6), 1393 (1991). [CrossRef]

33. W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt. **42**(9), 1710–1720 (2003). [CrossRef] [PubMed]

## References and links

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2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | P. Sheng, |

4. | E. Akkermans and G. Montambaux, |

5. | J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. |

6. | M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. |

7. | R. Lenke and G. Maret, “Multiple scattering of light: Coherent backscattering and transmission,” in |

8. | E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. |

9. | A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. |

10. | M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. |

11. | P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. |

12. | P. W. Anderson, “Absence of diffusion in certain random lattices,” |

13. | H. C. van de Hulst, |

14. | C. F. Bohren and D. R. Huffman, |

15. | E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. |

16. | M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. |

17. | M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E |

18. | R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. |

19. | R. Tweer, “Vielfachstreuung von licht in systemen dicht gepackter mie-streuer: Auf dem weg zur anderson-lokalisierung?” Ph.D. thesis (2002). |

20. | C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B |

21. | K. Busch and C. M. Soukoulis, “Transport properties of random media: A new effective medium theory,” Phys. Rev. Lett. |

22. | K. Busch and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B |

23. | S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. |

24. | D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B |

25. | J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. |

26. | W. J. Wiscombe, “Mie scattering calculations: Advances in technique and fast, vector-speed computer codes,” |

27. | Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. |

28. | H. Du, “Mie-scattering calculation,” Appl. Opt. |

29. | J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. |

30. | P. R. Conwell, P. W. Barber, and C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A |

31. | C. Pecharroman, T. G. C. no, J. E. Iglesias, “Average dielectric constant of coated spheres: Application to the ir absorption spectra of nio and mgo,” Appl. Spectrosc. |

32. | B. A. van Tiggelen and A. Lagendijk, “Rigorous Treatment of the Speed of Diffusing Classical Waves,” Europhys. Lett. |

33. | W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt. |

**OCIS Codes**

(260.3160) Physical optics : Interference

(290.1990) Scattering : Diffusion

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Scattering

**History**

Original Manuscript: October 25, 2010

Revised Manuscript: December 28, 2010

Manuscript Accepted: January 17, 2011

Published: February 1, 2011

**Virtual Issues**

Vol. 6, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Hao Zhang, Heyuan Zhu, and Min Xu, "Transport properties of light in a disordered medium composed of
two-layered dispersive spheres," Opt. Express **19**, 2928-2940 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-4-2928

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