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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 3 — Mar. 18, 2011
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Transport properties of light in a disordered medium composed of two-layered dispersive spheres

Hao Zhang, Heyuan Zhu, and Min Xu  »View Author Affiliations


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

The diffusive propagation of light in disordered media has attracted much attention since the theoretical predictions of strong localization of light [1

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

, 2

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

]. One reason of this interest is the realization that wave propagations in strongly disordered media exhibit quite different behavior from that associated with classical diffusions [3

3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

, 4

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

].

In a disordered medium, photons are scattered by the constituent microspheres carrying out a random walk, and the direction of propagation is randomized over a characteristic length called transport mean free path l*. The behaviors of diffusive propagation are then characterized by l* and a diffusion coefficient D = vel*/3, where ve is the velocity of electromagnetic energy. The value of 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

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]

], which is regarded as a constant, termed as Boltzmann value DB, in the classical diffusion theory in which wave interference is ignored [7

7. R. Lenke and G. Maret, “Multiple scattering of light: Coherent backscattering and transmission,” in Scattering in Polymeric and Colloidal Systems, W. Brown and K. Mortensen, eds. (Gordon and Breach Science Publishers, 2000).

].

In a multiple-scattering disordered medium, however, wave interference may result in the possibility of wave localization inside the sample. The major factor that determines the non-classical behaviors of light in such a medium is the turbidity of disordered media. One can use the value of 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

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]

], to characterize the turbidity [3

3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

, 9

9. A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

]. In experiments, the value of 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]

].

If 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

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]

], or weak localization of light, which is a precursor of strong localization. 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

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

]. At even higher turbidity, especially when kl* approaches unity, strongly scatterings occur [9

9. A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

]. 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]

], which means that the renormalized 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]

].

From the viewpoint of light scattering, another yardstick in the problem of multiple scattering in disordered media is the radius of the spheres R. If the wavelength λR, 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

13. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc. New York, 1981).

, 14

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

]. In the Mie-scattering regime, the scattering cross section manifests itself as several resonant points superimposed on a slowly-varying profile [13

13. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc. New York, 1981).

, 14

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

], and the homogeneous dielectric spheres act as microresonators at the resonant points, also called as Mie resonances. Such a resonant effect leads to a decrease of l* due to the increased scattering cross section [3

3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

], 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

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]

]. The transport velocity of electromagnetic energy, i.e ve, thus decreases due to the increased dwell time [16

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

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]

]. Therefore, when the Mie resonance occurs, both l* and ve decrease, leading to a reduction of light transport. It is necessary to separate the localization effects and resonant scattering, since only the reduction of l* signifies the wave localization according to the aforementioned discussions.

Due to the fact that the resonant scattering of Mie spheres is morphology-dependent, some investigations have been conducted on the transport properties of light within the disordered media consisiting of 15% – 20% polydispersed and irregularly-shaped TiO2 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]

], and within highly monodispersed polystyrene samples [18

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]

], and both have revealed the reduction of ve 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.

In this paper, we investigate the transport properties of light in a disordered medium consisting of two-layered homogeneous dielectric spheres, which is a potential system to realize strong localization close to the Ioffe-Regel criterion [9

9. A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

, 19

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

]. The sphere under considerations includes a dispersive core layer, which may influence the scattering cross section significantly if its size is large enough. We derive an effective-medium theory for treating disordered media consisting of two-layered spheres based on a generalized version of the coherent potential approximation (CPA) method [20

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

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]

], and then calculate ve for different packing fractions to study the influences on transport velocity by the dispersion, resonant scatterings and density of the spheres.

2. Configuration of the multilayered spheres with dispersive inclusions

In this paper, a disordered medium consisting of two-layered Mie homogeneous dielectric sphere in air is investigated. The core layer is assumed to be a typical polymer doped by colloidal quantum dots (QDs) whose average diameter is less than 10nm [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]

]. The mantel layer is silicon with a dielectric constant of ɛ2 = 12.0. Since the wavelength of incident ligth is in the order of 100nm 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 (ɛm = 2.56) can be written as [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]

],
ɛcore=ɛm(1+3ηβ(ω)1ηβ(ω))
(1)
where β(ω) = (ɛQDɛm)/(ɛQD + 2ɛm), and ɛQD is the effective dielectric constant of the QDs which can be described by the single-resonance Lorentz model [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]

, 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]

],
ɛQD=1+ωp2ω02ω2iγω
(2)
where ω0, ωp and 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]

], i.e, ω0rmant/2πc = 0.245, ωp = 0.8ω0 and γ = 0.01ω0, to calculate the effective ɛcore for different values of η. The results are shown in Fig. 1. Since the absorption of the admixture is provided by the QDs, the imaginary part of ɛcore decreases when the concentration of QDs decreases. Hereof we only consider the lossless cases by neglecting the imaginary part of ɛcore, which is approximately valid when η is very low.

Fig. 1 Effective dielectric constants for the dispersive material constructed by polymer doped with quantum dots for three different concentrations η = 0.001, 0.005, 0.01, respectively. r is the radius of the mantel layer.

3. Theory

The coated Coherent Potential Approximation method is adopted in the calculation of the effective index of the system. It is based on the physical idea that the distribution of the electromagnetic energy within a random medium should be homogeneous after averaged over the correlation length of the random media [20

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

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]

]. Therefore the averaged forward-scattering amplitude of a spherical region should be equal to zero. In addition, the structure factor in the multiple scattering system is taken into account by involving a coated sphere embedded in the effective medium as a scattering unit. The radius of the coated layer is given by Rc = r2f−1/3, where r2 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.

Fig. 2 Scheme of the coated CPA as an effective medium theory. Different colorful spheres in (a) indicate different materials with different refractive indices. The effective dielectric constant ɛ̄ is calculated by the effective medium theory, Ψl represents the vectorial field in lth layer, and Ψi represents the vectorial incident field. The dashed volume in (b) indicates the averaged counterpart of the coated spheres.

Since single scattering unit in the coated CPA theory produces scattering patterns similar to those by its averaged counterpart, the electromagnetic energy contained in the coated spheres in Fig. 2(a) should be equal to that contained in the dashed volume Fig. 2(b). So we have
0Rcd3rρE(1)(r)=0Rcd3rρE(2)(r)
(3)
where ρE(1) and ρE(2) represent the energy densities of the three-layered sphere in Fig. 2(a) and that of the averaged volume with an effective ɛ̄ in Fig. 2(b), respectively. The energy density for an electromagnetic vectorial field is given by,
ρE(r)=12[ɛ(r)|E(r)|2+μ|H(r)|2]
(4)

4. Numerical results and discussions

In order to obtain precise and stable numerical results of scattering amplitudes or the coefficients of cn and dn 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

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

28

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

], which are suitable to handle the multilayered sphere under the consideration.

When the disordered medium with Mie spheres is dilute, single Mie scattering plays a major role in determining the optical properties of the disordered medium where the structure factor 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]

]. As mentioned previously, the influence of the multiple scatterings may be treated effectively by adding a coated layer to the real spheres, whose radius is volume-fraction dependent.

In the following sections, we investigate the scattering cross sections of single Mie-scattering and the effective refractive index of the medium, respectively.

4.1. Scattering cross-section efficiencies of single scattering

The scattering cross section Csca 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 Qsca = Csca/πr2, 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]

] to calculate Qsca to reveal the influence of dispersion relation of the material on single-scattering events.

The scattering efficiencies are calculated for four different concentrations of QDs in the core layer with core radius ranging from 0.1r2 to 0.9r2 and incident wavelength from r2/0.27 to r2/0.24. The results are shown in Fig. 3. It is shown in Fig. 1 that near at r2/λ = 0.258, both the real and imaginary parts of ɛcore show the anomalous behavior of a pole resonance. When the concentration increases from η = 0 to η = 0.01, the calculated Qsca shows the similar behavior of a pole resonance to that in Fig. 1, while the background beyond the vicinity is approximately unchanged.

Fig. 3 Calculated scattering efficiency factors for two-layered spheres whose core layers are dispersive. The dielectric constant of the core layer, i.e ɛcore is determined by incident wavelengths and the concentration of QDs (a) η = 0, (b) η = 0.001, (c) η = 0.005 and (d) η = 0.01 respectively.

It has been shown that the efficiency factors for scattering Qscas manifest themselves as the spectra consisting of rapid oscillations superimposed on slowly varying profiles in the region k0r ≥ 1, which are the standard characteristics of the Mie scattering [14

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

], and the positions of the resonant peaks are sensitive to the morphology of the spheres [30

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]

]. For the spheres considered herein, the curves of Qscas also display the behavior of the slowly varying profiles but only three branches of Mie resonance appear due to the fact that r2 is several times smaller than λ. When the core size becomes smaller, e.g., r1 < 0.3r2, its contribution to the Qsca is trivial.

4.2. Effective refractive index in the long-wavelength limit

To check the coated CPA method for disordered media composed of two-layered spheres, we compare it with the Maxwell-Garnett theory in the long-wavelength limit. When λr2, 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

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

, 31

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]

],
ɛ¯=(1f)ɛ3+fS11f+fS2
(5)
where the parameters S1 and S2 are defined as follows,
S1=3ɛ3ɛ2(ɛ1(1+2ξ)+2ɛ2(1ξ))/Θ
(6)
S2=3ɛ3(ɛ1(1ξ)+ɛ2(2+ξ))/Θ
(7)
Θ=(ɛ1+2ɛ2)(ɛ2+2ɛ3)+2ξ(ɛ2ɛ3)(ɛ1ɛ2)
(8)
where ξ = (r1/r2)3 is the volume fraction occupied by the core layer compared to the whole two-layered sphere.

Figure 4 presents the effective refractive indices of the optical system composed of two-layered spheres calculated by the coated CPA method and Maxwell-Garnett theory respectively. Since λr2, so ɛ1 ∼ 2.56 regardless of η. We change the relative radius of the core layer r1/r2 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.

Fig. 4 Numerical comparisons between the Qscas calculated by the coated CPA method and those calculated by the Maxwell-Garnett theory.

Since comprehensive checks with experiments have been conducted for the disordered media composed of pure spheres for the wavelength comparable to the size of the spheres, which all lead to pretty good agreements [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]

, 20

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]

], we believe that the results obtained from the coated CPA method are faithful.

4.3. Transport velocities of electromagnetic energy

In Ref. [16

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]

], van Albada et. al. proposed a simplified treatement for the Bethe-Salpeter equation to obtain the expression of ve by involving single scattering of scalar waves only, which gives results in good agreement with the experiments for dilute disordered samples [16

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

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]

]. In Ref. [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]

], Störzer et. al. obtained ve, according to the Boltzmann expression of D = vel*/3. By measuring D from the time-resolved transmission and l* from the angular dependence of the albedo from TiO2 disordered samples, they found that the measured ve is in good agreement with that obtained by the coated CPA, i.e ve = c/n̄.

We use the expression of ve = 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%.

The scattering efficiency factors of single scattering are shown in Figs. 5(a–b), which indicates the influence of single scattering to the effective refractive indices. It is shown in Fig. 5(a) that the change of ɛcore by the concentration η does not result in obvious difference on Qsca. 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.

Fig. 5 Calculated scattering efficiency factors Qsca for two-layered spheres under considerations and transport velocities of electromagnetic energy for optical disordered medium composed of the two-layered spheres with f = 60%. The radia of the core layer are r1 = 0.5r2 in (a) and (c), r1 = 0.9r2 in (b) and (d), respectively. Numerical results for systems with different values of the concentration η, i.e η = 0, η = 0.001, η = 0.005 and η = 0.01 are indicated by the blue, red, black and green lines respectively.

Figure 5(a) shows that a resonant scattering occurs at r2 = 0.262λ, while the corresponding ve, 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.

If the disordered medium is dilute enough, e.g. f = 1%, the curve of transport velocity shows a valley (Fig. 6(b)) which coincides with the resonant peak of the Qsca 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

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

], hence a reduced ve. 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

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]

].

Fig. 6 (a) Calculated scattering efficiency factor for a single scattering of a two-layered sphere with r1 = 0.5r2 and η = 0.01. Figs. (b–d) show the transport velocities of electromagnetic energy for disordered media composed of two-layered spheres with r1 = 0.5r2, η = 0.01 and volume fractions f = 1%, 30%, 50%, respectively.

If the disordered medium is sufficiently dense, however, the resonant scattering-based valley on the ve 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 〈c [see Eq. (3.57) in Ref. [3

3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

]], 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 ve subsequently.

In addition, with the increase of f, the renormalized D as well as ve 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, increases with an increase of f or equivalently with a decrease of Rc. Therefore, the effective indices calculated by a standard CPA method are larger than those obtained by the coated CPA method [20

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]

], since the former one only considers single scatterings while neglects the correlation effects [3

3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

], and the corresponding scattering unit does not include the coated layer, or Rc = r2. The larger values of c/ve 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 f → 0, the standard CPA or van Albada’s method will give the same results of c/ve as that of the coated CPA method. Since in this situation their scattering configurations are essentially identical when Rc → ∞, that is why these three methods give approximately the same results for very dilute disordered media.

If the size r1 is large enough, e.g r1 = 0.9r2 in Fig. 5(b), the efficiency factor Qsca for single scattering is sensitive to ɛcore hence to the concentration η. The pole-resonance of Qsca is different from the Mie resonances, and it corresponds to the behavior of ɛcore near r2 = 0.258λ (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 r1 in Fig. 5(c) as well.

It should be noted that, according to Ref. [18

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]

], a random medium with polydispersity more than 10% would wash out the single-particle resonances, therefore the samples for experiments should be prepared well to be highly monodispersed, if one wants to conduct the corresponding experiments.

It concludes, in geneneal, that the effective indices of diordered media can be significantly modified by the dispersion relation of the constituent layer of the multilayered spheres, and for a densely disordered medium, the resonant effect may be smeared out by the correlation effects of multiple scattering.

5. Summary

We have derived the mathematical formulae of the coated CPA method for disordered media composed of two-layered dielectric spheres whose consituent materials exhibit anomalous dispersion.

For such spheres, the total scattering cross section for an incident plane wave displays similar pattern to that of the dispersion of the constituent material, and is sensitive to the concentration of QDs to the polymer that determines the dispersion relation of the material.

We have used the derived formulae to calculate the effective refractive indices of the disordered media composed of two-layered spheres in the long-wavelength limit. The results are in good agreement with those obtained by the Maxwell-Garnett theory.

Finally, we have investigated the transport velocities of light propagating in the disordered medium under considerations. The anomalous dispersion of the constituent material leads to pole-resonance pattern for both the total scattering efficiency factors and the transport velocity of electromagnetic energy. However, it should be noted that the contribution of resonant scattering may be smeared out by the correlation effects of multiple scattering occuring in a strongly scattering medium.

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

In this section, we perform a standard Mie theory to obtain the mathmatical expressions required for solving Eq. (3), where we follow the notation 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]

].

Since the isotropic material of the constituent layer investigated in this paper is linear and homogeneous, we can use spherical Bessel functions of the first (jn) and second (yn) kinds to expand (E⃗l, H⃗l) in terms of vectorial harmonics [13

13. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc. New York, 1981).

, 14

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

]. For the core layer, however, yn will be infinite at the center, therefore we only use jn as the expansion base. Suppose the incident field is a plane wave, E⃗i = E⃗0exp(ikz). The electromagnetic field in the core layer can be expanded as,
E1=n=1En(cnMo1n(1)idnNe1n(1))
(9)
H1=k1ωμ1n=1En(dnMe1n(1)+icnNo1n(1))
(10)
where
En=E0in2n+1n(n+1)
(11)
k1 = n1k0, and k⃗0(k0 = 2π/λ) is the wave vector in air. Mo1n(i) and No1n(i) are the corresponding vectorial harmonic functions. The superscript (i) means that the vectorial harmonic function has the radial dependence of the Bessel function of the i-th order.

In the l-th (l > 1) layer of the sphere, both jn and yn are finite, therefore the vectorial harmonics of the fields should include both of them,
El=n=1En(fn(l)Mo1n(1)ign(l)Ne1n(1)+vn(l)Mo1n(2)iwn(l)Ne1n(2))
(12)
Hl=klωμln=1En(gn(l)Me1n(1)+ifn(l)No1n(1)+wn(l)Me1n(2)+ivn(l)No1n(2))
(13)

The total energy 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 ρE(1) is expressed by Eq. (4). The total energy 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,
E(1)=E1(1)+E2(1)+E3(1)
(14)

Then, considering the expressions of the electromagnetic fields, i.e Eqs. (9,10) and the orthogonality properties of the vectorial harmonic functions [14

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

], we have the expression for the electromagnetic energy within the core layer as follows [22

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]

],
E1(1)=12ω2c2ɛ1n=1(|cn|2+|dn|2)1k130k1r1ρ2dρWn(jn,jn;ρ)
(15)
where ki=ɛiω/c, and i = 1,2, 3. The function Wn is defined as follows [22

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]

],
Wn(zn,zn¯;r)=(2n+1)zn(r)zn¯(r)+(n+1)zn1(r)zn1¯(r)+nzn+1(r)zn+1¯(r)
(16)

By using Eqs. (12,13), the expression for the electromagnetic energy distributed within the second constituent layer can be written as follows [22

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]

],
E2(1)=12ω2c2ɛ2n=11k23k2r1k2r2ρ2dρ[(|fn(2)|2+|gn(2)|2)Wn(jn,jn;ρ)+(|vn(2)|2+|wn(2)|2)Wn(nn,nn;ρ)+2(|fn(2)vn(2)|+|gn(2)wn(2)|)Wn(jn,nn;ρ)]
(17)
where the coefficients of the vectorial harmonic functions in the expansions of E⃗2 and H⃗2, i.e fn(2), vn(2), gn(2), wn(2) can be written as follows [22

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]

],
fn(2)=ψn(m1x)χn(m2x)[G1n(m1x)(m2/m1)G2n(m2x)]cnϕn(m1,m2;x)cn
(18)
vn(2)=ψn(m1x)ψn(m2x)[G1n(m1x)(m2/m1)G1n(m2x)]cnζn(m1,m2;x)cn
(19)
gn(2)=ψn(m1x)χn(m2x)[(m2/m1)G1n(m1x)G2n(m2x)]dnγn(m1,m2;x)dn
(20)
wn(2)=ψn(m1x)ψn(m2x)[(m2/m1)G1n(m1x)G1n(m2x)]dnηn(m1,m2;x)dn
(21)
where x = k0r1, y = k0r2, z = k0Rc, and mi = ni/. ni is the refractive index in the ith constituent layer of the multilayered spheres and is the effective refractive index of the disordered media, n¯=ɛ¯. ψn and χn are the Riccati-Bessel functions, and the ratios of the Riccati-Bessel functions, i.e G1n, G2n are defined as follows [14

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

],
G1n(m1x)=ψn(m1x)/ψn(m1x),G2n(m1x)=χn(m1x)/χn(m1x)

Therefore, by using the expressions of En(1) and the expressions for the amplitudes of the vectorial harmonic functions within the core layer, i.e cn and dn, which can be derived by a typical Mie theory considering the boundary conditions [14

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

], the total electromagnetic energy contained within the three-layer sphere in Fig. 2(a) can be calculated directly. Moreover, the expression for the electromagnetic energy distributed within the area surrouned by the dashed line in Fig. 2(b) can be calculated by Eq. (4) as well. In addition, it should be noted that the recursive relation for the coefficients of the vectorial harmonic functions in the 2nd and 3rd layers respectively, i.e Eq. (23) can be easily extended to the case of n-layer spheres (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

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

].

This work is supported by the Absorption and Innovation Projects of Introduced Technology of Shanghai (No. 2010CH-007), and the exchange-scholar program between University of Konstanz and Fudan University. Furthermore, We acknowledge fruitful discussions with Georg Maret, Christof Aegerter and Wolfgang Bührer.

References and links

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]

3.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.

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]

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]

7.

R. Lenke and G. Maret, “Multiple scattering of light: Coherent backscattering and transmission,” in Scattering in Polymeric and Colloidal Systems, W. Brown and K. Mortensen, eds. (Gordon and Breach Science Publishers, 2000).

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]

9.

A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

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]

12.

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

13.

H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc. New York, 1981).

14.

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

15.

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

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]

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]

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 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]

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]

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]

25.

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

26.

W. J. Wiscombe, “Mie scattering calculations: Advances in technique and fast, vector-speed computer codes,” Tech. rep., NCAR Technical Note NCAR/TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307) (1979).

27.

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

28.

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

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]

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]

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]

32.

B. A. van Tiggelen and A. Lagendijk, “Rigorous Treatment of the Speed of Diffusing Classical Waves,” Europhys. Lett. 23, 311–316 (1993). [CrossRef]

33.

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

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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References

  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]
  3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, Heidelberger, 2006), 2nd ed.
  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]
  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]
  7. R. Lenke, and G. Maret, “Multiple scattering of light: Coherent backscattering and transmission,” in Scattering in Polymeric and Colloidal Systems, W. Brown and K. Mortensen, eds. (Gordon and Breach Science Publishers, 2000).
  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]
  9. A. Ioffe, and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).
  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]
  12. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc. New York, 1981).
  14. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (JohnWiley & Sons, Inc, 1998). [CrossRef]
  15. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]
  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 Stat. Nonlin. Soft Matter Phys. 73, 065602 (2006). [CrossRef]
  18. R. Sapienza, P. D. Garc’ıa, 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]
  19. R. Tweer, “Vielfachstreuung von licht in systemen dicht gepackter mie-streuer: Auf dem weg zur andersonlokalisierung?” Ph.D. thesis (2002).
  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]
  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]
  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]
  25. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. 13, 302–313 (1969). [CrossRef]
  26. W. J. Wiscombe, “Mie scattering calculations: Advances in technique and fast, vector-speed computer codes,” Tech. rep., NCAR Technical Note NCAR/TN-140 + STR (National Center for Atmospheric Research, Boulder, Colo. 80307) (1979).
  27. Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. 26(6), 1393 (1991). [CrossRef]
  28. H. Du, “Mie-scattering calculation,” Appl. Opt. 43, 1951–1956 (2004). [CrossRef] [PubMed]
  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]
  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]
  31. C. Pecharroman, “T. G. C. no, and 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]
  32. B. A. van Tiggelen, and A. Lagendijk, “Rigorous Treatment of the Speed of Diffusing Classical Waves,” Europhys. Lett. 23, 311–316 (1993). [CrossRef]
  33. W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt. 42(9), 1710–1720 (2003). [CrossRef] [PubMed]

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