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
The diffusive propagation of light in disordered media has attracted much attention since the theoretical predictions of strong localization of light [1, 2]. 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, 4].
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, 6], which is regarded as a constant, termed as Boltzmann value DB, in the classical diffusion theory in which wave interference is ignored .
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, 8], to characterize the turbidity [3, 9]. In experiments, the value of kl* gives the corresponding value of l* .
If kl* is much larger than unity, meaning low turbidity, constructive interference on counterpropagating paths of multiple scattering leads to enhanced backscattering [10, 11], 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, 5]. At even higher turbidity, especially when kl* approaches unity, strongly scatterings occur . 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 , which means that the renormalized D becomes zero at finite length scales .
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 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]. In the Mie-scattering regime, the scattering cross section manifests itself as several resonant points superimposed on a slowly-varying profile [13, 14], 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 , as well as an increase of dwell time [15–18]. The transport velocity of electromagnetic energy, i.e ve, thus decreases due to the increased dwell time [16,17]. 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 , and within highly monodispersed polystyrene samples , 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, 19]. 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 [20–22], 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 . 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 ,23, 24], 24], 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.
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 [20,21]. 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.
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 haveFig. 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,
In this case we only deal with non-magnetic materials, so μ/μ0 = 1. The self-consistent Eq. (3) can be solved by an iterative procedure [20, 21]. The related formulae of the coated CPA method for disordered media consisting of two-layered spheres are given in Appendix A.
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 [25–28], 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 . 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.  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.
It has been shown that the efficiency factors for scattering Q′scas 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 , and the positions of the resonant peaks are sensitive to the morphology of the spheres . For the spheres considered herein, the curves of Q′scas 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 λ ≫ r2, 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, 31],
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 λ ≫ r2, 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.
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, 20], we believe that the results obtained from the coated CPA method are faithful.
4.3. Transport velocities of electromagnetic energy
In Ref. , 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, 18]. In Ref. , 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.
Strictly speaking, the rigorous definition of ve, i.e. ve ≡ 〈Se〉/〈ρe〉 , is related to the average intensity Green’s function, 〈GG*〉 , and therefore it should be obtained in the context of a Bethe-Salpeter equation for full-vector waves, and the correlation effects in multiple scatterings are thus taken into account in strongly scattering media. As mentioned above, the coated CPA theory involves both effects of single scattering and correlation by considering the energy distribution of the spheres coated by a layer with Rc and ɛair, therefore it gives better results of ve for strongly scattering media. However, the calculation of ve can be simplified in the lowest order according to Ref. .
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.
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 , 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. .
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 〈T̄〉c [see Eq. (3.57) in Ref. ], 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, n̄ 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 , since the former one only considers single scatterings while neglects the correlation effects , 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. , 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.
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
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, 14]. 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,
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,
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 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,
Then, considering the expressions of the electromagnetic fields, i.e Eqs. (9,10) and the orthogonality properties of the vectorial harmonic functions , we have the expression for the electromagnetic energy within the core layer as follows ,22], 22], 14],
Therefore, by using the expressions of 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 , 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, 33].
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.
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