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

Hao Zhang; Heyuan Zhu; Min Xu

doi:10.1364/OE.19.002928

1. Introduction

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 = v_el^*/3, where v_e 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 D_B, in the classical diffusion theory in which wave interference is ignored [7].

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^* [6].

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 [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], which means that the renormalized D becomes zero at finite length scales [4].

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 [3], as well as an increase of dwell time [15–18]. The transport velocity of electromagnetic energy, i.e v_e, thus decreases due to the increased dwell time [16,17]. Therefore, when the Mie resonance occurs, both l^* and v_e 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], and within highly monodispersed polystyrene samples [18], and both have revealed the reduction of v_e 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 v_e 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]. 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 (ɛ_m = 2.56) can be written as [24],

ɛ_{c o r e} = ɛ_{m} (1 + \frac{3 η β (ω)}{1 - η β (ω)})

where β(ω) = (ɛ_QD – ɛ_m)/(ɛ_QD + 2ɛ_m), and ɛ_QD is the effective dielectric constant of the QDs which can be described by the single-resonance Lorentz model [23, 24],

ɛ_{Q D} = 1 + \frac{ω_{p}^{2}}{ω_{0}^{2} - ω^{2} - i γ ω}

where ω₀, ω_p and γ represent the resonance frequency, the oscillator strength and the damping constant, respectively. We use the parameters provided in Ref. [24], i.e, ω₀r_mant/2πc = 0.245, ω_p = 0.8ω₀ and γ = 0.01ω₀, to calculate the effective ɛ_core 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 ɛ_core decreases when the concentration of QDs decreases. Hereof we only consider the lossless cases by neglecting the imaginary part of ɛ_core, 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.

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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 [20,21]. 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 R_c = 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, Ψ_l represents the vectorial field in l^th layer, and Ψ_i represents the vectorial incident field. The dashed volume in (b) indicates the averaged counterpart of the coated spheres.

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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

\int_{0}^{R_{c}} d^{3} r ρ_{E}^{(1)} (\vec{r}) = \int_{0}^{R_{c}} d^{3} r ρ_{E}^{(2)} (\vec{r})

where

ρ_{E}^{(1)}

and

ρ_{E}^{(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,

ρ_{E} (\vec{r}) = \frac{1}{2} [ɛ (\vec{r}) {| \vec{E} (\vec{r}) |}^{2} + μ {| \vec{H} (\vec{r}) |}^{2}]

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 c_n and d_n 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 [29]. 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 C_sca 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 Q_sca = C_sca/πr², where r is the radius of the real sphere. We use a standard Mie algorithm for a coated sphere described in Ref. [14] to calculate Q_sca 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 ɛ_core show the anomalous behavior of a pole resonance. When the concentration increases from η = 0 to η = 0.01, the calculated Q_sca 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 ɛ_core 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.

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

\bar{ɛ} = \frac{(1 - f) ɛ_{3} + f S_{1}}{1 - f + f S_{2}}

where the parameters S₁ and S₂ are defined as follows,

S_{1} = 3 ɛ_{3} ɛ_{2} (ɛ_{1} (1 + 2 ξ) + 2 ɛ_{2} (1 - ξ)) / Θ

S_{2} = 3 ɛ_{3} (ɛ_{1} (1 - ξ) + ɛ_{2} (2 + ξ)) / Θ

Θ = (ɛ_{1} + 2 ɛ_{2}) (ɛ_{2} + 2 ɛ_{3}) + 2 ξ (ɛ_{2} - ɛ_{3}) (ɛ_{1} - ɛ_{2})

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 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.

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

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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. [16], van Albada et. al. proposed a simplified treatement for the Bethe-Salpeter equation to obtain the expression of v_e 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. [17], Störzer et. al. obtained v_e, according to the Boltzmann expression of D = v_el^*/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 v_e is in good agreement with that obtained by the coated CPA, i.e v_e = c/n̄.

Strictly speaking, the rigorous definition of v_e, i.e. v_e ≡ 〈S_e〉/〈ρ_e〉 [32], is related to the average intensity Green’s function, 〈GG^*〉 [4], 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 R_c and ɛ_air, therefore it gives better results of v_e for strongly scattering media. However, the calculation of v_e can be simplified in the lowest order according to Ref. [16].

We use the expression of 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%.

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 ɛ_core by the concentration η does not result in obvious difference on Q_sca. 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 Q_sca 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.

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Figure 5(a) shows that a resonant scattering occurs at r₂ = 0.262λ, while the corresponding v_e, 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 Q_sca 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], hence a reduced v_e. 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].

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.

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If the disordered medium is sufficiently dense, however, the resonant scattering-based valley on the v_e 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. [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 v_e subsequently.

In addition, with the increase of f, the renormalized D as well as v_e 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 R_c. Therefore, the effective indices calculated by a standard CPA method are larger than those obtained by the coated CPA method [20], since the former one only considers single scatterings while neglects the correlation effects [3], and the corresponding scattering unit does not include the coated layer, or R_c = r₂. The larger values of c/v_e 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/v_e as that of the coated CPA method. Since in this situation their scattering configurations are essentially identical when R_c → ∞, 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 Q_sca for single scattering is sensitive to ɛ_core hence to the concentration η. The pole-resonance of Q_sca is different from the Mie resonances, and it corresponds to the behavior of ɛ_core 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], 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].

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 (j_n) and second (y_n) kinds to expand (E⃗_l, H⃗_l) in terms of vectorial harmonics [13, 14]. For the core layer, however, y_n will be infinite at the center, therefore we only use j_n as the expansion base. Suppose the incident field is a plane wave, E⃗_i = E⃗₀exp(ikz). The electromagnetic field in the core layer can be expanded as,

{\vec{E}}_{1} = \sum_{n = 1}^{\infty} E_{n} (c_{n} {\vec{M}}_{o 1 n}^{(1)} - i d_{n} {\vec{N}}_{e 1 n}^{(1)})

{\vec{H}}_{1} = - \frac{k_{1}}{ω μ_{1}} \sum_{n = 1}^{\infty} E_{n} (d_{n} {\vec{M}}_{e 1 n}^{(1)} + i c_{n} {\vec{N}}_{o 1 n}^{(1)})

where

E_{n} = E_{0} i^{n} \frac{2 n + 1}{n (n + 1)}

k₁ = n₁k₀, and k⃗₀(k₀ = 2π/λ) is the wave vector in air.

{\vec{M}}_{o 1 n}^{(i)}

and

{\vec{N}}_{o 1 n}^{(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 j_n and y_n are finite, therefore the vectorial harmonics of the fields should include both of them,

{\vec{E}}_{l} = \sum_{n = 1}^{\infty} E_{n} (f_{n}^{(l)} {\vec{M}}_{o 1 n}^{(1)} - i g_{n}^{(l)} {\vec{N}}_{e 1 n}^{(1)} + v_{n}^{(l)} {\vec{M}}_{o 1 n}^{(2)} - i w_{n}^{(l)} {\vec{N}}_{e 1 n}^{(2)})

{\vec{H}}_{l} = - \frac{k_{l}}{ω μ_{l}} \sum_{n = 1}^{\infty} E_{n} (g_{n}^{(l)} {\vec{M}}_{e 1 n}^{(1)} + i f_{n}^{(l)} {\vec{N}}_{o 1 n}^{(1)} + w_{n}^{(l)} {\vec{M}}_{e 1 n}^{(2)} + i v_{n}^{(l)} {\vec{N}}_{o 1 n}^{(2)})

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 $ρ_{E}^{(1)}$ 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)} = E_{1}^{(1)} + E_{2}^{(1)} + E_{3}^{(1)}

Then, considering the expressions of the electromagnetic fields, i.e Eqs. (9,10) and the orthogonality properties of the vectorial harmonic functions [14], we have the expression for the electromagnetic energy within the core layer as follows [22],

E_{1}^{(1)} = \frac{1}{2} \cdot \frac{ω^{2}}{c^{2}} ɛ_{1} \sum_{n = 1}^{\infty} ({| c_{n} |}^{2} + {| d_{n} |}^{2}) \frac{1}{k_{1}^{3}} \int_{0}^{k_{1} r_{1}} ρ^{2} d ρ W_{n} (j_{n}, j_{n}; ρ)

where

k_{i} = \sqrt{ɛ_{i}} ω / c

, and i = 1,2, 3. The function W_n is defined as follows [22],

W_{n} (z_{n}, \bar{z_{n}}; r) = (2 n + 1) z_{n} (r) \bar{z_{n}} (r) + (n + 1) z_{n - 1} (r) \bar{z_{n - 1}} (r) + n z_{n + 1} (r) \bar{z_{n + 1}} (r)

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

\begin{array}{r} E_{2}^{(1)} = \frac{1}{2} \cdot \frac{ω^{2}}{c^{2}} ɛ_{2} \sum_{n = 1}^{\infty} \frac{1}{k_{2}^{3}} \int_{k_{2} r_{1}}^{k_{2} r_{2}} ρ^{2} d ρ [({| f_{n}^{(2)} |}^{2} + {| g_{n}^{(2)} |}^{2}) W_{n} (j_{n}, j_{n}; ρ) \\ + ({| v_{n}^{(2)} |}^{2} + {| w_{n}^{(2)} |}^{2}) W_{n} (n_{n}, n_{n}; ρ) + 2 (| f_{n}^{(2)} \cdot v_{n}^{(2)} | + | g_{n}^{(2)} \cdot w_{n}^{(2)} |) W_{n} (j_{n}, n_{n}; ρ)] \end{array}

where the coefficients of the vectorial harmonic functions in the expansions of E⃗₂ and H⃗₂, i.e

f_{n}^{(2)}

,

v_{n}^{(2)}

,

g_{n}^{(2)}

,

w_{n}^{(2)}

can be written as follows [22],

f_{n}^{(2)} = ψ_{n} (m_{1} x) χ_{n} (m_{2} x) [G_{1 n} (m_{1} x) - (m_{2} / m_{1}) G_{2 n} (m_{2} x)] c_{n} \equiv ϕ_{n} (m_{1}, m_{2}; x) c_{n}

v_{n}^{(2)} = ψ_{n} (m_{1} x) ψ_{n} (m_{2} x) [G_{1 n} (m_{1} x) - (m_{2} / m_{1}) G_{1 n} (m_{2} x)] c_{n} \equiv ζ_{n} (m_{1}, m_{2}; x) c_{n}

g_{n}^{(2)} = ψ_{n} (m_{1} x) χ_{n} (m_{2} x) [(m_{2} / m_{1}) G_{1 n} (m_{1} x) - G_{2 n} (m_{2} x)] d_{n} \equiv γ_{n} (m_{1}, m_{2}; x) d_{n}

w_{n}^{(2)} = ψ_{n} (m_{1} x) ψ_{n} (m_{2} x) [(m_{2} / m_{1}) G_{1 n} (m_{1} x) - G_{1 n} (m_{2} x)] d_{n} \equiv η_{n} (m_{1}, m_{2}; x) d_{n}

where x = k₀r₁, y = k₀r₂, z = k₀R_c, and m_i = n_i/n̄. n_i is the refractive index in the i^th constituent layer of the multilayered spheres and n̄ is the effective refractive index of the disordered media,

\bar{n} = \sqrt{\bar{ɛ}}

. ψ_n and χ_n are the Riccati-Bessel functions, and the ratios of the Riccati-Bessel functions, i.e G_1n, G_2n are defined as follows [14],

G_{1 n} (m_{1} x) = ψ_{n}^{'} (m_{1} x) / ψ_{n} (m_{1} x), G_{2 n} (m_{1} x) = χ_{n}^{'} (m_{1} x) / χ_{n} (m_{1} x)

Similarly, the expression for the electromagnetic energy within the third constituent layer is obtained by using Eqs. (12,13) as well,

\begin{array}{r} E_{3}^{(1)} = \frac{1}{2} \cdot \frac{ω^{2}}{c^{3}} ɛ_{3} \sum_{n = 1}^{\infty} \frac{1}{k_{3}^{3}} \int_{k_{3} r_{2}}^{k_{3} R_{c}} ρ^{2} d ρ [({| f_{n}^{(3)} |}^{2} + {| g_{n}^{(3)} |}^{2}) W_{n} (j_{n}, j_{n}; ρ) \\ + ({| v_{n}^{(3)} |}^{2} + {| w_{n}^{(3)} |}^{2}) W_{n} (n_{n}, n_{n}; ρ) + 2 (| f_{n}^{(3)} \cdot v_{n}^{(3)} | + | g_{n}^{(3)} \cdot w_{n}^{(3)} |) W_{n} (j_{n}, n_{n}; ρ)] \end{array}

where ɛ₃ in the coated CPA theory is the dielectric constant of air. The coefficients of the vectorial harmonic functions in the expansions of E⃗₃ and H⃗₃ within the third constituent layer, i.e

f_{n}^{(3)}

,

v_{n}^{(3)}

,

g_{n}^{(3)}

,

w_{n}^{(3)}

are related to

f_{n}^{(2)}

,

v_{n}^{(2)}

,

g_{n}^{(2)}

,

w_{n}^{(2)}

by the following matrix equation,

(\begin{matrix} f_{n}^{(3)} \\ v_{n}^{(3)} \\ g_{n}^{(3)} \\ w_{n}^{(3)} \end{matrix}) = (\begin{matrix} ϕ_{n} (m_{2}, m_{3}; y) & ς_{n} (m_{2}, m_{3}; y) & 0 & 0 \\ ζ_{n} (m_{2}, m_{3}; y) & ϑ_{n} (m_{2}, m_{3}; y) & 0 & 0 \\ 0 & 0 & γ_{n} (m_{2}, m_{3}; y) & ϖ_{n} (m_{2}, m_{3}; y) \\ 0 & 0 & η_{n} (m_{2}, m_{3}; y) & ρ_{n} (m_{2}, m_{3}; y) \end{matrix}) (\begin{matrix} f_{n}^{(2)} \\ v_{n}^{(2)} \\ g_{n}^{(2)} \\ w_{n}^{(2)} \end{matrix})

where the functions of ς, ϑ, ϖ, ρ are defined as follows,

ς_{n} (m_{2}, m_{3}; y) \equiv χ_{n} (m_{2} y) χ_{n} (m_{3} y) [(m_{3} / m_{2}) G_{2 n} (m_{3} y) - G_{2 n} (m_{2} y)]

ϑ_{n} (m_{2}, m_{3}; y) \equiv χ_{n} (m_{2} y) ψ_{n} (m_{3} y) [(m_{3} / m_{2}) G_{1 n} (m_{3} y) - G_{2 n} (m_{2} y)]

ϖ_{n} (m_{2}, m_{3}; y) \equiv χ_{n} (m_{2} y) χ_{n} (m_{3} y) [G_{2 n} (m_{3} y) - (m_{3} / m_{2}) G_{2 n} (m_{2} y)]

ρ_{n} (m_{2}, m_{3}; y) \equiv χ_{n} (m_{2} y) ψ_{n} (m_{3} y) [G_{1 n} (m_{3} y) - (m_{3} / m_{2}) G_{2 n} (m_{2} y)])

Therefore, by using the expressions of $E_{n}^{(1)}$ and the expressions for the amplitudes of the vectorial harmonic functions within the core layer, i.e c_n and d_n, which can be derived by a typical Mie theory considering the boundary conditions [14], 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 2^nd and 3^rd 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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Transport properties of light in a disordered medium composed of two-layered dispersive spheres

Abstract

1. Introduction

2. Configuration of the multilayered spheres with dispersive inclusions

3. Theory

4. Numerical results and discussions

4.1. Scattering cross-section efficiencies of single scattering

4.2. Effective refractive index in the long-wavelength limit

4.3. Transport velocities of electromagnetic energy

5. Summary

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

References and links

Cited By

Figures (6)

Equations (28)

Optics Express