- Email: [email protected]
Effective medium theory for two-dimensional random media composed of core–shell cylinders
Optics Communications 306 (2013) 9–16
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Effective medium theory for two-dimensional random media composed of core–shell cylinders$ Hao Zhang a,b,n, Yongqiang Shen a, Yuchen Xu a, Heyuan Zhu a, Ming Lei b, Xiangchao Zhang a, Min Xu a a Shanghai Engineering Center for Ultra-Precision Optical Manufacturing, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China b State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi'an 710119, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 March 2013 Received in revised form 6 May 2013 Accepted 7 May 2013 Available online 29 May 2013
In this paper, based on the generalized coated coherent potential approximation method, we derive the mathematical formulae, for the extended effective medium theory, to investigate the optical properties of disordered media composed of core–shell cylinders. The effective indices of such media are obtained in the long-wavelength limit and in the Mie-scattering region. Moreover, we use this method to study optical properties of random media composed of core–shell cylinders with the core layer consisting of epsilon-less-than-one material. & 2013 Published by Elsevier B.V.
Keywords: Effective medium theory Mie scattering Diffusion
1. Introduction The propagation and scattering of electromagnetic waves has been a topic extensively investigated by both the physics and engineering communities for decades. It is now common knowledge that the properties of responses of matter to incident electromagnetic waves in the process of interactions depend greatly on the relative sizes of the composites of the matter to the wavelength of the electromagnetic waves, i.e. r=λ. For relatively small composites, i.e. r=λ 5 1, the electromagnetic responses of the composites can be regarded as electric dipoles microscopically, and the averaged collective responses of the microscopic dipoles, also known as the coherent field, lead to an overall dielectric response of the matter. For this case, the diffuse field is much smaller compared to the coherent field, and thus neglected. Therefore, the dielectric responses are the overall electromagnetic responses of the matter. Such theories are also called as the effective medium theories (EMT) [1]. Several EMT models have
☆ This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. n Corresponding author at: Shanghai Engineering Center for Ultra-Precision Optical Manufacturing, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China. Tel.: +86 2155665372. E-mail address: [email protected] (H. Zhang).
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.05.027
been proposed, e.g. the Maxwell-Garnett theory [2], the Bruggerman model [3], the asymptotic multiscale theory [4], and etc. [5]. However, when the wavelength is comparable to the size of the composites, i.e. r=λ≥1, the microstructure of the composites involving the dielectric constants of the constituent materials, boundaries and shapes, will influence the microscopic electromagnetic responses significantly, and the electric dipole model breaks down as a consequence. Mie has proposed a rigorous theory to describe the scattering properties of light incident on a sphere with its size comparable to λ [6,7]. According to the Mie-scattering theory, resonant modes will be formed within the sphere when nx≈l þ 1=2, in which n is the refractive index of the sphere, x ¼ 2πr=λ and l is an integer. As the Mie resonances occur, the scattering cross sections of the sphere reach peak values which means that local effects will dominate within the medium composed of Mie scatterers. Moreover, for dense medium composed of Mie spheres, the multiple scatterings taking place among clusters formed by neighboring spheres are dominant which means that the local diffuse field is not negligible compared to the coherent field. The inhomogeneity that caused by the nontrivial diffuse field makes the conventional EMT break down. There has been effort based on the scattering approach to extend the EMT to study random media composed of large scatterers, and such theories are known as the extended effective medium theories (EEMTs) [8,9]. The effective field approximation (EFA) is one of the EEMTs, which assumes that the response field at any of the scatterers is homogeneous after ensemble-averaging over different configurations [10], and the local variations of the
10
H. Zhang et al. / Optics Communications 306 (2013) 9–16
refractive index are subsequently averaged out. Busch et al. has proposed an EEMT based on the framework of the EFA to calculate the effective index neff of disordered media composed of large spheres, and some derived physical entities such as the effective propagation constant q ¼ neff ω=c, the mean free path l¼ 0.5 Im(q), and the renormalized wave vector k¼ Re(q) [11]. The method they proposed, also called as the energy-density coherent potential approximation method, leads to the calculated results which show reasonable agreement to experiments [12,13]. Recently, random medium composed of single-layer or core– shell cylinders has attracted much interest due to their potential applications in the fields of radar and metamaterials [14], and among them, the random systems composed of core–shell cylinders is the major concern since some novel effects can be realized in this structure [15]. Although much effort has been devoted on the investigation of dielectric responses of such random matter [8,16,17], to our knowledge, the optical properties of random medium composed of dielectric core–shell cylinders of infinite length still lack detailed investigations. In the present work, based on the generalized coherent potential approximation method Busch et al. proposed, we derive the mathematical formulation of the EEMT for the investigations of the optical properties of random media composed of core–shell dielectric cylinders of infinite length. In such two-dimensional random systems, the optical properties, such as the effective indices in the long-wavelength limit, where the conventional EMTs are valid, and in the Mie-scattering region, where the nonlocal effects are nontrivial, are under considerations. We also conduct this method to study the optical properties of two-dimensional random systems composed of core–shell cylinders with the core layer constructed by epsilon-less-than-one materials.
the electric polarization of the random medium ! ! ∑pi 〈P 〉¼ ΔV
ð1Þ
! ! ! 〈 D 〉 ¼ ϵ0 〈 E 〉 þ 〈 P 〉
ð2Þ
! According to the relation between electric displacement 〈 D 〉 ! and electric field E , i.e.
the effective dielectric constant to describe the macroscopic effects of the electromagnetic responses of the inclusions can be defined via ! ! 〈 D 〉 ¼ ϵeff 〈 E 〉
Moreover, the average electric displacement and electric field can be given by the weighted average determined by the corresponding values distributed within the respective layers h ! ! ! i ! 〈 D 〉 ¼ f ηϵ1 〈 E1 〉 þ ð1−ηÞϵ2 〈 E2 〉 þ ð1−f Þϵm 〈Ee 〉 ð4Þ h ! ! i ! ! 〈 E 〉 ¼ f η〈 E1 〉 þ ð1−ηÞ〈 E2 〉 þ ð1−f Þ〈Ee 〉
2.1. Maxwell-Garnett theory Consider a random medium composed of core–shell dielectric cylinders of infinite length, shown in Fig. 1, embedded in a background medium with a dielectric constant of ϵm . The volume fraction occupied by the inclusions is f and the structure of the core–shell cylinders is shown in the inset of Fig. 1. As known, when the incident wavelength λ is much larger than the size of the inclusions, the electromagnetic response of the ! inclusion can be regarded as a point dipole pi , and the summation over the total responses of the inclusions within a unit space gives
r2 r1
ε1
ε2
ð5Þ
where f is the volume fraction occupied by core–shell cylinders, ! ! ! η ¼ r 21 =r 22 , 〈 E1 〉; 〈 E2 〉 and 〈Ee 〉 are the electric field distributed within the core-, shell-layer of the cylinder, and the background medium, respectively. According to Eqs. (3)–(5) and the formulae for the electric fields Ei (i¼1,2) within the core–shell cylinders, generally, the formulae for the effective dielectric constant ϵeff for a random medium composed of core–shell cylinders of infinite length is approximately given by ϵeff ¼
2. Theory
ð3Þ
ϵm þ f ½ηϵ1 α1 þ ð1−ηÞϵ2 α2 −ϵm 1 þ f ½ηα1 þ ð1−ηÞα2 −1
ð6Þ
where the parameters α1 and α2 are defined as the ratios between ! ! 〈Ei 〉=〈Ee 〉 ði ¼ 1; 2Þ. Eq. (6) is also known as the Maxwell-Garnett formula. Considering the boundary conditions, the electric fields within the multilayered cylinder for S-mode waves are contin! ! ! uous, i.e. 〈 E1 〉 ¼ 〈 E2 〉 ¼ 〈Ee 〉, therefore, α1 ¼ α2 ¼ 1. Thus, according to Eq. (6), the effective dielectric constant ϵeff for S-mode waves can be written directly by ϵeff ¼ ð1−f Þϵm þ f ½ηϵ1 þ ð1−ηÞϵ2
ð7Þ
However, the case for P-mode waves is more complicated, and herein, the field distribution of the electric fields within the core– shell cylinders for P mode is derived in Appendix A.1. According to Eqs. (25) and (26), α1 and α2 are respectively given by α1 ¼
4ϵm ϵ2 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð8Þ
α2 ¼
2ϵm ðϵ1 þ ϵ2 Þ ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð9Þ
So the final solution of ϵeff for P-mode waves can be obtained subsequently. 2.2. Coated coherent potential approximation method
εm
Fig. 1. Schematic view of a random medium composed of core–shell cylinders of infinite length. The positions of the cylinders are random. The inset is the core–shell dielectric cylinders embedded in the background with a dielectric constant of ϵm .
The coated coherent potential approximation (CCPA) method is conducted as well to calculate the effective index of the random system composed of core–shell dielectric cylinders of infinite length. The mechanism of the CCPA method is based on the physical idea that the distribution of the electromagnetic energy within a random medium should be homogeneous after being configurationally averaged over the correlation length of the random media [10,18]. Therefore, the averaged forward-scattering amplitude of a cylindrical region within the random medium should be approximately
H. Zhang et al. / Optics Communications 306 (2013) 9–16
method can be used to calculate the effective dielectric constant of pffiffiffiffiffiffiffi the random media. The calculated results of neff ðneff ¼ ϵeff Þ for random media composed of core–shell dielectric cylinders are illustrated in Fig. 3. The sizes of the core- and shell-layer are changing by an interval of 100 nm, and the medium is changing from dilute one to dense one. As a comparison, the Maxwell-Garnett method, described as Eq. (6), is conducted as well to calculate neff for random media composed of corresponding sizes of the core–shell cylinders, which are shown as the red lines in Fig. 3. By comparisons, it is obvious that the results obtained by the CCPA method are in good consistency to those obtained by the Maxwell-Garnett method, which is in accordance with the numerical simulations for random media composed of core–shell spheres we have shown in Ref. [18].
equal to zero. In addition, in order to take into account the effects of the structural factor for the multiple scatterings occurring within the random medium, a coated layer is involved to the real cylinder, which is embedded in the effective medium as an effective scattering unit. The radius of the coated layer is given by rc ¼r2 f−1/2. The schematic view of the CCPA method is shown in Fig. 2. It has been shown by Busch et al. that the effective medium obtained by the CCPA method for random media composed of dielectric spheres are in reasonable agreement with experiments [10,11]. According to the CCPA theory which gives that the coated scattering unit illuminated by a plane wave has a scattering pattern similar to that of its averaged counterpart, thus, the electromagnetic energy contained in the coated cylinder in Fig. 2 (b) should be equal to that contained in the dashed volume in Fig. 2(c), which subsequently gives the main equation for the CCPA method as follows: Z rc Z rc ! ! 3! 3! d r ρð1Þ ð r Þ ¼ d r ρð2Þ ð10Þ E E ðr Þ 0
11
3.2. In the Mie-scattering region However, when the wavelength is comparable to the size of the dielectric inclusions, i.e. kr≥1, the Mie scatterings take place when the incident waves impinge on the inclusions, and the effective medium theory for the case of long-wavelength limit, e.g. the Maxwell-Garnett method, breaks down, since in such a region, i.e. the Mie-scattering region, the electromagnetic response of a dielectric inclusion to the incident light is local and the averagings of the electric displacement and the electric field are strongly affected by the distribution of the scatterers, which means that the random medium cannot be regarded as a uniform effective medium any longer.
0
ð2Þ where ρð1Þ E and ρE represent the electromagnetic energy density within the three-layered cylinder in Fig. 2(b) and that of the averaged volume with an effective ϵeff in Fig. 2(c), respectively. In addition, as known, the energy density for an electromagnetic vectorial field is given by ! ! ! ! ! ! ! ρE ð r Þ ¼ 12½ϵð r Þ E ð r Þj2 þ μð r Þ H ð r Þj2 ð11Þ
here we only deal with non-magnetic materials, so μ=μ0 ¼ 1. The self-consistent equation, i.e. Eq. (10), can be solved by an iterative procedure. The related formulae of the CCPA method for a random medium composed of core–shell dielectric cylinders of infinite length are derived in Appendix A.2.
2.2 r =100nm, CCPA 1
r =200nm, CCPA 1
r1=300nm, CCPA
2
Effective refractive indices [neff]
3. Results and discussions The random media composed of core–shell cylinders, whose constituent materials are SiO2 (n1 ¼1.44), ZnO (n2 ¼ 2.7) or epsilonless-than-one materials, are under investigations herein, and for the sake of clarity, only the S-polarized modes are under considerations. 3.1. In the long-wavelength limit In the long-wavelength limit, i.e. λ 5 r 2 , the average electromagnetic responses of the overall core–shell inclusions are approximately homogeneous within the random media [5]. Under such conditions, the product of the wavevector and the mean free path, i.e. kl which is also the Ioffe–Regel criterion for wave localization [19], has the value much larger than one, therefore the scattering events taking place within the random medium can be regarded as single-scattering events, which means that the formal CPA condition is valid [20], and the above-mentioned CCPA
r =400nm, CCPA 1
r1=500nm, CCPA r =100nm, MG
1.8
1
r =200nm, MG 1
r =300nm, MG 1
1.6
r =400nm, MG 1
r1=500nm, MG
1.4
1.2
1
0
0.1
0.3
0.4
0.5
0.6
Volume fraction [f] Fig. 3. Effective refractive indices for random media composed of infinite core– shell (SiO2/ZnO) cylinders. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
c m 2
0.2
c 2
1 1
m
eff
eff
Fig. 2. Schematic view of the CCPA method for random media composed of core–shell dielectric cylinders, illustrated in (a). The coated layer to the actual core–shell cylinders in (b) has the size of rc and the dielectric constant equal to ϵm . (c) The dashed region indicates the effective scattering unit described in the CCPA method.
12
H. Zhang et al. / Optics Communications 306 (2013) 9–16
Moreover, the influences from the neighbor inclusions when the Mie scattering takes place are not trivial when the volume fraction f is large enough, and the multiple scatterings occur for densely random media composed of core–shell dielectric cylinders. In such situations, the contribution to the overall scatterings can be divided into the form factor, which is attributed to single scatterings, and the structural factor, which is due to the correlation effects caused by multiple scatterings [21]. However, it has been reported by experiments that in the Mie-scattering region, the configurationally averaged velocities of the electromagnetic energy is in good qualitative agreement between the experiments and theoretical calculations using the CCPA method, which is attributed to the coated layer involving the correlation effects [12,22]. Therefore, even for dense media in which multiple-scattering events are not trivial, the CCPA method can be performed to obtain the effective refractive index neff, 5 f=5%
f=20%
f=40%
f=60%
Effective refractive indices [neff]
4.5 4
which can be used to describe the averaged electromagnetic responses. The calculated effective refractive index neff for random media composed of core–shell dielectric cylinders with different sizes and volume fractions is shown in Fig. 4(a). The resonant points of the neff curves when varying the size of the cylinder, i.e. the spikes in Fig. 4(a), are in coincidence with the resonant peaks in the calculated scattering cross-section curve in Fig. 4(b) for a corresponding core–shell dielectric cylinder with an identical structure. Such a coincidence can be understood by the fact that the resonant modes formed within the core–shell cylinders lead to larger values of the average electric displacement and neff as well, since the effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi refractive index neff which can be roughly obtained by 〈D〉=〈E〉, in which the distributed electric fields can be regarded as weighted factors of the dielectric constants in the formulae calculating neff, i.e. Eqs. (3)–(5). Thus, if resonant modes are formed within the core–shell cylinders, indicated by the peaks in Fig. 4(b), the contribution from the area with higher dielectric constants ðϵ1 and ϵ2 Þ to the neff would dominate, which leads to the peak values of neff, also indicated by the spikes in Fig. 4(a). In addition, the shift to a larger value of the effective dielectric constant when increasing the volume fraction, as shown in Fig. 4 (a), can be understood in a similar way. 3.3. For constituent material with ϵ-less-than-one
3.5 3 2.5 2 1.5
Qs
1 5
0 0.5
0.7
0.9
1.1
1.3
1.5
r2/λ
4.5
5
4
4.5
Scattering efficiency factor Qsca
Scattering efficiency factor Qsca
Fig. 4. (a) Effective refractive indices for random media composed of infinite core– shell (SiO2/ZnO) cylinders with different volume fractions. r1 ¼ 300 nm and r2 ¼600 nm. (b) Efficiency of the scattering cross-section Qs for the core–shell cylinders described in (a).
Scattering of multilayered composites has been attracted much interest due to the realization of transparency [23,24]. The constituent materials of multilayered composites can be anisotropic, birefringent materials or metamaterials. Epsilon-less-than-one material (ELTO material) is a class of metamaterials that propagate waves through small phase change and possess unusual optical properties that can be used to create novel optical elements, such as subwavelength waveguides [25], perfect bending waveguides [26], cloaking [27], etc. Generally, such materials can be found naturally at infrared or optical frequencies, and recently it has been shown that the ELTO material for ϵ near zero was properly synthesized by metastructures [15]. According to the Mie-scattering formulation [6], thep behavior ffiffiffiffiffiffiffiffiffiffiffiffiffi of the vector harmonics depends on variables of mi ðmi ¼ ϵi =ϵm Þ and ri, and in the core layer, only the first kind of the vector harmonics is available due to the divergence of the second kind of spherical Bessel function yn at the origin. For mi close to zero, similar divergent behavior of the second vector harmonics occurs even
3.5 3 2.5 2 1.5 1 0.5 0.1
P4
P1 P2
4
P3
3.5
P5
3 2.5 2 1.5 1
0.2
0.3
0.4
0.5
r1/r2
0.6
0.7
0.8
0.9
0.5
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Wavelength [μm]
Fig. 5. Calculated scattering efficiency factors Qsca for S-polarized incidence on core–shell cylinders containing epsilon-less-than-one material. The refractive indices of the background medium, n0 ¼1.0, and of the shell layer, n2 ¼ 3.0: (a) the incident wavelength is λ ¼ 2:5 μm and the radius of the shell layer is r 2 ¼ 2:0 μm and (b) r 1 ¼ 1:05 μm and r 2 ¼ 2:0 μm.
H. Zhang et al. / Optics Communications 306 (2013) 9–16
for the shell layers, therefore, special care should be taken when performing the standard Mie theory for multilayered cylinders containing ELTO material. In Fig. 5, the scattering efficiency factors for S-polarized waves normal incident on a core–shell cylinder of infinite length containing ELTO material are calculated. It is obvious that more resonant modes are formed within the core–shell cylinders when introducing the core layer, and the curves of Qsca for core layer with n ¼0.9, 0.5 and n ¼ 1.5 are quite similar in shape, though the positions of resonant points in the curve for n ¼1.5 are overall shifted. The wavelength-dependent scattering efficiency factors for core–shell cylinders with a fixed space structure is shown in Fig. 5(b). The results are certainly surprising at first sight, since the positions of many resonant points for cases of core layers with different values of n1 are nearly coincident, which are indicated by the arrows in Fig. 5(b). They can be understood, if one proves that the distribution of resonant modes locates dominantly within the
2.5
13
shell layer. Thus, it is the refractive index and shape of the shell layer that induce resonant modes, and the influence from moderate variations of n1 is negligible. The fractions of electromagnetic energy contained within the shell layer, i.e. E2, compared to the total energy contained within the core–shell cylinders, i.e. E1+E2, are calculated using a standard Mie-scattering algorithm, and the dependence on a continuous variation of r1/r2 is investigated. The results are shown in Fig. 6(d)–(f), and for comparison, Qscas for the corresponding core–shell cylinders are obtained as well, as shown in Fig. 6(a)–(c). When the refractive index of the core layer decreases to values less than one, e.g. n1 ¼ 0.5 in Fig. 6(f), the possibility to find the energy dominantly located within the shell layer increases, which can be understood partly by the fact that smaller ϵ leads to smaller values of energy, since as mentioned above, the energy density ρE ∝ϵjEj2 . As for the resonant points indicated by the arrows in Fig. 5(b), it is convinced in Figs. 6(d)–(f) that for r1/r2 ¼ 0.5, the resonant modes formed within the core–shell cylinders, which are
2.5
4.5
0.9
3.5 2
3 2.5 2
1.5 1.5
0.8
Wavelength [µm]
Wavelength [µm]
4
2
0.7 0.6 0.5
1.5
0.4 0.3
1
0.2
0.5 1 0.1
0.3
0.5
0.7
1 0.1
0.9
0.3
r1/r2
0.5
0.7
0.9
r1/r2 2.5
2.5
0.9
3
2
2.5 2
1.5
Wavelength [µm]
Wavelength [µm]
3.5
0.8 2
0.7 0.6 0.5
1.5
0.4
1.5
1 0.1
0.3 0.2
1 0.3
0.5
0.7
1 0.1
0.9
0.3
r1/r2
0.5
0.7
0.9
r1/r2 2.5
2.5
0.9
3
2
2.5 2
1.5
Wavelength [µm]
Wavelength [µm]
3.5
0.8 2
0.7 0.6 0.5
1.5
0.4
1.5
0.3 1 0.1
1 0.3
0.5
r1/r2
0.7
0.9
1 0.1
0.3
0.5
0.7
0.9
r1/r2
Fig. 6. Calculated scattering efficiency factor Qsca for core–shell cylinders with the refractive index of the shell layer n2 ¼ 3.0, and of the core layer: (a) n1 ¼ 1.5; (b) n1 ¼ 0.9; and (c) n1 ¼0.5. The radius of the shell layer is r 2 ¼ 2 μm. The relative fractions of energy contained within the shell layer, i.e. E2/(E1+E2) for the corresponding cylinders (a)–(c) are shown in (d)–(f) respectively.
14
H. Zhang et al. / Optics Communications 306 (2013) 9–16
2.1
2.2
2
Effective refractive index neff
Effective refractive index neff
2
1.8
1.6
1.4
1.2
1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
1.1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r1/r2
Volume fraction f
Fig. 7. Calculated effective refractive index neff of S-modes in the long-wavelength limit for random media composed of core–shell cylinders containing epsilon-less-thanone material. The refractive index of the background medium is n0 ¼1.0: (a) r 1 ¼ 1:05 μm and r 2 ¼ 2 μm and (b) the volume fraction of core–shell cylinders is f¼ 40%.
indicated as the resonant lines found in Fig. 6(a)–(c), are dominantly located within the shell layer. The calculated effective refractive index neff of S-polarized modes for random media composed of core–shell cylinders containing ELTO materials in the long-wavelength limit is shown in Fig. 7. The two above-mentioned methods, i.e. the MaxwellGarnett method and the CCPA method, are conducted with the purpose of comparisons, and again a good coincidence can be found.
4. Conclusions We have presented here an effective medium theory for twodimensional random media composed of core–shell cylinders of infinite length based on the generalization of coherent-potential approximation method. As a comparison, the traditional EMT of the Maxwell-Garnett method for such random media is derived, and by numerical calculations, it is clearly proved that these two methods produce nearly the same results in the long-wavelength limit. We also use the developed EMT method to obtain the effective index of such random media in the Mie-scattering region, which plays an important role in the description of transport properties of light propagating within such random media, such as the mean free path and so on. By performing a standard Mie scattering theory, we investigate the optical properties of core– shell cylinders containing ELTO materials, and find that when the core layer is constructed by ELTO material, the resonant modes formed within the core–shell cylinders dominantly locate within the shell layer, and the peak points in the curves of the scattering efficiency factors are therefore nearly coincident. Finally, we use the developed EMT method and the Maxwell-Garnett method to obtain the effective index of random media composed of core– shell cylinders containing epsilon-less-than-one material, and similar agreement is found.
Laboratory of Transient Optics and Photonics under Grant No. SKLST201111.
Appendix A A.1. Solution to the P-mode field distribution within a coated cylinder in a homogeneous electrostatic field The solution to the Laplace equation, i.e. ∇2 ψ ¼ 0, in cylindrical coordinates for a core–shell cylinder with an infinite length along the z-direction in a uniform field which is applied along the r^ direction, can be written as ∞
ψ ¼ ∑ ðAn r n þ Bn r −n Þ cos nθ
ð12Þ
n¼1
here only the p-mode (electric field along the z-axis) is under considerations and the s-mode (magnetic field along the z-axis) can be handled easily by considering the boundary conditions. Considering that the finite value of the potential at r ¼0 and the condition that, when r-∞, ψ ¼ −E0 r cos θ, the mathematical expressions for the potential within the respective layer of the core–shell cylinders, as shown in Fig. 1, are given by ∞
ψ 1 ðr; θÞ ¼ ∑ An r n cos nθ
ð13Þ
n¼1 ∞
ψ 2 ðr; θÞ ¼ ∑ ðBn r n þ C n r −n Þ cos nθ
ð14Þ
n¼1
∞
ψ 3 ðr; θÞ ¼ −E0 r cos θ þ ∑ Dn r −n cos nθ
ð15Þ
n¼1
The continuation of the electric potential and the normal component of the D at the boundaries of r ¼r1 and r ¼r2 gives that ∞
∞
n¼1
n¼1
∑ An r n1 cos nθ ¼ ∑ ðBn r n1 þ C n r −n 1 Þ cos nθ
Acknowledgments This work was supported in part by the National Science and Technology Major Project of the Ministry of Science and Technology of China under Grant no. 2011ZX02402, the National High Technology R&D Program of China (863 Program) under Grant No. 2012AA040406, and the Open Funding provided by State Key
∞
∞
n¼1
n¼1
−n−1 ϵ1 ∑ nAn r n−1 cos nθ ¼ ϵ2 ∑ ðnBn r n−1 Þ cos nθ 1 1 −nC n r 1
∞
∞
n¼1
n¼1
−n ∑ ðBn r n2 þ C n r −n 2 Þ cos nθ ¼ −E0 r 2 cos θ þ ∑ Dn r 2 cos nθ
ð16Þ
ð17Þ
ð18Þ
H. Zhang et al. / Optics Communications 306 (2013) 9–16
∞ ∞ ϵ2 ∑ ðnBn r 2n−1 −nC n r 2−n−1 Þ cos nθ ¼ ϵm −E0 cos θ− ∑ nDn r 2−n−1 cos nθ n¼1
n¼1
ð19Þ After a careful algebra, it is derived from Eqs. (16) to (19) that Al ¼Bl ¼Cl ¼Dl ¼0 for l 4 1, and only the first order of the expansion series survives. The coefficients of A1,B1 are derived as A1 ¼ −4ϵm ϵ2 E0 =Δ
ð20Þ
B1 ¼ −2ϵm ðϵ1 þ ϵ2 ÞE0 =Δ
ð21Þ
where the constant of Δ is defined as Δ ¼ ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð22Þ
where the ratio of η equals to r 21 =r 22 . The coefficients of Cl and Dl contribute trivially to the averaging field 〈E〉 of the total medium, which is defined by Z ! ! ! 〈E 〉¼ E dr ð23Þ ΔV
Therefore, they are not shown here. It is easily derived from Eqs. (20) to (21) that the averaging field within the core layer, i.e. 〈E1 〉, and that within the shell layer, i.e. 〈E2 〉, are given by 4ϵm ϵ2 E0 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð24Þ
〈E2 〉 ¼
2ϵm ðϵ1 þ ϵ2 Þ E0 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð25Þ
A.2. Mie theory for electromagnetic energy distribution among multilayered cylinders of infinite length As mentioned above, the corresponding equation for the CCPA method is shown as Eq. (1). The total energy E(1) contained within the multilayered cylinders when the cylinder is illuminated by a plane wave can be obtained following the Mie theory and the definition of the electromagnetic energy density, i.e. Eq. (2). Since the material of the constituent layers is isotropic and linear, according to the standard Mie theory, a series of the vectorial cylindrical-harmonic functions, which compose of the cylindrical Bessel functions, can be used to expand the vectorial !! electromagnetic fields of ð E ; H Þ distributed within the L-layer cylinders. For the core layer, however, since the second Bessel function of Yn is infinite at the center, therefore only the first kind of the cylindrical Bessel function of Jn is used as the expansion series functions. Supposing that the incident field is a plane wave, i.e. ! ! Ei ¼ E0 expðik0 zÞ, and the incident direction is perpendicular to the cylinder axis, the electromagnetic field in the core layer can be expanded as follows: ∞ ð1Þ !ð1Þ ð1Þ ! ∑ En ½t ð1Þ n N n;1 þ qn M n;1
n ¼ −∞
ð26Þ
ð1Þ !ð1Þ ! −ik1 ∞ ð1Þ ! ∑ En ½t ð1Þ ð27Þ H1 ¼ n N n;1 þ qn M n;1 ωμ n ¼ −∞ !ð1Þ !ð1Þ where M n;l ; N n;l are the mentioned vector cylindrical harmonics, ! and they are generated by the formulae as follows: M n ¼ ! ! ∇ ðe^z ψ n Þ; N n ¼ ð∇ M n Þ=k, where the function ψ n is the solution of the scalar wave equation 2
∇2 ψ þ k ψ ¼ 0
In the l-th ðl 41Þ layer of the cylinder, both Jn and Yn are finite, therefore the vectorial harmonic functions of the fields should include both of them which can be written as ! El ¼
∞ ð1Þ ð2Þ ð2Þ !ð1Þ ðlÞ ! ðlÞ ! ðlÞ ! ∑ En ½t ðlÞ n N n;l þ qn M n;l −pn N n;l −wn M n;l
n ¼ −∞
ð1Þ ð2Þ ð2Þ !ð1Þ ! −ik1 ∞ ðlÞ ! ðlÞ ! ðlÞ ! ∑ En ½t ðlÞ Hl ¼ n N n;l þ qn M n;l −pn N n;l −wn M n;l ωμ n ¼ −∞
En ¼ E0 ð−iÞ =kl where kl ¼nlk0 medium.
ð31Þ
The total energy E(1) for a coated core–shell cylinder with an infinite length under consideration, as shown in Fig. 2(b), can be obtained by summing up over the respective energy contained in the constituent layers, which can be easily written as ð1Þ Eð1Þ ¼ E1ð1Þ þ Eð1Þ 2 þ E3
ð32Þ
Considering the mathematical expressions of the electromagnetic fields, i.e. Eqs. (26) and (27), and the orthogonality relation of the vectorial harmonic functions, the mathematical expression for the electromagnetic energy within the core layer can be derived as follows: ð1Þ 2 2 Eð1Þ ¼ 12ϵ1 ∑En ðt ð1Þ ð33Þ n j þ qn j ÞI n ðx1 ; m1 Þ
ð28Þ
where ml ¼ nl/n0, xl ¼ k0 rl, and the definition of In(x1, m1) is given by Z ml xj n2 y dy 1 þ 2 J n ðyÞj2 þ J ′n ðyÞj2 I n ðxj ; ml Þ ¼ ð34Þ y 0 By using the analystic solution for the integral of the Bessel function of Jn, the solution for In (xj, ml) can be derived as follows: 1 1 1 þ 2 ½K n−1 ðml xj Þ−K nþ1 ðml xj Þ I n ðxj ; ml Þ ¼ ml xj j2 K n ðml xj Þ þ 2 4 4n ð35Þ and the definition of Kn(mlxj) is given by ! n2 ′ 2 J ðml xj Þj2 K n ðml xj Þ ¼ J n ðml xj Þj − 1− n jml xj j2
ð29Þ ! and k0 is the wave vector in the background
ð36Þ
Additionally, the electromagnetic energy within the l-th ðl 4 1Þ layer can be obtained as well, as follows: 1 2 ðlÞ 2 Elð1Þ ¼ ϵl ∑En ðt ðlÞ n j þ qn j Þ½I n ðxl ; ml Þ−I n ðxl−1 ; ml Þ 2 n Z ml xl 2 ðlÞ 2 1 2 w j Þ 1 þ n Y n ðyÞj2 þ Y ′ ðyÞj2 y dy∑En ðpðlÞ j þ þ ϵl n n n 2 y2 n
ml xl−1
ð37Þ In the following part, the expressions for the coefficients of t ðlÞ n , ðlÞ ðlÞ qðlÞ , n pn , wn were derived according to the boundary conditions, which can be written by an iterative relation as follows: 10 0 1 0 ðl−1Þ 1 ðlÞ Ω1 0 0 0 t nðl−1Þ CB t n C B ðl−1Þ C B Bp C B C 0 0 C Ω1ðl−1Þ CB pðlÞ n C B n C B 0 CB ð38Þ B ðl−1Þ C ¼ B B ðlÞ C ðl−1Þ B C B qn C B 0 C 0 CB 0 Ω2 @ A @ @ qn A A wnðl−1Þ wðlÞ 0 0 0 Ω2ðl−1Þ n where the elements of the matrix are given by J n ðml xl−1 Þ V ðl−1; lÞ−BðlÞ n Uðl−1; lÞ ml Y n ðml xl−1 Þ Ω1ðl−1Þ ¼ Wðl−1Þ ml−1 γ l
ð1Þ t ð1Þ n , qn are the coefficients, and the expression for En is given by 2
ð30Þ
n
〈E1 〉 ¼
! E1 ¼
15
Ω2ðl−1Þ ¼
ml ml−1 γ l
J n ðml xl−1 Þ V ðl−1; lÞ−AðlÞ n Uðl−1; lÞ Y n ðml xl−1 Þ Wðl−1Þ
where the functions of U, V and W are defined as follows:
ð39Þ
ð40Þ
16
H. Zhang et al. / Optics Communications 306 (2013) 9–16
Vðl−1; lÞ ¼ γ l−1 Dnð1Þ ðml xl−1 Þ−γ l Dnð2Þ ðml−1 xl−1 Þ
ð41Þ
Uðl−1; lÞ ¼
ð2Þ γ l−1 Dð2Þ n ðml xl−1 Þ−γ l Dn ðml−1 xl−1 Þ
ð42Þ
Wðl−1Þ ¼
J n ðml−1 xl−1 Þ ð1Þ ½D ðml−1 xl−1 Þ−Dð2Þ n ðml−1 xl−1 Þ Y n ðml xl−1 Þ n
ð43Þ
pffiffiffiffiffiffiffiffiffiffi where γ l is the wave impedance in the lth layer, and γ l ¼ μl =ϵl . The functions of DðiÞ n are the ratios between the Bessel functions and their derivatives, which are defined as follows: Dð1Þ n ðρÞ ¼
J ′n ðρÞ ; J n ðρÞ
Dð2Þ n ðρÞ ¼
Y ′n ðρÞ ; Y n ðρÞ
Dð3Þ n ðρÞ ¼
H nð1Þ′ ðρÞ H ð1Þ n ðρÞ
ð44Þ
It should be noted that these ratio-functions show good convergence behaviors even for multilayered cylinders with absorptive materials involved [28]. In addition, the variables of ðlÞ AðlÞ n and Bn in Eqs. (39) and (40) can be obtained by the following equations: AðlÞ n ¼
wðlÞ n qðlÞ n
J n ðml xl−1 Þγ l−1 Dnð1Þ ðml xl−1 Þ−γ l H An ðml−1 xl−1 Þ
¼
BnðlÞ ¼
ð45Þ
pðlÞ n t ðlÞ n
¼
B J n ðml xl−1 Þγ l−1 Dð1Þ n ðml xl−1 Þ−γ l H n ðml−1 xl−1 Þ
Y n ðml xl−1 Þγ l−1 Dnð1Þ ðml xl−1 Þ−γ l H Bn ðml−1 xl−1 Þ
Bnð1Þ ¼ 0
ð46Þ
where the functions of HnA,B are defined by J n ðml xl Þ ð1Þ ð2Þ D ðml xl Þ−AðlÞ n Dn ðml xl Þ Y n ðml xl Þ n ; J n ðml xl Þ ðlÞ −An Y n ðml xl Þ J n ðml xl Þ ð1Þ ð2Þ D ðml xl Þ−BðlÞ n Dn ðml xl Þ Y ðm x Þ n ; H Bn ðml ; xl Þ ¼ n l l J n ðml xl Þ ðlÞ −Bn Y n ðml xl Þ H An ðml ; xl Þ ¼
ðl≠1Þ
ðl≠1Þ
ð1Þ H A;B n ðm1 x1 Þ ¼ Dn ðm1 x1 Þ
ð47Þ
The coefficient of the outermost layer (l ¼L) can be expressed as " # J n ðxL Þ ð2Þ ð2Þ ð2Þ ½γ L Dð1Þ n ðxL Þ−Dn ðmL xL Þ−bnI ½γ L Dn ðxL Þ−Dn ðmL xL Þ H ð1Þ n ðxL Þ ðLÞ tn ¼ J ðmL xL Þ ð1Þ ½D ðmL xL Þ−Dð2Þ mL n n ðmL xL Þ Y n ðxL Þ n ð48Þ pnðLÞ ¼ BnðLÞ t ðLÞ n " qnðLÞ ¼
J n ðxL Þ
H nð1Þ ðxL Þ
ð49Þ # ð2Þ ð2Þ ð2Þ ½Dð1Þ n ðxL Þ−γ L Dn ðmL xL Þ−anII ½Dn ðxL Þ−γ L Dn ðmL xL Þ
mL
J n ðmL xL Þ ð1Þ ½D ðmL xL Þ−Dð2Þ n ðmL xL Þ Y n ðxL Þ n ð50Þ
wnðLÞ ¼ AnðLÞ qnðLÞ
ð51Þ
where bnI, anII are the scattering coefficient of the coated core–shell cylinders, which can be obtained as follows: anII ¼
A J n ðxL Þ γ L Dð1Þ n ðxL Þ−H n ðmL xL Þ
ð3Þ A H ð1Þ n ðxL Þ γ L Dn ðxL Þ−H n ðmL xL Þ
B J n ðxL Þ γ L Dð1Þ n ðxL Þ−H n ðmL xL Þ
B H nð1Þ ðxL Þ γ L Dð3Þ n ðxL Þ−H n ðmL xL Þ
ð53Þ
Moreover, it is easy to infer that [29], due to the normal incidence, the coefficients of qnðlÞ and wðlÞ n equal zero when anII ¼ 0, which is corresponding to the S-polarized incidence, and similarly, the coefficients of tn(l) and pn(l) equal zero when bnI ¼0, which is corresponding to the P-polarized incidence. Therefore, the coefficients for the respective layers can be obtained by the scattering coefficients, i.e. Eqs. (52) and (53) and the iterative equations, i.e. Eqs. (39) and (40). As a consequence, the electric and magnetic fields distributed within the respective layer can be calculated by the summation of the vectorial harmonics which are shown as Eqs. (26) and (27) and Eqs. (30) and (31).
References
A Y n ðml xl−1 Þγ l−1 Dð1Þ n ðml xl−1 Þ−γ l H n ðml−1 xl−1 Þ
Að1Þ n ¼0
bnI ¼
ð52Þ
[1] D.E. Aspnes, American Journal of Physics 50 (8) (1982) 704. [2] J.C. Maxwell Garnett, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203 (359–371) (1904) 385. [3] D.A.G. Bruggeman, Annalen der Physik 416 (7) (1935) 636. [4] O. Ouchetto, Cheng-Wei Qiu, S. Zouhdi, Le-Wei Li, A. Razek, IEEE Transactions on Microwave Theory and Techniques 54 (11) (2006) 3893. [5] Tuck C. Choy, Effective medium theory, Oxford University Press, 1999. [6] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., 1998. [7] H.C. van de Hulst, Light Scattering by Small Particles, Dover Publications Inc., New York, 1981. [8] Rubén G. Barrera, Augusto García-Valenzuela, Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach, Journal of the Optical Society of America A 20 (February (2)) (2003) 296. [9] Rubén G. Barrera, Alejandro Reyes-Coronado, Augusto García-Valenzuela, Physical Review B 75 (May (18)) (2007) 184202. [10] K. Busch, C.M. Soukoulis, Physical Review Letters 75 (November (19)) (1995) 3442. [11] K. Busch, C.M. Soukoulis, Physical Review B 54 (July (2)) (1996) 893. [12] Martin Störzer, Christof M. Aegerter, Georg Maret, Physical Review E 73 (June (6)) (2006) 065602. [13] R. Sapienza, P.D. García, J. Bertolotti, M.D. Martín, Á. Blanco, L. Viña, C. López, D.S. Wiersma, Physical Review Letters 99 (December (23)) (2007) 233902. [14] Yaxian Ni, Lei Gao, Cheng-Wei Qiu, Plasmonics 5 (2010) 251. [15] Ernst Jan R. Vesseur, Toon Coenen, Humeyra Caglayan, Nader Engheta, Albert Polman, Physical Review Letters 110 (January) (2013) 013902. [16] J.J. Niez, Annals of Physics 325 (2) (2010) 392. [17] Yukihisa Nanbu, Mitsuo Tateiba, Waves in Random Media 6 (4) (1996) 347. [18] Hao Zhang, Heyuan Zhu, Min Xu, Optics Express 19 (Febuary (4)) (2011) 2928. [19] A.F. Ioffe, A.R. Regel, Progress in Semiconductors 4 (1960) 237. [20] Ping Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd edition, Springer, Heidelberger, 2006. [21] Seth Fraden, Georg Maret, Physical Review Letters 65 (July) (1990) 512. [22] H.P. Schriemer, M.L. Cowan, J.H. Page, Ping Sheng, Zhengyou Liu, D.A. Weitz, Physical Review Letters 79 (October) (1997) 3166. [23] L. Gao, T.H. Fung, K.W. Yu, C.W. Qiu, Electromagnetic transparency by coated spheres with radial anisotropy, Physical Review E 78 (October) (2008) 046609. [24] D.L. Gao, L. Gao, C.W. Qiu, Europhysics Letters 95 (3) (2011) 34004. [25] Mário Silveirinha, Nader Engheta, Physical Review Letters 97 (2006) 157403. [26] Jie Luo, Ping Xu, Huanyang Chen, Bo Hou, Lei Gao, Yun Lai, Applied Physics Letters 100 (22) (2012) 221903. [27] Viet Cuong Nguyen, Lang Chen, Klaus Halterman, Physical Review Letters 105 (December) (2010) 233908. [28] W.J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes. Technical Report, NCAR Technical Note NCAR/ TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307), 1979. [29] Zhensen Wu, Lixin Guo, Suomin Cui, Acta Electronica Sinica 23 (1995) 114 (in Chinese).
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Effective medium theory for two-dimensional random media composed of core–shell cylinders$ Hao Zhang a,b,n, Yongqiang Shen a, Yuchen Xu a, Heyuan Zhu a, Ming Lei b, Xiangchao Zhang a, Min Xu a a Shanghai Engineering Center for Ultra-Precision Optical Manufacturing, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China b State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi'an 710119, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 March 2013 Received in revised form 6 May 2013 Accepted 7 May 2013 Available online 29 May 2013
In this paper, based on the generalized coated coherent potential approximation method, we derive the mathematical formulae, for the extended effective medium theory, to investigate the optical properties of disordered media composed of core–shell cylinders. The effective indices of such media are obtained in the long-wavelength limit and in the Mie-scattering region. Moreover, we use this method to study optical properties of random media composed of core–shell cylinders with the core layer consisting of epsilon-less-than-one material. & 2013 Published by Elsevier B.V.
Keywords: Effective medium theory Mie scattering Diffusion
1. Introduction The propagation and scattering of electromagnetic waves has been a topic extensively investigated by both the physics and engineering communities for decades. It is now common knowledge that the properties of responses of matter to incident electromagnetic waves in the process of interactions depend greatly on the relative sizes of the composites of the matter to the wavelength of the electromagnetic waves, i.e. r=λ. For relatively small composites, i.e. r=λ 5 1, the electromagnetic responses of the composites can be regarded as electric dipoles microscopically, and the averaged collective responses of the microscopic dipoles, also known as the coherent field, lead to an overall dielectric response of the matter. For this case, the diffuse field is much smaller compared to the coherent field, and thus neglected. Therefore, the dielectric responses are the overall electromagnetic responses of the matter. Such theories are also called as the effective medium theories (EMT) [1]. Several EMT models have
☆ This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. n Corresponding author at: Shanghai Engineering Center for Ultra-Precision Optical Manufacturing, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China. Tel.: +86 2155665372. E-mail address: [email protected] (H. Zhang).
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.05.027
been proposed, e.g. the Maxwell-Garnett theory [2], the Bruggerman model [3], the asymptotic multiscale theory [4], and etc. [5]. However, when the wavelength is comparable to the size of the composites, i.e. r=λ≥1, the microstructure of the composites involving the dielectric constants of the constituent materials, boundaries and shapes, will influence the microscopic electromagnetic responses significantly, and the electric dipole model breaks down as a consequence. Mie has proposed a rigorous theory to describe the scattering properties of light incident on a sphere with its size comparable to λ [6,7]. According to the Mie-scattering theory, resonant modes will be formed within the sphere when nx≈l þ 1=2, in which n is the refractive index of the sphere, x ¼ 2πr=λ and l is an integer. As the Mie resonances occur, the scattering cross sections of the sphere reach peak values which means that local effects will dominate within the medium composed of Mie scatterers. Moreover, for dense medium composed of Mie spheres, the multiple scatterings taking place among clusters formed by neighboring spheres are dominant which means that the local diffuse field is not negligible compared to the coherent field. The inhomogeneity that caused by the nontrivial diffuse field makes the conventional EMT break down. There has been effort based on the scattering approach to extend the EMT to study random media composed of large scatterers, and such theories are known as the extended effective medium theories (EEMTs) [8,9]. The effective field approximation (EFA) is one of the EEMTs, which assumes that the response field at any of the scatterers is homogeneous after ensemble-averaging over different configurations [10], and the local variations of the
10
H. Zhang et al. / Optics Communications 306 (2013) 9–16
refractive index are subsequently averaged out. Busch et al. has proposed an EEMT based on the framework of the EFA to calculate the effective index neff of disordered media composed of large spheres, and some derived physical entities such as the effective propagation constant q ¼ neff ω=c, the mean free path l¼ 0.5 Im(q), and the renormalized wave vector k¼ Re(q) [11]. The method they proposed, also called as the energy-density coherent potential approximation method, leads to the calculated results which show reasonable agreement to experiments [12,13]. Recently, random medium composed of single-layer or core– shell cylinders has attracted much interest due to their potential applications in the fields of radar and metamaterials [14], and among them, the random systems composed of core–shell cylinders is the major concern since some novel effects can be realized in this structure [15]. Although much effort has been devoted on the investigation of dielectric responses of such random matter [8,16,17], to our knowledge, the optical properties of random medium composed of dielectric core–shell cylinders of infinite length still lack detailed investigations. In the present work, based on the generalized coherent potential approximation method Busch et al. proposed, we derive the mathematical formulation of the EEMT for the investigations of the optical properties of random media composed of core–shell dielectric cylinders of infinite length. In such two-dimensional random systems, the optical properties, such as the effective indices in the long-wavelength limit, where the conventional EMTs are valid, and in the Mie-scattering region, where the nonlocal effects are nontrivial, are under considerations. We also conduct this method to study the optical properties of two-dimensional random systems composed of core–shell cylinders with the core layer constructed by epsilon-less-than-one materials.
the electric polarization of the random medium ! ! ∑pi 〈P 〉¼ ΔV
ð1Þ
! ! ! 〈 D 〉 ¼ ϵ0 〈 E 〉 þ 〈 P 〉
ð2Þ
! According to the relation between electric displacement 〈 D 〉 ! and electric field E , i.e.
the effective dielectric constant to describe the macroscopic effects of the electromagnetic responses of the inclusions can be defined via ! ! 〈 D 〉 ¼ ϵeff 〈 E 〉
Moreover, the average electric displacement and electric field can be given by the weighted average determined by the corresponding values distributed within the respective layers h ! ! ! i ! 〈 D 〉 ¼ f ηϵ1 〈 E1 〉 þ ð1−ηÞϵ2 〈 E2 〉 þ ð1−f Þϵm 〈Ee 〉 ð4Þ h ! ! i ! ! 〈 E 〉 ¼ f η〈 E1 〉 þ ð1−ηÞ〈 E2 〉 þ ð1−f Þ〈Ee 〉
2.1. Maxwell-Garnett theory Consider a random medium composed of core–shell dielectric cylinders of infinite length, shown in Fig. 1, embedded in a background medium with a dielectric constant of ϵm . The volume fraction occupied by the inclusions is f and the structure of the core–shell cylinders is shown in the inset of Fig. 1. As known, when the incident wavelength λ is much larger than the size of the inclusions, the electromagnetic response of the ! inclusion can be regarded as a point dipole pi , and the summation over the total responses of the inclusions within a unit space gives
r2 r1
ε1
ε2
ð5Þ
where f is the volume fraction occupied by core–shell cylinders, ! ! ! η ¼ r 21 =r 22 , 〈 E1 〉; 〈 E2 〉 and 〈Ee 〉 are the electric field distributed within the core-, shell-layer of the cylinder, and the background medium, respectively. According to Eqs. (3)–(5) and the formulae for the electric fields Ei (i¼1,2) within the core–shell cylinders, generally, the formulae for the effective dielectric constant ϵeff for a random medium composed of core–shell cylinders of infinite length is approximately given by ϵeff ¼
2. Theory
ð3Þ
ϵm þ f ½ηϵ1 α1 þ ð1−ηÞϵ2 α2 −ϵm 1 þ f ½ηα1 þ ð1−ηÞα2 −1
ð6Þ
where the parameters α1 and α2 are defined as the ratios between ! ! 〈Ei 〉=〈Ee 〉 ði ¼ 1; 2Þ. Eq. (6) is also known as the Maxwell-Garnett formula. Considering the boundary conditions, the electric fields within the multilayered cylinder for S-mode waves are contin! ! ! uous, i.e. 〈 E1 〉 ¼ 〈 E2 〉 ¼ 〈Ee 〉, therefore, α1 ¼ α2 ¼ 1. Thus, according to Eq. (6), the effective dielectric constant ϵeff for S-mode waves can be written directly by ϵeff ¼ ð1−f Þϵm þ f ½ηϵ1 þ ð1−ηÞϵ2
ð7Þ
However, the case for P-mode waves is more complicated, and herein, the field distribution of the electric fields within the core– shell cylinders for P mode is derived in Appendix A.1. According to Eqs. (25) and (26), α1 and α2 are respectively given by α1 ¼
4ϵm ϵ2 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð8Þ
α2 ¼
2ϵm ðϵ1 þ ϵ2 Þ ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð9Þ
So the final solution of ϵeff for P-mode waves can be obtained subsequently. 2.2. Coated coherent potential approximation method
εm
Fig. 1. Schematic view of a random medium composed of core–shell cylinders of infinite length. The positions of the cylinders are random. The inset is the core–shell dielectric cylinders embedded in the background with a dielectric constant of ϵm .
The coated coherent potential approximation (CCPA) method is conducted as well to calculate the effective index of the random system composed of core–shell dielectric cylinders of infinite length. The mechanism of the CCPA method is based on the physical idea that the distribution of the electromagnetic energy within a random medium should be homogeneous after being configurationally averaged over the correlation length of the random media [10,18]. Therefore, the averaged forward-scattering amplitude of a cylindrical region within the random medium should be approximately
H. Zhang et al. / Optics Communications 306 (2013) 9–16
method can be used to calculate the effective dielectric constant of pffiffiffiffiffiffiffi the random media. The calculated results of neff ðneff ¼ ϵeff Þ for random media composed of core–shell dielectric cylinders are illustrated in Fig. 3. The sizes of the core- and shell-layer are changing by an interval of 100 nm, and the medium is changing from dilute one to dense one. As a comparison, the Maxwell-Garnett method, described as Eq. (6), is conducted as well to calculate neff for random media composed of corresponding sizes of the core–shell cylinders, which are shown as the red lines in Fig. 3. By comparisons, it is obvious that the results obtained by the CCPA method are in good consistency to those obtained by the Maxwell-Garnett method, which is in accordance with the numerical simulations for random media composed of core–shell spheres we have shown in Ref. [18].
equal to zero. In addition, in order to take into account the effects of the structural factor for the multiple scatterings occurring within the random medium, a coated layer is involved to the real cylinder, which is embedded in the effective medium as an effective scattering unit. The radius of the coated layer is given by rc ¼r2 f−1/2. The schematic view of the CCPA method is shown in Fig. 2. It has been shown by Busch et al. that the effective medium obtained by the CCPA method for random media composed of dielectric spheres are in reasonable agreement with experiments [10,11]. According to the CCPA theory which gives that the coated scattering unit illuminated by a plane wave has a scattering pattern similar to that of its averaged counterpart, thus, the electromagnetic energy contained in the coated cylinder in Fig. 2 (b) should be equal to that contained in the dashed volume in Fig. 2(c), which subsequently gives the main equation for the CCPA method as follows: Z rc Z rc ! ! 3! 3! d r ρð1Þ ð r Þ ¼ d r ρð2Þ ð10Þ E E ðr Þ 0
11
3.2. In the Mie-scattering region However, when the wavelength is comparable to the size of the dielectric inclusions, i.e. kr≥1, the Mie scatterings take place when the incident waves impinge on the inclusions, and the effective medium theory for the case of long-wavelength limit, e.g. the Maxwell-Garnett method, breaks down, since in such a region, i.e. the Mie-scattering region, the electromagnetic response of a dielectric inclusion to the incident light is local and the averagings of the electric displacement and the electric field are strongly affected by the distribution of the scatterers, which means that the random medium cannot be regarded as a uniform effective medium any longer.
0
ð2Þ where ρð1Þ E and ρE represent the electromagnetic energy density within the three-layered cylinder in Fig. 2(b) and that of the averaged volume with an effective ϵeff in Fig. 2(c), respectively. In addition, as known, the energy density for an electromagnetic vectorial field is given by ! ! ! ! ! ! ! ρE ð r Þ ¼ 12½ϵð r Þ E ð r Þj2 þ μð r Þ H ð r Þj2 ð11Þ
here we only deal with non-magnetic materials, so μ=μ0 ¼ 1. The self-consistent equation, i.e. Eq. (10), can be solved by an iterative procedure. The related formulae of the CCPA method for a random medium composed of core–shell dielectric cylinders of infinite length are derived in Appendix A.2.
2.2 r =100nm, CCPA 1
r =200nm, CCPA 1
r1=300nm, CCPA
2
Effective refractive indices [neff]
3. Results and discussions The random media composed of core–shell cylinders, whose constituent materials are SiO2 (n1 ¼1.44), ZnO (n2 ¼ 2.7) or epsilonless-than-one materials, are under investigations herein, and for the sake of clarity, only the S-polarized modes are under considerations. 3.1. In the long-wavelength limit In the long-wavelength limit, i.e. λ 5 r 2 , the average electromagnetic responses of the overall core–shell inclusions are approximately homogeneous within the random media [5]. Under such conditions, the product of the wavevector and the mean free path, i.e. kl which is also the Ioffe–Regel criterion for wave localization [19], has the value much larger than one, therefore the scattering events taking place within the random medium can be regarded as single-scattering events, which means that the formal CPA condition is valid [20], and the above-mentioned CCPA
r =400nm, CCPA 1
r1=500nm, CCPA r =100nm, MG
1.8
1
r =200nm, MG 1
r =300nm, MG 1
1.6
r =400nm, MG 1
r1=500nm, MG
1.4
1.2
1
0
0.1
0.3
0.4
0.5
0.6
Volume fraction [f] Fig. 3. Effective refractive indices for random media composed of infinite core– shell (SiO2/ZnO) cylinders. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
c m 2
0.2
c 2
1 1
m
eff
eff
Fig. 2. Schematic view of the CCPA method for random media composed of core–shell dielectric cylinders, illustrated in (a). The coated layer to the actual core–shell cylinders in (b) has the size of rc and the dielectric constant equal to ϵm . (c) The dashed region indicates the effective scattering unit described in the CCPA method.
12
H. Zhang et al. / Optics Communications 306 (2013) 9–16
Moreover, the influences from the neighbor inclusions when the Mie scattering takes place are not trivial when the volume fraction f is large enough, and the multiple scatterings occur for densely random media composed of core–shell dielectric cylinders. In such situations, the contribution to the overall scatterings can be divided into the form factor, which is attributed to single scatterings, and the structural factor, which is due to the correlation effects caused by multiple scatterings [21]. However, it has been reported by experiments that in the Mie-scattering region, the configurationally averaged velocities of the electromagnetic energy is in good qualitative agreement between the experiments and theoretical calculations using the CCPA method, which is attributed to the coated layer involving the correlation effects [12,22]. Therefore, even for dense media in which multiple-scattering events are not trivial, the CCPA method can be performed to obtain the effective refractive index neff, 5 f=5%
f=20%
f=40%
f=60%
Effective refractive indices [neff]
4.5 4
which can be used to describe the averaged electromagnetic responses. The calculated effective refractive index neff for random media composed of core–shell dielectric cylinders with different sizes and volume fractions is shown in Fig. 4(a). The resonant points of the neff curves when varying the size of the cylinder, i.e. the spikes in Fig. 4(a), are in coincidence with the resonant peaks in the calculated scattering cross-section curve in Fig. 4(b) for a corresponding core–shell dielectric cylinder with an identical structure. Such a coincidence can be understood by the fact that the resonant modes formed within the core–shell cylinders lead to larger values of the average electric displacement and neff as well, since the effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi refractive index neff which can be roughly obtained by 〈D〉=〈E〉, in which the distributed electric fields can be regarded as weighted factors of the dielectric constants in the formulae calculating neff, i.e. Eqs. (3)–(5). Thus, if resonant modes are formed within the core–shell cylinders, indicated by the peaks in Fig. 4(b), the contribution from the area with higher dielectric constants ðϵ1 and ϵ2 Þ to the neff would dominate, which leads to the peak values of neff, also indicated by the spikes in Fig. 4(a). In addition, the shift to a larger value of the effective dielectric constant when increasing the volume fraction, as shown in Fig. 4 (a), can be understood in a similar way. 3.3. For constituent material with ϵ-less-than-one
3.5 3 2.5 2 1.5
Qs
1 5
0 0.5
0.7
0.9
1.1
1.3
1.5
r2/λ
4.5
5
4
4.5
Scattering efficiency factor Qsca
Scattering efficiency factor Qsca
Fig. 4. (a) Effective refractive indices for random media composed of infinite core– shell (SiO2/ZnO) cylinders with different volume fractions. r1 ¼ 300 nm and r2 ¼600 nm. (b) Efficiency of the scattering cross-section Qs for the core–shell cylinders described in (a).
Scattering of multilayered composites has been attracted much interest due to the realization of transparency [23,24]. The constituent materials of multilayered composites can be anisotropic, birefringent materials or metamaterials. Epsilon-less-than-one material (ELTO material) is a class of metamaterials that propagate waves through small phase change and possess unusual optical properties that can be used to create novel optical elements, such as subwavelength waveguides [25], perfect bending waveguides [26], cloaking [27], etc. Generally, such materials can be found naturally at infrared or optical frequencies, and recently it has been shown that the ELTO material for ϵ near zero was properly synthesized by metastructures [15]. According to the Mie-scattering formulation [6], thep behavior ffiffiffiffiffiffiffiffiffiffiffiffiffi of the vector harmonics depends on variables of mi ðmi ¼ ϵi =ϵm Þ and ri, and in the core layer, only the first kind of the vector harmonics is available due to the divergence of the second kind of spherical Bessel function yn at the origin. For mi close to zero, similar divergent behavior of the second vector harmonics occurs even
3.5 3 2.5 2 1.5 1 0.5 0.1
P4
P1 P2
4
P3
3.5
P5
3 2.5 2 1.5 1
0.2
0.3
0.4
0.5
r1/r2
0.6
0.7
0.8
0.9
0.5
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Wavelength [μm]
Fig. 5. Calculated scattering efficiency factors Qsca for S-polarized incidence on core–shell cylinders containing epsilon-less-than-one material. The refractive indices of the background medium, n0 ¼1.0, and of the shell layer, n2 ¼ 3.0: (a) the incident wavelength is λ ¼ 2:5 μm and the radius of the shell layer is r 2 ¼ 2:0 μm and (b) r 1 ¼ 1:05 μm and r 2 ¼ 2:0 μm.
H. Zhang et al. / Optics Communications 306 (2013) 9–16
for the shell layers, therefore, special care should be taken when performing the standard Mie theory for multilayered cylinders containing ELTO material. In Fig. 5, the scattering efficiency factors for S-polarized waves normal incident on a core–shell cylinder of infinite length containing ELTO material are calculated. It is obvious that more resonant modes are formed within the core–shell cylinders when introducing the core layer, and the curves of Qsca for core layer with n ¼0.9, 0.5 and n ¼ 1.5 are quite similar in shape, though the positions of resonant points in the curve for n ¼1.5 are overall shifted. The wavelength-dependent scattering efficiency factors for core–shell cylinders with a fixed space structure is shown in Fig. 5(b). The results are certainly surprising at first sight, since the positions of many resonant points for cases of core layers with different values of n1 are nearly coincident, which are indicated by the arrows in Fig. 5(b). They can be understood, if one proves that the distribution of resonant modes locates dominantly within the
2.5
13
shell layer. Thus, it is the refractive index and shape of the shell layer that induce resonant modes, and the influence from moderate variations of n1 is negligible. The fractions of electromagnetic energy contained within the shell layer, i.e. E2, compared to the total energy contained within the core–shell cylinders, i.e. E1+E2, are calculated using a standard Mie-scattering algorithm, and the dependence on a continuous variation of r1/r2 is investigated. The results are shown in Fig. 6(d)–(f), and for comparison, Qscas for the corresponding core–shell cylinders are obtained as well, as shown in Fig. 6(a)–(c). When the refractive index of the core layer decreases to values less than one, e.g. n1 ¼ 0.5 in Fig. 6(f), the possibility to find the energy dominantly located within the shell layer increases, which can be understood partly by the fact that smaller ϵ leads to smaller values of energy, since as mentioned above, the energy density ρE ∝ϵjEj2 . As for the resonant points indicated by the arrows in Fig. 5(b), it is convinced in Figs. 6(d)–(f) that for r1/r2 ¼ 0.5, the resonant modes formed within the core–shell cylinders, which are
2.5
4.5
0.9
3.5 2
3 2.5 2
1.5 1.5
0.8
Wavelength [µm]
Wavelength [µm]
4
2
0.7 0.6 0.5
1.5
0.4 0.3
1
0.2
0.5 1 0.1
0.3
0.5
0.7
1 0.1
0.9
0.3
r1/r2
0.5
0.7
0.9
r1/r2 2.5
2.5
0.9
3
2
2.5 2
1.5
Wavelength [µm]
Wavelength [µm]
3.5
0.8 2
0.7 0.6 0.5
1.5
0.4
1.5
1 0.1
0.3 0.2
1 0.3
0.5
0.7
1 0.1
0.9
0.3
r1/r2
0.5
0.7
0.9
r1/r2 2.5
2.5
0.9
3
2
2.5 2
1.5
Wavelength [µm]
Wavelength [µm]
3.5
0.8 2
0.7 0.6 0.5
1.5
0.4
1.5
0.3 1 0.1
1 0.3
0.5
r1/r2
0.7
0.9
1 0.1
0.3
0.5
0.7
0.9
r1/r2
Fig. 6. Calculated scattering efficiency factor Qsca for core–shell cylinders with the refractive index of the shell layer n2 ¼ 3.0, and of the core layer: (a) n1 ¼ 1.5; (b) n1 ¼ 0.9; and (c) n1 ¼0.5. The radius of the shell layer is r 2 ¼ 2 μm. The relative fractions of energy contained within the shell layer, i.e. E2/(E1+E2) for the corresponding cylinders (a)–(c) are shown in (d)–(f) respectively.
14
H. Zhang et al. / Optics Communications 306 (2013) 9–16
2.1
2.2
2
Effective refractive index neff
Effective refractive index neff
2
1.8
1.6
1.4
1.2
1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
1.1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r1/r2
Volume fraction f
Fig. 7. Calculated effective refractive index neff of S-modes in the long-wavelength limit for random media composed of core–shell cylinders containing epsilon-less-thanone material. The refractive index of the background medium is n0 ¼1.0: (a) r 1 ¼ 1:05 μm and r 2 ¼ 2 μm and (b) the volume fraction of core–shell cylinders is f¼ 40%.
indicated as the resonant lines found in Fig. 6(a)–(c), are dominantly located within the shell layer. The calculated effective refractive index neff of S-polarized modes for random media composed of core–shell cylinders containing ELTO materials in the long-wavelength limit is shown in Fig. 7. The two above-mentioned methods, i.e. the MaxwellGarnett method and the CCPA method, are conducted with the purpose of comparisons, and again a good coincidence can be found.
4. Conclusions We have presented here an effective medium theory for twodimensional random media composed of core–shell cylinders of infinite length based on the generalization of coherent-potential approximation method. As a comparison, the traditional EMT of the Maxwell-Garnett method for such random media is derived, and by numerical calculations, it is clearly proved that these two methods produce nearly the same results in the long-wavelength limit. We also use the developed EMT method to obtain the effective index of such random media in the Mie-scattering region, which plays an important role in the description of transport properties of light propagating within such random media, such as the mean free path and so on. By performing a standard Mie scattering theory, we investigate the optical properties of core– shell cylinders containing ELTO materials, and find that when the core layer is constructed by ELTO material, the resonant modes formed within the core–shell cylinders dominantly locate within the shell layer, and the peak points in the curves of the scattering efficiency factors are therefore nearly coincident. Finally, we use the developed EMT method and the Maxwell-Garnett method to obtain the effective index of random media composed of core– shell cylinders containing epsilon-less-than-one material, and similar agreement is found.
Laboratory of Transient Optics and Photonics under Grant No. SKLST201111.
Appendix A A.1. Solution to the P-mode field distribution within a coated cylinder in a homogeneous electrostatic field The solution to the Laplace equation, i.e. ∇2 ψ ¼ 0, in cylindrical coordinates for a core–shell cylinder with an infinite length along the z-direction in a uniform field which is applied along the r^ direction, can be written as ∞
ψ ¼ ∑ ðAn r n þ Bn r −n Þ cos nθ
ð12Þ
n¼1
here only the p-mode (electric field along the z-axis) is under considerations and the s-mode (magnetic field along the z-axis) can be handled easily by considering the boundary conditions. Considering that the finite value of the potential at r ¼0 and the condition that, when r-∞, ψ ¼ −E0 r cos θ, the mathematical expressions for the potential within the respective layer of the core–shell cylinders, as shown in Fig. 1, are given by ∞
ψ 1 ðr; θÞ ¼ ∑ An r n cos nθ
ð13Þ
n¼1 ∞
ψ 2 ðr; θÞ ¼ ∑ ðBn r n þ C n r −n Þ cos nθ
ð14Þ
n¼1
∞
ψ 3 ðr; θÞ ¼ −E0 r cos θ þ ∑ Dn r −n cos nθ
ð15Þ
n¼1
The continuation of the electric potential and the normal component of the D at the boundaries of r ¼r1 and r ¼r2 gives that ∞
∞
n¼1
n¼1
∑ An r n1 cos nθ ¼ ∑ ðBn r n1 þ C n r −n 1 Þ cos nθ
Acknowledgments This work was supported in part by the National Science and Technology Major Project of the Ministry of Science and Technology of China under Grant no. 2011ZX02402, the National High Technology R&D Program of China (863 Program) under Grant No. 2012AA040406, and the Open Funding provided by State Key
∞
∞
n¼1
n¼1
−n−1 ϵ1 ∑ nAn r n−1 cos nθ ¼ ϵ2 ∑ ðnBn r n−1 Þ cos nθ 1 1 −nC n r 1
∞
∞
n¼1
n¼1
−n ∑ ðBn r n2 þ C n r −n 2 Þ cos nθ ¼ −E0 r 2 cos θ þ ∑ Dn r 2 cos nθ
ð16Þ
ð17Þ
ð18Þ
H. Zhang et al. / Optics Communications 306 (2013) 9–16
∞ ∞ ϵ2 ∑ ðnBn r 2n−1 −nC n r 2−n−1 Þ cos nθ ¼ ϵm −E0 cos θ− ∑ nDn r 2−n−1 cos nθ n¼1
n¼1
ð19Þ After a careful algebra, it is derived from Eqs. (16) to (19) that Al ¼Bl ¼Cl ¼Dl ¼0 for l 4 1, and only the first order of the expansion series survives. The coefficients of A1,B1 are derived as A1 ¼ −4ϵm ϵ2 E0 =Δ
ð20Þ
B1 ¼ −2ϵm ðϵ1 þ ϵ2 ÞE0 =Δ
ð21Þ
where the constant of Δ is defined as Δ ¼ ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð22Þ
where the ratio of η equals to r 21 =r 22 . The coefficients of Cl and Dl contribute trivially to the averaging field 〈E〉 of the total medium, which is defined by Z ! ! ! 〈E 〉¼ E dr ð23Þ ΔV
Therefore, they are not shown here. It is easily derived from Eqs. (20) to (21) that the averaging field within the core layer, i.e. 〈E1 〉, and that within the shell layer, i.e. 〈E2 〉, are given by 4ϵm ϵ2 E0 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð24Þ
〈E2 〉 ¼
2ϵm ðϵ1 þ ϵ2 Þ E0 ðϵ1 þ ϵ2 Þðϵ2 þ ϵm Þ−ηðϵ2 −ϵ1 Þðϵ2 −ϵm Þ
ð25Þ
A.2. Mie theory for electromagnetic energy distribution among multilayered cylinders of infinite length As mentioned above, the corresponding equation for the CCPA method is shown as Eq. (1). The total energy E(1) contained within the multilayered cylinders when the cylinder is illuminated by a plane wave can be obtained following the Mie theory and the definition of the electromagnetic energy density, i.e. Eq. (2). Since the material of the constituent layers is isotropic and linear, according to the standard Mie theory, a series of the vectorial cylindrical-harmonic functions, which compose of the cylindrical Bessel functions, can be used to expand the vectorial !! electromagnetic fields of ð E ; H Þ distributed within the L-layer cylinders. For the core layer, however, since the second Bessel function of Yn is infinite at the center, therefore only the first kind of the cylindrical Bessel function of Jn is used as the expansion series functions. Supposing that the incident field is a plane wave, i.e. ! ! Ei ¼ E0 expðik0 zÞ, and the incident direction is perpendicular to the cylinder axis, the electromagnetic field in the core layer can be expanded as follows: ∞ ð1Þ !ð1Þ ð1Þ ! ∑ En ½t ð1Þ n N n;1 þ qn M n;1
n ¼ −∞
ð26Þ
ð1Þ !ð1Þ ! −ik1 ∞ ð1Þ ! ∑ En ½t ð1Þ ð27Þ H1 ¼ n N n;1 þ qn M n;1 ωμ n ¼ −∞ !ð1Þ !ð1Þ where M n;l ; N n;l are the mentioned vector cylindrical harmonics, ! and they are generated by the formulae as follows: M n ¼ ! ! ∇ ðe^z ψ n Þ; N n ¼ ð∇ M n Þ=k, where the function ψ n is the solution of the scalar wave equation 2
∇2 ψ þ k ψ ¼ 0
In the l-th ðl 41Þ layer of the cylinder, both Jn and Yn are finite, therefore the vectorial harmonic functions of the fields should include both of them which can be written as ! El ¼
∞ ð1Þ ð2Þ ð2Þ !ð1Þ ðlÞ ! ðlÞ ! ðlÞ ! ∑ En ½t ðlÞ n N n;l þ qn M n;l −pn N n;l −wn M n;l
n ¼ −∞
ð1Þ ð2Þ ð2Þ !ð1Þ ! −ik1 ∞ ðlÞ ! ðlÞ ! ðlÞ ! ∑ En ½t ðlÞ Hl ¼ n N n;l þ qn M n;l −pn N n;l −wn M n;l ωμ n ¼ −∞
En ¼ E0 ð−iÞ =kl where kl ¼nlk0 medium.
ð31Þ
The total energy E(1) for a coated core–shell cylinder with an infinite length under consideration, as shown in Fig. 2(b), can be obtained by summing up over the respective energy contained in the constituent layers, which can be easily written as ð1Þ Eð1Þ ¼ E1ð1Þ þ Eð1Þ 2 þ E3
ð32Þ
Considering the mathematical expressions of the electromagnetic fields, i.e. Eqs. (26) and (27), and the orthogonality relation of the vectorial harmonic functions, the mathematical expression for the electromagnetic energy within the core layer can be derived as follows: ð1Þ 2 2 Eð1Þ ¼ 12ϵ1 ∑En ðt ð1Þ ð33Þ n j þ qn j ÞI n ðx1 ; m1 Þ
ð28Þ
where ml ¼ nl/n0, xl ¼ k0 rl, and the definition of In(x1, m1) is given by Z ml xj n2 y dy 1 þ 2 J n ðyÞj2 þ J ′n ðyÞj2 I n ðxj ; ml Þ ¼ ð34Þ y 0 By using the analystic solution for the integral of the Bessel function of Jn, the solution for In (xj, ml) can be derived as follows: 1 1 1 þ 2 ½K n−1 ðml xj Þ−K nþ1 ðml xj Þ I n ðxj ; ml Þ ¼ ml xj j2 K n ðml xj Þ þ 2 4 4n ð35Þ and the definition of Kn(mlxj) is given by ! n2 ′ 2 J ðml xj Þj2 K n ðml xj Þ ¼ J n ðml xj Þj − 1− n jml xj j2
ð29Þ ! and k0 is the wave vector in the background
ð36Þ
Additionally, the electromagnetic energy within the l-th ðl 4 1Þ layer can be obtained as well, as follows: 1 2 ðlÞ 2 Elð1Þ ¼ ϵl ∑En ðt ðlÞ n j þ qn j Þ½I n ðxl ; ml Þ−I n ðxl−1 ; ml Þ 2 n Z ml xl 2 ðlÞ 2 1 2 w j Þ 1 þ n Y n ðyÞj2 þ Y ′ ðyÞj2 y dy∑En ðpðlÞ j þ þ ϵl n n n 2 y2 n
ml xl−1
ð37Þ In the following part, the expressions for the coefficients of t ðlÞ n , ðlÞ ðlÞ qðlÞ , n pn , wn were derived according to the boundary conditions, which can be written by an iterative relation as follows: 10 0 1 0 ðl−1Þ 1 ðlÞ Ω1 0 0 0 t nðl−1Þ CB t n C B ðl−1Þ C B Bp C B C 0 0 C Ω1ðl−1Þ CB pðlÞ n C B n C B 0 CB ð38Þ B ðl−1Þ C ¼ B B ðlÞ C ðl−1Þ B C B qn C B 0 C 0 CB 0 Ω2 @ A @ @ qn A A wnðl−1Þ wðlÞ 0 0 0 Ω2ðl−1Þ n where the elements of the matrix are given by J n ðml xl−1 Þ V ðl−1; lÞ−BðlÞ n Uðl−1; lÞ ml Y n ðml xl−1 Þ Ω1ðl−1Þ ¼ Wðl−1Þ ml−1 γ l
ð1Þ t ð1Þ n , qn are the coefficients, and the expression for En is given by 2
ð30Þ
n
〈E1 〉 ¼
! E1 ¼
15
Ω2ðl−1Þ ¼
ml ml−1 γ l
J n ðml xl−1 Þ V ðl−1; lÞ−AðlÞ n Uðl−1; lÞ Y n ðml xl−1 Þ Wðl−1Þ
where the functions of U, V and W are defined as follows:
ð39Þ
ð40Þ
16
H. Zhang et al. / Optics Communications 306 (2013) 9–16
Vðl−1; lÞ ¼ γ l−1 Dnð1Þ ðml xl−1 Þ−γ l Dnð2Þ ðml−1 xl−1 Þ
ð41Þ
Uðl−1; lÞ ¼
ð2Þ γ l−1 Dð2Þ n ðml xl−1 Þ−γ l Dn ðml−1 xl−1 Þ
ð42Þ
Wðl−1Þ ¼
J n ðml−1 xl−1 Þ ð1Þ ½D ðml−1 xl−1 Þ−Dð2Þ n ðml−1 xl−1 Þ Y n ðml xl−1 Þ n
ð43Þ
pffiffiffiffiffiffiffiffiffiffi where γ l is the wave impedance in the lth layer, and γ l ¼ μl =ϵl . The functions of DðiÞ n are the ratios between the Bessel functions and their derivatives, which are defined as follows: Dð1Þ n ðρÞ ¼
J ′n ðρÞ ; J n ðρÞ
Dð2Þ n ðρÞ ¼
Y ′n ðρÞ ; Y n ðρÞ
Dð3Þ n ðρÞ ¼
H nð1Þ′ ðρÞ H ð1Þ n ðρÞ
ð44Þ
It should be noted that these ratio-functions show good convergence behaviors even for multilayered cylinders with absorptive materials involved [28]. In addition, the variables of ðlÞ AðlÞ n and Bn in Eqs. (39) and (40) can be obtained by the following equations: AðlÞ n ¼
wðlÞ n qðlÞ n
J n ðml xl−1 Þγ l−1 Dnð1Þ ðml xl−1 Þ−γ l H An ðml−1 xl−1 Þ
¼
BnðlÞ ¼
ð45Þ
pðlÞ n t ðlÞ n
¼
B J n ðml xl−1 Þγ l−1 Dð1Þ n ðml xl−1 Þ−γ l H n ðml−1 xl−1 Þ
Y n ðml xl−1 Þγ l−1 Dnð1Þ ðml xl−1 Þ−γ l H Bn ðml−1 xl−1 Þ
Bnð1Þ ¼ 0
ð46Þ
where the functions of HnA,B are defined by J n ðml xl Þ ð1Þ ð2Þ D ðml xl Þ−AðlÞ n Dn ðml xl Þ Y n ðml xl Þ n ; J n ðml xl Þ ðlÞ −An Y n ðml xl Þ J n ðml xl Þ ð1Þ ð2Þ D ðml xl Þ−BðlÞ n Dn ðml xl Þ Y ðm x Þ n ; H Bn ðml ; xl Þ ¼ n l l J n ðml xl Þ ðlÞ −Bn Y n ðml xl Þ H An ðml ; xl Þ ¼
ðl≠1Þ
ðl≠1Þ
ð1Þ H A;B n ðm1 x1 Þ ¼ Dn ðm1 x1 Þ
ð47Þ
The coefficient of the outermost layer (l ¼L) can be expressed as " # J n ðxL Þ ð2Þ ð2Þ ð2Þ ½γ L Dð1Þ n ðxL Þ−Dn ðmL xL Þ−bnI ½γ L Dn ðxL Þ−Dn ðmL xL Þ H ð1Þ n ðxL Þ ðLÞ tn ¼ J ðmL xL Þ ð1Þ ½D ðmL xL Þ−Dð2Þ mL n n ðmL xL Þ Y n ðxL Þ n ð48Þ pnðLÞ ¼ BnðLÞ t ðLÞ n " qnðLÞ ¼
J n ðxL Þ
H nð1Þ ðxL Þ
ð49Þ # ð2Þ ð2Þ ð2Þ ½Dð1Þ n ðxL Þ−γ L Dn ðmL xL Þ−anII ½Dn ðxL Þ−γ L Dn ðmL xL Þ
mL
J n ðmL xL Þ ð1Þ ½D ðmL xL Þ−Dð2Þ n ðmL xL Þ Y n ðxL Þ n ð50Þ
wnðLÞ ¼ AnðLÞ qnðLÞ
ð51Þ
where bnI, anII are the scattering coefficient of the coated core–shell cylinders, which can be obtained as follows: anII ¼
A J n ðxL Þ γ L Dð1Þ n ðxL Þ−H n ðmL xL Þ
ð3Þ A H ð1Þ n ðxL Þ γ L Dn ðxL Þ−H n ðmL xL Þ
B J n ðxL Þ γ L Dð1Þ n ðxL Þ−H n ðmL xL Þ
B H nð1Þ ðxL Þ γ L Dð3Þ n ðxL Þ−H n ðmL xL Þ
ð53Þ
Moreover, it is easy to infer that [29], due to the normal incidence, the coefficients of qnðlÞ and wðlÞ n equal zero when anII ¼ 0, which is corresponding to the S-polarized incidence, and similarly, the coefficients of tn(l) and pn(l) equal zero when bnI ¼0, which is corresponding to the P-polarized incidence. Therefore, the coefficients for the respective layers can be obtained by the scattering coefficients, i.e. Eqs. (52) and (53) and the iterative equations, i.e. Eqs. (39) and (40). As a consequence, the electric and magnetic fields distributed within the respective layer can be calculated by the summation of the vectorial harmonics which are shown as Eqs. (26) and (27) and Eqs. (30) and (31).
References
A Y n ðml xl−1 Þγ l−1 Dð1Þ n ðml xl−1 Þ−γ l H n ðml−1 xl−1 Þ
Að1Þ n ¼0
bnI ¼
ð52Þ
[1] D.E. Aspnes, American Journal of Physics 50 (8) (1982) 704. [2] J.C. Maxwell Garnett, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203 (359–371) (1904) 385. [3] D.A.G. Bruggeman, Annalen der Physik 416 (7) (1935) 636. [4] O. Ouchetto, Cheng-Wei Qiu, S. Zouhdi, Le-Wei Li, A. Razek, IEEE Transactions on Microwave Theory and Techniques 54 (11) (2006) 3893. [5] Tuck C. Choy, Effective medium theory, Oxford University Press, 1999. [6] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., 1998. [7] H.C. van de Hulst, Light Scattering by Small Particles, Dover Publications Inc., New York, 1981. [8] Rubén G. Barrera, Augusto García-Valenzuela, Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach, Journal of the Optical Society of America A 20 (February (2)) (2003) 296. [9] Rubén G. Barrera, Alejandro Reyes-Coronado, Augusto García-Valenzuela, Physical Review B 75 (May (18)) (2007) 184202. [10] K. Busch, C.M. Soukoulis, Physical Review Letters 75 (November (19)) (1995) 3442. [11] K. Busch, C.M. Soukoulis, Physical Review B 54 (July (2)) (1996) 893. [12] Martin Störzer, Christof M. Aegerter, Georg Maret, Physical Review E 73 (June (6)) (2006) 065602. [13] R. Sapienza, P.D. García, J. Bertolotti, M.D. Martín, Á. Blanco, L. Viña, C. López, D.S. Wiersma, Physical Review Letters 99 (December (23)) (2007) 233902. [14] Yaxian Ni, Lei Gao, Cheng-Wei Qiu, Plasmonics 5 (2010) 251. [15] Ernst Jan R. Vesseur, Toon Coenen, Humeyra Caglayan, Nader Engheta, Albert Polman, Physical Review Letters 110 (January) (2013) 013902. [16] J.J. Niez, Annals of Physics 325 (2) (2010) 392. [17] Yukihisa Nanbu, Mitsuo Tateiba, Waves in Random Media 6 (4) (1996) 347. [18] Hao Zhang, Heyuan Zhu, Min Xu, Optics Express 19 (Febuary (4)) (2011) 2928. [19] A.F. Ioffe, A.R. Regel, Progress in Semiconductors 4 (1960) 237. [20] Ping Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd edition, Springer, Heidelberger, 2006. [21] Seth Fraden, Georg Maret, Physical Review Letters 65 (July) (1990) 512. [22] H.P. Schriemer, M.L. Cowan, J.H. Page, Ping Sheng, Zhengyou Liu, D.A. Weitz, Physical Review Letters 79 (October) (1997) 3166. [23] L. Gao, T.H. Fung, K.W. Yu, C.W. Qiu, Electromagnetic transparency by coated spheres with radial anisotropy, Physical Review E 78 (October) (2008) 046609. [24] D.L. Gao, L. Gao, C.W. Qiu, Europhysics Letters 95 (3) (2011) 34004. [25] Mário Silveirinha, Nader Engheta, Physical Review Letters 97 (2006) 157403. [26] Jie Luo, Ping Xu, Huanyang Chen, Bo Hou, Lei Gao, Yun Lai, Applied Physics Letters 100 (22) (2012) 221903. [27] Viet Cuong Nguyen, Lang Chen, Klaus Halterman, Physical Review Letters 105 (December) (2010) 233908. [28] W.J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes. Technical Report, NCAR Technical Note NCAR/ TN-140+STR (National Center for Atmospheric Research, Boulder, Colo. 80307), 1979. [29] Zhensen Wu, Lixin Guo, Suomin Cui, Acta Electronica Sinica 23 (1995) 114 (in Chinese).