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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26027–26043
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Complex modes and effective refractive index in 3D periodic arrays of plasmonic nanospheres

Salvatore Campione, Sergiy Steshenko, Matteo Albani, and Filippo Capolino  »View Author Affiliations


Optics Express, Vol. 19, Issue 27, pp. 26027-26043 (2011)
http://dx.doi.org/10.1364/OE.19.026027


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Abstract

We characterize the modes with complex wavenumber for both longitudinal and transverse polarization states (with respect to the mode traveling direction) in three dimensional (3D) periodic arrays of plasmonic nanospheres, including metal losses. The Ewald representation of the required dyadic periodic Green’s function to represent the field in 3D periodic arrays is derived from the scalar case, which can be analytically continued into the complex wavenumber space. We observe the presence of one longitudinal mode and two transverse modes, one forward and one backward. Despite the presence of two modes for transverse polarization, we notice that the forward one is “dominant” (i.e., it contributes most to the field in the array). Therefore, in case of transverse polarization, we describe the composite material in terms of a homogenized effective refractive index, comparing results from (i) modal analysis, (ii) Maxwell Garnett theory, (iii) Nicolson-Ross-Weir retrieval method from scattering parameters for finite thickness structures (considering different thicknesses, showing consistency of results), and (iv) the fitting of the fields obtained through HFSS simulations. The agreement among the different methods justifies the performed homogenization procedure in case of transverse polarization.

© 2011 OSA

1. Introduction

2. Simulation model

The structure under analysis is the 3D periodic array of metallic nanospheres reported in Fig. 1
Fig. 1 3D periodic array of metallic nanospheres embedded in a homogeneous medium with permittivityεh. The radius of each nanosphere is r; and a, b and c are the periodicities along x-, y- and z-direction, respectively.
. The monochromatic time harmonic convention, exp(iωt), is assumed here and throughout the paper, and is therefore suppressed hereafter.

We model each nanosphere as a single electric dipole. As such, for a metallic spherical particle the induced dipole moment is
p=αeeEloc,
(1)
where αee is the isotropic electric polarizability of the nanosphere, Eloc is the local field produced by all the nanospheres of the array except the considered nanosphere, plus the external incident field to the array, if present. Bold fonts refer to vector quantities. The SDA is a good approximation when the metallic nanospheres are used close to their fundamental plasmonic frequency, when particle dimensions are much smaller than the wavelength, and when the edge-to-edge spacing d between spheres is larger than the spheres’ radius r (i.e., dr). However, even for smaller distances the SDA may provide satisfactory approximated results [15

15. A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express 19(3), 2754–2772 (2011).

]. In general, for a spacing between the spheres smaller than their radius (i.e., 0<d<r), more accurate results would involve multipole field contributions [10

10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

,16

16. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

18

18. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994).

]. According to Mie theory, the electric polarizability for a nanosphere is [10

10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

,11

11. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1.

]
αee=6πiεhε0k3a1=6πiεhε0k3mrψ1(mrkr)ψ1(kr)ψ1(kr)ψ1(mrkr)mrψ1(mrkr)ξ1(kr)ξ1(kr)ψ1(mrkr),
(2)
where εh is the relative permittivity of the host medium, ε0 is the absolute permittivity of free space, k=ωεh/c0=k0εh is the host medium wavenumber, with k0 denoting the free space wavenumber, c0 the speed of light in free space and a1 the electric-dipole Mie scattering coefficient as in [10

10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

]. Moreover, ψ1(ρ)=ρj1(ρ)=sinρ/ρcosρ and ξ1(ρ)=ρh1(1)(ρ)=(i/ρ1)eiρ are the Riccati-Bessel functions [19

19. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

], and mr=εm/εh is the metallic nanosphere relative refractive index, with εm denoting the relative permittivity of the metallic nanospheres, which is described by the Drude model εm=εωp2/[ω(ω+iγ)], where ε is a high-frequency fitting parameter, ωp is the plasma frequency of the metal (expressed in rad/s) andγis the damping factor (expressed in 1/s). In general, this model provides a reasonably accurate description of the dielectric properties of the metal across the infrared and optical frequency ranges. In the UV regime, the effect of interband transitions should be taken into account in the metal permittivity response, employing for example the Lorentz-Drude model as in [20

20. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).

]. However in this frequency region this more accurate model would not change the wave physics described in this paper. In the case the array is in a dense host material, it would resonate in the visible region and the Drude model would provide a good approximation.

Consider now a 3D periodic array of nanospheres as in Fig. 1, immersed in a homogeneous background, with relative permittivity εh, for which each spherical nanoparticle is placed at positions rn=r0+dn, where n=(n1,n2,n3) is a triple index, and dn=n1ax^+n2by^+n3cz^ (where a caret on top of a bold letter refers to unit vector quantities), with n1,n2,n3=0,±1,±2,..., r0=x0x^+y0y^+z0z^, and a, b and c are the periodicities along x-, y- and z-direction, respectively [11

11. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1.

,21

21. S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express 1(6), 1077–1089 (2011).

].

Suppose that the array is either excited or a mode (a periodic field) is present, with wavevector kB=kxx^+kyy^+kzz^. Consequently, each nanosphere will have an electric dipole moment equal to pn=p0eikBdn (the polarization direction is fixed for symmetry reasons). Then, the local electric field acting on a nanosphere at position r0 is given by
E0loc(r0,kB)=Einc(r0)+G¯(r0,r0,kB)p0,
(3)
where G¯(r0,r0,kB)p0 is the electric field produced by all the other nanospheres but the one at position r0, and G¯(r0,r0,kB) represents the regularized periodic dyadic GF. This is defined as G¯(r,r0,kB)=G¯(r,r0,kB)G¯(r,r0), which is not singular at r=r0, and
G¯(r,r0,kB)=nG¯(r,rn)eikBdn
(4)
is the electric-field dyadic GF for the phased periodic array of nanospheres, and a bar under a bold letter refers to dyadic quantities. Here G¯(r,r) denotes the dyadic GF for a single dipole in the homogeneous background and is given by
G¯(r,r)=eikR4πεhε0[(k2R+ikR21R3)I¯(k2R+3ikR23R3)R^R^],
(5)
where R=RR^=rr is the vector from the source at r to the observer at r, and I¯ is the identity dyad, and R=|rr|. Substituting then the expression for the local field given in Eq. (3) into Eq. (1), one obtains p0=αeeEinc(r0)+αeeG¯(r0,r0,kB)p0, which leads to the linear system

A¯(kB)p0=αeeEinc(r0),A¯(kB)=I¯αeeG¯(r0,r0,kB).
(6)

3. Dispersion diagrams for 3D periodic arrays of plasmonic nanospheres

3.1 Transverse polarization (T-pol)

The dispersion diagrams for the Structure I outlined in Table 1 are shown in Fig. 2
Fig. 2 Dispersion diagram for T-pol. (a) Real part and (b) imaginary part of the wavenumber kz=βz+iαz, only for modes whose power flow is toward the positive z-direction, i.e., αz0. The black dotted curves show the behavior of each mode in the lossless case, i.e., when setting γ=0 in the silver constitutive relation.
for both the real and the imaginary parts of the wavenumber kz=βz+iαz with respect to the host wavenumber k in the case of T-pol. Only modes with αz0, i.e., those with power flow toward the positive z-direction, are shown.

Note that the presence of two transverse modes with moderately low attenuation constant αz is in agreement with what previously predicted in [9

9. A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

] by using the nano-transmission line network concept, and analytically in [25

25. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

] for an ideal lossless case.

3.2 Longitudinal polarization (L-pol)

The dispersion diagrams for the Structure I outlined in Table 1 are shown in Fig. 4
Fig. 4 As in Fig. 2, for L-pol.
for both the real and the imaginary parts of the wavenumber kz=βz+iαz in the case of L-pol (longitudinal with respect to the mode traveling direction z). Only the modes with αz0, i.e., whose power flow is toward the positive z-direction, are shown. Notice again that other modes with normalized attenuation constant larger than αzc/π=2 are present but not reported here since guided modes can travel a significant distance only when their attenuation constant is small, or αz<<k. The case relative to ideal lossless silver nanospheres is also reported in Fig. 4 as a black dotted line. The almost flat portion of the curve in Fig. 4(a) reveals a certain degree of non-locality, and thus spatial dispersion.

Note again that the presence of one longitudinal mode with moderately low attenuation constant αz is in agreement with what previously predicted in [9

9. A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

,25

25. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

].

4. Parametric analysis: comparison of modal dispersion diagrams

4.1 Transverse polarization (T-pol)

We compare in Fig. 6
Fig. 6 Comparison between the wavenumbers kz=βz+iαz of the Mode 1, T-pol, in the three structures in Table 1 versus frequency, normalized to their respective z-periodicity c. (a) Real part and (b) imaginary part.
the real and the imaginary parts of the wavenumber kz of Mode 1, with T-pol, computed for the three arrays in Table 1.

It can be easily noticed that increasing the periodicities in the directions orthogonal to the mode traveling direction (i.e., Structure III) highly reduces the wavenumber imaginary part, thus allowing the mode to travel longer distances for some frequency ranges, as explained in Sec. 4.3. For the case of Structure II, the phase propagation constant reaches the edge of the BZ (this does not happen for the two other structures). Furthermore, the bandgap is wider for Structure II.

4.2 Longitudinal polarization (L-pol)

We compare in Fig. 7
Fig. 7 As in Fig. 6, for L-pol.
the real and the imaginary parts of the wavenumber kz for Mode 1 (L-pol) computed for the three arrays in Table 1.

4.3 Figure of merit

In accord to the modes shown in Secs. 4.1 and 4.2, we are here interested in quantifying how far a mode can travel in terms of guided wavelengths. Therefore, it is convenient to define the figure of merit

F=βzαz.
(7)

The higher the figure of merit F is, the longer the traveled distance in terms of guided wavelength is, relatively to the attenuation. We show in Fig. 8
Fig. 8 Comparison between the figure of merit F of the Mode 1 in the three structures in Table 1 versus frequency. (a) Transversal and (b) longitudinal polarization.
the figure of merit versus frequency of the transverse and longitudinal cases analyzed in Secs. 4.1 and 4.2, comparing F of the Mode 1 for both T-pol and L-pol, for the three structures in Table 1. In both polarizations, the modes that show the best performance are the ones in Structure III in Table 1, for which a=b=2c: for the T-pol case, the peak of F is about 7×104, whereas it is about 19 for the L-pol case. Also notice how the figure of merit for Structure III for T-pol is large for a wide frequency range analyzed (i.e., about 100-650 THz), whereas for L-pol is large only for a narrow frequency range (i.e., 790-850 THz). In Structure III, modes with T-pol and L-pol can travel longer distances in terms of guided wavelengths than in the other two arrays, as can be inferred by the graphs in Fig. 8.

5. 3D lattice with finite thickness along z: transmission, reflection and absorption

We analyze propagation in Structure I in Table 1 (Fig. 1) made of 10 layers of silver nanospheres stacked along the z-direction (Fig. 9
Fig. 9 Magnitude of transmission, reflection and absorption coefficients for a stack of 10 layers of Structure I (only a transverse cut in the xz plane of a 3D array is shown). Results obtained by using the SDA and HFSS are in good agreement.
), by using (i) HFSS and (ii) the SDA with the Mie polarization expression in Eq. (2). The stack is illuminated by a normally incident plane wave traveling toward + z, and the magnitude of transmission T and reflection R of the stack are shown in Fig. 9, together with the absorption A=1|T|2|R|2, evaluated in dB as 10Log A. Results in Fig. 9 show good agreement between the HFSS full-wave simulation and the SDA theoretical results (also the phase of R and T obtained from the two methods are in good agreement, not shown). The strong stop-band between 730 THz (kc/π=0.365) and 890 THz (kc/π=0.445) is due to the large imaginary part of the attenuation constant αz of Mode 1 shown in Fig. 2. Notice also that absorption A is particularly strong at the edges of the stop-band. In the evaluation of the SDA results we have used the Ewald representation of the dyadic GF for 2D periodic arrays reported in [27

27. S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011).

].

6. Homogenization: effective refractive index

In general, considering only modes with a moderately low attenuation constant αz, a plane wave impinging on the composite material in Fig. 9 could excite any of the permitted modes, i.e., two modes with T-pol and one mode with L-pol, with different amplitudes (as also briefly discussed in [9

9. A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

]). However, for a normal incident plane wave, only modes with T-pol could be excited. We will show in Sec. 6.2 that even though the analyzed structure has a certain degree of spatial dispersion (as mentioned in Sec. 3), Mode 1 (T-pol) in Sec. 3.1 is “dominant”. Accordingly, a wave in the composite material, in the case of transverse polarization, could be described with good approximation as a TEM wave in a homogeneous material, which can in turn be represented by an effective refractive index. Though, strictly speaking, the effective permittivity characterizing the crystal as a homogenized material has a small degree of non-locality (spatial dispersion) [9

9. A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

,25

25. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

].

6.1 NRW retrieval method and comparison with theoretical results

Transmission and reflection coefficients for a stack of layers are here used to retrieve the effective refractive index of the metamaterial by using the Nicholson-Ross-Weir (NRW) method [28

28. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

33

33. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).

]. Treating the composite slab as a uniform continuous medium with same thickness t, according to NRW, the complex effective refractive index can be retrieved by
neff=±cos1(1R2+T22T)k0t+2πqk0t,
(8)
where q is an integer to be determined, and t=Nc, with N denoting the number of layers and c the separation between two contiguous layers. We address the reader to [31

31. D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

,32

32. C. R. Simovski, “On the extraction of local material parameters of metamaterials from experimental or simulated data,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 11.1.

] for guidelines on how to choose q and +/− in Eq. (8).

In Fig. 11
Fig. 11 Effective refractive index retrieved by using NRW from HFSS for different number of layers (4, 6 and 10), and by mode analysis.
we show the results obtained by using the NRW method (with HFSS simulations) for N = 4 and N = 6 layers of metallic nanospheres stacked along the z-direction in comparison to the result in Fig. 10. Observe that both real and imaginary parts agree well, although the imaginary part of the NRW-HFSS method is smaller around the bandgap for increasing number of layers. This is indeed in the region where |T| is close to 0. In this region, the HFSS results show some discrepancies probably due to the sensitivity of Eq. (8) to small perturbations of |T|, as |T|0 [37

37. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

] (which are mainly due to the HFSS numerical precision). To understand if this is true, in Fig. 12
Fig. 12 As in Fig. 11, but using R and T computed through SDA.
we show the calculations for the same stack of layers by using the SDA, which is more stable numerically than HFSS, especially for small |T|. Results from the NRW-SDA method are in good agreement to those obtained with mode analysis for both 4 (where the lowest |T|70dB) and 6 layers (where the lowest |T|110dB) (in Fig. 11 the simulation with 6 layers was already in disagreement). However, in the case of 10 layers, in the frequency region of extremely low value of |T|, also the NRW-SDA provides inaccurate results in the bandgap, due to the limited SDA numerical precision in the numerical algorithm for the estimation of the transmission coefficient and to the high sensitivity of Eq. (8) explained above.

6.2 Field fitting retrieval method

From the HFSS full-wave simulation of 10 layers of plasmonic nanospheres shown in Sec. 5, illuminated by a normally incident plane wave traveling toward +z, with electric field polarized along y, we extract the y component of the electric field (1 point per layer, at the center of each sphere) at two representative frequencies of the curves reported in Fig. 10: first at 745 THz (kc/π=0.3725), corresponding to large Re[neff] and low Im[neff], and then at 875 THz (kc/π=0.4375), corresponding to low Re[neff] and low Im[neff].

At each of the two frequency points the total field in each sphere is represented as the superposition of a direct (E+) and a reflected (E) wave, pertaining to a single mode (T-pol) with complex wavenumber kz, traveling along the ± z directions (see Fig. 9) as follows
Ey(n)=E+eikz(n12)c+Eeikz(n12)c,
(9)
where n=1,2,...,10, and E+,E,andkz are all unknown complex valued. The Ey(n) behavior obtained via HFSS is shown Figs. 13
Fig. 13 Field in 10 layers of plasmonic nanospheres. Comparison between the HFSS full-wave field (Simulation) and the fitting result at 745 THz (i.e., large Re[neff] and low Im[neff]).
and 14
Fig. 14 As in Fig. 13, at 875 THz (i.e., low Re[neff] and low Im[neff]).
(blue circles), at 745 and 875 THz, respectively. We then perform a curve fitting of the extracted data by using Eq. (9), and we report the field matching curves (both magnitude and phase) in comparison to the extracted full-wave simulation fields in Figs. 13 and 14 (blue curves). It can be observed that the fitting curves are in good agreement with the extracted fields, thus we can conclude that the field can be represented as the superposition of a direct and a reflected wave pertaining to a single mode. Moreover, from the fitting, we extract kz/k02.15+0.07i at 745 THz (which is compared with the mode solution in Fig. 10 where kz/k02.13+0.07i) andkz/k00.40+0.07i at 875 THz (which is compared with the mode solution in Fig. 10 where kz/k00.40+0.07i). A good agreement is then observed. Also, looking at the phase information in Figs. 13(b) and 14(b), it can be easily noticed that the phase accumulated by the wave at 745 THz after the 10 layers is much larger than the one at 875 THz, showing that the latter travels very slowly inside the structure, confirming that at this frequency we have an ENZ composite material. Moreover, the ratio between the reflected and the direct wave Ψ(n)=Ψ(0)e2ikz(n12)c (with Ψ(0)=E/E+) has been verified to be consistent for every n, where Ψ(0)=0.13ei0.078 at 745 THz and Ψ(0)=0.06ei1.55 at 875 THz. As a consequence, we can conclude that Mode 1 for Structure I, reported in the dispersion diagrams in Fig. 2, is dominant.

7. Conclusion

Appendix A: Ewald representation for the dyadic GF for 3D periodic arrays

The 3D array regularized dyadic GF reported above Eq. (4) can be computed as

G¯(r,r0,kB)=1εhε0[k2I¯+]G(r,r0,kB),
(10)

where G(r,r0,kB) is the regularized scalar GF, not singular at r=r0, defined as

G(r,r0,kB)=14πn(0,0,0)eikRnRneikBdn,
(11)

with n=(n1,n2,n3) a triple index, dn=n1ax^+n2by^+n3cz^, Rn=rr0dn, Rn=|Rn|, and denotes the Hessian dyadic differential operator. According to the Ewald representation [38

38. P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).

,39

39. K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

] the dyadic GF is split into the sum

G¯(r,r0,kB)=G¯spectral(r,r0,kB)+G¯spatial(r,r0,kB)
(12)

of a spectral and a spatial dyadic part, which are obtained by using in Eq. (10) an analogous split of the scalar GF G(r,r0,kB)=Gspectral(r,r0,kB)+Gspatial(r,r0,kB). The expressions of these latter two scalar terms, the spectral Gspectral(r,r0,kB) and the regularized spatial Gspatial(r,r0,kB) can be found in [13

13. I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007).

,14

14. G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

], and in the particular case where r=r0=0, as required in Eq. (6), it follows that

Gspectral(r0,r0,kB)=1abcneγn24E2γn2,
(13)
Gspatial(r0,r0,kB)=18πn(0,0,0)eikBdnRnf(Rn)+f(0)2ik8π,
(14)

where γn2=|kB+kn|2k2, with kn=(2πn1/a)x^+(2πn2/b)y^+(2πn3/c)z^, and

f(Rn)=eikRnerfc(β)+e+ikRnerfc(β+).
(15)

Here erfc(β±) denotes the complementary error function of argument β±=RnE±ik/(2E), where E is the Ewald parameter as in [40

40. A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

], which is in general chosen as

E=[π2(1/a2+1/b2+1/c2)a2+b2+c2]1/4.
(16)

Also, in Eq. (14) a prime (f) denotes a derivative of f with respect to its argument.

Differentiating Eqs. (13) and (14) in rectangular and spherical coordinates, respectively, leads to the explicit representations for the Hessian dyads

Gspectral(r0,r0,kB)=1abcn(kB+kn)(kB+kn)eγn24E2γn2,
(17)
Gspatial(r0,r0,kB)=18πn(0,0,0)eikBdnF¯spatial,n+18πf(0)+2ik33I¯,
(18)

where

F¯spatial,n=(f(Rn)Rn2f(Rn)Rn3)I¯+(f(Rn)Rn3f(Rn)Rn2+3f(Rn)Rn3)R^nR^n,
(19)

with R^n=Rn/Rn.

In Eqs. (19) and (18), a double and a triple prime (f and f) denote the second and third derivatives of f in Eq. (15) with respect to its argument, respectively.

As a last remark, we want to mention that there is a difference between the scalar and the dyadic GF in terms of number of elements needed for convergence in Eqs. (13)-(14) and (17)-(18): namely, we have observed that it was sufficient to use 11 terms for each index (i.e., n1,n2,n3=5,...,5) to obtain a converging dyadic GF, whereas 3 terms for each index (i.e., n1,n2,n3=1,0,1) were usually enough for the scalar GF to converge.

Acknowledgment

The authors acknowledge partial support from the National Science Foundation, award NSF-CMMI #1101074, and from the European Commission FP7/2008, research area “NMP-2008-2.2-2 Nanostructured meta-materials”, grant “METACHEM”, no. 228762. The authors also thank Ansys for providing HFSS that was instrumental in this analysis.

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A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B 28(6), 1446–1458 (2011).

8.

S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: Fabrication, optical properties and applications,” in Selected Topics in Photonic Crystals and Metamaterials, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 143–196.

9.

A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

10.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

11.

S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1.

12.

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

13.

I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007).

14.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

15.

A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express 19(3), 2754–2772 (2011).

16.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

17.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

18.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994).

19.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

20.

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).

21.

S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express 1(6), 1077–1089 (2011).

22.

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006).

23.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

24.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

25.

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

26.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).

27.

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011).

28.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

29.

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974).

30.

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997).

31.

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

32.

C. R. Simovski, “On the extraction of local material parameters of metamaterials from experimental or simulated data,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 11.1.

33.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).

34.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).

35.

A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 9.1.

36.

S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Characterization of the optical modes in 3D-periodic arrays of metallic nanospheres,” in URSI General Assembly and Scientific Symposium (Istanbul, Turkey, 2011).

37.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

38.

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).

39.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

40.

A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: September 13, 2011
Revised Manuscript: November 16, 2011
Manuscript Accepted: November 16, 2011
Published: December 7, 2011

Citation
Salvatore Campione, Sergiy Steshenko, Matteo Albani, and Filippo Capolino, "Complex modes and effective refractive index in 3D periodic arrays of plasmonic nanospheres," Opt. Express 19, 26027-26043 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26027


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References

  1. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B74(20), 205436 (2006).
  2. R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
  3. R. Sainidou and G. F. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express16(7), 4499–4506 (2008).
  4. D. Van Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett.34(4), 422–424 (2009).
  5. R. A. Shore and A. D. Yaghjian, Complex Waves on 1D, 2D, and 3D Periodic Arrays of Lossy and Lossless Magnetodielectric Spheres (Air Force Research Laboratory, Hanscom, AFB, MA, 2010; revised, 2011).
  6. S. Campione, S. Steshenko, and F. Capolino, “Complex bound and leaky modes in chains of plasmonic nanospheres,” Opt. Express19(19), 18345–18363 (2011).
  7. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B28(6), 1446–1458 (2011).
  8. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: Fabrication, optical properties and applications,” in Selected Topics in Photonic Crystals and Metamaterials, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 143–196.
  9. A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B75(2), 024304 (2007).
  10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  11. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1.
  12. P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett.8(3), 124–126 (1998).
  13. I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett.49(6), 1353–1357 (2007).
  14. G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech.56(9), 2069–2075 (2008).
  15. A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express19(3), 2754–2772 (2011).
  16. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  17. V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B70(5), 054202 (2004).
  18. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A11(11), 2851–2861 (1994).
  19. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).
  20. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37(22), 5271–5283 (1998).
  21. S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
  22. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14(4), 1557–1567 (2006).
  23. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B62(23), 15299–15302 (2000).
  24. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
  25. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
  26. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
  27. S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1 Pt 2), 016607 (2011).
  28. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.19(4), 377–382 (1970).
  29. W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62(1), 33–36 (1974).
  30. A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech.45(1), 52–57 (1997).
  31. D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
  32. C. R. Simovski, “On the extraction of local material parameters of metamaterials from experimental or simulated data,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 11.1.
  33. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).
  34. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).
  35. A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 9.1.
  36. S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Characterization of the optical modes in 3D-periodic arrays of metallic nanospheres,” in URSI General Assembly and Scientific Symposium (Istanbul, Turkey, 2011).
  37. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1 Pt 2), 016608 (2004).
  38. P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin)369(3), 253–287 (1921).
  39. K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys.63(1), 222–235 (1986).
  40. A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett.10(5), 168–170 (2000).

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