## Photonic band structures of periodic arrays of pores in a metallic host: tight-binding beyond the quasistatic approximation |

Optics Express, Vol. 21, Issue 17, pp. 19834-19849 (2013)

http://dx.doi.org/10.1364/OE.21.019834

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### Abstract

We have calculated the photonic band structures of metallic inverse opals and of periodic linear chains of spherical pores in a metallic host, below a plasma frequency *ω*_{p}. In both cases, we use a tight-binding approximation, assuming a Drude dielectric function for the metallic component, but without making the quasistatic approximation. The tight-binding modes are linear combinations of the single-cavity transverse magnetic (TM) modes. For the inverse-opal structures, the lowest modes are analogous to those constructed from the three degenerate atomic *p*-states in fcc crystals. For the linear chains, in the limit of small spheres compared to a wavelength, the results bear some qualitative resemblance to the dispersion relation for metal spheres in an insulating host, as calculated by Brongersma *et al.* [Phys. Rev. B **62**, R16356 (2000)]. Because the electromagnetic fields of these modes decay exponentially in the metal, there are no radiative losses, in contrast to the case of arrays of metallic spheres in air. We suggest that this tight-binding approach to photonic band structures of such metallic inverse materials may be a useful approach for studying photonic crystals containing metallic components, even beyond the quasistatic approximation.

© 2013 OSA

## 1. Introduction

*ω*and Bloch vector

**k**in media in which the dielectric constant is a periodic function of position. A major reason for such interest is the possibility of producing photonic band gaps, i.e., frequency regions, extending through all

**k**-space, where electromagnetic waves cannot propagate through the medium. Such media have many potentially valuable applications, including possible use as filters and in films with rejection-wavelength tuning [1

1. J. Q. Xia, Y. R. Ying, and S. H. Foulger, “Electric-Field-Induced Rejection-Wavelength Tuning of Photonic-Bandgap Composites,” Adv. Mater. **17**, 2463–2467 (2005) [CrossRef] .

2. Kurt Busch and Sajeev John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. **83**, 967–970 (1999) [CrossRef] .

3. Eli Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987) [CrossRef] [PubMed] .

4. A. Scherer, O. Painter, B. D’Urso, R. Lee, and A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B **16**, 3906 (1998) [CrossRef] .

5. Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides, ” Phys. Rev. Lett. **77**, 3787–3790 (1996) [CrossRef] [PubMed] .

6. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science **284**, 1819–1821 (1999) [CrossRef] [PubMed] .

7. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber,” Science **298**, 399–402 (2002) [CrossRef] [PubMed] .

8. Y. Cao, J. O. Schenk, and M. A. Fiddy, “Third order nonlinear effect near a degenerate band edge,” Opt. Photonics Lett. **1**, 1–7 (2008) [CrossRef] .

*et al.*[9

9. Arthur R. McGurn and Alexei A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B **48**, 17576–17579 (1993) [CrossRef] .

*f*(i.e., volume fraction of metal spheres or cylinders) satisfied

*f*≤ 0.1%.

*et al.*[10

10. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B **50**, 16835–16844 (1994) [CrossRef] .

*f*and

*ω*>

*ω*

_{p}, the calculated photonic band structures are just slightly perturbed versions of the dispersion curves for electromagnetic waves in vacuum. However, for

*ω*<

*ω*

_{p}and

**H**-polarized waves (magnetic field

**H**parallel to the cylinders), they obtained many nearly flat bands for

*ω*<

*ω*

_{p}; these bands were found to converge very slowly with increasing numbers of plane waves. They later extended this work to systems with dissipation [11

11. V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B **55**, 7427–7444 (1997) [CrossRef] .

*et al.*[12

12. I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B **48**, 5004–5012 (1993) [CrossRef] .

2. Kurt Busch and Sajeev John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. **83**, 967–970 (1999) [CrossRef] .

*et al.*[13

13. Mark L. Brongersma, John W. Hartman, and Harry A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**, R16356–R16359 (2000) [CrossRef] .

14. Stefan A. Maier, Pieter G. Kik, Harry A. Atwater, Sheffer Meltzer, Elad Harel, Bruce E. Koel, and Ari A.G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003) [CrossRef] [PubMed] .

27. Matthew D. Arnold, Martin G. Blaber, Michael J. Ford, and Nadine Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express **18**, 7528–7542 (2010) [CrossRef] [PubMed] .

28. Sung Yong Park and David Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B **69**, 125418 (2004) [CrossRef] .

**E**= 0, where

**E**is the electric field.

29. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004) [CrossRef] .

*et al.*[30

30. D. Gaillot, T. Yamashita, and C. J. Summers, “Photonic band gaps in highly conformal inverse-opal based photonic crystals,” Phys. Rev. B **72**, 205109 (2005) [CrossRef] .

30. D. Gaillot, T. Yamashita, and C. J. Summers, “Photonic band gaps in highly conformal inverse-opal based photonic crystals,” Phys. Rev. B **72**, 205109 (2005) [CrossRef] .

*f*of air voids using 3D finite difference time domain (FDTD) method. It was found that for certain values of

*f*, a complete band gap opens up between the eighth and ninth bands.

30. D. Gaillot, T. Yamashita, and C. J. Summers, “Photonic band gaps in highly conformal inverse-opal based photonic crystals,” Phys. Rev. B **72**, 205109 (2005) [CrossRef] .

*et al.*[9

9. Arthur R. McGurn and Alexei A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B **48**, 17576–17579 (1993) [CrossRef] .

31. Ali E. Aliev, Sergey B. Lee, Anvar A. Zakhidov, and Ray H. Baughman, “Superconductivity in Pb inverse opal,” Physica C **453**, 15–23 (2007) [CrossRef] .

13. Mark L. Brongersma, John W. Hartman, and Harry A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**, R16356–R16359 (2000) [CrossRef] .

*ω*

_{p}is a tight-binding approximation which is valid even in the non-quasistatic regime. Because the analogs of the tight-binding atomic states decay exponentially in the metallic host medium, the resulting tight-binding waves do not lose energy radiatively, as do the corresponding waves along 1D chains of metallic nanoparticles in air. The absence of radiative decay has been previously noted in a multiple-scattering calculation of the band structure of a periodic array of pores in a host metal [32

32. G. Gantzounis and N. Stefanou, “Cavity-plasmon waveguides: Multiple scattering calculations of dispersion in weakly coupled dielectric nanocavities in a metallic host material,” Phys. Rev. B **74**, 085102 (2006) [CrossRef] .

*ω*<

*ω*

_{p}. In Section 3, we give the numerical results for the TM and TE modes of a single cavity and those of the tight-binding method for the metal inverse opals and the linear chain of nanopores. Section 4 presents a summary and discussion.

## 2. Formalism

*R*. We first write down the equations for the TM and TE modes of a spherical cavity in a Drude metal. Then, we present a tight-binding method for

*ω*<

*ω*

_{p}.

### 2.1. Spherical Cavity

#### 2.1.1. TM Modes

**B**field. To that end, we combine the two homogeneous Maxwell equations

**B**. We express the position- and frequency-dependent dielectric function

*ε*(

**x**,

*ω*) as

*θ*(

**x**) = 1 inside the metallic region and

*θ*(

**x**) = 0 elsewhere. Then, after a little algebra, we obtain This expression gives rise to different equations inside and outside the void.

**B**and

**E**are given in the standard literature [33]. The coefficients of the solutions inside and outside the cavity can then be determined from the boundary conditions at

*r*=

*R*. The allowed frequencies for

*ω*<

*ω*

_{p}are then found to satisfy where

*k*=

*ω/c*and

*j*is the spherical Bessel function, and

_{ℓ}*k*is the modified spherical Bessel function (note that this

_{ℓ}*k*is different from the wave vectors

_{ℓ}*k*and

*k′*). In the limit

*kR*≫ 1 and

*k′R*≪ 1, we can readily obtain the asymptotic forms of the solutions to Eq. (2). The result is

*k′*

^{2}(

*ℓ*+ 1) =

*k*

^{2}

*ℓ*, or, for a Drude metal, The largest value,

*ℓ*= 1 and the limiting value for large

*ℓ*is

### 2.2. Tight-Binding Approach to Modes for ω < ω_{p}

*ω*<

*ω*

_{p}. We apply the resulting method, first, to an fcc lattice of pores, and then to a linear chain of spherical pores in a metallic host.

*λ*th mode by

**E**

*(*

_{λ}**x**). This field satisfies where

*𝒪*is a Hermitian operator, the eigenstates corresponding to unequal eigenvalues

**E**

*(*

_{λ}**x**) is real for

*ω*<

*ω*

_{p}, the complex conjugation is, in fact, unnecessary.

*ℓ*= 1, and there should be three of these. For a spherical cavity, all three are degenerate, i.e., all three have the same eigenfrequencies. Even though the three modes have equal frequencies, one can always choose an orthonormal set, with electric fields

**E**

_{1},

**E**

_{2}, and

**E**

_{3}satisfying the orthonormality relation in Eq. (6).

**R**. Here,

*𝒪*is the “Hamiltonian” of the system as defined implicitly in Eq. (5).

*ℓ*= 1 single-cavity modes. In order to do this, we first make the standard tight-binding assumption that the “atomic” states corresponding to different cavities are orthogonal: This orthogonality of states on different cavities is reasonable since the fields fall off exponentially with separation.

**k**is a Bloch vector, and the

**R**’s are the Bravais lattice vectors. In writing Eq. (9), we have assumed that there are

*N*identical spherical cavities, and that the Bloch states satisfy the usual periodic boundary conditions of Born-von Karman type. We also introduce the elements of the “Hamiltonian” matrix

*ω*(

**k**) by diagonalizing a 3 × 3 matrix as follows: where

*ω*

_{at}is the eigenvalue of a single-cavity mode. The solutions to these equations give the three

*p*-bands for a periodic lattice of cavities in a metallic host. This procedure is analogous to that used in the well-known procedure for obtaining tight-binding bands from three degenerate

*p*-bands in the electronic structure of conventional solids (see, e.g., [34]).

*𝒪*is just a little trickier. We first note that

*𝒪*=

*𝒪*

**+**

_{R}*𝒪′*, where

*𝒪*

**is the single-cavity operator:**

_{R}*𝒪*

**= ∇ × (∇×) if**

_{R}**x**is inside the

**R**th cavity and

**E**

*is an eigenstate of*

_{β}*𝒪*

**with an eigenvalue**

_{R}*𝒪*

**does not contribute to the matrix element**

_{R}*M*, which is therefore just given by We can also write where is a step function which is unity inside the cavity centered at

_{α,β}**R′**and is zero otherwise.

**R′**= 0. In this case, we finally will get where the integral runs just over the cavity centered at the origin. As a further approximation, we can just replace

**E**

*(*

_{β}**x**−

**R**) by the value of this function at the origin, i.e.,

**E**

*(−*

_{β}**R**). Then this field can be taken outside the integral and we just have where once again the integral runs over the cavity centered at the origin.

*Cartesian*components of the normalized eigenfunctions

**E**

*(*

_{α}**x**) are readily calculated from the solutions discussed in Sec. 2.1.1 and the integral in Eq. (17). In fact, it turns out that all the integrals entering the matrix element can be obtained analytically. Given this matrix element, the computation of the full tight-binding band structure of the three

*p*-bands is also straightforward.

## 3. Numerical Results

*R/d*, the ratio of the void radius to the fcc lattice constant, and

*ω*

_{p}

*d/c*. Of course, other combinations of these parameters would serve equally well. Given these parameters, the scaled frequencies

*ω/ω*

_{p}are functions of the scaled wave vector

**k**

*d*. In what follows, we consider only those scaled units.

*R*to the nearest neighbor distance

*f*= 0.160. In our calculations, we also arbitrarily use the value

*ω*

_{p}

*d/c*= 1. For typical metallic values of

*ω*

_{p}, this would correspond to

*d*of order 20 nm. For the linear chain of nanopores (see below), we use

*d*to denote the separation between the centers of two adjacent nanopores and

*R*to denote the radius of a nanopore as in Fig. 1(b); we take the ratio

*R/d*= 1/3 at first, then change it later.

**k**values at symmetry points in the Brillouin zone. These are Γ = (0, 0, 0),

*X*= (2

*π/d*)(0, 0, 1),

*U*= (2

*π/d*)(1/4, 1/4, 1),

*L*= (2

*π/d*)(1/2, 1/2, 1/2),

*W*= (2

*π/d*)(1/2, 0, 1), and

*K*= (2

*π/d*)(3/4, 0, 3/4).

*ω*

_{p}is the plasma frequency of the conduction electrons.

*ε*(

*ω*) < 0 when

*ω*<

*ω*

_{p}, while

*ε*(

*ω*) > 0 when

*ω*>

*ω*

_{p}. Our calculations are thus carried out assuming that the Drude relaxation time

*τ*→

**∞**. For a metal in its normal state,

*n*is the conduction electron density and

*m*is the electron mass. Note that with this choice of dielectric function, the entire band structure can be expressed in scaled form of

*ω/ω*

_{p}.

*single*cavity, which could be considered a single “atom” of the void lattice. We show only results for

*ω*<

*ω*

_{p}, since these are the results most relevant to possible narrow-band photonic states in the inverse opal structure. Our results for

*ω*<

*ω*

_{p}for an isolated spherical cavity in an infinite medium, and when

*kR*≪ 1 and

*k′R*≪ 1 are given in Table 1. These two inequalities are reasonable for the choice of “inverse opal” system parameters

*ω*

_{p}

*d/c*= 1, because

*ω*axis for

*ℓ*> 5, so that it is difficult to get eigenfrequencies for

*ℓ*> 5 in the isolated spherical cavity. However the eigenfrequencies continue to exist even for

*ℓ*> 5 when

*kR*≪ 1 and

*k′R*≪ 1.

*ω*<

*ω*

_{p}with

*ω*

_{p}

*d/c*= 1. This fact is consistent with that the eigenvalues for

*ω*<

*ω*

_{p}do not exist for TE modes when

*kR*≪ 1 and

*k′R*≪ 1.

**R**= (

*d*/2)(±1, ±1, 0), (

*d*/2)(±1, ∓1, 0), (

*d*/2)(±1, 0, ±1), (

*d*/2)(±1, 0, ∓1), (

*d*/2)(0, ±1, ±1), and (

*d*/2)(0, ±1, ∓1). Assuming

*ω*

_{p}

*d/c*= 1.0 and using

*ω*

_{at}

*d*/(2

*πc*) = 0.1296 (

*ω*

_{at}= 0.8143

*ω*

_{p}) for

*ℓ*= 1 in an infinite medium, we get the tight-binding results in Fig. 2. This Figure shows three separate bands in the

*X*-

*U*-

*L*region and

*X*-

*W*-

*K*region, which behave as expected for the

*p*-bands. The bandwidth is relatively small as

*M*(

_{α,β}**R**)

*d*

^{2}∼ 0.001, which proves the general relation between the bandwidth and the overlap integral [34]. All three bands are degenerate at

**k**= 0 (the Γ point). In addition, there is a double degeneracy when

**k**is directed along either a cube axis (Γ-

*X*) or a cube body diagonal (Γ-

*L*), the higher (concave upward) bands being degenerate in both cases. The lower two bands have a band gap at the

*U*point, and these bands cross at the

*W*point.

*ω*

_{p}

*d/c*= 1.0, so this exponential decay does not fully set in until a fairly large number of shells is included. For a larger value of

*ω*

_{p}

*d/c*, the eigenfrequencies will converge much more quickly with number of shells. This problem is worse in 3D than in 1D (see below), because the magnitude of the hopping integral, as a function of separation

*r*, varies as 1/

*r*

^{3}for small

*ω*

_{p}

*r/c*, while the number of terms in each shell increases as

*r*

^{2}. Convergence is assured, however, with a sufficient number of shells, because of the exponential decay which sets in at large

*r*. As is seen below, the convergence is much faster in 1D.

**R**=

*d*(0, 0, ±

*n*), where ±

*n*is the

*n*th nearest-neighbor,

*d*is the separation between two nanopores and we assume that the chain is directed along the

*z*axis. We can calculate the tight-binding band structure including as many sets of neighbors ±

*n*as we wish. We use the ratio

*R/d*= 1/3 and the dimensionless parameter

*ω*

_{p}

*d/c*= 0.35. These are arbitrarily chosen to be the same as used in [13

13. Mark L. Brongersma, John W. Hartman, and Harry A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**, R16356–R16359 (2000) [CrossRef] .

*ω*

_{at}

*d*/(2

*πc*) = 0.0454 (

*ω*

_{at}= 0.8150

*ω*

_{p}) for

*ℓ*= 1 in an infinite medium. Our resulting tight-binding dispersion relations are shown by open triangles in Fig. 4 with only nearest-neighbors included. The transverse (T) branches are twofold degenerate, while the longitudinal (L) branch is non-degenerate. As we increase the number of nearest-neighbors (nn’s) included, the separation between the L and T branches increases at the zone center but decreases at the zone boundary, as shown in Fig. 4. The sum also converges quickly, so there is only a slight difference between the dispersion relation including through the next-nearest-neighbors and that including through the 5th nearest-neighbors. However, if one includes more than nearest-neighbor overlap, the L and T branches no longer cross exactly at

*k*= ±

*π*/(2

*d*). [32

32. G. Gantzounis and N. Stefanou, “Cavity-plasmon waveguides: Multiple scattering calculations of dispersion in weakly coupled dielectric nanocavities in a metallic host material,” Phys. Rev. B **74**, 085102 (2006) [CrossRef] .

*ℓ*= 1 dispersion relations crossed at

*k*= ±

*π*/(2

*d*) if only nearest-neighbor hopping is included.

*ω*at

*k*= 0 since the difference between different numbers of nn’s is the most evident there. For

*ω*

_{p}

*d/c*= 1.0 and

*ω*

_{at}

*d*/(2

*πc*) = 0.1291 (

*ω*

_{at}= 0.8112

*ω*

_{p}), we find that both the L and T frequencies change by less than 0.3% in going to the 5th nn shell, and are unchanged to within 0.05% thereafter, up to 10 nn shells. This convergence is quicker than in 3D and can be readily seen in the

*ω*(

*k*) plots.

*ω*

_{p}

*d/c*, namely 1.0, 2.0, and 5.0. Such calculations are possible here because our calculations are non-quasistatic, so that the overlap integral between neighboring spheres falls off exponentially with separation. The results, and the corresponding results including more overlap integrals, for a typical example,

*ω*

_{p}

*d/c*= 5.0, are shown in Fig. 5, since the results for

*ω*

_{p}

*d/c*= 1.0 and 2.0 are similar to Fig. 4 except for the increase of

*ω*along the

*y*-axis and

*ω*

_{at}. It is also striking that, as

*ω*

_{p}

*d/c*increases in going from Fig. 4 to 5, the ratio

*r*

_{LT}of the width of the L band to that of the T band decreases. In Fig. 4,

*r*

_{LT}> 1, while in Fig. 5,

*r*

_{LT}< 1.

*ω*

_{at}. For the nn case, the T and L bands cross at ±

*π*/(2

*d*), while they cross at smaller values than |

*π*/(2

*d*)| when further neighbors are included, as can be seen in Figs. 4 and 5, but the crossing points get closer to ±

*π*/(2

*d*) as

*ω*

_{p}

*d/c*increases. Also, the effects of including further neighbors become smaller as

*ω*

_{p}

*d/c*increases; they are smallest at

*ω*

_{p}

*d/c*= 5.0, as can be seen in Fig. 5.

*R/d*other than 1/3, but still keeping the same value of

*ω*

_{p}

*d/c*= 0.35 and including up to the fifth nearest-neighbors. For a smaller

*R/d*= 0.25, the variation of the band energies with

*k*becomes smaller, as seen by open triangles in Fig. 6, than it is in Fig. 4, but the crossing points between the L and T branches still occur at values of |

*k*| slightly less than |

*π*/(2

*d*)|. This behavior becomes clearer when the results for more values of

*R/d*are plotted together as in Fig. 6. As

*R/d*increases, the variation of the band energies with

*k*, and the separation between the L and T branches at both the zone center and zone boundary, increase, but the L and T branches still cross at values of |

*k*| slightly less than |

*π*/(2

*d*)|. Furthermore, the separation between the L and T bands increases slightly at

*k*= 0, but decreases slightly at

*k*= ±

*π/d*compared to the results with only nn’s included. We show only

*R/d*up to 0.4 in this Figure because, in the quasistatic limit, there is evidence that for larger values of

*R/d*the dispersion relations are significantly modified by higher values of

*ℓ*[28

28. Sung Yong Park and David Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B **69**, 125418 (2004) [CrossRef] .

## 4. Discussion

*ω*

_{p}, when

*ℓ*= 1 using a tight-binding approximation. In both cases, we include only the

*ℓ*= 1 “atomic” states of the voids. As a possible point of comparison, we have also computed the same band structures using the asymptotic forms of the spherical and modified spherical Bessel functions for small void radius. In this asymptotic region, there are only TM modes. The results for the linear chain of voids somewhat resemble those of [13

**62**, R16356–R16359 (2000) [CrossRef] .

*et al.*[35

35. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-Binding Parametrization for Photonic Band Gap Materials,” Phys. Rev. Lett. **81**, 1405–1408 (1998) [CrossRef] .

35. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-Binding Parametrization for Photonic Band Gap Materials,” Phys. Rev. Lett. **81**, 1405–1408 (1998) [CrossRef] .

35. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-Binding Parametrization for Photonic Band Gap Materials,” Phys. Rev. Lett. **81**, 1405–1408 (1998) [CrossRef] .

**81**, 1405–1408 (1998) [CrossRef] .

32. G. Gantzounis and N. Stefanou, “Cavity-plasmon waveguides: Multiple scattering calculations of dispersion in weakly coupled dielectric nanocavities in a metallic host material,” Phys. Rev. B **74**, 085102 (2006) [CrossRef] .

36. N. Stefanou, A. Modinos, and V. Yannopapas, “Optical transparency of mesoporous metals,” Solid State Commun. **118**, 69–73 (2001) [CrossRef] .

*X*direction in the Brillouin zone with those presented in [36

36. N. Stefanou, A. Modinos, and V. Yannopapas, “Optical transparency of mesoporous metals,” Solid State Commun. **118**, 69–73 (2001) [CrossRef] .

36. N. Stefanou, A. Modinos, and V. Yannopapas, “Optical transparency of mesoporous metals,” Solid State Commun. **118**, 69–73 (2001) [CrossRef] .

*ε*= 11.9 in [36

**118**, 69–73 (2001) [CrossRef] .

*ε*= 1 in our calculations). Nevertheless, the doubly degenerate bands in our calculation appear to have the same general shape as the corresponding bands shown in their Fig. 1(a), that is, an increase with increasing

*k*starting from the Γ point followed by a flattening of these bands as the point

_{z}*X*is approached. The non-degenerate band along Γ −

*X*in our calculations falls monotonically starting from the Γ point (lower band along the line Γ −

*X*in our Fig. 2), whereas the corresponding band shown in their Fig. 1(a) seems to be nearly flat. We tentatively attribute this difference primarily to the difference in the parameters of the two calculations.

*n*of nearest neighbor shells included in the calculations. In order to answer this question, we show in Fig. 3 the calculated energies at Γ as functions of

*n*, for the inverse opal structure. As mentioned earlier, the frequency will certainly converge extremely well for a sufficient number of neighbor shells, because the hopping integral will eventually fall off exponentially with separation. This convergence is faster in 1D than in 3D because the magnitude of the hopping integral varies as 1/

*r*

^{3}for small

*ω*

_{p}

*r/c*, while the number of terms in each shell increases as

*r*

^{2}.

**74**, 085102 (2006) [CrossRef] .

*ka*≪ 1 (where

*a*is the cavity radius), leads to a dispersion relation of the tight-binding form

*ω*=

*ω*

_{0}+

*ω*

_{1}cos(

*kd*), where

*d*is distance between the sphere centers. This result is obtained provided that one neglects interactions other than between nearest neighbor cavities, and also that one disregards interactions between the lowest (

*ℓ*= 1) plasmon band and all the higher bands. In our present tight-binding calculation, we do not need to assume

*ka*≪ 1, and we are also able to go beyond nearest neighbor hopping, though we do include only the

*ℓ*= 1 bands. Thus, our approach is somewhat different from that of [32

**74**, 085102 (2006) [CrossRef] .

*ℓ*> 1, our approach could readily be extended to do so. Finally, our bands do not cross exactly at |

*k*|

*d*=

*π*/2 when we include more than one shell of neighbors in the tight-binding calculation; so our dispersion relations are not exactly of the form

*ω*

_{0}+

*ω*

_{1}cos(

*kd*) found in [32

**74**, 085102 (2006) [CrossRef] .

**74**, 085102 (2006) [CrossRef] .

**R**. This neglect is reasonable, for our problem, because the basis functions decay exponentially into the metal, and thus the overlap should be small.

37. B. A. McKinnon and T. C. Choy, “Significance of nonorthogonality in tight-binding models,” Phys. Rev. B **52**, 14531–14538 (1995) [CrossRef] .

38. Madhu Menon and K. R. Subbaswamy, “Transferable nonorthogonal tight-binding scheme for silicon,” Phys. Rev. B **50**, 11577–11582 (1994) [CrossRef] .

39. Per-Olov Löwdin, “On the NonOrthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals,” J. Chem. Phys. **18**, 365–375 (1950) [CrossRef] .

*H*with an orthonormal basis from the the original Hamiltonian

*ℋ*with a nonorthogonal basis by means of the transformation where

*S*is a matrix whose elements are given by Eq. (20), and

*I*is the identity matrix. To carry out this procedure, one would simply need to calculate the overlap matrix elements

*S*(

_{λμ}**R**) and execute the transformation in Eq. (21). We believe that the resulting corrections to the band structure would be small, because, as in the analogous electronic problem for narrow tight-binding bands, the off-diagonal elements of the overlap matrix are expected to be small for exponentially decaying basis functions, compared to those of the original Hamiltonian

*ℋ*.

*R/d*is greater than about 0.4, it becomes important to include more than just

*ℓ*= 1, as in [28

28. Sung Yong Park and David Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B **69**, 125418 (2004) [CrossRef] .

*ℓ*’s might be rather difficult in the present dynamical case, though it would be straightforward in the quasistatic limit for 1D chains of spherical nanopores.

## Acknowledgments

## References and links

1. | J. Q. Xia, Y. R. Ying, and S. H. Foulger, “Electric-Field-Induced Rejection-Wavelength Tuning of Photonic-Bandgap Composites,” Adv. Mater. |

2. | Kurt Busch and Sajeev John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. |

3. | Eli Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

4. | A. Scherer, O. Painter, B. D’Urso, R. Lee, and A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B |

5. | Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides, ” Phys. Rev. Lett. |

6. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science |

7. | F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber,” Science |

8. | Y. Cao, J. O. Schenk, and M. A. Fiddy, “Third order nonlinear effect near a degenerate band edge,” Opt. Photonics Lett. |

9. | Arthur R. McGurn and Alexei A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B |

10. | V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B |

11. | V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B |

12. | I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B |

13. | Mark L. Brongersma, John W. Hartman, and Harry A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

14. | Stefan A. Maier, Pieter G. Kik, Harry A. Atwater, Sheffer Meltzer, Elad Harel, Bruce E. Koel, and Ari A.G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. |

15. | S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics–A Route to Nanoscale Optical Devices,” Adv. Mater. |

16. | Prashant K. Jain, Wenyu Huang, and Mostafa A. El-Sayed, “On the Universal Scaling Behavior of the Distance Decay of Plasmon Coupling in Metal Nanoparticle Pairs: A Plasmon Ruler Equation,” Nano Lett. |

17. | LinLin Zhao, K. Lance Kelly, and George C. Schatz, “The Extinction Spectra of Silver Nanoparticle Arrays: Influence of Array Structure on Plasmon Resonance Wavelength and Width,” J. Phys. Chem. B |

18. | Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. |

19. | L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, and A. Polman, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B |

20. | Shengli Zou and George C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. |

21. | A. F. Koenderink and Albert Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B |

22. | Andrea Alù and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

23. | Alexandre Bouhelier, Renaud Bachelot, Jin Seo Im, Gary P. Wiederrecht, Gilles Lerondel, Sergei Kostcheev, and Pascal Royer, “Electromagnetic Interactions in Plasmonic Nanoparticle Arrays,” J. Phys. Chem. B |

24. | Vadim A. Markel and Andrey K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B |

25. | Yu-Rong Zhen, Kin Hung Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B |

26. | W. Jacak, J. Krasnyj, J. Jacak, A. Chepok, L. Jacak, W. Donderowicz, D. Z. Hu, and D. M. Schaadt, “Undamped collective surface plasmon oscillations along metallic nanosphere chains,” J. Appl. Phys. |

27. | Matthew D. Arnold, Martin G. Blaber, Michael J. Ford, and Nadine Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express |

28. | Sung Yong Park and David Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B |

29. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

30. | D. Gaillot, T. Yamashita, and C. J. Summers, “Photonic band gaps in highly conformal inverse-opal based photonic crystals,” Phys. Rev. B |

31. | Ali E. Aliev, Sergey B. Lee, Anvar A. Zakhidov, and Ray H. Baughman, “Superconductivity in Pb inverse opal,” Physica C |

32. | G. Gantzounis and N. Stefanou, “Cavity-plasmon waveguides: Multiple scattering calculations of dispersion in weakly coupled dielectric nanocavities in a metallic host material,” Phys. Rev. B |

33. | See, e.g.,J. D. Jackson, “Earth and Ionosphere as a Resonant Cavity: Schumann Resonances,” in |

34. | See, e.g.,N. W. Ashcroft and N. D. Mermin, “Problem 2. Tight-Binding |

35. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-Binding Parametrization for Photonic Band Gap Materials,” Phys. Rev. Lett. |

36. | N. Stefanou, A. Modinos, and V. Yannopapas, “Optical transparency of mesoporous metals,” Solid State Commun. |

37. | B. A. McKinnon and T. C. Choy, “Significance of nonorthogonality in tight-binding models,” Phys. Rev. B |

38. | Madhu Menon and K. R. Subbaswamy, “Transferable nonorthogonal tight-binding scheme for silicon,” Phys. Rev. B |

39. | Per-Olov Löwdin, “On the NonOrthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals,” J. Chem. Phys. |

**OCIS Codes**

(160.5293) Materials : Photonic bandgap materials

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 21, 2013

Revised Manuscript: July 12, 2013

Manuscript Accepted: July 21, 2013

Published: August 16, 2013

**Citation**

Kwangmoo Kim and D. Stroud, "Photonic band structures of periodic arrays of pores in a metallic host: tight-binding beyond the quasistatic approximation," Opt. Express **21**, 19834-19849 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19834

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### References

- J. Q. Xia, Y. R. Ying, S. H. Foulger, “Electric-Field-Induced Rejection-Wavelength Tuning of Photonic-Bandgap Composites,” Adv. Mater. 17, 2463–2467 (2005). [CrossRef]
- Kurt Busch, Sajeev John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999). [CrossRef]
- Eli Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
- A. Scherer, O. Painter, B. D’Urso, R. Lee, A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B 16, 3906 (1998). [CrossRef]
- Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides, ” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]
- F. Benabid, J. C. Knight, G. Antonopoulos, P. St. J. Russell, “Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]
- Y. Cao, J. O. Schenk, M. A. Fiddy, “Third order nonlinear effect near a degenerate band edge,” Opt. Photonics Lett. 1, 1–7 (2008). [CrossRef]
- Arthur R. McGurn, Alexei A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B 48, 17576–17579 (1993). [CrossRef]
- V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994). [CrossRef]
- V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997). [CrossRef]
- I. H. H. Zabel, D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48, 5004–5012 (1993). [CrossRef]
- Mark L. Brongersma, John W. Hartman, Harry A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000). [CrossRef]
- Stefan A. Maier, Pieter G. Kik, Harry A. Atwater, Sheffer Meltzer, Elad Harel, Bruce E. Koel, Ari A.G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]
- S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, H. A. Atwater, “Plasmonics–A Route to Nanoscale Optical Devices,” Adv. Mater. 13, 1501–1505 (2001). [CrossRef]
- Prashant K. Jain, Wenyu Huang, Mostafa A. El-Sayed, “On the Universal Scaling Behavior of the Distance Decay of Plasmon Coupling in Metal Nanoparticle Pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]
- LinLin Zhao, K. Lance Kelly, George C. Schatz, “The Extinction Spectra of Silver Nanoparticle Arrays: Influence of Array Structure on Plasmon Resonance Wavelength and Width,” J. Phys. Chem. B 107, 7343–7350 (2003). [CrossRef]
- Q.-H. Wei, K.-H. Su, S. Durant, X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004). [CrossRef]
- L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, A. Polman, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005). [CrossRef]
- Shengli Zou, George C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. 121, 12606–12612 (2004). [CrossRef] [PubMed]
- A. F. Koenderink, Albert Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]
- Andrea Alù, Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006). [CrossRef]
- Alexandre Bouhelier, Renaud Bachelot, Jin Seo Im, Gary P. Wiederrecht, Gilles Lerondel, Sergei Kostcheev, Pascal Royer, “Electromagnetic Interactions in Plasmonic Nanoparticle Arrays,” J. Phys. Chem. B 109, 3195–3198 (2005). [CrossRef]
- Vadim A. Markel, Andrey K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B 75, 085426 (2007). [CrossRef]
- Yu-Rong Zhen, Kin Hung Fung, C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008). [CrossRef]
- W. Jacak, J. Krasnyj, J. Jacak, A. Chepok, L. Jacak, W. Donderowicz, D. Z. Hu, D. M. Schaadt, “Undamped collective surface plasmon oscillations along metallic nanosphere chains,” J. Appl. Phys. 108, 084304 (2010). [CrossRef]
- Matthew D. Arnold, Martin G. Blaber, Michael J. Ford, Nadine Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express 18, 7528–7542 (2010). [CrossRef] [PubMed]
- Sung Yong Park, David Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004). [CrossRef]
- W. H. Weber, G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004). [CrossRef]
- D. Gaillot, T. Yamashita, C. J. Summers, “Photonic band gaps in highly conformal inverse-opal based photonic crystals,” Phys. Rev. B 72, 205109 (2005). [CrossRef]
- Ali E. Aliev, Sergey B. Lee, Anvar A. Zakhidov, Ray H. Baughman, “Superconductivity in Pb inverse opal,” Physica C 453, 15–23 (2007). [CrossRef]
- G. Gantzounis, N. Stefanou, “Cavity-plasmon waveguides: Multiple scattering calculations of dispersion in weakly coupled dielectric nanocavities in a metallic host material,” Phys. Rev. B 74, 085102 (2006). [CrossRef]
- See, e.g.,J. D. Jackson, “Earth and Ionosphere as a Resonant Cavity: Schumann Resonances,” in Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 374–376.
- See, e.g.,N. W. Ashcroft, N. D. Mermin, “Problem 2. Tight-Binding p-Bands in Cubic Crystals” & “General remarks on the tight-binding method,” in Solid State Physics (Saunders College Publishing, Orlando, 1976), pp. 189–190& pp. 184–185.
- E. Lidorikis, M. M. Sigalas, E. N. Economou, C. M. Soukoulis, “Tight-Binding Parametrization for Photonic Band Gap Materials,” Phys. Rev. Lett. 81, 1405–1408 (1998). [CrossRef]
- N. Stefanou, A. Modinos, V. Yannopapas, “Optical transparency of mesoporous metals,” Solid State Commun. 118, 69–73 (2001). [CrossRef]
- B. A. McKinnon, T. C. Choy, “Significance of nonorthogonality in tight-binding models,” Phys. Rev. B 52, 14531–14538 (1995). [CrossRef]
- Madhu Menon, K. R. Subbaswamy, “Transferable nonorthogonal tight-binding scheme for silicon,” Phys. Rev. B 50, 11577–11582 (1994). [CrossRef]
- Per-Olov Löwdin, “On the NonOrthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals,” J. Chem. Phys. 18, 365–375 (1950). [CrossRef]

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