Complex waves on periodic arrays of lossy and lossless permeable spheres: 2. Numerical results

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Abstract

[1] This is the second part of a two-part series dealing with complex dipolar waves propagating along the axes of 1D, 2D and 3D infinite periodic arrays of small lossless and lossy permeable spheres. Shore and Yaghjian (2012) provide the theory of the complex waves and the dispersion (kβ) equations for their propagation constants. In this paper we present and discuss representative dispersion diagrams for arrays of magnetodielectric spheres (spheres with appreciable permittivity and permeability), diamond spheres, and silver nanospheres.

1. Introduction

[2] Shore and Yaghjian [2012] (hereinafter referred to as part 1) centered on the theory of kdβd equations of complex dipolar waves supported by 1D, 2D, and 3D periodic arrays of lossy or lossless permeable spheres, where βd is the traveling wave electrical separation distance of the array elements in the direction parallel to a specified array axis, and kdis the corresponding free-space electrical separation distance. We showed how the dispersion equations for complex propagation constants could be obtained from the corresponding equations for real propagation constants by an analytic continuation procedure. As an example, we derived the 1Dkdβd equation, and listed all the other kdβd equations, referring to Shore and Yaghjian [2010] for their derivation. It was also shown how for sufficiently small values of kd and |βd|, a 3D array could be regarded as a continuous homogeneous medium with effective permittivity and permeability given by simple expressions obtainable from solving the dispersion equation.

[3] In this paper we present a variety of kdβddiagrams obtained with high-accuracy computer solutions of the dispersion equations given inpart 1. Our objective is primarily to display a representative selection of kdβd diagrams without attempting to give detailed physical explanations of the behavior shown in these diagrams. In a series of papers, Engheta et al. [2005] and Alù and Engheta [2006a, 2006b, 2007] have made an important contribution toward obtaining physical understanding of the behavior of dispersion diagrams of periodic arrays of electric dipoles in terms of nanocircuit and nanotransmission lines. Further work along these lines is needed for arrays of permeable spheres for which both electric and magnetic dipoles and their interactions must be taken into account.

[4] The examples of dispersion diagrams we show here are for the most part designed to demonstrate what happens when the dispersion diagrams obtained in our previous papers and reports [Shore and Yaghjian, 2004, 2005a, 2005b, 2006, 2007] under the assumption that βd is real, are extended to allow for complex βd. (In most of our previous work we used the term “magnetodielectric” to describe inclusions with arbitrary permittivity and permeability, including purely dielectric inclusions. Here it is more convenient to use “permeable” instead of “magnetodielectric” and to reserve “magnetodielectric” for inclusions with appreciable permittivity and permeability.) Some important notations and conventions used in our plots of the dispersion diagrams are as follows. We consistently use solid and dashed lines to plot inline image(βd) and inline image(βd), respectively, and use dotted lines to plot light lines (kd =  inline image(βd)). Harmonic time dependence of exp(−iωt), ω > 0 is assumed. In the description of the arrays, a denotes the radius of the spherical inclusions, and d denotes the distance between the centers of adjacent array elements. We plot inline image(βd) in the interval [0, π]. Thus the sign of inline image(βd) corresponding to this positive (+z direction) phase velocity determines whether the wave is a “forward” [ inline image(βd) > 0] or “backward” [ inline image(βd) < 0] complex wave [Capolino et al., 2009]. If we required that inline image(βd) always be positive, then inline image(βd) would be negative for “backward” waves and inline image(βd) would have to be plotted in the interval [−ππ] instead of in the interval [0, π]. It should be noted that for lossless arrays, if β is a solution of the dispersion equation then β* is also a solution with the same positive real part of β. Thus, for lossless arrays, forward and backward waves always come in pairs. In Figures 1b, 1c, 2b, 3b, 3c, 4b, 5a, 6b, 6c, 7b, 8b–8d, 9b, 10b–10d, 11b, and 14d we show the imaginary part of β for only one of the waves in the pair.

[5] The group velocity [cd(kd)/dℜ(βd)] is not necessarily in the direction of decay of the complex wave because the proof of the equality between group and energy-transport velocities requires thatβ be real [Yaghjian et al., 2009]. Moreover, for proper complex waves on lossless arrays, the total energy transport across an xy plane (z= constant plane) is zero. Also, the proof of the equality between group and energy-transport velocities applies to continuous media and thus requires not only thatβ be real, but also that βd ≪ 1 and kd ≪ 1. Therefore, in reference to complex waves, the terms “forward” and “backward” simply denote that the phase velocity and exponential decay are in the same or opposite directions, respectively, and not necessarily that the phase and group velocities are in the same and opposite directions. For the portions of the kdβd diagrams where β is real, the wave corresponding to a positive (negative) slope of the kdβd diagram is often referred to as a “forward” (“backward”) wave. However, in this paper we will reserve the terms “forward” and “backward” wave solely for complex waves as defined above, and use positive and negative group velocity as synonymous with positive and negative slope of the portions of the kβ diagram where β is real. As we have stated in section 1 in part 1, our analysis does not enable us to tell which of the modes are physical (excitable by localized sources) and which are not, and which of the 1D and 2D modes are proper (decay in the transverse direction away from the array axis) and which are not.

[6] We have noted in section 4 in part 1 that the polylogarithm functions in the 1D array kdβd equations, and the square root functions in the 2D array kdβd equations are multivalued, and we have drawn attention to the large number of alternative branches of the 1D and 2D array dispersion equations that result from the presence of multivalued functions in the dispersion equation. This is especially true for 1D arrays. Time and space do not allow us to calculate and show the dispersion diagrams for all choices of the branches of these multivalued functions. However, as we have noted, even if we begin with a kdβd diagram for a 1D or 2D array obtained by solving the dispersion equation calculated with the principal values of the multivalued functions, it may be necessary to use a choice of alternative branches of the multivalued functions to continue the kdβd curves across the light line. The question then arises as to how it is possible in a relatively short amount of time to find a particular choice of an alternative branch of the dispersion equation that continues a kdβd curve across the light line. The procedure we use is as follows. If in the course of calculating a kdβd curve with, say, the principal branch of the dispersion equation, we are unable to continue the curve across the light line, we note the values of kd and the real and imaginary parts of βd. We then use a separate program that enables us to fix kd at a value slightly larger or smaller as the case may be than the value where the curve terminated at the light line, and to then cycle through all the alternative branches of the dispersion equation searching for a branch (that is, a choice of the n-coefficients in section 4 inpart 1) that yields a solution differing only very slightly from the solution at the light line obtained with the principal branch of the dispersion equation. This can be done quite rapidly because the search region for the solution can be made very narrow. When such a solution is found it is written into a file. After cycling through all the dispersion equation branches the file will contain one solution, or at most a very small number of solutions that have been found which satisfy the continuity requirement. In the discussion of light line crossings of the kdβd curves in Figures 1b, 1c, 2b, 3c, 6c, 7b, 8d, 9a, 12b, 13b, 14b, 15b, and 16b, the alternative branches will be specified using the notation of the n-coefficients given in section 4 inpart 1.

[7] It is important to comment on the procedure used to solve the kdβd equations for complex βd. In our previous work on traveling waves with real propagation constants, all the kdβd equations encountered were real functions of the real argument βd and so a simple search procedure with secant algorithm refinement sufficed. In the kdβd equations encountered here in our extension of our earlier work to traveling waves with complex propagation constants, the equations are complex functions of the complex argument βd and the procedure for solving the equations for βd is necessarily more complicated. Starting with an idea of a relatively large region in the complex βd plane where the zero of the kdβd equation is thought to lie (this can often be obtained by starting with the kdβd diagram for the real case), we regard the function of βd, f(βdkd), whose zero we are attempting to find, as a function of the two real variables inline image(βd), inline image(βd) and vary the real and imaginary parts of βd separately with a relatively coarse grid, to locate the point in the region that gives a minimum value of |f(βdkd)|. The search region and grid are then reduced in size and the procedure repeated to obtain a more refined estimate of the zero. The value of βd that minimizes |f(βdkd)| obtained after several repetitions of this procedure is then used as the starting guess for a quasi-Newton method with a finite difference gradient [Dennis and Schnabel, 1983, Appendix A], implemented in the IMSL subroutine UMINF, or for Müller's Method [Müller, 1956; Press et al., 1992], implemented in the IMSL subroutine ZANLY, to minimize |f(βdkd)| and thereby obtain a highly accurate value for the zero of f(βdkd). It should be noted that it is difficult to be certain that all complex zeroes of the kdβd equation have been found in a given region of the complex βd plane, so that not all kdβd curves may be present in the kdβd diagrams shown in section 2.

[8] In this paper we show and discuss dispersion curves for representative examples of 1D, 2D, and 3D arrays of magnetodielectric spheres (section 2.1), diamond spheres (section 2.2), and silver nanosphere arrays (section 2.3). Section 2.4 is devoted largely to a consideration of the general behavior of dispersion curves of 1D and 2D arrays in the vicinity of the light line. Figures additional to those in this paper are shown by Shore and Yaghjian [2010].

2. Numerical Results

2.1. Magnetodielectric Sphere Arrays

2.1.1. 1D Arrays of Magnetodielectric Spheres

[9] In this section we consider a linear arrays of magnetodielectric spheres (spheres with appreciable permittivity and permeability). For a linear array of magnetodielectric spheres with ϵr = μr = 10, a/d = 0.45, and the dipoles transverse to the array axis, the kdβd diagram for βd real is given by Shore and Yaghjian [2004]. In Figure 1awe show the first branch of this diagram with a positive-slope segment forkd from 0.724 to 0.884 in which βd increases from the light line to π at kd = 0.884 (ka = 0.398) close to the resonance frequency of kd = 0.900 (ka = 0.405), followed by a negative-slope segment forkd from 0.884 to 0.959 where βd decreases from πto the light line. The extension of this diagram into the fast-wave region is shown inFigure 1b. The dispersion curve of Figure 1a continues across the light line as a complex wave with inline image(βd) decreasing to zero at kd = 0.960. It then increases and recrosses the light line at kd = 0.998. The crossing of the light line at kd = 0.998 is made possible by moving from the principal branch of the dispersion equation to the one specified by choosing n3p = 1 and all the other n-coefficients equal to zero. Askd increases from 0.960, inline image(βd) decreases from 0 to −1.28 as inline image(βd) decreases from the light line to 0 (the backward-wave segment), then changes sign and increases to about 2.3 atkd = 1.0 (the forward-wave segment). We have arbitrarily terminated the plot atkd = 1 but note that a new branch, not shown, of the kdβd diagram begins at kd ≈ 1.22. There is also a second kdβd curve, Mode 2 in Figure 1b. Mode 2 begins as a highly attenuated backward wave, at first nearly vertical until it reaches the inline image(βd) curve of Figure 1awith which it is then almost identical for a narrow frequency range. The negative-slope segment then takes off from the inline image(βd) curve of Figure 1a at kd ≈ 0.872, and decreases to 0 at kd = 0.916, following which there is a forward-wave segment that crosses at the light line atkd ≈ 0.930. The continuation of the forward-wave segment past the light line is made possible by moving from the principal branch of the dispersion equation to the one specified by choosingn3m = 1 and all the other n-coefficients equal to zero. What is particularly interesting for Mode 2 is that there is a small section of the backward-wave segment in the slow-wave region whereβd is complex, a transition region referred to by Lampariello et al. [1990]as “the winding down of the leaky-wave solution.” (The structure analyzed byLampariello et al. [1990]is different from ours being a dielectric-filled rectangular waveguide with an asymmetric slit in its top wall and with an air-filled parallel plate stub guide above, and the frequency behavior is the opposite of ours.) Despite the similarity between this backward-wave segment ofFigure 1b and the transition region of Lampariello et al. [1990, Figures 3 and 4] we do not want to assert that the backward-wave segment in the fast-wave region ofFigure 1bcorresponds to a leaky wave; that is, an outward-energy-propagating improper complex wave. (A detailed treatment of leaky waves and leaky-wave antennas is given byJackson and Oliner [2008].)In this paper we consider only the longitudinal propagation constants of waves propagating in the direction of the array axis and do not attempt in general to investigate the behavior of these waves in the direction(s) transverse to the array axis.

Figure 1.

kdβd diagram for a 1D periodic array of magnetodielectric spheres with ϵr = μr = 10, and a/d = 0.45; dipoles perpendicular to array axis. (a) βd real, (b) βd complex, and (c) βd complex; multiple modes shown.

[10] In Figure 1c we show the kdβd diagram for the same array but with an additional mode displayed, demonstrating the complexity of the kdβd diagram for complex waves. Mode 3 in Figure 1c was obtained using equation (13) in part 1 instead of equation (8a) in part 1 for calculating F1+(kdβd). The crossing of the light line by Mode 3 is made possible by moving onto the branch of the dispersion equation specified by choosing n1p = 1, n3p = 1, and all the other n-coefficients equal to zero.

[11] For a linear array of magnetodielectric spheres with ϵr = μr = 20, a/d = 0.45, and the electric dipoles parallel to the array axis, the kdβd diagram for βd real is given by Shore and Yaghjian [2006, Figure 41] and Shore and Yaghjian [2007, Figure 29] which we reproduce here as Figure 2a. The extension of this diagram for complex βd is shown in Figure 2b. The kdβd diagram for real βdis seen to be a continuation of a complex mode that begins in the slow-wave region forkd < 0.407 with the alternative branch of the dispersion equation specified by n2pm = n3m = 1 and all other n-coefficients equal to 0. The mode then crosses the light line into the fast-wave region, with the continuation made possible by moving to the principal branch of the dispersion equation. Forkd between 0.407 and 0.453, inline image(βd) describes a fish hook from the light line into the fast-wave region and then back again to the light line. Note also that the bottom portion of the fish hook is a negative-slope wave but inline image(βd) is positive. For kd between 0.453 and 0.484, βd is real, increasing from the light line to π close to the resonance frequency of the spheres at ka = 0.214 (kd = 0.475). This portion of the kdβd diagram is identical to that of Figure 2a. For values of kd > 0.484, there is a strongly attenuated wave with inline image(βd) remaining equal to π and inline image(βd) increasing rapidly.

Figure 2.

kdβd diagram for a 1D periodic array of magnetodielectric spheres with ϵr = μr = 20 and a/d = 0.45; dipole moments parallel to propagation direction. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 41] and Shore and Yaghjian [2007, Figure 29]) and (b) βd complex.

2.1.2. 2D Arrays of Magnetodielectric Spheres

[12] For βd real, the kdβd diagram for a 2D array of magnetodielectric spheres with ϵr = μr = 20, a/d = 0.45, and the electric dipoles parallel or perpendicular to the plane of the array, is given by Shore and Yaghjian [2006, Figure 36] and Shore and Yaghjian [2007, Figure 24] and is reproduced here as Figure 3a. Its extension for complex βd is shown in Figure 3b. The lower branch of the diagram goes to π at kd = 0.449 (ka = 0.202) fairly close to the resonance frequency of the spheres at kd = 0.475 (ka = 0.213) while the negative-slope branch of the diagram starts atkd = 0.472(ka = 0.212). The bandgap between kd = 0.4487 and 0.4721 in Figure 3a is seen to correspond to an attenuated wave with inline image(βd) = π and inline image(βd) increasing from 0.0 to 1.35 at the center of the bandgap. The behavior of the dispersion diagram from kd = 0.486 to 0.495 is especially interesting, and we show a detail of this region in Figure 3c. We have encountered behavior like this before in Figure 1bwhere a fast-wave region upper segment with inline image(βd) increasing from 0 to the light line is joined to the curve of inline image(βd) in the slow-wave region by a backward-wave segment which includes a small transition interval in the slow-wave region whereβd is complex. At the light line at kd = 0.495, the forward-wave fast-wave segment, calculated with the principal value of the dispersion equationn01 = n03 = n1 = n2 = 1 is continued into the slow-wave region by the alternative branch of the dispersion equation specified byn01 = n03 = n1 = n2 = −1. It is interesting that this continuation into the slow-wave region withβd complex, the upper part of Mode 1a in Figure 3c, meets up with another mode, Mode 2 in Figure 3c, at kd = 0.497 when inline image(βd) = 0. Mode 2 is also calculated with n01 = n03 = n1 = n2 = −1.

Figure 3.

kdβddiagram for a 2D square-lattice array of magnetodielectric spheres withϵr = μr = 20 and a/d = .45; electric dipole moments parallel or perpendicular to the array plane. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 36] and Shore and Yaghjian [2007, Figure 24]), (b) βd complex, and (c) βd complex; detail.

2.1.3. 3D Arrays of Magnetodielectric Spheres

[13] For βd real, the kdβd diagram for a 3D array of magnetodielectric spheres with ϵr = μr = 20, a/d = 0.45, and the electric and magnetic dipoles normal to the array axis is given by Shore and Yaghjian [2006, Figure 37] and Shore and Yaghjian [2007, Figure 25] and is reproduced here as Figure 4a. When complex solutions of the kdβd equation are allowed we obtain the kdβd diagram of Figure 4b. Mode 1 is identical to the dispersion curve of Figure 4a, and an additional mode with high attenuation is shown, also obtained as a solution to the kdβd equation. Mode 1 goes to π at kd = 0.450 (ka = 0.203) close to the resonance frequency of the spheres at kd = 0.475 (ka = 0.213). Figure 4b demonstrates that complex modes can be supported by lossless 3D structures. No total power is carried by these modes, however.

Figure 4.

kdβddiagram for a 3D cubic-lattice array of magnetodielectric spheres, dipole moments normal to the propagation direction, witha/d = .45. (a) ϵr = μr = 20, βd real (reproduced from Shore and Yaghjian [2006, Figure 37] and Shore and Yaghjian [2007, Figure 25]), (b) ϵr = μr = 20, βd complex; multiple modes shown, and (c) (ϵr = 20.0, μr = 20.0 + 0.04i) and (ϵr = 20.0, μr = 20.0 + 0.4i), βd complex.

[14] In Figure 4c we show the dispersion diagrams for the principal mode of Figure 4b when the permeability of the spheres has a loss tangent of 0.002 and 0.02. The real part of the dispersion diagrams are so close to the dispersion diagram when the loss tangent is zero that even though the three curves are plotted in Figure 4c it is impossible to distinguish them. The permeability loss tangents of 0.002 and 0.02 do, however, result in an imaginary part of the propagation constant βdas shown in the dashed line for the 0.002 loss tangent and the dot-dashed line for the 0.02 loss tangent. For the 0.002 loss tangent, the imaginary part ofβd results in a power loss per wavelength [−20log10(exp(−2π| inline image(βd)|/kd)) = −20(2π| inline image(βd)|/kd)ln10 ] of between 2.04 and 4.96 dB per wavelength in the negative-slope segment of the dispersion diagram, while for the 0.02 loss tangent, the imaginary part ofβd results in a proportionally greater maximum power loss of between 20.1 and 50.1 dB per wavelength. For a linear array with ϵr =  inline image(μr) = 10 (not shown), the power loss for a 0.002 permeability loss tangent lies between 1.00 and 1.34 dB per wavelength, and for a 0.02 permeability loss tangent the power loss lies between 10.0 and 13.4 dB per wavelength. Currently we are not aware of any magnetodielectric material at frequencies above a few hundred MHz with both relative permittivity and permeability having real parts reasonably close to the same value of about 10 or greater, and a permeability loss tangent sufficiently small, that can be used for the fabrication of low-loss, isotropic doubly negative (DNG) materials. (For values of the real parts of the permittivity and permeability appreciably less than 10, it does not appear possible to obtain solutions of thekdβd equation with both |βd| and kd < 1 as required for the array to reasonably approximate a DNG homogeneous isotropic medium.)

2.1.4. Effective Parameters for 3D Arrays of Magnetodielectric Spheres

[15] In Figures 5a–5c we show the kdβd diagrams and plots of the real and imaginary parts of the corresponding effective parameters, ϵreff and μreff, for two 3D cubic-lattice arrays of magnetodielectric spheres: 1) a lossless array withϵr = (13.8, 0.0) and μr = (11.0, 0.0); and 2) a lossy array with ϵr = (13.8, 0.1) and μr = (11.0, 0.0). In both arrays the dipoles are perpendicular to the array axis and a/d = 0.4. βd is complex for the lossless as well as for the lossy array. For clarity the plots in Figures 5a–5c have been limited to the range 0.7 ≤ kd ≤ 0.9. In the real βddispersion diagram for the first array there are stop-bands in the intervalskd = [0.7893, 0.8078] and kd = [0.8738, 0.8812] where no wave with a real β can propagate. In Figure 5a, in these frequency intervals βd is complex so that the wave attenuates. In the interval kd = [0.7893, 0.8078] inline image(βd) = π and inline image(βd) increases from 0 to 0.301 and then decreases to 0. In the interval kd = [0.8738, 0.8812], inline image(βd) = 0 and | inline image(βd)| increases from 0 to 0.091 and then decreases to 0 and thus is too small to be seen in the scale of the plot (as are the corresponding imaginary parts of the effective parameters). In the first and second positive-slope branches of thekdβd diagram, kd = [0.7, 0.7893] and [0.8738, 0.9], it is seen from Figure 5b that ϵreff and μreffare real and positive, while in the negative-slope branch,kd = [0.8078, 0.8738], the effective parameters are both negative, an example of how a DNG medium can be formed from a periodic array of magnetodielectric spheres. Although in the interval kd = [0.7893, 0.8078], |βd| is too large for the array to behave as a continuous, homogeneous, isotropic medium, nevertheless viewed formally the complex values of the effective parameters are consistent with the complex value of βd in this interval and can yield a reflection coefficient for a slab of the array that is a reasonable approximation to the actual reflection coefficient. A consequence of the formal extension of the bulk parameters beyond the region where they are physically meaningful is, however, that the imaginary parts of ϵreff and μreff can become negative in this interval.

Figure 5.

kdβdcurves and effective parameters for 3D cubic-lattice arrays of magnetodielectric spheres; 1)ϵr = (13.8, 0.0), μr = (11.0, 0.0); 2) ϵr = (13.8, 0.1), μr = (11.0, 0.0); dipole moments perpendicular to array axis; a/d = 0.4 (a) kdβd curves, (b) effective permittivity and permeability (lossless), and (c) effective permittivity and permeability (lossy).

[16] The effect of loss on the behavior of inline image(ϵreff) is seen from Figure 5c to be significant only in the bandgap region kd = [0.7893, 0.8078]. Whereas for the lossless array, inline image(ϵreff) has a positive and negative infinite asymptote at kd = 0.80875, for the lossy array inline image(ϵreff) is continuous (it drops down to −12.7, not shown in Figure 5c, before increasing again) with a rather curious “V-shaped” dip at aroundkd = 0.8013. In the bandgap region the behavior of inline image(ϵreff) is very different for the lossless and lossy arrays. For the lossless array, inline image(ϵreff) is negative in the entire bandgap region, decreasing to −6.7 at kd = 0.8066 before increasing to 0 at kd = 0.8078. In contrast, for the lossy array inline image(ϵreff) first decreases to −3.1 at kd = 0.80, then jumps suddenly to around 5 and increases rapidly to 19 at kd = 0.80875, and then decreases to 0.06 at kd = 0.9.

[17] The effect of loss on the behavior of inline image(μreff) is seen from Figures 5b and 5c to be significant only in the bandgap region, but the effect of loss is much less pronounced than it is on inline image(ϵreff). The general shape of inline image(μreff) for the lossless and lossy arrays is similar, but the peak value for the lossless array is about 15 as compared with about 6 for the lossy array. There is much more of an effect of loss on inline image(μreff) than on inline image(μreff). For the lossless array inline image(μreff) is positive throughout with a peak value of 7.8 reached rather abruptly at kd = 0.790 decreasing steadily to 0 at kd = 0.8078. For the lossy array, inline image(μreff) increases steadily to a peak value of 4.2 at kd = 0.7938, then decreases rapidly to −2.9 at kd = 0.8013, after which it increases steadily to positive values at kd = 0.8038.

2.2. Diamond Spheres

2.2.1. Linear (1D) Arrays of Diamond Spheres

[18] In this section we consider two linear arrays of diamond spheres, the first with the dipoles normal to the array axis, and the second with the magnetic dipoles parallel to the array axis. For both arrays, ϵr = 5.84, μr = 1, a/d = 0.45. For the array with the dipoles transverse to the array axis, the dispersion diagram for real βd is shown by Shore and Yaghjian [2006, Figure 18] which we reproduce here as Figure 6a. In Figure 6bwe show the extension of this dispersion diagram into the fast-wave region, calculated with the principal value of the dispersion equation. The bandgap inFigure 6a, starting at kd = 2.5733 close to the first magnetic dipole resonance frequency of kd = 2.787 (ka = 1.254), is shown in Figure 6b to correspond to a segment with inline image(βd) = πand a small non-zero imaginary part. In the fast-wave region, inline image(βd) decreases from the light line to 0 at kd = 4.10 and then starts to increase. (We have truncated the curve at kd = 4.22.) inline image(βd) steadily increases in magnitude in the fast-wave region, negative in the backward-wave region and switching to positive when inline image(βd) starts to increase from 0. In Figure 6c we show the kdβd diagram for the same array with two additional modes shown, both with high attenuation. The continuation of Mode 2 from where it first intersects the light line close to kd = 3.144 to kd = 3.985 where it intersects the light line again, is calculated from the dispersion equation with the alternative mode specified by n1p = n1m = 0, n2pp = n2pm = n2mm = 1, and n2mp = n3p = −1. Its continuation for higher values of kd is calculated from the dispersion equation with the alternative mode specified by n1p = 1, n1m = −1, n2pp = n2mp = 0 , and n2pm = n2mm = n3p = 1.

Figure 6.

kdβd diagram for a 1D periodic array of diamond spheres; dipole moments normal to the propagation direction; a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 18]), (b) βd complex, and (c) βd complex; multiple modes shown.

[19] As noted in section 3.1.2 in part 1, for an array of dipoles oriented parallel to the array axis we need only consider electric or magnetic dipoles separately since there is no coupling between the electric and magnetic dipoles. For a 1D array of diamond spheres with the dipoles parallel to the array axis, we consider only the dispersion diagram for the magnetic dipoles parallel to the array axis, since the first Mie resonance at kd = 2.79 is for magnetic dipoles and the first electric dipole resonance at kd = 3.52 falls outside the frequency range of interest. The kdβd diagram for a 1D array of diamond spheres with the magnetic dipoles parallel to the array axis is given by Shore and Yaghjian [2006, Figure 25] which we reproduce here as Figure 7a. The extension of this diagram for complex βd is shown in Figure 7b. The dispersion curve begins in the slow-wave region as a highly attenuated wave whenkd = 1.28. As kd increases, inline image(βd) increases to π at kd = 1.74 with inline image(βd decreasing from −3.76 at kd = 1.28 to 2.62 at kd = 1.74. As kd continues to increase, inline image(βd) decreases until it crosses the light line when kd = 2.406. This section of the dispersion diagram is calculated using the alternative mode specified by n2pm = n3m = 1 and n2pp = n3p = 0. The kdβdcurve then crosses the light line into the fast-wave region with the crossing made possible by switching to the principal value of the dispersion equation. We see that inline image(βd) then increases rather slowly in the fast-wave region with inline image(βd) positive until kd = 2.815 where it crosses the light line again, becoming identical with the dispersion diagram of Figure 7a with inline image(βd) = 0, and continues to increase in the slow-wave region until it reaches the value ofπ close to the magnetic dipole resonance frequency of kd = 2.787 (ka = 1.254). After inline image(βd) reaches π it stays there for higher values of kd with inline image(βd) then increasing so that the wave attenuates. We have been unable to find a solution to the dispersion equation for kd < 1.28 so that the kdβddiagram begins curiously abruptly there. There is thus a stop-band interval fromkd = 0 to kd = 1.28 where there appears to be no solution to the kdβd equation, unless our numerical search routine for complex zeroes of the kdβd equation has been deficient. It appears that in this frequency range with no solution of the kdβd equation, the only fields that can exist are the “space waves” from the continuous spectra corresponding to integrals in the complex βd plane along the branch cuts of the kdβd equation [see Shore and Yaghjian, 2010, Appendix A]. These space waves require external sources for their excitation. If, for example, a 1D array of magnetodielectric spheres is excited by an external source, then in addition to discrete traveling waves that can channel some of the incident energy along the array, there will also be a continuous spectrum of scattered fields in other directions, the space waves. In certain frequency ranges external sources can excite only these space waves. These are exactly the frequency ranges where there are no solutions to the kdβd equation.

Figure 7.

kdβd diagram for a 1D periodic array of diamond spheres; magnetic dipole moments parallel to the propagation direction; a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 25]) and (b) βd complex.

2.2.2. 2D Arrays of Diamond Spheres

[20] In this section we consider a 2D array of diamond spheres with the electric dipoles in the plane of the array. As in the previous section ϵr = 5.84, μr = 1, and a/d = 0.45. The dispersion diagram for a 2D array of diamond spheres with the electric dipoles in the plane of the array and βd real, is given by Shore and Yaghjian [2006, Figure 19] and Shore and Yaghjian [2007, Figure 10] which is reproduced here as Figure 8a and the corresponding dispersion diagram for complex βd is shown in Figure 8b. inline image(βd) starts out very close to the light line, increasing to π at kd = 2.220(ka = 1.000), somewhat lower than the first magnetic dipole resonance frequency of kd = 2.787 (ka = 1.254). This section of the curve is identical to the lower curve of Figure 8a. The bandgap in Figure 8a is shown in Figure 8b to correspond to a segment with inline image(βd) = π and inline image(βd) negative. At the upper end of the bandgap at kd = 2.507 (ka = 1.128), inline image(βd) goes to zero where it remains until inline image(βd) crosses the light line. This section of the curve is identical to the upper curve of Figure 8a. As kd continues to increase, inline image(βd) enters the fast-wave region and inline image(βd) becomes very slightly negative (backward wave). With further increase of kd, βd becomes zero, and then as inline image(βd) starts to increase very slowly, inline image(βd) becomes positive (forward wave) and the wave becomes increasingly attenuated. In Figure 8c we show the kdβd diagram for the same array with four additional modes shown, all with high attenuation. Figure 8d is a detail of Figure 8c. The continuation of Mode 5 across the light line into the slow-wave region uses the alternative branch of the dispersion equation specified byn01 = n03 = 1, and n1 = n2 = −1. Corresponding figures for a 2D array of diamond spheres with the electric dipoles perpendicular to the array plane are given by Shore and Yaghjian [2010].

Figure 8.

kdβddiagram for a 2D square-lattice array of diamond spheres; electric dipole moments parallel to the array plane;a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 19] and Shore and Yaghjian [2007, Figure 10]), (b) βd complex, (c) βd complex; multiple modes shown, and (d) βd complex; multiple modes shown, detail.

[21] The kdβd diagram for a 2D array of diamond spheres with the magnetic dipoles parallel to the array axis and βd real is shown by Shore and Yaghjian [2006, Figure 26] and Shore and Yaghjian [2007, Figure 14] and is reproduced here as Figure 9a, with the extension for complex waves given in Figure 9b. Mode 1 remains unchanged from Figure 9a. Notice that Mode 1 does not extend into the fast-wave region, unlike the dispersion diagram for a 1D array of diamond spheres with magnetic dipoles parallel to the array axis,Figure 7b. Although the dispersion diagram for βd real, Figure 9a, resembles the corresponding 1D dispersion diagram of Figure 7awhich does have an extension into the fast-wave region, the two dispersion diagrams are actually quite different in their behavior at the light line. InFigure 9a, βd approaches tangency to the light line, whereas in Figure 7a βdis incident on the light line at a non-zero angle. In addition to Mode 1 we show the dispersion curve of a second mode, Mode 2, calculated with alternative branches of the dispersion equation. Mode 2 begins atkd = 0 with a real portion in the slow-wave region that starts out almost tangent to the light line. (We have truncated the curve atkd = 1.6.) In the real, forward-wave portion of the curve, calculated with the alternative branch of the dispersion equation specified byn01 = 1, n2 = −1, βd increases to 2.81 at kd = 2.25, where βdbecomes complex and a backward-wave portion of the curve begins, calculated with the same branch of the dispersion equation, which intersects the light line atkd = 2.48. The curve then crosses the light line into the fast-wave region, calculated with the branch of the dispersion equation specified byn01 = − 1, n2 = 1, and terminates when both inline image(βd) and inline image(βd) become zero at kd = 3.21.

Figure 9.

kdβddiagram for a 2D square-lattice array of diamond spheres; magnetic dipole moments parallel to the propagation direction;a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 26] and Shore and Yaghjian [2007, Figure 14]) and (b) βd complex.

2.2.3. 3D Arrays of Diamond Spheres

[22] In this section we consider 3D arrays of diamond spheres with the dipoles transverse and parallel to the propagation direction. For transverse dipoles the dispersion diagram for real βd is given by Shore and Yaghjian [2006, Figure 21] and Shore and Yaghjian [2007, Figure 12], reproduced here as Figure 10a, and the dispersion diagram for complex βd is shown in Figure 10b. The lower branch of the diagram goes to π at kd = 2.024 (ka = 0.911) considerably below the magnetic dipole resonance frequency of the spheres at kd = 2.787 (ka = 1.254). It is seen that the bandgaps in Figure 10a where there is no unattenuated traveling wave correspond in Figure 10b to regions where inline image(βd) ≠ 0 and hence the wave is attenuated. The region 3.50 < kd < 3.77 is interesting because there is a complex wave ( inline image[βd] > 0) in the middle of the bandgap. In Figure 10c shows the kdβd diagram for the same array with five additional high attenuation modes. A detail of Figure 10c is shown in Figure 10d.

Figure 10.

kdβddiagram for a 3D cubic-lattice array of diamond spheres; dipole moments normal to the propagation direction;a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 21] and Shore and Yaghjian [2007, Figure 12]), (b) βd complex, (c) βd complex; multiple modes shown, and (d) βd complex; multiple modes shown, detail.

[23] The kdβd curves for 3D arrays of diamond spheres with the electric or magnetic dipoles parallel to the array axis and βd real are shown by Shore and Yaghjian [2006, Figure 31] and Shore and Yaghjian [2007, Figure 15], reproduced here as Figure 11a, and the corresponding curves for complex βd are shown in Figure 11b. It is seen that for values of kd greater than the values where inline image(βd) goes to π, inline image(βd) increases steadily with inline image(βd) remaining equal to π. (The curves continue upwards along inline image(βd) = πbut we have truncated them for clarity of presentation.) Both curves have lower cut-off frequencies.

Figure 11.

kdβddiagram for a 3D cubic-lattice array of diamond spheres; magnetic or electric dipole moments parallel to the propagation direction;a/d = .45. (a) βd real (reproduced from Shore and Yaghjian [2006, Figure 31] and Shore and Yaghjian [2007, Figure 15]) and (b) βd complex.

2.3. Silver Nanosphere Arrays

2.3.1. Linear (1D) Arrays of Silver Nanospheres

[24] In this section we consider 1D arrays of silver nanospheres. As discussed by Shore and Yaghjian [2006, section 12.4] and Shore and Yaghjian [2007, section 6.3] the dispersion diagrams were computed using the following Drude model for relative permittivity which agrees quite well with the values of relative permittivity measured by Johnson and Christy [1972] over the visible range of frequencies where lowest order traveling waves exist:

display math

with the plasma frequency ωp = 1.72 × 1016 and the loss parameter γ = 8.35 × 1013. To conform to the parameters used by Sweatlock et al. [2005], the radius a of the spheres was chosen to be 5 nm and the spheres were embedded in glass with a dielectric constant equal to 2.56 (k = 1.6 × ω/c). With these parameters the relative permittivity of the spheres in glass can be written from the Drude equation in (1) as

display math

Our formulation in the works by Shore and Yaghjian [2006, 2007] assumed real permittivity and real βd, and so only the real part of (2) was used in the calculations to obtain the kdβd diagram. Here we use the complex value of ϵr since βd is assumed to be complex.

[25] The kdβdcurves for a 1D array of glass-embedded silver nanospheres and electric and magnetic dipoles perpendicular to the array axis, using only the real part of the permittivity and restrictingβd to be real, are given by Shore and Yaghjian [2006, Figure 28] and Shore and Yaghjian [2007, Figure 16] which we reproduce here as Figure 12a, with the corresponding curves for glass-embedded silver nanospheres using the complex values of the permittivity and complexβd are shown in Figure 12b. The parameter sis the free-space distance between the spheres so thatd = s + 2a. Curves are shown for s = 1 nm (a/d = 0.4545) and s = 4 nm (a/d = 0.3571). The slow-wave region portions of the curves for inline image(βd) from close to the light line to close to π are quite similar for the real and complex permittivity cases. Close to the light line, however, the behavior of the complex curves is very different from what it is for the real permittivity curves. Whereas βdfor the real permittivity curves decreases along the light line in the slow-wave region, in the complex permittivity dispersion curves inline imageβd) crosses the light line into the fast-wave region, decreases to zero, and then increases again to meet the light line, crossing over into the slow-wave region. Up to this point the curves have been calculated with the principal branch of the dispersion equation. In order to cross the light line into the slow wave region, the curves cannot be calculated using the principal branch of the dispersion equation but are calculated with the alternative branch of the dispersion equation specified byn1p = n2pp = n3p = 1 and all other n-coefficients equal to zero. The imaginary part ofβdshifts from negative to positive when the backward-wave portion of the curves changes to a forward-wave portion. The propagation constants also differ significantly for inline image(βd) close to π. For the real permittivity nanospheres there is a sharp cut-off, while for the complex permittivity nanospheres for very low frequencies there is a strongly attenuated wave that gradually becomes less attenuated as the frequency increases and a region with small attenuation begins.

Figure 12.

kdβdcurves for a 1D periodic array of glass-embedded silver nanospheres; dipole moments normal to the propagation direction;a = 5, s = 1, 4 nm. (a) Permittivity real, βd real (reproduced from Shore and Yaghjian [2006, Figure 28] and Shore and Yaghjian [2007, Figure 16]) and (b) permittivity complex, βd complex.

[26] As noted in section 3.1.2 in part 1, for an array of dipoles oriented parallel to the array axis we need only consider electric or magnetic dipoles separately since there is no coupling between the electric and magnetic dipoles. For a 1D array of silver nanospheres with the dipoles parallel to the array axis, we consider only the dispersion diagram for the electric dipoles parallel to the array axis, since the first Mie electric dipole resonance is at ka = 0.120 and the first magnetic dipole resonance is at the much larger value of ka close to 4.0. The kdβdcurves for a 1D array of glass-embedded silver nanospheres and electric dipoles parallel to the array axis with only the real part of the permittivity used andβd real are given by Shore and Yaghjian [2006, Figure 32] and Shore and Yaghjian [2007, Figure 20] which we reproduce here as Figure 13a. The corresponding curves for glass-embedded silver nanospheres using complex permittivity and complexβd are shown in Figure 13b. For both s = 1 and s = 4 there are two modes. Mode 1 begins in the slow-wave region for smallkd. This portion of the Mode 1 curve is calculated with the alternative branch of the dispersion equation specified by n2pp = n3p = 1 and n2pm = n3m = 0. As kd increases inline image(βdcrosses the light line into the fast-wave region where it describes a “fish-hook” and then crosses the light line again into the slow-wave region. The fast-wave portion of Mode 1 is calculated with the principal value of the dispersion equation. The second slow-wave portion of Mode 1 is calculated with the alternative branch of the dispersion equation specified byn2pp = n3p = 0 and n2pm = n3m = 1. As kd continues to increase inline image(βd increases as a forward wave to approximately π whereupon it becomes a backward wave and terminates abruptly. We have been unable to find a continuation of Mode 1 beyond this point. The imaginary part of βd for Mode 1 decreases until where the backward wave begins, whereupon it changes sign and increases to approximately 3.5 for both s = 1 and s = 4. (We have truncated these curves at −1.0 and π for clarity.) Since inline image(βd) is negative over most of the interval where inline image(βd) is increasing (forward wave), and positive where inline image(βd) is decreasing (backward wave), Mode 1 can therefore be called an improper mode. The Mode 2 inline image(βd) curves begin at inline image(βd) = 0 and increase in the fast-wave region until the light line is crossed and the curves enter the slow-wave region. The fast-wave region of the Mode 2 curves is calculated with the alternative branch of the dispersion equation specifiedn2pp = n3p = −1 and n2pm = n3m = 0 and the slow-wave region portion of the curves using the principal branch of the dispersion equation,n2pp = n2pm = n3p = n3m = 0. In the slow-wave region inline image(βdincreases as a long forward-wave segment to close toπ while inline image(βd) remains close to zero. As kd continues to increase inline image(βd) begins to increase steadily. First there is a frequency range for which inline image(βd) remains fairly close to π, after which with increasing frequency inline image(βd) decreases as a negative-slope wave to the light line with inline image(βd) steadily increasing to very large values (not shown in Figure 13). We have been unable to find a continuation of these curves beyond this point into the fast-wave region. (See the discussion of space-waves at the end ofsection 2.2.1.)

Figure 13.

kdβdcurves for a 1D periodic array of glass-embedded silver nanospheres; electric dipole moments parallel to the direction of propagation;a = 5, s = 1, 4 nm. (a) Permittivity real, βd real (reproduced from Shore and Yaghjian [2006, Figure 32] and Shore and Yaghjian [2007, Figure 20]) and (b) permittivity complex, βd complex.

2.3.2. 2D Arrays of Silver Nanospheres

[27] The kdβdcurves for 2D arrays of glass-embedded lossless silver nanospheres with electric and magnetic dipoles perpendicular to the array axis, the electric dipoles parallel to the array plane, only the real part of the permittivity used, andβd real, are given by Shore and Yaghjian [2006, Figure 29] and Shore and Yaghjian [2007, Figure 17] which we reproduce here as Figure 14a. The corresponding dispersion diagram for complex βd and lossy silver nanospheres and s = 1 nm (a/d = 0.4545) is shown in Figure 14b. The behavior of the lossy dispersion diagram is different in important respects from that of the real case. There are two distinct branches of the diagram. Mode 1, calculated entirely with the principal branch of the dispersion equation, begins for low frequencies with inline image(βd) close to π and the mode strongly attenuated. As the frequency increases inline image(βd) decreases, crosses the light line at kd ≈ 0.23, and then decreases asymptotically to zero as the frequency continues to increase. The attenuation decreases until kd ≈ 0.21 following which the attenuation increases steadily again. For low frequencies Mode 2 starts out very close to the light line just as it does for the lossless case with inline image(βd) = 0, but then as kdcontinues to increase there is a small region in the slow-wave region (kd from ≈ 0.182 to 0.216) where there is a complex wave with positive inline image(βd). The crossing of the light line at kd = 0.217 into the fast-wave region is calculated with the alternative branch of the dispersion diagram specified by (n0n03n1n2) = (−1, 1, 1, −1) (see section 4.2 in part 1). Mode 2 comes to an end when inline image(βd) = 0 at kd = 0.222. In Figure 14c we show the kdβd diagram for the same array but with multiple modes displayed. The corresponding dispersion diagrams for the larger sphere separation distance, s = 4 nm, are similar in all important respects to those of the s = 1 dispersion diagram apart from an expected displacement of the curve to higher frequencies, and are not shown here.

Figure 14.

kdβdcurves for a 2D square-lattice array of glass-embedded silver nanospheres; electric dipole moments parallel to the array plane;a = 5. (a) Permittivity real, βd real, s = 1, 4 nm (reproduced from Shore and Yaghjian [2006, Figure 29] and Shore and Yaghjian [2007, Figure 17]), (b) permittivity complex, βd complex, s = 1 nm, (c) permittivity complex, βd complex, s = 1 nm, multiple modes shown, and (d) permittivity real, βd complex, s = 1 nm.

[28] For comparison with Figure 14b, in Figure 14d we show the extension of the kdβd diagram of Figure 14a for complex βd but using only the real part of the permittivity instead of the complex permittivity used to obtain the dispersion diagram of Figure 14b. Comparing Figure 14d with Figure 14a and Figure 14b, we see that in addition to the real curve of Figure 14a we obtain an upper branch similar to that of Mode 1 in Figure 14bwith a complex wave starting out from the real curve in the slow-wave region, crossing over into the fast-wave region, and inline image(βd) asymptotically approaching zero as kdincreases. We have encountered similar complex slow-region curves before, referred to as “the winding down of the leaky-wave solution.” (SeeFigure 1b.) Almost all the portion of the dispersion diagram of Figure 14b down along inline image(βd) ≈ π is eliminated.

[29] The kdβdcurves for 2D arrays of glass-embedded lossless silver nanospheres with electric and magnetic dipoles perpendicular to the array axis and the electric dipoles perpendicular to the array plane are given byShore and Yaghjian [2006, Figure 29] and Shore and Yaghjian [2007, Figure 29] which we reproduce here as Figure 15a. The corresponding curve for lossy silver nanospheres and s = 1 nm (a/d = 0.4545) is shown in Figure 15b. Again there are two modes of the lossy dispersion diagram. The behavior of Mode 1 begins very similarly to that of Mode 1 in Figure 14b and is calculated with the principal branch of the dispersion equation, but then inline image(βd) crosses the light line, decreases to zero, and then increases steadily as kdcontinues to increase, crossing the light line again into the slow-wave region. The upper crossing of the light line into the slow-wave region is calculated with the alternative branch of the dispersion equation specified (n0, n03, n1, n2) = (1, −1, −1, −1). The initial behavior of Mode 2 of the lossy dispersion diagram resembles that of Figure 14b, starting out for low frequencies very close to the light line with inline image(βd) close to zero, and then as kdincreases entering a small region in the slow-wave region (kd from ≈ 0.324 to 0.326) where there is a complex wave with positive inline image(βd). The real part of βd then crosses the light line into the fast wave region as in Figure 14b but then, instead of going to zero, reverses direction, recrosses the light line, and then asymptotically approaches the light line for higher values of kd. The lower crossing of the light line by Mode 2 and the remainder of the curve is calculated with the alternative branch of the dispersion equation specified by (n0, n03, n1, n2) = (−1, −1, −1, −1). In Figure 15c we show the kdβd diagram for the same array but with multiple modes displayed. For the larger separation distance, s = 4 nm, the dispersion diagram shown in Figure 15d is completely similar to that of the smaller separation diagram in Figure 15b apart from an expected shift in the direction of higher frequency.

Figure 15.

kdβdcurves for a 2D square-lattice array of glass-embedded silver nanospheres; electric dipole moments perpendicular to the array plane;a = 5. (a) Permittivity real, βd real, s = 1, 4 nm (reproduced from Shore and Yaghjian [2006, Figure 29] and Shore and Yaghjian [2007, Figure 29]), (b) permittivity complex, βd complex, s = 1 nm, (c) permittivity complex, βd complex, s = 1 nm, multiple modes shown, and (d) permittivity complex, βd complex, s = 4 nm.

[30] The kdβdcurves for 2D arrays of glass-embedded silver nanospheres with electric dipoles parallel to the array axis, only the real part of the permittivity used, andβd real, are given by Shore and Yaghjian [2006, Figure 21] and Shore and Yaghjian [2007, Figure 33] which we reproduce here as Figure 16a. The corresponding curves for complex βd and lossy silver nanospheres are shown in Figure 16b. In Figure 16b the curves have been truncated at kd = 0.8. There is a strong similarity between the curves in Figure 16b and the corresponding 1D Mode 2 curves shown in Figure 13b. The lower slow-wave portion of these curves and the small fast-wave “fish hooks” are calculated with the alternative branch of the dispersion equation specified byn0 = n2 = −1 and the upper slow-wave portions of these curves are calculated with the principal branch of the dispersion equation,n0 = n2 = 1.

Figure 16.

kdβdcurves for a 2D square-lattice array of glass-embedded silver nanospheres; electric dipole moments parallel to the direction of propagation;a = 5, s = 1, 4 nm. (a) Permittivity real, βd real (reproduced from Shore and Yaghjian [2006, Figure 21] and Shore and Yaghjian [2007, Figure 33]) and (b) permittivity complex, βd complex.

2.3.3. 3D Arrays of Silver Nanospheres

[31] The kdβdcurves for 3D arrays of glass-embedded silver nanospheres with electric and magnetic dipoles perpendicular to the array axis, only the real part of the permittivity used, andβd real, are given by Shore and Yaghjian [2006, Figure 31] and Shore and Yaghjian [2007, Figure 19], reproduced here as Figure 17a, and the corresponding curves for lossy nanospheres and complex βd are shown in Figure 17b. As for the 1D and 2D cases, the behavior of the 3D lossy curves differ strongly from the corresponding real permittivity curves. For the complex permittivity case, as for the real permittivity case, the curves begin close to the light line, but inline image(βd) instead of continuing to π increases only a little beyond 1.0 followed by an interval of increasing kd in which it crosses the light line and decreases to close to zero. After this, with still further increase in frequency, inline image(βd) increases again, re-crosses the light line, and then remains quite close to the light line. inline image(βd) is close to zero during the initial increase of inline image(βd), then increases very rapidly to around 1.0 after which it decreases with increasing frequency to close to zero where inline image(βd) begins its increase from close to zero, and remains increasingly close to zero thereafter.

Figure 17.

kdβdcurves for a 3D cubic-lattice array of glass-embedded silver nanospheres; dipole moments normal to the propagation direction;a = 5, s = 1, 4 nm. (a) Permittivity real, βd real (reproduced from Shore and Yaghjian [2006, Figure 31] and Shore and Yaghjian [2007, Figure 19]) and (b) permittivity complex, βd complex.

[32] The kdβdcurves for 3D arrays of glass-embedded silver nanospheres with the electric dipoles parallel to the array axis, only the real part of the permittivity used, andβd real, are given by Shore and Yaghjian [2006,Figure 34] and Shore and Yaghjian [2007, Figure 22], reproduced here as Figure 18a, and the corresponding curves for lossy nanospheres and complex βd are shown in Figure 18b. In Figure 18b the curves have been truncated at kd = 0.6. The lossy dispersion diagrams differ greatly from the corresponding lossless diagrams. Instead of βd going from 0 to π in an extremely narrow frequency range, here for very low frequencies inline image(βd) begins close to zero with inline image(βd) very large. As the frequency increases inline image(βd) begins to increase, crosses the light line, and continues to increase until close to π. The portion of the curves from a little before they cross the light line to where they reach close to π correspond to the curves for the lossless case. The imaginary part of βd is fairly small in this portion of the dispersion curves. As kd continues to increase inline image(βd) increases rapidly. inline image(βd) stays close to π for awhile, and then decreases, somewhat rapidly at first and then much more slowly.

Figure 18.

kdβdcurves for a 3D cubic-lattice array of glass-embedded silver nanospheres; electric dipole moments parallel to the direction of propagation;a = 5, s = 1, 4 nm. (a) Permittivity real, βd real (reproduced from Shore and Yaghjian [2006, Figure 34] and Shore and Yaghjian [2007, Figure 22]) and (b) permittivity complex, βd complex.

2.4. Behavior of Dispersion Curves

[33] In our above discussions of dispersion diagrams of 1D and 2D arrays, we have seen examples where the dispersion curves continue across the light line with no change of branch of the dispersion equation needed, and other examples where a change of branch of the dispersion equation is required to continue the curves across the light line. (See the discussion of alternative branches of the dispersion equations in section 4 in part 1.) We have also encountered examples where a dispersion curve appears to end abruptly at the light line or elsewhere with no continuation found. What can account for these different types of behavior?

[34] Let us begin by considering the behavior of the curves in the vicinity of the light lines, inline image(βd) = kd. The reason why the light lines are often critical places for the dispersion curves is that the dispersion equations for 1D and 2D arrays include terms for which the multivalued functions in the equation are functions of βd − kd; at the light lines inline image(βd) − kd goes through a zero and changes sign with the result that there can often be significant changes in the corresponding multivalued functions in crossing the light lines. As an example consider the dispersion curve shown in Figure 2b calculated with the dispersion equation (equation (15) in part 1). The curve for inline image(βd) crosses the light line twice, at kd = 0.407 and at 0.453. At kd = 0.453 the principal branch of the dispersion equation is used for both the fast-wave and slow-wave regions. If the dilogarithm and trilogarithm functions are calculated on either side of the light line, it is found that even though inline image(Li2[exp(i(kd − βd)]) and inline image(Li3[exp(i(kd − βd)]) change sign moving from one side of the light line to the other, the magnitudes of these terms are very small, and so these sign changes have very little effect on the quantities entering directly into the dispersion equation. Consequently the dispersion equation is essentially continuous moving from one side of the light line to the other and no change of branch is need to continue the dispersion curve across the light line. In contrast, at kd = 0.407 it is found that the magnitudes of these terms is no longer small and their sign change when the light line is crossed has a significant effect on the quantities entering directly into the dispersion equation. Consequently, the dispersion equation is discontinuous if the same mode is used to calculate it on both sides of the light line, and a change of branch must be used to continue the dispersion curve across the light line.

[35] Much the same considerations apply to the kdβd diagrams for the 1D arrays with electric and magnetic dipoles perpendicular to the array axis, calculated with dispersion equation (equation (3) in part 1). For example, the Mode 2 dispersion curve of Figure 1b crosses the light line twice, at kd = 0.886 where the principal branch of the dispersion equation is used on both sides of the light line, and at kd = 0.930 where the crossing of the light line into the slow-wave region is made possible by switching from the principal branch of the dispersion equation to an alternative branch. It is again found that if the imaginary parts of the polylogarithm functions of exp[i(kd − βd)] are small as the light line is approached, the same branch of the dispersion equation can be used on both sides of the light line, whereas if the imaginary parts of the polylogarithm functions of exp[i(kd − βd)] are large in magnitude as the light line is approached, the values of the dispersion equation change significantly from one side of the light line to the other, and consequently the dispersion equation must be calculated with two different branches on the two sides of the light line.

[36] Now let us consider a different kind of behavior of a dispersion curve at the light line. In Figure 13b of section 2.3.1 we have seen that the upper ends of the Mode 2 curves for both s = 1 and s = 4 end abruptly at the light line because we were unable to find continuations of these curves into the fast-wave region. Our inability to find continuations of these curves across the light line is attributable to the extremely large values of inline image(βd) of these curves near the light line, resulting in correspondingly large values of both the real and imaginary parts of Li2[exp(i(kd − βd)] and Li3[exp(i(kd − βd)]. When the light line is crossed, the imaginary parts of these polylogarithm functions change sign resulting in an extremely large change in the dispersion equation. Consequently none of the alternative branches we tried was successful in continuing the dispersion curves into the fast-wave region. Even though it is conceivable that a continuation might be possible if higher-order branches of the polylogarithm functions are considered, what is clear is that the very large magnitudes of the imaginary parts of the dilogarithm and trilogarithm functions at the light line make finding a continuation of the dispersion curves much more difficult than when the imaginary parts are much smaller in magnitude.

[37] In Figure 13b, we have noted in section 2.3.1 that the upper ends of the Mode 1 curves for both s = 1 and s = 4 end abruptly, not at the light line, because we were unable to find continuations for higher values of kd. What is found here is that the magnitudes of the real parts of Li2[exp(i(kd − βd)] and Li3[exp(i(kd − βd)] are both large and change sign as kd increases beyond its values at the termination points. This results in a very large changes in the value of the dispersion equation, making a continuation possible, if at all, only with branches of the polylogarithm functions of higher order than those considered.

[38] So far we have been concerned with understanding why 1D dispersion curves can sometimes be continued across the light line with no change of branch of the dispersion equation, and other times require a change of branch of the dispersion equation, and why sometimes the dispersion curves cannot be continued for kdgreater than a cut-off value, either at the light line or elsewhere. We now consider the behavior of 2D array dispersion curves. As an example consider the dispersion diagram ofFigure 3c. We have seen in section 2.1.2 that the lower crossing of the light line by Mode 1a at kd ≈ 0.488 from the fast-wave region to the slow-wave region is obtained using the principal branch of the dispersion equation, but that the upper crossing of the light line atkd ≈ 0.495 from the fast-wave region to the slow-wave region by Mode 1a requires a change of branch of the dispersion equation. The explanation for this difference in behavior at the lower and upper light line crossings is simple. Referring to equation (26) inpart 1, the branch cuts for inline image in the complex β0 plane run from 1 to 1 + i and from −1 to −1 − i, or equivalently the branch cuts run from kd to kd + i and from −kd to −kd − i in the complex β plane, with the light line given by the vertical line inline image(βd) = kd in the complex β plane. If inline image(βd) < 0, as it is for the lower light line crossing of the backward-wave Mode 1a inFigure 3c, the branch cut from kd to kd + i is not crossed and hence no change in the branch of the dispersion equation is required to cross the light line. If, however, inline image(βd) > 0, as it is for the upper light line crossing of the forward-wave Mode 1a inFigure 3c, the branch cut from kd to kd + i is crossed when the light line is crossed and hence a change in branch of the dispersion equation is required to cross the light line. This analysis of the light line crossings in Figure 3c applies to 2D dispersion curve light line crossings in general: a change of branch of the dispersion equation is required or not as the light line is crossed depending simply on whether or not inline image(βd) > 0. Additional examples of this can be seen in the change of dispersion equation branch required in the crossing of the light line by Mode 5 in Figure 8c, Mode 2 in Figure 9b, and Mode 2 in Figure 14b, and in the absence of a dispersion equation branch change in the crossing of the light line by Mode 1 in Figure 8c and Figure 14b.

3. Conclusion

[39] In this second of two companion papers treating complex dipolar waves propagating along the axes of 1D, 2D, and 3D infinite periodic arrays of small lossless and lossy spherical inclusions, we have presented and discussed a variety of dispersion (kdβd) diagrams obtained with high-accuracy computer solutions of the dispersion equations given inpart 1. The examples of dispersion diagrams shown are for the most part designed to demonstrate the changes when the dispersion diagrams obtained in our previous papers and reports [Shore and Yaghjian, 2004, 2005a, 2005b, 2006, 2007] under the assumption that βd is real, are extended to allow for complex βd. Representative dispersion diagrams are shown and discussed for 1D, 2D, and 3D arrays of permeable spheres with appreciable permittivity and permeability, of diamond spheres, and of silver nanospheres. We have shown that continuation of 1D and 2D array dispersion curves across the light line is frequently possible only if the multivalued functions in the corresponding dispersion equations are evaluated with branches other than the principal branch when the light line is crossed, and we have explained why recourse to alternative branches is necessary in some light line crossings and not in others, as well as instances when the 1D dispersion curves end abruptly at the light line or elsewhere.

[40] In addition to the kdβd curves themselves we have also shown examples of plots of the bulk permittivity and permeability for a 3D array regarded as a homogeneous isotropic medium, obtained from parameters readily available from the solutions of the corresponding dispersion equations, and have demonstrated that in certain frequency ranges the array behaves as a doubly negative (DNG) medium.

Acknowledgments

[41] This work was supported by the U.S. Air Force Office of Scientific Research (AFOSR) through Arje Nachman. The authors wish to express their deep appreciation to Filippo Capolino and Salvatore Campione of the University of California Irvine for having suggested to us there was a strong possibility that many of the 1D and 2D array dispersion curves in the 2010 version of Shore and Yaghjian [2010] which stopped at the light line, could actually be continued across the light line, as they had found in one instance using a calculation method different from ours. It was this observation that led us to realize the importance of using branches of the multivalued functions in the 1D and 2D array dispersion equations other than the principal branches in calculating the dispersion curves for these arrays, as discussed in detail in section 4 in part 1 and in the comments on the dispersion diagrams shown in this paper.

Ancillary