## 1. Introduction

[2] *Shore and Yaghjian* [2012] (hereinafter referred to as part 1) centered on the theory of *kd*–*βd* 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 *kd*is 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 1D*kd*–*βd* equation, and listed all the other *kd*–*βd* 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 *kd*–*βd*diagrams obtained with high-accuracy computer solutions of the dispersion equations given inpart 1. Our objective is primarily to display a representative selection of *kd*–*βd* 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 (*βd*) and (*βd*), respectively, and use dotted lines to plot light lines (*kd* = (*βd*)). Harmonic time dependence of exp(−i*ωt*), *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 (*βd*) in the interval [0, *π*]. Thus the sign of (*βd*) corresponding to this positive (+*z* direction) phase velocity determines whether the wave is a “forward” [ (*βd*) > 0] or “backward” [ (*βd*) < 0] complex wave [*Capolino et al.*, 2009]. If we required that (*βd*) always be positive, then (*βd*) would be negative for “backward” waves and (*βd*) would have to be plotted in the interval [−*π*, *π*] 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 that*β* 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 that*β* 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 *kd*–*βd* diagrams where *β* is real, the wave corresponding to a positive (negative) slope of the *kd*–*βd* 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 *k*–*β* 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 *kd*–*βd* equations, and the square root functions in the 2D array *kd*–*βd* 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 *kd*–*βd* 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 *kd*–*βd* 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 *kd*–*βd* curve across the light line. The procedure we use is as follows. If in the course of calculating a *kd*–*βd* 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 *kd*–*βd* 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 *kd*–*βd* equations for complex *βd*. In our previous work on traveling waves with real propagation constants, all the *kd*–*βd* equations encountered were real functions of the real argument *βd* and so a simple search procedure with secant algorithm refinement sufficed. In the *kd*–*βd* 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 *kd*–*βd* equation is thought to lie (this can often be obtained by starting with the *kd*–*βd* diagram for the real case), we regard the function of *βd*, *f*(*βd*, *kd*), whose zero we are attempting to find, as a function of the two real variables (*βd*), (*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*(*βd*, *kd*)|. 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*(*βd*, *kd*)| 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*(*βd*, *kd*)| and thereby obtain a highly accurate value for the zero of *f*(*βd*, *kd*). It should be noted that it is difficult to be certain that all complex zeroes of the *kd*–*βd* equation have been found in a given region of the complex *βd* plane, so that not all *kd*–*βd* curves may be present in the *kd*–*βd* 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].