## 1. Introduction and Summary of Results

[2] The subject of this paper is traveling waves on two-dimensional (2-D) and three-dimensional (3-D) periodic arrays of lossless scatterers. Our investigation of these arrays is motivated in part by the recent theoretical demonstration by *Holloway et al.* [2003] that a doubly negative (DNG) material can be formed by embedding an array of spherical particles in a background matrix. The work of Holloway et al. is based on mixing formulas, obtained by *Lewin* [1947], related to the Clausius-Mossotti mixing formulas. In contrast, our work, which corroborates the findings of Holloway et al., is based on an analysis of the *kd*–*βd* equations for traveling waves on periodic arrays and has the advantages of not only yielding the *kd*–*βd* diagrams for all the arrays studied, but also yielding expressions for the effective (bulk) permittivity and permeability of 3-D arrays that are more accurate than the Clausius-Mossotti type formulas over a larger range of separation of the array elements. The work described here builds on and extends our earlier investigations of traveling waves on linear (one-dimensional (1-D)) periodic arrays of acoustic monopoles [*Yaghjian*, 2002], electric dipoles [*Shore and Yaghjian*, 2004a], and magnetodielectric spheres [*Shore and Yaghjian*, 2004b, 2005a, 2005b], using a spherical wave source scattering matrix formulation. The reader is referred to the technical report by *Shore and Yaghjian* [2006] for details and derivations that space does not permit us to give here. Although conceptually our treatment of traveling waves on 2-D and 3-D arrays is identical with our treatment of traveling waves on linear arrays, mathematically it is considerably more complicated because of the necessity of converting to rapidly convergent forms the double or triple summations that play a central role in the analysis.

[3] The class of problems we consider can be described as follows. We have a periodic array of identical elements each characterized by a scattering coefficient that relates the field scattered from the element to the field incident on the element. As in our previous related work, it is assumed that only the fields of the lowest order spherical multipoles (acoustic monopoles, electromagnetic dipoles) are significant in analyzing scattering from the array elements. In the case of a 2-D array, the array can be thought of as a linear array whose “elements” are equispaced element columns normal to the array axis (the array axis being the direction of propagation), and in the case of a 3-D array it can be helpful to regard the array as a linear array whose elements are equispaced element planes normal to the array axis. The spacing of elements in the direction parallel to the array axis is denoted by *d* and in the direction or directions (for 3-D arrays) normal to the array axis by *h*. Our interest is in unattenuated (lossless) traveling waves (with real propagation constants *β*) that can be supported by the array in the direction parallel to the array axis. (*Alù and Engheta* [2006] have recently used analytic continuation arguments to determine the complex propagation constants for attenuated traveling waves (leaky and absorptive) on 1-D infinite arrays of scattering elements. *Shore and Yaghjian* [2007] have also used analytic continuation to obtain the complex propagation constants for attenuated traveling waves (leaky and absorptive) on 1-D and 2-D infinite periodic arrays of lossless and lossy magnetodielectric spheres, and the complex propagation constants for attenuated traveling waves on 3-D infinite periodic arrays of lossy magnetodielectric spheres. They obtain the complex propagation constants for the corresponding arrays of electric or magnetic dipoles separately as special cases of the complex propagation constants for the magnetodielectric sphere arrays.) The focus of our attention is the so-called *k*–*β* equation (or diagram) (in our work more properly referred to as the *kd*–*βd* equation (or diagram)) that relates the traveling wave electrical (or acoustical) separation distance *βd* of the array elements in the direction parallel to the array axis, to the corresponding free-space electrical (or acoustical) separation distance *kd*, where *k* = *ω*/*c* is the free-space wave number with *ω* > 0 the angular frequency and *c* the free-space speed of light. In this paper we do not treat waves traveling in directions other than parallel to the array axis.

[4] Although this paper centers on traveling waves supported by infinite periodic arrays, for closely spaced 3-D arrays of magnetodielectric spheres the solution of the *kd*–*βd* equation can be used to obtain an effective permittivity and permeability of the array, which in turn can be used as the basis for an approximate treatment of the exciting of traveling waves in partially finite 3-D arrays (arrays that are finite in the direction of the array axis and infinite in the directions transverse to the array axis). Additionally, the analyses we have performed to obtain the *kd*–*βd* equations for infinite periodic 3-D arrays of lossless acoustic monopoles, electric dipoles perpendicular to the array axis, and magnetodielectric spheres with the electric and magnetic dipoles oriented perpendicular to the array axis, can be used to obtain exact computable expressions for the fields of partially finite periodic arrays of these kinds of elements, both lossless and lossy, when the arrays are illuminated by a plane wave propagating in a direction parallel to the array axis, that is, with the propagation vector of the plane wave normal to the interface between the array and free space.

[5] Some basic properties of the *kd*–*βd* diagram may be noted here. *Ishimaru* [1991, chapter 7] shows that the dependence of the *kd*–*βd* diagram on *βd* is periodic in *βd* with a period of 2*π*. *Yaghjian* [2007] proved that if a periodic array of reciprocal elements supports a traveling wave with propagation constant *β* it also supports a corresponding traveling wave with propagation constant −*β*. Therefore, for periodic arrays of reciprocal elements, as are all the arrays considered in this paper, *kd* is an even function of *βd*. It follows that for *βd* in the interval *π* < *βd* < 2*π*

where we have written *kd* as a function of *βd*. Hence we need only consider *βd* in the interval

*Yaghjian* [2002] showed that for a general infinite linear periodic array of lossless passive electrically small scatterers, *kd* ≤ *βd*, which coupled with (2) gives

[6] Thus the wavelengths of the lossless traveling waves are equal to or less than the free-space wavelength, and the traveling waves are slow waves compared with a free-space plane wave. The proof of (3) for linear (1-D) arrays given by *Yaghjian* [2002] is easily seen to be valid for traveling waves on 2-D arrays as well. (The essence of the proof is as follows. A linear or 2-D periodic array with separation *d* between adjacent elements (for a 2-D array the elements are periodic line sources) supporting a lossless traveling wave in the direction of the array axis with real propagation constant *β* can be regarded as a phased array with a phase shift of *βd* between adjacent elements. For *βd* < *kd* the array will radiate power into space in the direction θ = cos^{−1} (*βd*/*kd*) measured from the array axis (assuming that the embedded line element pattern does not have a null at θ). From considerations of conservation of power this result is inconsistent with the assumption that the traveling wave is lossless.) However, it is not true in general that *kd* < *βd* for traveling waves on 3-D arrays of lossless scatterers, nor is it necessarily true that *kd* ≤ *π*. Nevertheless, for 3-D arrays we can, without loss of generality, still limit our consideration of traveling waves to those for which 0 < *βd* ≤ *π*. However, for traveling waves on 3-D periodic arrays, *kd* can be greater than *π* and both fast and slow unattenuated waves can be supported.

[7] Although there is no upper limit on the transverse interelement spacing *h* for either 2-D or 3-D periodic arrays, the expressions we give for the rapidly convergent summations in the *kd*–*βd* equations are valid only for *kh* < 2*π*, that is, for *h* less than a wavelength. This restriction on the size of *h* is not an essential limitation of either the transverse element separation or of the analyses we perform. It is, rather, a matter of our not wanting to unnecessarily complicate the form of the rapidly convergent expressions we give by making them independent of the range of *kh* since in most practical applications the transverse element spacing can be expected to be less than a wavelength. Examples of how the range of *kh* can be extended are given in [*Shore and Yaghjian*, 2006]. Also of interest are the limiting values of the *kd*–*βd* equations as *kh* → 2*π* since some of the individual terms of the rapidly convergent expressions in the *kd*–*βd* equations are singular at *kh* = 2*π*. Closer analysis shows [*Shore and Yaghjian*, 2006], however, that the singularities of the various terms in the *kd*–*βd* equation cancel one another and hence the *kd*–*βd* equations remain nonsingular at *kh* = 2*π*.

[8] The arrays considered in this investigation are 2-D and 3-D arrays of acoustic monopoles, electric or magnetic dipoles, and magnetodielectric spheres. For the electric dipole arrays, we consider both arrays with the electric dipoles polarized transverse to the array axis, and arrays with the dipoles oriented parallel to the array axis. Although we refer to arrays of “magnetodielectric spheres,” the analyses performed are equally applicable to any array elements that can be modeled by a pair of crossed electric and magnetic dipoles transverse to the array axis. The electric and magnetic dipoles of the individual array elements are assumed to be uncoupled (as indeed they are for spheres whose permittivity and permeability are radially symmetric) so that an incident electric field at the element center in the direction of the electric dipole excites only the electric dipole field, and an incident magnetic field at the element center in the direction of the magnetic dipole excites only the magnetic dipole field. The electric and magnetic dipoles of different array elements are coupled, however, because the field scattered from an electric dipole has a component of the magnetic field parallel to the magnetic dipoles of the array elements, and the field scattered from a magnetic dipole has a component of the electric field parallel to the electric dipoles of the array elements. (The only restrictive assumption for spherical scatterers (with radially symmetric permittivity and permeability) are that they are either small enough, or the frequency is such, that all scattered multipoles of higher order than dipoles are negligible. Then with respect to the center of each sphere, orthogonality relations demand that only the incident Bessel function dipolar fields couple to the scattered Hankel function scattered fields. Moreover, only the incident Bessel function electric (magnetic) dipole has a nonzero electric (magnetic) field at the center of each sphere. The fields of all higher-order Bessel function multipoles are zero at the center of each sphere.) For 2-D arrays of magnetodielectric spheres we treat two sets of transverse arrays, one with the electric dipoles in the array plane, and one with the electric dipoles normal to the array plane.

[9] For all the arrays considered, an initial form of the *kd*–*βd* equation is obtained very simply by assuming a traveling wave excitation of the array and summing the acoustic, electric, or electromagnetic fields incident on a reference element from all the other elements of the array. (The field incident on a reference element from all the other elements of the array is sometimes called the “interaction field” in the literature [*Collin*, 1991, chapter 12].) This form of the *kd*–*βd* equation consists of summations of an infinite number of terms of the form exp(i*kr*)/(*kr*), exp(i*kr*)/(*kr*)^{2}, or exp(i*kr*)/(*kr*)^{3}. With only one exception, none of these summations can be expressed in closed form. Furthermore, the summations converge so slowly as to make the initial forms of the *kd*–*βd* equations useless for calculation purposes. Accordingly, it is necessary to obtain rapidly convergent expressions for all the slowly convergent summations that are encountered. (Some of the summations over the array elements in columns and planes transverse to the array axis encountered in the paper are not absolutely convergent. For certain values of the parameter *kh* they may not even be conditionally convergent. However, these summations can always be made absolutely convergent by the stratagem of adding a small positive imaginary part to the free-space propagation constant *k*. Physically, this is equivalent to assuming that the contributions of array elements at increasingly great distances from the reference element attenuate to zero. The analyses can then be performed rigorously and the small imaginary part allowed to go to zero at the end. In this paper and the associated technical report [*Shore and Yaghjian*, 2006] we do not enact this stratagem explicitly in our analyses but proceed formally assuming that all the summations we treat are made absolutely convergent by this stratagem.)

[10] To obtain rapidly convergent expressions, we make use of Floquet mode expansions, leading to a second form of the *kd*–*βd* equation that invariably includes series of the form ∑_{j}*a*_{j}*Z*_{n}(*jx*) known as Schlömilch series [*Watson*, 1962, chapter 19; *Erdélyi*, 1953, sections 7.10.3 and 7.15], where *a*_{j} can be a trigonometric function and *Z*_{n} is a Bessel function of order *n*. While the Schlömilch series involving modified Bessel functions *K*_{n} converge very rapidly because of the exponential decay of these functions, the Schlömilch series involving Hankel functions (or equivalently ordinary Bessel functions, *J*_{n}, and Neumann functions, *Y*_{n}) converge very slowly. Thus the second form of the *kd*–*βd* equation would leave us not all that much better off than we were with the initial form were it not for the fact that rapidly convergent expressions are available for all the Schlömilch series involving Hankel functions that we encounter. These expressions are given in Appendix A. Because of space limitations we give detailed derivations of the computationally tractable forms of the *kd*–*βd* equations for only two of the 3-D arrays considered, the 3-D acoustic monopole array and the 3-D magnetodielectric sphere array, and omit detailed derivations for the remaining 3-D arrays and for all the 2-D arrays. Full derivations for all the arrays, 2-D and 3-D, are, however, given by *Shore and Yaghjian* [2006]. The computationally tractable forms of the *kd*–*βd* equations for all the arrays, 2-D as well as 3-D, are given in section 4.

[11] The outline of the paper is as follows. In section 2 we derive the *kd*–*βd* equation for a 3-D array of acoustic monopoles. In section 3.1 we make use of this derivation to obtain the *kd*–*βd* equation for a 3-D array of magnetodielectric spheres with electric and magnetic dipoles oriented normal to the array axis. If the magnetodielectric sphere elements are sufficiently close together, the array can be regarded as a medium with an effective permittivity and permeability that determine the propagation characteristics of a traveling wave supported by the array. In section 3.2 we show how the solution to the *kd*–*βd* equation for a traveling wave can be used to obtain the effective permittivity and permeability. We also describe a second method, based on the Clausius-Mossotti relation and independent of the *kd*–*βd* equation, for obtaining the effective permittivity and permeability.

[12] In section 4 we give without derivation computationally tractable forms of the *kd*–*βd* equations for all the arrays considered, 2-D and 3-D. In section 5 we make use of rapidly convergent expressions obtained in section 3 and by *Shore and Yaghjian* [2006, section 5] to obtain computationally efficient expressions for the field of a partially finite periodic array of lossless or lossy magnetodielectric spheres (an array that is finite in the direction of the array axis and is of infinite extent in the directions transverse to the array axis), when the array is illuminated by a plane wave propagating in a direction parallel to the array axis; that is, with the propagation vector of the plane wave normal to the interface between the array and free space.

[13] Section 6 is devoted to presenting and discussing numerically computed *kd*–*βd* diagrams for 2-D and 3-D periodic arrays of short electric dipoles and for 2-D and 3-D periodic arrays of magnetodielectric spheres. For the sake of comparison, the *kd*–*βd* diagrams are also shown for some 1-D arrays.

[14] In Appendix A we give rapidly convergent expressions for all the Schlömilch series encountered in the analysis part of the paper, and in Appendix B we list important summation formulas.

[15] Although we do not explicitly consider periodic arrays of magnetic dipoles, it should be noted that the *kd*–*βd* equations for 2-D and 3-D arrays of magnetic dipoles are identical with the *kd*–*βd* equations obtained for 2-D and 3-D arrays of electric dipoles. Thus, for example, the *kd*–*βd* equation for a 2-D or 3-D array of magnetodielectric spheres in a frequency range where the magnetic dipole scattering coefficient is much larger than the electric dipole coefficient, can be obtained very accurately by solving the transcendental equation for the corresponding array of electric dipoles oriented in the direction of the magnetodielectric sphere magnetic dipoles, but with the phase of the normalized (see section 1) Mie magnetic dipole scattering coefficient replacing the phase of the electric dipole scattering coefficient in the transcendental equation.

[16] Since Tretyakov and his coworkers have also devoted considerable effort to studying traveling waves on periodic structures [*Yatsenko et al.*, 2000; *Tretyakov and Vitanen*, 2000; *Tretyakov*, 2003; *Vitanen and Tretyakov*, 2005] (hereinafter referred to as Tretyakov et al.), it is important to point out similarities and differences between their work and ours. Similarly to us, they focus on the *kd*–*βd* equation (which they call the eigenvalue equation) supported by periodic arrays of scatterers, and like us they assume that scattering from the array elements is adequately described by considering the elements to be dipoles. The first step for both of us in obtaining the *kd*–*βd* equation is to assume a traveling wave excitation of the array. Whereas Tretyakov and coworkers obtain their eigenvalue equation from an approximate expression for the dipole moment induced in the reference dipole by the interaction field, we use the exact induced dipole moment obtained from the Mie solution. Another significant difference between their work and ours is how the interaction field is evaluated. To obtain the interaction field from a plane (normal to the array axis) of array elements, they use an approximation technique in which they divide the plane into a small circular region (“hole”) of radius *R*_{0} centered on the array axis, and the region outside this circle. The contributions of dipoles inside the circular region to the interaction field are considered individually, while the contributions of the dipoles outside the circular region are obtained by replacing the dipoles by a homogeneous polarization sheet with an average dipole moment per unit area from which an equivalent averaged current density is obtained by multiplication by −i*ω*. The contribution to the interaction field of this equivalent current sheet is then obtained from the standard integral expression for the electric field radiated by a current distribution. For transverse element separations *kh* < 1, they take *R*_{0} = *h*/1.438, a value obtained from static considerations, so that only the contributions of the dipoles on the array axis are considered individually, the remainder of the array dipoles being treated as equivalent current sheets. In contrast with this approximate method, our approach is to obtain the interaction field by rigorously evaluating all the summations using either the Poisson summation formula or a Floquet mode expansion method, the results of which are then combined with Schlömilch series expressions to convert the initial numerically intractable summations to rapidly convergent forms. Our resulting *kd*–*βd* equation is valid for transverse element separations *kh* < 2*π* rather than *kh* < 1 for the approximation method used by Tretyakov et al. and can, as noted above, be extended if desired for any transverse element separations.

[17] Several important advantages also accrue to our approach for obtaining the *kd*–*βd* equations. One important advantage is that the frequency dependence for all frequencies of the scattering coefficients of array sphere elements is incorporated in the Mie scattering coefficients, whereas Tretyakov et al. use an approximate expression for the polarizability which is valid for *ka* ≪ 1 where *a* is the sphere radius. A second advantage of our approach is that it facilitates consideration of coupling between electric and magnetic dipoles. As we have noted, in periodic arrays of magnetodielectric spheres the electric and magnetic dipoles are coupled because the field scattered from an electric (magnetic) dipole has a component of the magnetic (electric) field parallel to the magnetic (electric) dipoles of the array elements. This coupling is fully taken into account in our framework and analysis but is neglected by Tretyakov et al. Finally, the scattering matrix framework that underlies our work makes it easy to generalize our approach to scattered modes higher than dipole modes, whereas the polarization-centered approach used by Tretyakov et al. would require intensive reworking to incorporate higher-order modes.