3.1. The kd–βd Equation for 3-D Magnetodielectric Sphere Arrays
 In this section we consider traveling waves supported by 3-D periodic arrays of lossless magnetodielectric spheres. It is assumed that the spheres can be modeled by pairs of crossed electric and magnetic dipoles, each of the dipoles perpendicular to the array axis. (It is unnecessary to consider 3-D arrays of electric and magnetic dipoles with the electric (magnetic) dipoles in the direction of the array axis and the magnetic (electric) dipoles perpendicular to the array axis, or 3-D arrays of electric and magnetic dipoles with all dipoles oriented in the direction of the array axis, because an electric (magnetic) dipole has no radial or longitudinal magnetic (electric) field [Stratton, 1941, sections 8.5 and 8.6] and so there is no coupling of the electric dipoles with the magnetic dipoles of such arrays.) The analysis performed here is equally applicable to any 3-D periodic array whose elements can be modeled by a pair of crossed electric and magnetic dipoles at right angles to each other such that only an incident electric (magnetic) field at the element center in the direction of the electric (magnetic) dipole excites only the electric (magnetic) dipole field. We choose the array axis to be the z axis of a Cartesian coordinate system with equispaced planes of magnetodielectric spheres normal to the z axis located at z = nd, n = 0, ±1, ±2, ⋯. In each plane the spheres are centered at x = mh, y = lh, l, m = 0, ±1, ±2, ⋯. The electric and magnetic dipole components of each sphere are oriented in the x and y direction, respectively. We assume an excitation of the array with the electric field parallel to the x axis and the magnetic field parallel to the y axis, and such that all the spheres in any plane of the array normal to the array axis are excited identically. Let E00 and H00 be the electric and magnetic field, respectively, incident on the sphere at the location x = 0, y = 0, z = 0 from all the other spheres in the array. As will be seen (see (31)) E00 has an x component only, and H00 has a y component only. Let E00mln and H00mln be the electric and magnetic field, respectively, incident on the reference sphere from the sphere at the location (x, y, z) = (mh, lh, nd) so that
 The quantities in (27) are defined with reference to a local spherical polar coordinate system with origin at (x, y, z) = (mh, lh, nd) (in turn defined with reference to a local Cartesian coordinate system with the same origin whose axes are parallel to those of the global Cartesian coordinate system). The distance from the (m, l, n) sphere to the (0, 0, 0) sphere, rmln0, is given by
and the unit vector in the direction from the (m, l, n) sphere to the (0, 0, 0) sphere, mln0, is
 The constants b−n and b+n are related to the x component of the electric field and the y component of the magnetic field, respectively, incident on any sphere in the nth plane by the scattering equations [Shore and Yaghjian, 2004b, equation (31)]
where S− and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively. “Normalized” means that b−n (b+n) is the coefficient of exp(ikr)/(kr) in the outgoing electric (magnetic) dipole field in response to the incident field E0x0n (H0y0n/Y0) at the center of the x (y) directed electric (magnetic) dipole. (As we noted at the beginning of this section of the paper, although we refer to the array elements as “magnetodielectric spheres,” our analysis applies equally well to any array elements that can be modeled as a pair of crossed electric and magnetic dipoles such that an incident electric (magnetic) field in the direction of the electric (magnetic) dipole excites only the electric (magnetic) dipole field. If the array elements are indeed spheres then S− and S+ are the normalized Mie dipole scattering coefficients [Shore and Yaghjian, 2004b, equations (30a) and (30b)], S− = −3ib1sc/2 and S+ = −3ia1sc/2, where b1sc and a1sc are the electric and magnetic Mie dipole scattering coefficients defined by Stratton [1941, equations (11) and (10), section 9.25]. If the array elements are not magnetodielectric spheres then S− and S+ must be known for the results of this section of the paper to be applied.) Substituting (30) in (27) along with explicit algebraic expressions in m, l, and nd/h for all the quantities in (27), we obtain for n ≠ 0 (see Shore and Yaghjian [2006, equations (9.3)–(9.28)] for details)
where we have let
 The corresponding expressions for the self-plane n = 0 are obtained directly from (31) omitting the m = l = 0 term of the summations. The total x directed electric field and y directed magnetic field incident on the sphere in the (m, l) = (0, 0) position of the n = 0 plane are given by
 We now assume that the array is excited by a traveling wave in the z direction with real propagation constant β. Then the constants b−n and b+n are equal to b−0 and b+0, respectively, apart from a phase shift given by
 Substituting (34) in (31), using (from (30)) b−0 = S−E0x0 and b+0 = S+H0y0/Y0, substituting (31) (and the corresponding self-plane expressions) in (33), and multiplying by (kh)3 we obtain
and ρmln is given by (32). By eliminating q from (35a) and (35b) the kd–βd equation is obtained that determines the normalized traveling wave propagation constant βd in terms of kh, d/h, and the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients S− and S+. This will be done below (see (46)–(53)).
 The summations in (35) are useless for computational purposes and must be converted to rapidly convergent forms. We first note that (35a) and (35b), without the cross-term sums multiplied by −q and −1/q, are uncoupled and are simply the kd–βd equations for 3-D arrays of electric and magnetic dipoles transverse to the array axis, respectively, and furthermore, that these two equations are then identical apart from the scattering coefficient, S− or S+. (Note that the self-plane sums in (35a) and (35b) are identical as can be seen by interchanging the indices m and l.) Hence the noncross-term summations in (35a) and (35b) are identical with the summations treated by Shore and Yaghjian [2006, section 5] dealing with 3-D arrays of electric dipoles perpendicular to the array axis. Thus from Shore and Yaghjian [2006, equations (5.65) and (5.68)] we have
with the Clausen functions Cl2 and Cl3 defined and approximated by equations (B7), and with 0 < kh < 2π.
 Now let us treat the sum in (35a) and (35b) multiplied by −q and −1/q
 This sum is proportional to the y directed magnetic field incident on the (0, 0, 0) sphere scattered from the x directed electric dipoles of all the other spheres in the array, or to the x directed electric field incident on the (0, 0, 0) sphere scattered from the y directed magnetic dipoles of all the other spheres in the array. We can use the plane wave Floquet mode expansion of the 2-D vector Green's function to help transform this slowly convergent sum to a rapidly convergent form. Since we have already obtained the scalar counterpart of this expansion in (12), we can take the curl of (12) and (11) rewritten with arbitrary (x, y, z) and multiplied by the unit vector to obtain [Shore and Yaghjian, 2006, equation (9.74)]
for 0 < kh < 2π. However, then in (38),
 Substituting (37) and (41) in (35a) and (35b) we can then write these equations in the form
where assuming that q is real, an assumption that is verified below (53), ℜ−, the real part of the quantity within the brackets of (35a) with the original summations replaced by the new expressions we have derived, is given by
−, the imaginary part of the quantity within the brackets of (35a), is given by
ℜ+, the real part of the quantity within the brackets of (35b) with the original summations replaced by the rapidly convergent expressions we have derived, equals ℜ− with 1/q substituted for q; and +, the imaginary part of the quantity within the brackets of (35b), equals −. Using (A6),
 If we write the scattering coefficients S− and S+ as S− = ∣S−∣ and S+ = ∣S+∣ and equate imaginary parts in (42), we obtain the relations
where ψ− and ψ+ are the phases of the scattering coefficients S− and S+, respectively. The properties of the scattering coefficients (45) were derived independently by Shore and Yaghjian [2004a] from reciprocity and power conservation principles, and our obtaining them here thereby serves as an important check on the validity of our analysis.
 To obtain the kd–βd equation determining βd as a function of kd, d/h, and the scattering coefficients S− and S+, we write (42) as
where from (43) and (44),
 It is straightforward to show that the pair of equations in (46) whose solution gives the normalized traveling wave constant βd implies the important relations in (45) without having to assume that q is real as was done when we obtained (45) above. For, from (46)
where we have let q = qr + iqi, Σ1 = Σ1r + iΣ1i, and noted that Σ2 is real. It is then simple to solve (49) for ∣S−∣ and sin ψ− and, using the fact that Σ1i = −2/3 (kh)3, to obtain
 It follows immediately from (50) and (46) that
 However, the quantities within the parentheses of (52) depend only on properties of the individual array elements whereas ∣q∣2 varies with the array parameters d and h. It follows that (52) implies (45). Thus the relations (45) must be satisfied if the array is to support a lossless dipolar traveling wave. (The converse is, of course, not true. The relations (45) do not guarantee that there will be a solution to equations (46) that must be satisfied if a lossless traveling wave can be supported by the array.) Shore and Yaghjian [2004b] noted that (45) are indeed satisfied by the normalized electric and magnetic Mie scattering coefficients defined in section 1. Here we have shown that (45) must be satisfied by any element of a 3-D periodic array that supports a lossless traveling wave if the element can be modeled by a pair of crossed electric and magnetic dipoles at right angles to each other such that an incident electric (magnetic) field at the element center in the direction of the electric (magnetic) dipole excites only the electric (magnetic) dipole field. Since equations (46) also hold for 1-D periodic arrays [Shore and Yaghjian, 2004b, equations (51a) and (51b)] and 2-D periodic arrays [Shore and Yaghjian, 2006, equations (8.44a) and (8.44b)] of such elements, this conclusion applies to the elements of 1-D and 2-D periodic arrays as well. (In the 1-D case, h is replaced by d. In the 2-D case Σ1 in (46) is replaced by Σ3 ≠ Σ1 but the conclusion remains the same since Σ3i = −2/3 (kh)3.)
 Solving for −q in (46) and equating the resulting expressions we obtain the kd–βd equation
 We note that the kd–βdequation (53) is an equation of real quantities, for using the expression for −q given by the first equality of (53), the fact that the imaginary part of Σ1 is −(kh)3, the fact that Σ2 is real, and (45),
and similarly for the right-hand side of (53). It is simple to solve (53) numerically for βd given values of kd, kh, S−, and S+, using, for example, a simple search procedure with secant algorithm refinement.
 To facilitate calculations of Σ1 and Σ2, we note a rapidly convergent expression for the slowly convergent Schlömilch series ∑Y0(lkh) is given by (A7), and that all series involving negative exponentials and the modified Bessel function K0 (which decays exponentially) converge very rapidly so that only a few terms of these series gives sufficient accuracy.
3.2. Effective Permittivity and Permeability of the Array
 So far we have focused exclusively on obtaining the kd–βd equations for the various arrays considered. If the magnetodielectric sphere elements of a 3-D periodic array are sufficiently close together so that both the following inequalities hold
where d is the spacing in the direction of propagation, then the array can be regarded macroscopically as a medium (which we will refer to as the array medium) with effective or bulk relative permittivity εreff and effective relative permeability μreff that determine the propagation constant of a traveling wave in the direction of the array axis perpendicular to the orientations of the crossed electric and magnetic dipoles by which the spheres are modeled. (It should be noted that in general the array medium is anisotropic and that εreff and μreff do not determine the propagation of waves traveling in directions other than along the array axis. If the array elements are homogeneous magnetodielectric spheres then the directions of the electric and magnetic dipoles are established by the traveling wave, and as the number of spheres per unit volume becomes large (kd ≪ 1 as well as βd ≪ 1) the array medium for a cubic lattice becomes increasingly isotropic. We have, however, called attention to the fact that our analyses for arrays of magnetodielectric spheres apply equally well to any array elements if each of the elements can be modeled by a pair of crossed electric and magnetic dipoles. If the directions of the electric and magnetic dipoles of the array elements are fixed independently of the traveling wave, as they are for split ring resonators for example, then the array medium is anisotropic no matter how closely spaced the elements.) We will now show how εreff and μreff can be obtained from the parameters available to us in solving the kd–βdequation (53). To begin with, the propagation constant β can be expressed in terms of εreff and μreff by the equation
 For a 3-D periodic array of magnetodielectric spheres with εr = μr, the effective permittivity and permeability of the array medium are equal and we obtain immediately from (56)
where plus (minus) is taken accordingly as the group velocity is positive (negative).
 If εr ≠ μr we proceed as follows. The magnetic and dielectric properties of a dipolar medium are characterized by a magnetic polarization or magnetization M and an electric polarization P where M (P) is the magnetic (electric) dipole moment per unit volume of the medium. For the array medium with βd ≪ 1, we can approximate M and P by
where N is the number of magnetic (electric) dipoles per unit volume contributing to M (P) and m (p) is the magnetic (electric) moment of each elementary magnetic (electric) dipole. For a d × h × h rectangular lattice, N = 1/(dh2) and for βd ≪ 1, M and P are well defined because there are many dipoles per unit wavelength of the traveling wave.
 The magnetization and polarization are related to the “average magnetic and electric fields,” H and E, by the scalar constitutive equations [Stratton, 1941, section 1.6]
where ε0 is the permittivity of free space. (At this point in the derivation, (59) can be considered as defining the average magnetic and electric fields.) As we have noted above (see the remark just following (29)), the symmetry of the array results in the incident magnetic (electric) field at each magnetic (electric) dipole being in the same direction as the dipole in agreement with (59) above. In (58), m (p) is in the same direction as M (P), and in (59)M (P) is in the same or opposite direction as H (E) accordingly as μreff − 1 (εreff − 1) is positive or negative, respectively. If in (58) and (59)M, m, H (P, p, E) are written as M, m, H (P, p, E) respectively, multiplied by unit vectors, then the unit vectors are identical and cancel. We can therefore replace the vector quantities in (58) and (59) by their respective scalar quantities and obtain
 If in addition to βd ≪ 1, we have βh ≪ 1, kd ≪ 1, and kh ≪ 1, the average magnetic and electric fields are “macroscopic fields” that obey Maxwell's equations [Jackson, 1999, chapter 6] and thus for plane wavefields H/E is determined by the effective admittance of the array medium; that is
where μ0 is the permeability of free space and the plus (minus) sign corresponds to μreff and εreff both positive (negative) [Pendry, 2004]. Hence
 The ratio of m to p can be related to the parameter q defined by (36) which is known as a result of solving the kd–βdequation (53). We do this by comparing the magnetic and electric far field multiplied by b+0 and b−0, respectively, with the magnetic far field of a magnetic dipole of moment m and the electric far field of an electric dipole of moment p. The magnetic far field multiplied by b+0 is obtained by taking the 1/kr term of the field multiplied by b+n in (27b) and dropping all the subscripts yielding
while the electric far field multiplied by b−0 is obtained by taking the 1/kr term of the field multiplied by b−n in (27a) and dropping all the subscripts yielding
 The magnetic far field of a magnetic dipole of magnetic dipole moment m and the electric far field of an electric dipole of electric dipole moment p are given respectively by [Jackson, 1999, sections 9.2 and 9.3]
where q is the real number defined by (36) and obtained from (53).
 From (62) we then have
 Equations (56) and (67) form a pair of simultaneous equations which can be solved for the two unknowns εreff and μreff. Letting ℛ = βd/(kd) we obtain
 These expressions for the effective permittivity and permeability of the array medium are easily computed from the values of ℛ and q that are found from solving the transcendental equation (53) for the kd–βd diagram of the array. If there are only electric dipole scattered fields (no magnetic dipole scattered fields) of each array element, q = 0 and (68) reduce to
 Similarly, with only magnetic dipole scattered fields (no electric dipole scattered fields), q = ∞ and (68) gives
 Although we will not be concerned in this report with the practical details of exciting traveling waves, it is worth noting here that if the relative permittivity and permeability of the magnetodielectric sphere array elements are equal, then the effective relative permittivity and permeability of the array medium are also equal, and, from (61), the effective admittance of the array medium equals the admittance of free space. It is therefore likely that it will be easier to excite a lossless traveling wave in a slab of the array medium than it will be if the relative permittivity and permeability of the magnetodielectric sphere array elements differ appreciably.
 An alternative and, as we shall explain below, less satisfactory method of obtaining expressions for εreff and μreff under the conditions in (55) is to make use of the Clausius-Mossotti relation [Ishimaru, 1991, section 8-1; Panofsky and Phillips, 1962, section 2–4]. This method, unlike the procedure we have described above, makes no use of the solution to the kd–βd equation (53). Since the usual form of the Clausius-Mossotti relation is based on the assumption of a cubic lattice, we shall apply it only to arrays for which the transverse element spacing h equals the spacing d in the direction of the array axis. If the inequalities (55) are satisfied, then the array can be regarded macroscopically as a medium with effective relative permittivity εreff and effective relative permeability μreff that determine the propagation constant of a traveling wave in the direction of the array axis perpendicular to the orientations of the crossed electric and magnetic dipoles by which the spheres are modeled. We will focus on obtaining εreff because, as will be seen, an expression for μreff can be obtained almost immediately from the expression for εreff.
 We consider the electric polarizability P of a cubic volume of the array medium, taking the axes of the cube to be parallel to the axes of the cubic array lattice. The polarization of the array medium is given from the Clausius-Mossotti relation [Ishimaru, 1991, equation 8-1] as
where p is the moment of each electric dipole. (An easy way to derive (71) for a cubic lattice is to begin with the constitutive relation D = εE = ε0E + P so that
where E is the macroscopic electric field satisfying Maxwell's equations, and P can be expressed as
 To find the local field applied to one dipole from all the other dipoles, remove that one dipole and consider the free-space cubical cavity formed by surfaces that contain the dipoles adjacent to the one removed dipole. Outside this cubical cavity approximate the average polarization density P as a continuum of polarization density. For this continuum containing the small cubical cavity, the electric field at the center of the cavity (the point where the center of the removed dipole was located) is the local field E0 given by (in the limit as the maximum breadth of the cavity becomes infinitesimally small, much smaller than a free-space or traveling-wave wavelength will suffice)
where is the depolarization dyadic for the center of the cube [Yaghjian, 1980]. It is given by = /3, the same value as inside a spherical cavity [Yaghjian, 1980; Sihvola, 1999, section 3.3.1]. Thus, E0 = E + P/(3ε0), which combines with (72) and (73) to yield (71). In reality the dipole scatterers are separated by a finite distance so that (74) is an approximation to the actual local field. N is the number of dipoles per unit volume, and E0 is the local electric field incident on a dipole from all the other dipoles of the array, both electric and magnetic. For a cubic lattice N = 1/d3. If, similarly to what we did above in obtaining (60), P, p, and E0 in (71) are written as P, p, and E0, respectively, multiplied by unit vectors, then the unit vectors are identical and cancel. We can therefore replace the vector quantities in (71) by their respective scalar quantities and obtain
 Solving (75) for εreff,CM yields
 We can find p by equating the expression for the far field radiated by an electric dipole of moment p with the expression for the far field of an electric dipole excited by an incident field E0 in the direction of the dipole at the center of the dipole. From Stratton [1941, section 8.5, equation (30)] the far-field radiated by an electric dipole of moment p is
where θ is the spherical polar angle measured from the direction of the dipole. From the scattering equation (30) giving the coefficient of exp(ikr)/(kr) in the electric dipole field scattered from a dipole in response to the incident field E0 we see that the far scattered field is, referring to Shore and Yaghjian [2004a, equation (40)],
where for a magnetodielectric sphere, S− is the normalized electric dipole scattering coefficient given by S− = −ib1sc [Shore and Yaghjian, 2004b, equation (30a)] with b1sc the Mie electric dipole scattering coefficient given by Stratton [1941, section 9.25, equation (11)]. Equating (77) and (78) then yields
and hence from the second equation in (76) and N = 1/d3
so that the relative permittivity of the array medium is now known from the first equation in (76).
 A similar analysis performed for the magnetic dipoles of the array with the magnetization M, the magnetic dipole moment m, and the incident magnetic field H0, paralleling P, p, and ε0E0, respectively, gives us an expression for the relative permeability of the array medium
where a1sc is the Mie magnetic dipole scattering coefficient given by Stratton [1941, section 9.5, equation (10)]. The expressions for εreff and μreff that we have obtained by using the Clausius-Mossotti relation can then be used to obtain an approximate kd–βd equation when the inequalities (55) are satisfied,
 For our arrays of magnetodielectric spheres, it is obvious that the Clausius-Mossotti relations give values of approximate effective permittivity and permeability, and consequently approximate values for the propagation constant β via (82) because all three of these quantities have imaginary parts when B and A are inserted from (80) and (81), whereas the exact values of β, and thus effective values of εreff and μreff in (68) are real. Nonetheless, when the inequalities in (55) are satisfied, the real parts of the approximate values of βCM, εreff,CM, and μreff,CM, will agree closely with the values of β, εreff, and μreff, respectively. In the figures of this report showing the effective constitutive parameters obtained from numerical results, we plot only the real parts of εreff,CM and μreff,CM. A clear example of where the Clausius-Mossotti relations give very inaccurate results is in the resonance frequency range of an individual scatterer. In a resonance region, a lossless scatterer radiates significant energy and yet the fields of the 3-D array of scatterers can combine to give lossless propagation, whereas the Clausius-Mossotti relations falsely predict constitutive parameters with high loss (large imaginary part) in a resonance region.
 There are at least two reasons to prefer the formulas (68) for the effective permittivity and permeability of the array medium to the Clausius-Mossotti formulas, (76) and (81). First, the formulas (68) correctly predict that the values of effective permittivity and permeability are real for lossless scatterers, whereas the effective permittivity and permeability in (76) and (81) generally have imaginary parts as well as real parts even for lossless scatterers. (Of course, these imaginary parts become small for βd ≪ 1 and kd ≪ 1.) Second, in the important case of magnetodielectric spheres made of materials with μr = εr, the admittance equation (61) holds exactly, the value of q equals ±1, and
with the plus or minus sign occurring if the group velocity is positive or negative, respectively. (If kd ≳ 1 or βd ≳ 1, the group velocity does not necessarily determine the direction of the energy flow in the traveling wave with respect to the direction of the phase velocity of the traveling wave [Tretyakov, 2003, section 5.3].) Significantly, we found in all of our numerical results that the effective permittivity and permeability computed by the “Shore-Yaghjian formulas,” (68), agreed well with the real parts of the Clausius-Mossotti formulas, (76) and (81), in the regions of the kd–βd diagrams where both βd ≲ 1 and kd ≲ 1. This indicates that the cubic arrays can be well characterized macroscopically as homogeneous isotropic material with scalar μreff and εreff if both βd ≲ 1 and kd ≲ 1.