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Keywords:

  • traveling waves;
  • periodic arrays;
  • lossless scatterers

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[1] The kdβd (dispersion) equations are found for traveling waves on two- and three-dimensional infinite periodic arrays of small lossless acoustic monopoles, electric or magnetic dipoles, and magnetodielectric spheres. Using Floquet mode expansions and then expressions for the rapid summation of Schlömilch series, prohibitively slowly convergent summations are converted to forms that can be used for the efficient calculation of the kdβd equations. Computer programs have been written to obtain the kdβd diagrams for all the arrays treated, and representative numerical results are presented and discussed. Expressions, more accurate than the Clausius-Mossotti relations, are obtained for the effective or bulk permittivity and permeability of the arrays utilizing quantities readily available from the solutions of the kdβd equations. Exact computable expressions for the fields of three-dimensional lossless or lossy magnetodielectric sphere arrays that are finite in the direction of the array axis, illuminated by a plane wave parallel to the array axis, are obtained from the analyses performed to obtain the kdβd curves for the infinite arrays.

1. Introduction and Summary of Results

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[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 kdβd equations for traveling waves on periodic arrays and has the advantages of not only yielding the kdβd 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 kβ equation (or diagram) (in our work more properly referred to as the kdβd 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 kdβd 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 kdβd 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 kdβd diagram may be noted here. Ishimaru [1991, chapter 7] shows that the dependence of the kdβd diagram on βd is periodic in βd with a period of 2π. 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 < 2π

  • equation image

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

  • equation image

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

  • equation image

[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 kdπ. Nevertheless, for 3-D arrays we can, without loss of generality, still limit our consideration of traveling waves to those for which 0 < βdπ. 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 kdβd equations are valid only for kh < 2π, 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 kdβd equations as kh [RIGHTWARDS ARROW] 2π since some of the individual terms of the rapidly convergent expressions in the kdβd equations are singular at kh = 2π. Closer analysis shows [Shore and Yaghjian, 2006], however, that the singularities of the various terms in the kdβd equation cancel one another and hence the kdβd equations remain nonsingular at kh = 2π.

[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 kdβd 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 kdβd equation consists of summations of an infinite number of terms of the form exp(ikr)/(kr), exp(ikr)/(kr)2, or exp(ikr)/(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 kdβd 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 kdβd equation that invariably includes series of the form ∑jajZn(jx) known as Schlömilch series [Watson, 1962, chapter 19; Erdélyi, 1953, sections 7.10.3 and 7.15], where aj can be a trigonometric function and Zn is a Bessel function of order n. While the Schlömilch series involving modified Bessel functions Kn converge very rapidly because of the exponential decay of these functions, the Schlömilch series involving Hankel functions (or equivalently ordinary Bessel functions, Jn, and Neumann functions, Yn) converge very slowly. Thus the second form of the kdβd 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 kdβd 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 kdβd 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 kdβd equation for a 3-D array of acoustic monopoles. In section 3.1 we make use of this derivation to obtain the kdβd 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 kdβd 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 kdβd equation, for obtaining the effective permittivity and permeability.

[12] In section 4 we give without derivation computationally tractable forms of the kdβd 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 kdβd 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 kdβd 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 kdβd equations for 2-D and 3-D arrays of magnetic dipoles are identical with the kdβd equations obtained for 2-D and 3-D arrays of electric dipoles. Thus, for example, the kdβd 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 kdβd 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 kdβd 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 R0 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 −iω. 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 R0 = 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 kdβd equation is valid for transverse element separations kh < 2π 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 kdβd 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.

2. Three-Dimensional Acoustic Monopole Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[18] In this section we investigate traveling waves supported by 3-D periodic arrays of lossless acoustic monopoles. The z axis of a Cartesian coordinate system is taken to be the array axis and equispaced planes of acoustic monopoles parallel to the xy plane are located at z = nd, n = 0, ±1, ±2 ⋯. In each plane the monopoles are located at (x, y) = (mh, lh), m, l = 0, ±1, ±2, ⋯. We assume an excitation of the array such that all monopoles in any plane of the array are excited identically. The pressure p00 incident on the monopole at the location x = 0, y = 0, z = 0 (the subscript 0 is used here and throughout the paper to indicate an incident pressure or field, while the superscript is used to indicate location) from all the other monopoles in the array is given by

  • equation image

where rmln is the distance from the (m, l, n)th monopole to the (0, 0, 0) monopole,

  • equation image

and k = ω/c = 2π/λ is the propagation constant, λ is the acoustic wavelength, and c is the acoustic wave speed. In the “self-plane” (the plane containing the reference (0, 0, 0) monopole), n = 0 and

  • equation image

[19] The constants bn are related to the pressure incident on any monopole in the nth plane, p0n, by

  • equation image

where S is the scattering coefficient of an acoustic monopole.

[20] We assume that the array is excited by a traveling wave in the z direction with real propagation constant β. Then the constants bn in (4) are identical apart from a phase shift given by

  • equation image

and

  • equation image

[21] Since b0 = Sp00, we then have, multiplying through by kh

  • equation image

[22] Equation (10) is the kdβd equation that determines the normalized traveling wave propagation constant βd in terms of kh, d/h, and the acoustic monopole scattering coefficient S. If the triple summation over n, m, and l, and the self-plane summation over m and l converged rapidly we could use (10) to calculate kdβd diagrams for 3-D acoustic monopole arrays. Unfortunately, however, the summations converge very slowly and hence must be transformed to rapidly convergent forms if (10) is to become practical for purposes of calculating βd as a function of kh, d/h, and S.

[23] Beginning with the triple summation in (10), we let

  • equation image

and use the Floquet mode expansion method to convert S1(n) to a rapidly convergent form. (It is a simple matter to use the Poisson summation formula to convert S1(n) to a rapidly convergent form. We emphasize the Floquet mode expansion method here because it is a more general procedure not dependent on the availability of tabulated integrals for its successful execution. For details of the Poisson summation treatment of S1(n) see Shore and Yaghjian [2006, equations (3.9)–(3.17)].) In particular, a plane wave Floquet mode expansion of the 2-D periodic scalar Green's function shows that [Shore and Yaghjian, 2006, equations (3.11)–(3.43)]

  • equation image

[24] Next we consider the self-plane sum in (10), which we can write as

  • equation image

where we have let

  • equation image

[25] The sum S2(l) can be regarded as a special case of a summation encountered in the treatment of the 2-D acoustic monopole array in [Shore and Yaghjian, 2006] which is converted to a rapidly convergent form using both the Poisson summation formula and the Floquet mode expansion method [see Shore and Yaghjian, 2006, equations (3.46), (2.9)–(2.16), and (2.27)–(2.43)]. The result is

  • equation image

[26] The self-column sum over m in (13) can be evaluated in closed form using (B1)

  • equation image

[27] Substituting (12), (15), and (16) in (10) we obtain the transformed kdβd equation

  • equation image

[28] From (B5)

  • equation image

[29] The sum

  • equation image

converges very rapidly because of the negative exponential so that it is necessary to include only a few terms in the sum, for example n from 1 to 2 and m, l from −2 to 2, for sufficient accuracy. Alternately an approximation to the sum can be obtained by first performing the summation over n from 1 to ∞ in closed form using (B4) and then including only terms in the summation over m and l from −1 to 1. When this is done we obtain

  • equation image

where r1 = equation image, and r2 = equation image. The slowly convergent Schlömilch series equation imageH0(1)(lkh) can be evaluated using (A6) and (A7). The series equation imageK0(lequation image) converges extremely rapidly because of the exponential decay of K0 (for example, for l = m = 2, and kh < 2π, K0(lequation image) < 9.5 × 10−11) so that only a few terms of the series need be included.

[30] It is useful in obtaining a convenient form of the kdβdequation (17) for calculation purposes to first write the equation as

  • equation image

where from (17), (18), and (16), Re, the real part of the expression within the brackets of (17) is given by

  • equation image

and equation image, the imaginary part of the expression within the brackets of (17), is given by

  • equation image

using (A6). If we write the scattering coefficient as S = ∣Seiψ and then equate imaginary parts in (21), we obtain the relation

  • equation image

[31] This relation was derived by Yaghjian [2002] using reciprocity and power conservation relations, and has been shown here to also be a necessary condition for a 3-D array of lossless acoustic monopoles to support a traveling wave. The derivation of (24) here thus serves as an important check on the correctness of our analysis. Substituting (24) in (21) and equating real parts we then obtain the form of the kdβd equation that is used to calculate βd as a function of kh, d/h, and the phase ψ of the scattering coefficient

  • equation image

with ℜ given by (22) and kh < 2π. Equation (25) is easily solved numerically for βd given values of kd, kh, and ψ, using, for example, a simple search procedure with secant algorithm refinement. In calculating Re the sum of the exponentials is either truncated in accordance with the remark following (19) or evaluated using (20), the Neumann function sum is evaluated using (A7), and the modified Bessel function sum is truncated in accordance with the remark following (20).

3. Three-Dimensional Magnetodielectric Sphere Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[32] This section consists of sections 3.1 and 3.2. In section 3.1 we obtain the kdβd equation for traveling waves supported by 3-D infinite periodic arrays of lossless magnetodielectric spheres. A vector form of the Floquet mode expansion method is used to convert the original form of the kdβd equation containing extremely slowly convergent summations to a form suitable for calculations. If the array elements are sufficiently close together, the array can be regarded as a form of homogeneous medium characterized by an effective permittivity and permeability. In section 3.2 we describe two methods for obtaining this effective permittivity and permeability. The first method obtains the permittivity and permeability directly from the solution to the kdβd equation. The second method, based on the Clausius-Mossotti relation, is completely independent of the kdβd equation and is more restrictive than the first method.

3.1. The kdβd Equation for 3-D Magnetodielectric Sphere Arrays

[33] 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

  • equation image
  • equation image

[35] 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

  • equation image

and the unit vector in the direction from the (m, l, n) sphere to the (0, 0, 0) sphere, equation imagemln0, is

  • equation image

[36] It is straightforward to obtain algebraic expressions for all the quantities in (27), and it then follows immediately by considering the summations over m and l from −∞ to ∞ of odd terms in m and l that the equation image and equation image components of the electric field vanish, and the equation image and equation image components of the magnetic field vanish so that we are left with an x directed electric field and a y directed magnetic field incident on the reference sphere.

[37] The constants bn 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)]

  • equation image

where S and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively. “Normalized” means that bn (b+n) is the coefficient of exp(ikr)/(kr) in the outgoing electric (magnetic) dipole field in response to the incident field E0x0nequation image (H0y0n/Y0equation image) 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)

  • equation image

and

  • equation image

where we have let

  • equation image

[38] 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

  • equation image

[39] We now assume that the array is excited by a traveling wave in the z direction with real propagation constant β. Then the constants bn and b+n are equal to b−0 and b+0, respectively, apart from a phase shift given by

  • equation image

[40] Substituting (34) in (31), using (from (30)) b−0 = SE0x0 and b+0 = S+H0y0/Y0, substituting (31) (and the corresponding self-plane expressions) in (33), and multiplying by (kh)3 we obtain

  • equation image

and

  • equation image

where

  • equation image

and ρmln is given by (32). By eliminating q from (35a) and (35b) the kdβd 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)).

[41] 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 kdβd 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

  • equation image

with the Clausen functions Cl2 and Cl3 defined and approximated by equations (B7), and with 0 < kh < 2π.

[42] Now let us treat the sum in (35a) and (35b) multiplied by −q and −1/q

  • equation image

[43] 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 equation image to obtain [Shore and Yaghjian, 2006, equation (9.74)]

  • equation image

for 0 < kh < 2π. However, then in (38),

  • equation image

[44] Thus, using (B6),

  • equation image

[45] Substituting (37) and (41) in (35a) and (35b) we can then write these equations in the form

  • equation image

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

  • equation image

equation image, the imaginary part of the quantity within the brackets of (35a), is given by

  • equation image

+, 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 equation image+, the imaginary part of the quantity within the brackets of (35b), equals equation image. Using (A6),

  • equation image

[46] If we write the scattering coefficients S and S+ as S = ∣Sequation image and S+ = ∣S+equation image and equate imaginary parts in (42), we obtain the relations

  • equation image

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.

[47] To obtain the kdβd equation determining βd as a function of kd, d/h, and the scattering coefficients S and S+, we write (42) as

  • equation image

where from (43) and (44),

  • equation image

and

  • equation image

[48] 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)

  • equation image

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

  • equation image

[49] It follows immediately from (50) and (46) that

  • equation image

[50] Adding (50) and (51) then gives

  • equation image

[51] However, the quantities within the parentheses of (52) depend only on properties of the individual array elements whereas ∣q2 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.)

[52] Solving for −q in (46) and equating the resulting expressions we obtain the kdβd equation

  • equation image

[53] We note that the kdβdequation (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 −equation image(kh)3, the fact that Σ2 is real, and (45),

  • equation image

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.

[54] 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

[55] So far we have focused exclusively on obtaining the kdβd 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

  • equation image

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 kdβdequation (53). To begin with, the propagation constant β can be expressed in terms of εreff and μreff by the equation

  • equation image

[56] 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)

  • equation image

where plus (minus) is taken accordingly as the group velocity is positive (negative).

[57] If εrμr 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

  • equation image

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.

[58] 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]

  • equation image

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

  • equation image

[59] 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

  • equation image

where μ0 is the permeability of free space and the plus (minus) sign corresponds to μreff and εreff both positive (negative) [Pendry, 2004]. Hence

  • equation image

[60] 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 kdβdequation (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

  • equation image

while the electric far field multiplied by b−0 is obtained by taking the 1/kr term of the field multiplied by bn in (27a) and dropping all the subscripts yielding

  • equation image

[61] The magnetic far field of a magnetic dipole of magnetic dipole moment mequation image and the electric far field of an electric dipole of electric dipole moment pequation image are given respectively by [Jackson, 1999, sections 9.2 and 9.3]

  • equation image

[62] Letting equation image = equation image and equation image = equation image and equating (63) with (64) then yields

  • equation image

[63] Thus

  • equation image

where q is the real number defined by (36) and obtained from (53).

[64] From (62) we then have

  • equation image

[65] 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

  • equation image

[66] 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 kdβd 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

  • equation image

[67] Similarly, with only magnetic dipole scattered fields (no electric dipole scattered fields), q = ∞ and (68) gives

  • equation image

[68] 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.

[69] 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 kdβd 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.

[70] 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

  • equation image

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

  • equation image

where E is the macroscopic electric field satisfying Maxwell's equations, and P can be expressed as

  • equation image

[71] 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)

  • equation image

where equation image is the depolarization dyadic for the center of the cube [Yaghjian, 1980]. It is given by equation image = equation image/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

  • equation image

[72] Solving (75) for εreff,CM yields

  • equation image

[73] 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

  • equation image

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)],

  • equation image

where for a magnetodielectric sphere, S is the normalized electric dipole scattering coefficient given by S = −iequation imageb1sc [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

  • equation image

and hence from the second equation in (76) and N = 1/d3

  • equation image

so that the relative permittivity of the array medium is now known from the first equation in (76).

[74] 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

  • equation image

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 kdβd equation when the inequalities (55) are satisfied,

  • equation image

[75] 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.

[76] 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

  • equation image

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 kdβd 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.

4. List of kdβd Equations for 2-D and 3-D Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[77] In this section we list for easy reference without derivations the rapidly convergent forms of the kdβd equations for all the 2-D and 3-D arrays investigated. Derivations of (86) and (106) are given in sections 2 and 3, respectively, and derivations of all the kdβd equations are given by Shore and Yaghjian [2006].

4.1. Two-Dimensional Acoustic Monopole Arrays

[78] 

  • equation image
  • equation image

[79] The Neumann function sum is evaluated using (A1), and the modified Bessel function sum converges extremely rapidly because of the exponential decay of K0.

4.2. Three-Dimensional Acoustic Monopole Arrays

[80] 

  • equation image
  • equation image

[81] The sum of exponentials converges very rapidly because of the negative exponential so that it is necessary to include only a few terms in the sum, for example n from 1 to 2 and m, l from −2 to 2 for sufficient accuracy. Alternately an approximation to the sum can be obtained by first performing the summation over n from 1 to ∞ in closed form using (B4) and then including only terms in the summation over m and l from −1 to 1. This yields

  • equation image

where r1 = equation image, and r2 = equation image. The Neumann function sum is evaluated using (A7), and the modified Bessel function sum converges extremely rapidly because of the exponential decay of K0.

4.3. Two-Dimensional Electric Dipole Arrays With Dipoles Perpendicular to the Array Axis

4.3.1. Electric Dipoles Parallel to the Array Plane

[82] 

  • equation image
  • equation image

[83] The Neumann function sum is evaluated using (A1), and the modified Bessel function sum converges extremely rapidly because of the exponential decay of K0. The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

4.3.2. Electric Dipoles Perpendicular to the Array Plane

[84] 

  • equation image
  • equation image

[85] The sum ∑ cos(nβd) Y0(nkd) can be evaluated using (A1), the sum ∑ cos(nβd)Y2(nkd) can be evaluated very efficiently by using (A5), and the K0 and K2 series converge extremely rapidly because of the exponential decay of K0 and K2.

4.4. Three-Dimensional Electric Dipole Arrays With Dipoles Perpendicular to the Array Axis

[86] 

  • equation image
  • equation image

[87] The sum of exponentials converges very rapidly because of the negative exponential so that it is necessary to include only a few terms in the sum, for example n from 1 to 2 and m, l from −2 to 2 for sufficient accuracy, the Neumann function sum is evaluated using (A7), and the modified Bessel function sum converges extremely rapidly because of the exponential decay of K0. The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

4.5. Two-Dimensional Electric Dipole Arrays With Dipoles Parallel to the Array Axis

[88] 

  • equation image
  • equation image

[89] The sum ∑ cos(nβd) Y0(nkd) can be evaluated using (A1), the sum ∑ cos(nβd) Y2(nkd) can be evaluated very efficiently by using (A5), and the modified Bessel function K0 and K2 series converge extremely rapidly because of the exponential decay of K0 and K2. The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

4.6. Three-Dimensional Electric Dipole Arrays With Dipoles Parallel to the Array Axis

[90] 

  • equation image
  • equation image

[91] The first triple sum in (98) converges very rapidly because of the negative exponential so that it is necessary to include only a few terms in the sum, for example, n from 1 to 2 and m, l from −2 to 2, for sufficient accuracy. Alternately an approximation to the sum can be obtained by first performing the summation over n from 1 to ∞ in closed form using (B4) and then including only terms in the summation over m and l from −1 to 1. When this is done we obtain

  • equation image

where r1 = equation image, and r2 = equation image. Accelerated convergence expressions for the Schlömilch series ∑Y0(lkh) and Y2(lkh) are given in (A7) and (A9), respectively. The modified Bessel function series in (98) converge extremely rapidly because of the exponential decay of K0 and K1 so that only a few terms of the series need be included. The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

4.7. Two-Dimensional Magnetodielectric Sphere Arrays

4.7.1. Electric Dipoles Parallel to the Array Plane

[92] 

  • equation image

S and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively,

  • equation image

where b1sc and a1sc are the electric and magnetic Mie dipole scattering coefficients defined by Stratton [1941, equations (11) and (10), section 9.25]:

  • equation image
  • equation image

and

  • equation image

[93] Rapidly convergent expressions for the slowly convergent Schlömilch series ∑ cos(nβd) Y0(nkd) and ∑ cos(nβd) Y2(nkd) are given in (A1) and (A5), respectively. All series involving the modified Bessel functions K0, K1, and K2, converge very rapidly because of the exponential decay of these functions so that only a few terms of the series give sufficient accuracy. The convergence of the series ∑ sin(nβd) Y1(nkd) can be greatly accelerated by using (A4). The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

4.7.2. Electric Dipoles Perpendicular to the Array Plane

[94] 

  • equation image

S and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively, given by (101), and Σ1, Σ2, and Σ3 are given by (102), (103), and (104), respectively.

4.8. Three-Dimensional Magnetodielectric Sphere Arrays

[95] 

  • equation image

S and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively, given by (101)

  • equation image

and

  • equation image

[96] A rapidly convergent expression for the slowly convergent Schlömilch series ∑ Y0(lkh) is given by (A7), and 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. Alternately, approximate closed form expressions for the summations involving negative exponentials can be obtained by first performing the summation over n from 1 to ∞ using (B4) and then including only terms in the summations over m and l for which ∣m∣ ≤ 1 and ∣l∣ ≤ 1, thus yielding

  • equation image

where r1 = equation image, and r2 = equation image. The corresponding approximate closed form expression for the sum of negative exponentials in Σ2 is

  • equation image

where r1 = equation image, and r2 = equation image. The Clausen functions Cl2 and Cl3 are defined and approximated by (B7).

5. Partially Finite 3-D Array of Magnetodielectric Spheres

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[97] In this section we show that the analysis we have performed in section 3 to obtain the kdβd equation for an infinite periodic 3-D array of lossless magnetodielectric spheres with electric and magnetic dipoles oriented perpendicular to the array axis, can be used to obtain expressions for the field of a partially finite periodic array of these elements (an array that is finite in the direction of the array axis and of infinite extent in the directions transverse to the array axis). 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 free space and the array. The procedure followed uses a method due to Foldy [1945]. It should be noted that βd, the normalized propagation constant for the traveling wave supported by the infinite array, does not appear in any of the expressions used in this section. Hence the results of this section of the paper are valid not only for arrays of lossless scatterers for which βd is real as previously assumed, but also for arrays of lossy scatterers for which βd is complex.

[98] We investigate the field excited by a plane wave incident from free space on a 3-D periodic array of lossless or lossy magnetodielectric spheres. The array is finite in the direction of the array axis and infinite in the directions transverse to the array axis. The direction of incidence of the illuminating plane wave is parallel to the array axis, normal to the interface between the array and free space. As in section 3, 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. The z axis of a Cartesian coordinate system is taken to be the array axis and N + 1 equispaced planes parallel to the xy plane of magnetodielectric spheres are located at x = nd, n = 0, 1, 2, ⋯, N. In each plane the spheres are centered at (x, y) = (mh, lh), l, m = 0, ±1, ±2, ⋯ with the electric and magnetic dipoles oriented in the x and y direction, respectively. The electric and magnetic field vectors of the incident plane wave illuminating the array from the left are

  • equation image

so that all spheres in any plane of the array are excited identically. As shown in section 3, the electric field at a point on the array axis has an x component only, and the magnetic field has a y component only. Hence, at a point on the array axis not coinciding with an array element, the electric field is given by (see (31))

  • equation image

and

  • equation image

where

  • equation image

[99] That is, the total field is equal to the incident field plus the sum of the waves scattered from all the elements of the array. The coefficients bn and b+n of the scattered waves are given by

  • equation image

where S and S+ are the normalized magnetodielectric sphere electric and magnetic dipole scattering coefficients, respectively, given in section 1; Exn(zn) and Hyn(zn) are the external electric and magnetic fields, respectively, incident on a sphere in the nth plane; and zn = nd, n = 0, 1, 2, ⋯, N. Hence, from (112) and (114), the total electric field at a point on the array axis not coinciding with an array element is

  • equation image

and the total magnetic field at a point on the array axis not coinciding with an array element is

  • equation image

[100] Since all the spheres in any plane of the array normal to the array axis are excited identically, Exn(zn) and Hyn(zn)/Y0 are equal, respectively, to the external electric and magnetic fields incident on the sphere on the z axis at z = zn = nd. Expressions for Exn(zn) and Hyn(zn)/Y0 ceequation image and the contribution of the fields scattered from all the array elements other than the (0, 0, zn) sphere

  • equation image

and

  • equation image

where

  • equation image
  • equation image

and

  • equation image

where

  • equation image

[101] Thus we have a system of 2(N + 1) equations for the 2(N + 1) unknowns Exn(zn), Hyn(zn)/Y0, n = 0, 1, 2, ⋯, N

  • equation image
  • equation image

[102] Rapidly convergent expressions are available for σ11 and σ2 from the analyses performed for 3-D arrays of electric dipoles perpendicular to the array axis of Shore and Yaghjian [2006, equations (5.64) and (5.68)] and we give the results here:

  • equation image

and

  • equation image

with the Clausen functions Cl2 and Cl3 defined and approximated by equations (B7). Using (A6) and (A7),

  • equation image

[103] Finally, from (39), we have a rapidly convergent expression for σ12

  • equation image

[104] It is then straightforward to write a computer program to solve the system of equations (120) for the values of Ezn(zn) and Hyn(zn)/Y0. To calculate the field at any point on the array axis (other than at elements of the array), we then use (115), which can be written as

  • equation image
  • equation image

with σ11(∣ndz∣) and σ12(ndz) given by (117) with j and n replaced by n and z/d, respectively. The rapidly convergent expressions (121) and (124) are used for calculating σ11(∣ndz∣) and σ12(ndz) with n replaced by nx/d.

[105] Given values of Ex(z) we can calculate the reflection coefficient of the wave scattered back in the negative z direction for z < 0 as well as the transmission coefficient of the wave traveling in the positive z direction for z > Nd. Since the amplitude of the plane wave incident on the partially finite array is 1, the reflection coefficient, R, is the complex coefficient of the wave e−ikz for z < 0 with R obtained from the equation

  • equation image

[106] In practice, z should not be chosen too close to the origin in order for the transient scattered waves to die out, say z/d < −10. The square of the magnitude of the transmission coefficient, ∣T2, is, of course, equal to 1 − ∣R2, since the array is assumed to be lossless. If the complex transmission coefficient is desired, it can be obtained from the equation

  • equation image

[107] In practice, z should not be chosen too close to the end of the array at z = Nd, say z/d > N + 10.

6. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[108] In this section we present the results obtained by numerically solving the transcendental equations derived in sections 2–5 for the real propagation constants β of traveling waves on 2-D and 3-D infinite periodic arrays of lossless scatterers whose only significant scattered fields are those of electric and/or magnetic dipoles. For the sake of comparison, we also find the propagation constants on the corresponding 1-D periodic arrays of some scatterers from their transcendental equations given in previous reports [Shore and Yaghjian, 2004a, 2004b]. All these transcendental equations involve only well-known functions or rapidly convergent summations. They are readily solved on a personal computer to efficiently obtain kdβd curves (diagrams) for the traveling waves.

[109] As explained in section 1, traveling waves for βd > π can be reexpressed as traveling waves with −π < βd < 0. Moreover, for periodic arrays of elements composed of reciprocal material only, it is proven by Yaghjian [2007] that every traveling wave is bidirectional, that is, for every traveling wave with propagation constant β there exists a corresponding traveling wave with propagation constant −β. Thus the kdβd diagrams in this section require βd to cover only the domain [0, π] because all the arrays considered in this section are composed of reciprocal elements.

[110] Also, as explained in section 1, unattenuated traveling waves (real β) on 1-D and 2-D infinite periodic arrays cannot exist for kd > βd. In other words, all fast waves on 1-D and 2-D lossless periodic arrays are leaky waves; see section 1. Therefore all the kd axes on the kdβd diagrams for the 1-D and 2-D arrays in this section span the range only from 0 to π. Since unattenuated traveling waves on lossless 3-D arrays can exist for kd > βd, the kdβd diagrams for the 3-D periodic arrays considered in this section may have their kd axes extend to values larger than π. In all kdβd diagrams, however, the range of kd is restricted to the values of kd at which the quadrupole moments appear (the first spherical multipoles of higher order than the electric and magnetic dipoles), since our analysis assumes that all scattered fields are negligible except the electric and magnetic dipole fields.

[111] For generally shaped scatterers, the analysis assumes that the scattered electric and magnetic dipoles of any one element are determined only by the values of the incident field (of all the other elements) at the center of that element, and thus there may be some inaccuracy introduced if the elements are too closely packed. Emphatically, however, for spherical scatterers we have proven (as part of the foregoing analysis and the analysis in the previous reports [Shore and Yaghjian, 2004a, 2004b]), using the orthogonality of the spherical harmonics and the field values at the center of the incident spherical waves, that this assumption of the scattered dipole fields being determined solely by the central incident field values holds exactly. Consequently, for spherical scatterers there is no loss of accuracy introduced into our equations by packing the scatterers as tightly as possible as long as the predominant scattering is that of electric and magnetic dipoles.

[112] Once the kdβd diagram is found for a 3-D infinite periodic array, we use the formulas derived in section 3.2 (referred to herein as the Shore-Yaghjian formulas) for determining the effective (bulk) permittivity and permeability of the array from the parameters in the transcendental equation. In addition, these bulk parameters are also determined from the Clausius-Mossotti relations, which, in general, are not as accurate as those determined from the Shore-Yaghjian formulas. As a rule of thumb, the bulk parameters are not accurate predictors of propagation characteristics and reflection or transmission coefficients unless both kd and βd are less than about unity. Thus we show the values of the bulk permittivity and permeability in the following figures for kd no greater than about unity.

[113] All the elements of the 2-D and 3-D arrays considered in this paper are arranged in rectangular and rectangular parallelepiped lattices, respectively. Furthermore, all the 2-D and 3-D arrays considered in this section on the numerical results have the elements arranged in square and cubic lattices, respectively, that is, h = d. If only electric (magnetic) dipoles are present in a traveling wave, the transcendental equation for βd depends only on kd and the phase ψ of the scattering coefficient. Although the phase ψ of the scattering coefficient of any given scatterer will generally change with frequency (that is, with kd for a fixed separation distance d), it is revealing to observe the plots of the family of kdβd curves determined by different values of the phase ψ for 1-D, 2-D, and 3-D arrays of acoustic monopoles, electric (magnetic) dipoles, and magnetodielectric spheres given by Yaghjian [2002] and Shore and Yaghjian [2004a, 2004b, 2006]. The 1-D, 2-D, and 3-D family of kdβd curves for magnetic dipoles are identical to those for electric dipoles.

[114] In sections 6.1–6.4, kdβd diagrams and effective permittivity and permeability curves (for 3-D arrays) are given for representative scatterers, namely, for short perfectly electrically conducting (PEC) wires, for PEC spheres, for diamond spheres, for silver nanospheres, and for magnetodielectric spheres. Plots of the reflection coefficient for partially finite arrays of magnetodielectric spheres are given in section 6.5.

6.1. PEC Short Wires and PEC Spheres

[115] In this section, the kdβd diagrams and relative permittivity and permeability of perfectly electrically conducting (PEC) short wires and PEC spheres are determined from the relevant transcendental equations.

6.1.1. PEC Short Wires

[116] The kdβd diagram for an infinite linear (that is, 1-D) periodic array of short parallel PEC wires (electric dipoles), normal to the propagation axis, with different ratios of wire length to separation distance (2h/d) is given in Figure 4 of Shore and Yaghjian [2004a]. (Here the symbol h denotes the half length of the wires and should not be confused with the 2-D and 3-D array transverse separation distance denoted by the same symbol h, which is set equal to the longitudinal separation distance d along the propagation axis in all the numerical examples of this section.) The ratio of the radius of each wire to its length [ρ/(2h)] is equal to 0.1, and the phase ψ of the scattering coefficient S was computed at each kh with the Numerical Electromagnetics Code (NEC) [Burke, 1992]. The corresponding kdβd diagrams for traveling waves on 2-D and 3-D short parallel PEC wires (electric dipoles) normal to the propagation axis are computed from the transcendental equations in (89), (91), and (93) and are shown in Figures 1, 2, and 3, all three of which display the same general variation for the 2-D and 3-D arrays of parallel wires as for the 1-D array in Figure 4 of Shore and Yaghjian [2004a]. Until the values of kd become significantly greater than 1, the values of β are fairly close to the values of k and thus the traveling waves on these short wires will be loosely coupled to the wires except for values of βd fairly close to π, where the 3-D array does not well approximate a homogeneous medium characterized by bulk permittivity. Incidentally, the curves for these short wires do not continue to very small values of kd simply because we didn't have the NEC-determined values of ψ readily available for these small values of kd. Also, the range of the kd axis in the 3-D array in Figure 3 is restricted to π because significant multipoles of higher order than electric dipoles are excited on the wires for kd > π.

image

Figure 1. The kdβd curves for 2-D array of short-wire electric dipoles (parallel to the array plane) with ψ obtained from NEC code.

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image

Figure 2. The kdβd curves for 2-D array of short-wire electric dipoles (perpendicular to the array plane) with ψ obtained from NEC code.

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image

Figure 3. The kdβd curves for 3-D array of short-wire electric dipoles (normal to the propagation direction) with ψ obtained from NEC code.

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[117] The numerical solution to the transcendental equations in (95) and (97) showed that there are no unattenuated traveling waves on 2-D and 3-D arrays (kd < π) of short PEC wires oriented parallel to the propagation axis, as was also found in the case of 1-D arrays of wires parallel to the array axis [Shore and Yaghjian, 2004a]. It is noted, however, that the constant ψ curves [Shore and Yaghjian, 2006, Figures 6 and 7] imply that if the short wires were passively loaded to change the values of their scattering coefficients, that is, change the values of ψ, traveling waves could be supported by these longitudinally oriented wires [Vitanen and Tretyakov, 2005] The effective relative permittivity versus kd of the 3-D array of transverse dipoles corresponding to the kdβd curves of Figure 3 is computed from the Shore-Yaghjian formula in (69) and from the real part of the Clausius-Mossotti relation (76) and is shown in Figure 4. (As explained in section 3.2, the Clausius-Mossotti relations, unlike the Shore-Yaghjian formulas, erroneously predict imaginary parts for the effective permittivity and permeability of these unattenuated traveling waves on lossless arrays. The imaginary parts are usually much smaller than the real parts, however, if kd and βd are equation image1.) In the region kdequation image 1, the values of βd are equation image 1, and thus as would be expected, the real part of the Clausius-Mossotti relation predicts a relative permittivity that agrees quite well with the Shore-Yaghjian results obtained from the exact transcendental equation. Of course, since there are no appreciable magnetic dipole moments on short wires, the relative permeability of the 3-D arrays is equal to unity.

image

Figure 4. Effective relative permittivity for 3-D array of short-wire electric dipoles (normal to the propagation direction) with ψ obtained from NEC code.

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6.1.2. PEC Spheres

[118] The fields scattered by a PEC sphere illuminated by an incident plane wave contains a significant magnetic dipole moment as well as an electric dipole moment even as the frequency (kd) approaches zero. Thus the transcendental equations for magnetodielectric spheres must be used to find the kdβd diagrams for the traveling waves on 1-D, 2-D, and 3-D arrays of PEC spheres. Specifically, the kdβd diagrams for the transverse traveling waves (that is, traveling waves with orthogonal electric and magnetic dipole moments normal to the direction of propagation) on square lattice 2-D and cubic lattice 3-D arrays of PEC spheres are determined from (100) and (105) (for 2-D arrays) and from (106) (for 3-D arrays) and are shown in Figures 5–7 for values of a/d = 0.3, 0.4, and 0.45, where a is the radius of the spheres. The electric and magnetic dipole scattering coefficients, S and S+, required by the transcendental equations are obtained from the electric and magnetic dipole coefficients in the Mie solution; see section 3 and Shore and Yaghjian [2004b, equations (30a) and (30b)]. The resonance of the first quadrupole moment of the PEC sphere occurs at a value of ka = 2.3 and thus the 3-D curves in Figure 7 are truncated at the corresponding values of kd.

image

Figure 5. The kdβd curves for 2-D array of PEC spheres (electric dipole moments parallel to the array plane) with dipole scattering coefficients obtained from Mie solution.

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image

Figure 6. The kdβd curves for 2-D array of PEC spheres (electric dipole moments perpendicular to the array plane) with dipole scattering coefficients obtained from Mie solution.

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image

Figure 7. The kdβd curves for 3-D array of PEC spheres (dipole moments normal to the propagation direction) with dipole scattering coefficients obtained from Mie solution.

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[119] All the Figures 5–7 for PEC spheres exhibit kdβd curves similar to those by Shore and Yaghjian [2004a, Figure 4] and Figures 1–3 for the short PEC wires. In particular, traveling waves on the PEC spheres do not travel with speeds much less than the free-space speed of light (and thus are not strongly coupled to the spheres) until the values of βd are fairly close to π, where the 3-D array does not well approximate a homogeneous media characterized by a bulk permittivity and permeability. We also note that the 3-D array curves in Figure 7 have regions of negative group velocity for values of kd > π. However, because kd > π in this region, the negative group velocity does not imply that the direction of power flow in the traveling wave is opposite the direction of the phase velocity; see section 3.2 and Tretyakov [2003, section 5.3].

[120] The effective relative permittivity and permeability versus kd of the 3-D array of orthogonal transverse electric and magnetic dipoles corresponding to the kdβd curves of Figure 7 are computed from the Shore-Yaghjian formulas in (68) and from the real parts (see section 6.1.1) of the Clausius-Mossotti relations in (76) and (81) and are shown in Figure 8. In the region kd ≲ 1, the values of βd are ≲1, and thus as would be expected, the real parts of the Clausius-Mossotti relations predict a relative permittivity and permeability that agree quite well with the Shore-Yaghjian formulas obtained from the exact transcendental equation. The values of the effective relative permittivity and permeability in Figure 8 are commensurate with the facts that for ka ≪ 1 the electric and magnetic dipole moments in the Mie solution to the PEC sphere are in the ratio of two to one and have opposite signs.

image

Figure 8. Effective relative permittivity and permeability for 3-D array of PEC spheres (dipole moments normal to the propagation direction) with dipole scattering coefficients obtained from Mie solution.

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[121] The numerical solution to the transcendental equations (96) of Shore and Yaghjian [2004a], (95), and (97) for longitudinal traveling waves (that is, traveling waves with either electric or magnetic dipoles along the direction of propagation) show that none exist on 1-D and 2-D arrays of PEC spheres and that only an electric dipole longitudinal wave exists on a 3-D array of PEC spheres, as shown in Figure 9, and then only for the case of a/d = 0.3 and values of kd greater than 5.5.

image

Figure 9. The kdβd curve for 3-D array of PEC spheres (electric dipole moments parallel to the propagation direction) with electric dipole scattering coefficients obtained from Mie solution.

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6.2. Diamond Spheres

[122] Diamond has a nearly constant relative permittivity of 5.84 with very low loss at optical frequencies and a relative permeability equal to unity. We shall assume this value of relative permittivity for diamond in all the numerical computations for arrays of diamond spheres of radius a. We shall also assume a single packing ratio of a/d = 0.45. The first resonance of the diamond sphere is a magnetic dipole resonance that occurs at a ka = 1.25 even though the electric dipole moment dominates at lower values of ka. (In general, the first resonance of a positive permittivity sphere is a magnetic dipole and the first resonance of a negative permittivity sphere is an electric dipole.) Therefore, to obtain the kdβd curves for the transverse waves on 2-D and 3-D arrays of diamond spheres, one must use the magnetodielectric transcendental equations (100) and (105) (for 2-D arrays) and (106) (for 3-D arrays). The electric and magnetic dipole scattering coefficients, S and S+, required by the transcendental equations are obtained from the electric and magnetic dipole coefficients in the Mie solution; see section 3 and Shore and Yaghjian [2004b, equations (30a) and (30b)]. The numerical solutions to these transcendental equations are plotted in the kdβd diagrams of Figures 10–12. The curves in Figures 10–12 for the slow traveling waves are similar to those for the traveling waves on PEC wires and PEC spheres except for the second branches of the curves produced by the magnetic dipole resonance of the diamond sphere. On these second branches the group velocity is negative (opposite the phase velocity). However, the 3-D array of these diamonds would not be meaningfully characterized by negative effective permittivity and permeability in this region of negative group velocity because kd ≫ 1 in this region. In the region kd < 1 the effective relative permeability of the 3-D array is approximately equal to unity and the effective relative permittivity computed from both the Shore-Yaghjian formula in (68) and the real part (see section 6.1.1) of the Clausius-Mossotti relation (76) is shown in Figure 13, which shows that the bulk relative permittivity of the 3-D array of closely packed diamond spheres in this small kd region is about 2, a value between that of diamond (5.84) and that of free space (1.0). This is reflected in the fact that in contrast to the 2-D kdβd diagrams of Figures 10 and 11, the 3-D kdβd curve of Figure 12 has a constant slope of less than one in the low-frequency (large wavelength) limit since the slope of the kdβd line approaching the origin is equal to (see (57))

  • equation image

which in this case is approximately equal to 1/equation image ≈ 0.7. The range of the values of kd in Figure 12 for transverse traveling waves on the 3-D array of diamond spheres is limited to 4 because the first quadrupole resonance occurs at about this value.

image

Figure 10. The kdβd diagram for 2-D array of diamond spheres (electric dipole moments parallel to the array plane) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 11. The kdβd diagram for 2-D array of diamond spheres (electric dipole moments perpendicular to the array plane) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 12. The kdβd diagram for 3-D array of diamond spheres (dipole moments normal to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 13. Effective relative permittivity for 3-D array of diamond spheres (dipole moments normal to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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[123] The kdβd curves for longitudinal traveling waves of electric or magnetic dipoles parallel to the direction of propagation are computed for the diamond spheres from the transcendental equations in (95) and (97) for 2-D and 3-D arrays, respectively, and are plotted in Figures 14 and 15. Figure 14 shows that on the 2-D array of diamond spheres there are only magnetic dipole longitudinal waves and then only in a narrow bandwidth near the first resonant frequency of the spheres. Figure 15 shows that both electric (for kd > π) and magnetic dipole longitudinal traveling waves exist on 3-D arrays of diamond spheres, again in fairly narrow bandwidths.

image

Figure 14. The kdβd diagram for 2-D array of diamond spheres (magnetic dipole moments parallel to the propagation direction) with a/d = .45 and magnetic dipole scattering coefficients obtained from Mie solution.

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image

Figure 15. The kdβd diagram for 3-D array of diamond spheres (magnetic dipole moments parallel to the propagation direction) with a/d = .45 and magnetic dipole scattering coefficients obtained from Mie solution.

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6.3. Silver Nanospheres

[124] The kdβd diagrams for the unattenuated traveling waves on PEC wires, PEC spheres, and diamond spheres have shown that βdequation imagekd and thus the fields of these traveling waves are not strongly coupled or tightly confined to the elements of the arrays. That is, the preponderance of the power in the fields is not concentrated within a distance appreciably less than a wavelength from the scattering elements of the arrays. For use in optical circuitry, it is desirable to produce traveling waves at optical frequencies that are confined to a small fraction of a wavelength from array elements that are also small fractions of a wavelength across. One possibility for producing such traveling waves is to use linear (1-D) chains of silver nanospheres [Maier et al., 2003; Park and Stroud, 2004; Citrin, 2004; Burin et al., 2004], since at optical frequencies silver behaves as a plasma with a negative dielectric constant (relative permittivity). (Another possibility for producing tightly confined traveling waves would be to use spheres with high values (≳100) of permittivity or permeability. However, low-loss materials apparently do not exist that have such high values of permittivity or permeability at optical frequencies.) Such plasmas support surface waves at the silver interface which lead to electric dipole resonances of the silver spheres with fields confined to a small fraction of a wavelength from the spheres. Therefore, in this section we shall use our transcendental equations to compute the kdβd diagrams for traveling waves on 1-D, 2-D, and 3-D arrays of silver nanospheres [Shore and Yaghjian, 2005a]. Unlike spheres with a positive dielectric constant (like diamond spheres) whose first resonance is that of a magnetic dipole, the first resonance of “plasmonic” spheres with negative dielectric constant is that of an electric dipole. Consequently, for silver nanospheres the transcendental equations for purely electric dipoles would produce accurate kdβd curves through the first dipole resonance. Nonetheless, we used the magnetodielectric transcendental equation (52) of Shore and Yaghjian [2004b] or equation (18) of Shore and Yaghjian [2005b] for 1-D arrays, (100) and (105) for 2-D arrays, and (106) for 3-D arrays of silver nanospheres.

[125] The kdβd diagrams for transverse traveling waves on 1-D, 2-D, and 3-D silver nanospheres are shown in Figures 16–19. The curves 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:

  • equation image

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 nanometers 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 (129) as

  • equation image
image

Figure 16. The kdβd curves for 1-D array of glass-embedded silver nanospheres (dipole moments normal to the propagation direction) with a = 5 nm and dipole scattering coefficients obtained from Mie solution.

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image

Figure 17. The kdβd curves for 2-D array of glass-embedded silver nanospheres (electric dipole moments parallel to the array plane) with a = 5 nm and dipole scattering coefficients obtained from Mie solution.

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image

Figure 18. The kdβd curves for 2-D array of glass-embedded silver nanospheres (electric dipole moments perpendicular to the array plane) with a = 5 nm and dipole scattering coefficients obtained from Mie solution.

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image

Figure 19. The kdβd curves for 3-D array of glass-embedded silver nanospheres (dipole moments normal to the propagation direction) with a = 5 nm and dipole scattering coefficients obtained from Mie solution.

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[126] Since our formulation assumes lossless array elements, and the permittivity in (129)–(130) contains loss (an imaginary part), we inserted just the real part of (130) into the transcendental equations used to obtain Figures 16–19. This approximation should give reasonably accurate values for the real parts of the propagation constants (βd) of the traveling waves because the imaginary part of the permittivity in (130) is a small fraction of the real part in the visible range of frequencies where the lowest-order traveling waves exist. The electric and magnetic dipole scattering coefficients, S and S+, required by the transcendental equations are obtained from the electric and magnetic dipole coefficients in the Mie solution; see section 3 and Shore and Yaghjian [2004b, equations (30a) and (30b)]. The parameter s in Figures 16–19 is the free-space distance between the spheres so that d = s + 2a, and traveling waves are found in Figures 16–19 for s equal to 1 nanometer and 4 nanometers, again to conform to the values chosen by Sweatlock et al. [2005].

[127] The electric dipole longitudinal traveling waves on 1-D, 2-D, and 3-D arrays of the same silver nanospheres embedded in glass are computed from [Shore and Yaghjian, 2004a, equation (96)], (95), and (97), respectively, and are shown in Figures 20, 21, and 22.

image

Figure 20. The kdβd curves for 1-D array of glass-embedded silver nanospheres (electric dipole moments parallel to the direction of propagation) with a = 5 nm and electric dipole scattering coefficients obtained from Mie solution.

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image

Figure 21. The kdβd curves for 2-D array of glass-embedded silver nanospheres (electric dipole moments parallel to the direction of propagation) with a = 5 nm and electric dipole scattering coefficients obtained from Mie solution.

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image

Figure 22. The kdβd curves for 3-D array of glass-embedded silver nanospheres (electric dipole moments parallel to the direction of propagation) with a = 5 nm and electric dipole scattering coefficients obtained from Mie solution.

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[128] The kdβd curves for the transverse traveling waves on silver nanospheres in Figures 16–19 and for the longitudinal traveling waves in Figures 20–22 show that much of the kdβd curves of all the traveling waves exist in fairly narrow frequency bands and over much of each of these narrow frequency bands βdkd. This implies that these plasmonic traveling waves, as expected, can have most of their power confined to within a small fraction of a free-space wavelength from the spheres. The 1-D and 2-D kdβd curves in Figures 16–18 for the transverse traveling waves on the silver nanospheres asymptotically approach the origin along the kd = βd line, that is, the “light line.” In contrast, the 3-D kdβd curves in Figure 19 asymptotically approach the origin along lines of constant slope less than one. This is because, similarly to the behavior of the kdβd curve of the 3-D diamond sphere array discussed in section 6.2, in the low-frequency (large wavelength) limit the 3-D silver nanosphere array behaves as a medium with an effective relative permeability approximately equal to unity and an effective relative permittivity greater than one (see below) with the slope of the kdβd line approaching the origin given by (128).

[129] The kdβd curves in Figures 20 and 21 for the slow longitudinal traveling waves on 1-D and 2-D arrays of silver nanospheres, in contrast to the kdβd curves for the transverse traveling waves, end abruptly on the kd = βd light line.

[130] Concentrating on the kdβd curves in Figures 16–19 for the transverse traveling waves on the 1-D, 2-D, and 3-D arrays of silver nanospheres shows that for each curve the group velocity becomes zero and then the curve remains fairly flat to the right of this maximum. This implies that for a traveling wave containing a spectrum of frequencies covering this maximum and the continuation of the kdβd curve to the right of the maximum, there exists a superposition of a “frozen-mode” field and a spectrum of traveling waves moving with very slow group velocities [Simovski et al., 2005]. It should be noted that the slightly negative group velocities in Figure 19 for the 3-D array occur for βd significantly greater than unity and thus these negative group velocities do not imply meaningful bulk relative permittivities and permeabilities with values both less than zero. Of course, since the magnetic dipole moments of the silver nanospheres are negligible, the bulk relative permeability cannot have a meaningful value other than unity.

[131] The effective relative permeability of the 3-D array of glass-embedded silver nanospheres is approximately equal to unity and the effective relative permittivity computed from both the Shore-Yaghjian formula (68) and the real part (see section 6.1.1) of the Clausius-Mossotti relation (76) is shown in Figure 23. Figure 23 reveals for the more loosely packed 3-D array of silver nanospheres (s = 4 nm) in the region [kd < 0.3, βd < 1] that the bulk relative permittivity is less than 10 and approaches a constant value of about 1.8 as kd [RIGHTWARDS ARROW] 0. For the more tightly packed nanospheres (s = 1 nm) in the region [kd < 0.2, βd < 1], the bulk relative permittivity is less than 12 and approaches a constant value of about 3.0 as kd [RIGHTWARDS ARROW] 0.

image

Figure 23. Effective relative permittivity for 3-D array of glass-embedded silver nanospheres (dipole moments normal to the propagation direction) with a = 5 nm and dipole scattering coefficients obtained from Mie solution.

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6.4. Magnetodielectric Spheres

[132] Using expressions derived by Lewin [1947] that are closely related to the Clausius-Mossotti relations for spherical scatterers, Holloway et al. [2003] have shown that 3-D arrays of magnetodielectric spheres with equal (and unequal) values of relative permittivity and permeability exhibit frequency bands (near the resonances of these spheres) in which the bulk permittivity and permeability are both negative. We concentrate on the case of equal relative permittivity and permeability because the impedance of a 3-D array of such spheres will be close to the impedance of free space. To confirm these predictions of “double negative” (DNG) “metamaterials,” we apply the magnetodielectric transcendental equations of Shore and Yaghjian [2004b, equation (52)] or Shore and Yaghjian [2005b, equation (18)] (for 1-D arrays), (100) and (105) (for 2-D arrays), and (53) (for 3-D arrays) to obtain the kdβd diagrams shown in Figures 24 and 25 [Shore and Yaghjian, 2004b, Figure 15; Shore and Yaghjian, 2005b, Figure 8] for the transverse traveling waves on spheres with relative permittivity and permeability equal to 20. In all these figures, the value of a/d = 0.45 is used for the sphere radius to spacing ratio and the range of kd is restricted to values less than the first quadrupole resonance that occurs at ka = 0.28. There is only one figure for the 2-D array because the transverse traveling waves with the electric dipole moments perpendicular and parallel to the plane of the 2-D array have the same kdβd diagram if the permittivity and permeability of the spheres have the same value.

image

Figure 24. The kdβd diagram for 2-D array of εr = μr = 20 magnetodielectric spheres (electric dipole moments parallel or perpendicular to the array plane) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 25. The kdβd diagram for 3-D array of εr = μr = 20 magnetodielectric spheres (dipole moments normal to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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[133] The shape of the kdβd curves in Figure 15 of Shore and Yaghjian [2004b] or Figure 8 of Shore and Yaghjian [2005b] and Figure 24 for the transverse traveling waves on the 1-D and 2-D arrays of μr = εr = 20 magnetodielectric spheres are qualitatively the same but with the first branch of the 1-D curve having two values of kd with zero group velocity and the second branch of the 2-D curve having one value of kd with zero group velocity. The kdβd curve for the 3-D array, which is of greatest interest as a possible DNG material, crosses the light line and has no values of kd with zero group velocity.

[134] The effective (bulk) relative permittivity and permeability of this 3-D metamaterial is computed from the Shore-Yaghjian formulas in (68) and from the real parts (see section 6.1.1) of the Clausius-Mossotti relations in (76) and (81), and are shown in Figure 26. The Clausius-Mossotti curve in Figure 26 is indistinguishable from the corresponding curve computed from the Lewin formulas of Holloway et al. [2003, Figure 7]. The results of both the Shore-Yaghjian formulas and the Clausius-Mossotti relations plotted in Figure 26 confirm the results obtained by Holloway et al. [2003] that there exists a frequency band (near the first resonance of the magnetodielectric spheres) in which both μreff and εreff are negative, namely in the approximately 10% fractional bandwidth between kd equal to about 0.45 and 0.50. Moreover, βd is also less than 1 over a sizable portion of this DNG frequency band so that the material should behave as a fairly homogeneous medium at these frequencies. The bulk properties of this 3-D magnetodielectric array medium are reflected in the slope of the kdβd curve of Figure 25 as the origin is approached. The slope of the curve in the low-frequency (large wavelength) limit is given by (see (57))

  • equation image
image

Figure 26. Effective relative permittivity and permeability for 3-D array of εr = μr = 20 magnetodielectric spheres (dipole moments normal to the propagation direction) with dipole scattering coefficients obtained from Mie solution.

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[135] Low-loss magnetodielectric material is commercially available at low microwave frequencies with εr = 13.8 and μr = 11.0 (J. S. Derov, Hanscom AFB, private communication, 2006). Our computations with spheres made from this magnetodielectric material result in the kdβd diagram of Figure 27 and the effective permittivity and permeability curves shown in Figure 28. Again there is a frequency band (between kd approximately equal to 0.7 and 0.8) where both μreff and εreff are negative. Also, βd is also less than 1 over a sizable portion of this DNG frequency band so that the material should behave as a fairly homogeneous medium at these frequencies.

image

Figure 27. The kdβd diagram for 3-D array of εr = 13.8, μr = 11.0 magnetodielectric spheres (dipole moments normal to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 28. Effective relative permittivity and permeability for 3-D array of εr = 13.8, μr = 11.0 magnetodielectric spheres (dipole moments normal to the propagation direction) with dipole scattering coefficients obtained from Mie solution.

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[136] We also looked for longitudinal traveling waves (electric or magnetic dipoles aligned with the direction of propagation) on the μr = εr = 20 magnetodielectric sphere separated with a/d = 0.45. None exist on the 2-D array in the range of kd up to the first quadrupole resonance, but the 1-D and 3-D arrays do support electric or magnetic dipole longitudinal traveling waves in the extremely narrow frequency bands shown in Figures 29 and 30. As seen in Figure 29, the 1-D kdβd longitudinal curve terminates on the kd = βd light line, whereas the 3-D longitudinal curve in Figure 30 continues unperturbed through the light line.

image

Figure 29. The kdβd diagram for 1-D array of εr = μr = 20 magnetodielectric spheres (dipole moments parallel to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 30. The kdβd diagram for 3-D array of εr = μr = 20 magnetodielectric spheres (dipole moments parallel to the propagation direction) with a/d = .45 and dipole scattering coefficients obtained from Mie solution.

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6.5. Partially Finite Magnetodielectric Sphere Array Reflection Coefficients

[137] In this section we show examples of reflection coefficient curves for partially finite arrays of magnetodielectric spheres. Plots for two kinds of spheres are shown: diamond spheres with εr = 5.84, μr = 1, and spheres composed of low-loss commercially available material with εr = 13.0, μr = 11.8. The values of the reflection coefficients, R, for the partially finite arrays are obtained from (126) and (125a). In all cases the value of N is 100 (that is, there are 101 equispaced infinite planes of spheres normal to the array axis), and the ratio of the radius of the spheres, a, to the separation of adjacent sphere centers, d, is 0.45. For both kinds of spheres we show plots for two cases, one where there is no loss, and one with loss inserted into the propagation constant of the incident plane wave when calculating the values of Exn(zn) and Hyn(zn)/Y0, n = 0,1, ⋯, N from (120) that enter into (125a). The value of the loss constant, ɛ, is chosen via the equation

  • equation image

with P = 1. Given the value of ɛ from (132) the values of the incident plane wave, eequation image, at the locations z = zn = nd, n = 0,1, ⋯, N in the right-hand side of (120) are then multiplied by the respective factors e−ɛnkd, n = 0,1, ⋯, N. (This exponential decay inserted into the incident field on the right-hand sides of (120) does not satisfy Maxwell's equations in lossless free space. Nonetheless, this mathematical ansatz serves to reduce the multiple interactions between the leading and trailing interfaces of the partially finite array, thereby producing a reflection coefficient that is nearly equal to that of the leading interface alone. This loss in the incident plane wave does, however, reduce the magnitude of the reflection coefficient to a value slightly less than unity in the stop bands.) The purpose of inserting loss is to reduce the reflections at the far end of the partially finite arrays and so make them behave more like semi-infinite arrays with no reflections at the far end. It is then possible to compare the reflection coefficients obtained from the partially finite array equation (126) with the reflection coefficients obtained from the calculations of bulk permittivity and permeability given by the Shore-Yaghjian formulas (76) and (81) using the expression [Jackson, 1999, equation (7.42)]

  • equation image

[138] In Figure 31 we show a plot of the magnitude of the reflection coefficient for the partially finite lossless diamond array. The pronounced oscillations of the pattern are the result of reflections between the two ends of the array. Thus the array behaves somewhat like a Fabry-Perot resonator. Note that the intervals of the plot where the magnitude of the reflection coefficient equals one correspond exactly to the gaps in the kdβd diagram of Figure 12, that is, the intervals of kd where no traveling wave exists to convey power from one end of the array to the other.

image

Figure 31. Reflection coefficient of a lossless partially finite 3-D array of diamond spheres (dipole moments normal to the propagation direction) with εr = 5.84, μr = 1, a/d = 0.45, and dipole scattering coefficients obtained from Mie solution.

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[139] In Figure 32 we show the plot of the magnitude of the reflection coefficient for the partially finite diamond array with loss inserted into the incident plane wave, together with a plot of the magnitude of the reflection coefficient obtained from the Shore-Yaghjian bulk parameter equations. The oscillations of Figure 31 have been considerably reduced. Note that for values of kd less than one there is excellent agreement between the lossy partially finite array reflection coefficient and the Shore-Yaghjian coefficient given by (133) apart from a small interval of kd between zero and about 0.1. This good agreement for kd < 1 is to be expected since the derivation of the Shore-Yaghjian bulk parameter expressions assumes a separation of the array elements sufficiently small so that the array can be regarded as a homogeneous medium. For larger values of kd there is a kind of rough qualitative agreement but the Shore-Yaghjian reflection coefficient values are in general not highly accurate. As kd becomes smaller than 0.1, the total thickness of the slab with decaying incident field (simulating a loss) becomes smaller than a free-space wavelength and its scattering becomes weaker.

image

Figure 32. Reflection coefficient of a lossy partially finite 3-D array of diamond spheres (dipole moments normal to the propagation direction) with εr = 5.84, μr = 1, a/d = 0.45, and dipole scattering coefficients obtained from Mie solution; and the Shore-Yaghjian reflection coefficient.

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[140] In Figure 33 we show a plot of the magnitude of the reflection coefficient for the partially finite lossless array of εr = 13.0, μr = 11.8 spheres over an extended range of kd. As with the lossless diamond sphere array reflection coefficient curve, the intervals of kd for which the magnitude of the reflection coefficient equals one correspond to intervals of the extended kdβd diagram of Figure 34 where no traveling wave exists. The portions of the curve between adjacent spikes correspond to the backward traveling wave branches of the kdβd diagram.

image

Figure 33. Reflection coefficient of a lossless partially finite 3-D array of εr = 13.8, μr = 11 magnetodielectric spheres (dipole moments normal to the array axis) with a/d = 0.45 and dipole scattering coefficients obtained from Mie solution.

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image

Figure 34. Extended kdβd diagram for an infinite 3-D array of εr = 13.8, μr = 11 magnetodielectric spheres, (dipoles normal to the array axis) with a/d = 0.45 and dipole scattering coefficients obtained from Mie solution.

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[141] In Figure 35 we show a corresponding plot of the magnitude of the reflection coefficient for the partially finite array of εr = 13.0, μr = 11.8 spheres with loss inserted into the incident plane wave, together with a plot of the magnitude of the reflection coefficient obtained from the Shore-Yaghjian bulk parameters in (133). Here the agreement between the two reflection coefficient curves is surprisingly good even for larger values of kd. There is, however, one feature of the partially finite array curve that is not reproduced in the Shore-Yaghjian curve, namely the smaller spike between each pair of principal spikes at the two ends of the backward traveling wave intervals. These smaller spikes occur exactly where the kdβd curve in Figure 34 crosses the kd = βd light line and, since they occur in the backward traveling wave intervals, correspond to places where the traveling wave has exactly the negative of the incident wave phase dependence on z. Some numerical experimentation shows that these smaller spikes are attributable to reflections from only the first few planes of the partially finite array, and so cannot be found in the reflection coefficient curve obtained from the Shore-Yaghjian bulk parameter expressions. It is also worth noting that for values of kd > 1, a reflection coefficient curve obtained from the Clausius-Mossotti mixing formula bulk parameter expressions (not shown here) does not track the partially finite array curve nearly as well as the curve obtained from the Shore-Yaghjian expressions. This is not surprising in view of the fact that the Clausius-Mossotti bulk parameter expressions are much more tightly bound to the assumption of small array element separations than are the Shore-Yaghjian bulk parameter expressions.

image

Figure 35. Reflection coefficient of a lossy partially finite 3-D array of εr = 13.8, μr = 11 magnetodielectric spheres, (dipoles normal to the array axis) with a/d = 0.45 and dipole scattering coefficients obtained from Mie solution; and the Shore-Yaghjian reflection coefficient.

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[142] In closing this section we note that the reflection coefficient for a partially finite array of magnetodielectric spheres with εr = μr = 20 was found to be very close to zero for all values of kd, a result that is consistent with εr = μr in (133).

Appendix A:: Rapidly Convergent Expressions for Schlömilch Series

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[143] In this appendix we collect the rapidly convergent expressions for Schlömilch series used in this paper.

[144] See Gradshteyn and Ryzhik [1994, equation 8.524(3)]:

  • equation image

In (A1), γ = 0.577215665…, referred to as C by Gradshteyn and Ryzhik [1994], is the Euler constant [Abramowitz and Stegun, 1965, Table 1.1], and J0 and Y0 are the Bessel and Neumann functions of order 0, respectively. We have found that truncating the expression on the right-hand side of (A1) at l = 10 yields sufficient accuracy for our purposes.

[145] See Gradshteyn and Ryzhik [1994, equation 8.526(1)]:

  • equation image

In (A2), K0 is the modified Bessel function of order 0.

[146] See Linton [2005, 2006] and Twersky [1961]:

  • equation image
  • equation image

where sinh qm = equation image, βm = (βd + 2)/(kd), λ1 = [2/(kd)] B1(βd/2π) and B1 is the Bernoulli polynomial [Gradshteyn and Ryzhik, 1994, equation 9.62], B1(x) = x − 1/2.

[147] See Linton [2005, 2006] and Twersky [1961]:

  • equation image

where sinh qm = equation image, βm = (βd + 2)/(kd), λ2 = [B0(βd/(2π)) − 2(2π/(kd))2B2(βd/(2π))]/(2π) and B0 and B2 are the Bernoulli polynomials [Gradshteyn and Ryzhik, 1994, equation 9.62] B0(x) = 1, B2(x) = x2x + 1/6.

[148] See Gradshteyn and Ryzhik [1994, equations 8.521(1) and 8.522(3)]:

  • equation image
  • equation image

with γ the Euler constant given above. Truncating the series on the right-hand side of (A7) at l = 10 gives sufficient accuracy for our purposes.

[149] See Linton [2005, 2006] and Twersky [1961]:

  • equation image
  • equation image

where sinh qm = equation image, βm = 2/(kh), λ2 = [B0(0) − 2(2π/(kh))2B2(0)]/(2π) = [1 − (2π/(kh))2/3]/(2π), and B0 and B2 are the Bernoulli polynomials given above.

Appendix B:: Summation Formulas

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References

[150] In this appendix we collect a number of summation formulas used frequently in this paper.

[151] See Gradshteyn and Ryzhik [1994, equations 1.441(1) and 1.441(2)]:

  • equation image

The sums

  • equation image
  • equation image

can be evaluated in closed form by formally using the formula for the sum of an infinite geometric progression

  • equation image

even though ∣z∣ = 1 here (rigorously, k can be thought of as having a very small imaginary part, consistent with the implicit harmonic time dependence eiωt, which is allowed to go to zero at the end) yielding

  • equation image
  • equation image

[152] See Shore and Yaghjian [2004a, Appendix C] and Lewin [1981]:

  • equation image
  • equation image
  • equation image
  • equation image

From (B7a)

  • equation image

The Clausen functions Cl2 and Cl3 are also given by the integral expressions [Lewin, 1981]

  • equation image

References

  1. Top of page
  2. Abstract
  3. 1. Introduction and Summary of Results
  4. 2. Three-Dimensional Acoustic Monopole Arrays
  5. 3. Three-Dimensional Magnetodielectric Sphere Arrays
  6. 4. List of kdβd Equations for 2-D and 3-D Arrays
  7. 5. Partially Finite 3-D Array of Magnetodielectric Spheres
  8. 6. Numerical Results
  9. Appendix A:: Rapidly Convergent Expressions for Schlömilch Series
  10. Appendix B:: Summation Formulas
  11. Acknowledgments
  12. References
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