^{1,a)}and Nicola Bowler

^{2}

### Abstract

Analytical expressions describing the variability of effective constitutive parameters of non-metallic metamaterials, as a function of the constituent geometric and material parameters and their variations, have been developed from the total differential of Clausius-Mossotti expressions (using Mie dipole polarizabilities) for the effective (bulk) constitutive parameters of the metamaterial. In practice, these expressions are important for estimating the performance of a metamaterial with particular variations in the parameters of its constituents that arise during the fabrication process, and can be used to guard against extinction of desired double negative (DNG) behavior. With the derived expressions, the effects of parameter variations on effective constitutive parameters of non-metallic metamaterials have been analyzed for three types of metamaterials: (i) cubic arrays of identical magnetodielectric spheres; (ii) cubic arrays of dielectric spheres with equal radius but two different permittivities; and (iii) cubic arrays of dielectric spheres with equal permittivity but two different radii. These effects are evaluated in terms of the calculated variations in values of the effective constitutive parameters of the metamaterial in the vicinity of the DNG or single negative (SNG) band for particular geometric and material parameters and their variations. Results show that variation in the following parameters impacts DNG bandwidth. Listed in order from greatest to least influence: (i) sphere radius; (ii) sphere permittivity and permeability; (iii) lattice constant of the array; and (iv) the constitutive parameters of the array medium, all impact the width of the achievable DNG band. For particular cases studied here, results also show that the DNG behavior may be extinguished if there are 0.78%, 0.016%, and 0.016% variations in all parameters of metamaterial types (i), (ii), and (iii), respectively, as defined above. For the design of non-metallic metamaterials with inclusions, having arbitrary material parameters, in either periodic or random arrangement, the presented results can give a qualitative guide on the level of fabrication tolerances that should be achieved in order to observe the predicted SNG or DNG behavior experimentally.

This material is based upon work supported by the Air Force Research Laboratory under Contract No. FA8650-04-C-5228 at Iowa State University's Center for NDE. The authors wish to express their deep gratitude to Dr. Robert A. Shore (Air Force Research Laboratory, Hanscom AFB, MA) and Mr. Xing-Xiang Liu (The University of Texas at Austin) for very helpful discussions and communications.

I. INTRODUCTION

II. THEORY

A. Cubic arrays of identical magnetodielectric spheres

B. Cubic arrays of two different magnetodielectric spheres

III. VERIFICATION

A. Clausius-Mossotti formulas

B. Expressions for the variabilities of effective constitutive parameters

IV. RESULTS

A. Cubic arrays of identical magnetodielectric spheres

B. Cubic arrays of dielectric spheres with equal radius but two different permittivities

C. Cubic arrays of dielectric spheres with equal permittivity but two different radii

V. CONCLUSION

### Key Topics

- Metamaterials
- 51.0
- Permittivity
- 32.0
- Dielectrics
- 17.0
- Mie scattering
- 16.0
- Lattice constants
- 9.0

## Figures

An array of identical spheres and unit cell geometry.

An array of identical spheres and unit cell geometry.

Two-sphere array and unit cell geometry.

Two-sphere array and unit cell geometry.

Comparison of dispersion diagrams for an array of identical spheres, Fig. 1 , obtained by formulas presented here, Eqs. (3) , (5) , and (30) , with that calculated by MPB. ^{ 35 } The 25 lowest bands computed by MPB are shown. In this calculation, , and *a/d* = 0.2672.

Comparison of dispersion diagrams for a two-sphere array, Fig. 2 , obtained by formulas presented here, Eqs. (3) , (19) , and (30) , with that calculated by MPB. ^{ 35 } The 60 lowest bands computed by MPB are shown. In this calculation, , and .

Comparison of variabilities of effective constitutive parameters in the vicinity of the DNG band ( ) of a metamaterial consisting of an array of identical spheres, Fig. 1 , computed by the formula presented herein Eq. (6) , with those calculated by expressions developed by Mathematica. In this calculation, , and *a/d* = 0.45; with , and .

Comparison of variabilities of effective constitutive parameters in the vicinity of the DNG band ( ) of a metamaterial consisting of an array of identical spheres, Fig. 1 , computed by the formula presented herein Eq. (6) , with those calculated by expressions developed by Mathematica. In this calculation, , and *a/d* = 0.45; with , and .

Comparisons of variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a metamaterial consisting of a two-sphere array, Fig. 2 , computed by the formula presented herein Eq. (6) , with those calculated by expressions developed by Mathematica. In this calculation, , and with , and .

Comparisons of variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a metamaterial consisting of a two-sphere array, Fig. 2 , computed by the formula presented herein Eq. (6) , with those calculated by expressions developed by Mathematica. In this calculation, , and with , and .

Variabilities of effective relative permittivity in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of identical magnetodielectric spheres, Fig. 1 , in each calculation, Table I . Parameters of this array are as in Fig. 5 . The equivalent plot for is not shown here since the only difference is that the effects of variations in and those of variations in are interchanged.

Variabilities of effective relative permittivity in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of identical magnetodielectric spheres, Fig. 1 , in each calculation, Table I . Parameters of this array are as in Fig. 5 . The equivalent plot for is not shown here since the only difference is that the effects of variations in and those of variations in are interchanged.

Ideal values and variation ranges of the effective relative permittivity and permeability for a non-metallic metamaterial consisting of a cubic array of identical magnetodielectric spheres, Fig. 1 . Dashed line: ideal values of and ; dark, medium, and light shaded areas: variation ranges for , 3%, and 5% with , and . Other parameters are as in Fig. 5 .

Ideal values and variation ranges of the effective relative permittivity and permeability for a non-metallic metamaterial consisting of a cubic array of identical magnetodielectric spheres, Fig. 1 . Dashed line: ideal values of and ; dark, medium, and light shaded areas: variation ranges for , 3%, and 5% with , and . Other parameters are as in Fig. 5 .

Variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal radius but two different permittivities, Fig. 2 , in each calculation, Table II . Parameters of this array are as in Fig. 6 .

Variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal radius but two different permittivities, Fig. 2 , in each calculation, Table II . Parameters of this array are as in Fig. 6 .

Ideal values and variation ranges of the effective relative permittivity (a), and permeability (b), for a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal radius but two different permittivities, Fig. 2 , with six combinations of parameter variations. Dashed line: ideal values of (a), and (b); dark, medium, and light shaded areas: variation ranges for , 0.03%, and 0.1% (a), 1.2%, 3%, and 5% (b) with , and . Other parameters are as in Fig. 6 .

Ideal values and variation ranges of the effective relative permittivity (a), and permeability (b), for a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal radius but two different permittivities, Fig. 2 , with six combinations of parameter variations. Dashed line: ideal values of (a), and (b); dark, medium, and light shaded areas: variation ranges for , 0.03%, and 0.1% (a), 1.2%, 3%, and 5% (b) with , and . Other parameters are as in Fig. 6 .

Variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal permittivity but two different radii, Fig. 2 , in each calculation, Table II . In these calculations, , and .

Variabilities of effective relative permittivity (a), and permeability (b), in the vicinity of the DNG band ( ) of a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal permittivity but two different radii, Fig. 2 , in each calculation, Table II . In these calculations, , and .

Ideal values and variation ranges of the effective relative permittivity (a), and permeability (b), for a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal permittivity but two different radii, Fig. 2 , with six combinations of parameter variations. Dashed line: ideal values of (a), and (b); dark, medium, and light shaded areas: variation ranges for , 0.03%, and 0.1% (a), 0.4%, 1%, and 5% (b) with , and . Other parameters are as in Fig. 11 .

Ideal values and variation ranges of the effective relative permittivity (a), and permeability (b), for a non-metallic metamaterial consisting of a cubic array of dielectric spheres with equal permittivity but two different radii, Fig. 2 , with six combinations of parameter variations. Dashed line: ideal values of (a), and (b); dark, medium, and light shaded areas: variation ranges for , 0.03%, and 0.1% (a), 0.4%, 1%, and 5% (b) with , and . Other parameters are as in Fig. 11 .

## Tables

The parameter with 5% variation (while others have no variation) in each calculation of variability of effective constitutive parameters of a non-metallic metamaterial consisting of an array of identical spheres, Fig. 1 .

The parameter with 5% variation (while others have no variation) in each calculation of variability of effective constitutive parameters of a non-metallic metamaterial consisting of an array of identical spheres, Fig. 1 .

The parameter with 5% variation (while others have no variation) in each calculation of variabilities of effective constitutive parameters of a non-metallic metamaterial consisting of a two-sphere array, Fig. 2 .

The parameter with 5% variation (while others have no variation) in each calculation of variabilities of effective constitutive parameters of a non-metallic metamaterial consisting of a two-sphere array, Fig. 2 .

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