Complex Waves on 1D, 2D, and 3D Periodic Arrays of Lossy and Lossless Magnetodielectric Spheres

Abstract

Complex waves propagating along the axes of 1D, 2D and 3D infinite periodic arrays of small lossless and lossy magnetodielectric spheres are investigated. The focus is on obtaining the kd-BETAd equations (diagrams) characterizing dipolar complex waves. The kd-BETAd (dispersion) equation for the complex propagation constants of a given array is obtained from the corresponding equation previously obtained for the real propagation constants by rewriting the real propagation dispersion equation in a form that can be analytically continued into the complex BETAd plane. This equation reduces correctly to the real BETAd dispersion equation and enables complex values of BETAd to be found as a function of kd and the array element parameters. By allowing for all the possible branches of the multivalued homogeneous dispersion equation analytically continued into the complex BETAd plane, the propagation constants of all the improper as well as proper complex waves supported by the 1D, 2D, and 3D arrays are found from the homogeneous solutions for these arrays. Green's functions are not required to find the propagation constants of the complex waves supported by the arrays. The kd-BETA equations for complex BETAd are solved by searching a given region of the complex BETAd plane with a progressively finer grid for the value of BETAd that minimizes the absolute value of the dispersion equation. A final very accurate value for the zero is then obtained using an IMSL implementation of a quasi-Newton algorithm or of Mueller's Method. Computer programs have been written to obtain the kd-BETAd diagrams for all the arrays treated, and extensive numerical results are presented and discussed. For 3D arrays of magnetodielectric spheres, it is possible in certain regions of the kd-BETAd diagrams to regard the arrays as media characterized by bulk or effective permittivities and permeabilities.

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Document Details

Document Type
Technical Report
Publication Date
May 16, 2010
Accession Number
ADA534784

Entities

People

  • Arthur Yaghjian
  • Robert A. Shore

Organizations

  • Air Force Research Laboratory

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  • Advanced Electronics
  • Air Platforms
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  • Air Force
  • Air Force Research Laboratories
  • Algorithms
  • Boundary Value Problems
  • Composite Materials
  • Crystal Lattices
  • Dielectric Permittivity
  • Electromagnetic Fields
  • Electromagnetic Metamaterials
  • Equations
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  • Materials Laboratories
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  • Three Dimensional
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  • Microwave Engineering.
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  • Wave Propagation and Nonlinear Chaotic Dynamics.