dielectric metamaterials

8/10/2019 Dielectric Metamaterials

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MIE RESONANCE-BASED DIELECTRIC METAMATERIALS

1.

Mie Scattering Theory

Mie developed an exact solution of the scattering of the electromagnetic field by a spherical particle.The result is based on a series expansion into spherical harmonics of the incident, scattered and internal

field derived from the wave equation in spherical coordinates. The coefficients a, b, c and d for the

scattering and internal fields, are obtained by the boundary conditions between the particle and the

surrounding medium (Bohren C., 1983).

Where , , , , for an orthonormal basis represented by the sine ("o" subindex) orcosine ("e" subindex) functions, Legendre and Bessel functions. The "i" index represent the kind of

Bessel Function, being i=1 for Bessel functions of first kind "", and i=3, for Bessel Functions of thirdkind or Hankel Functions "". The electric field patterns for the first two coefficient are showed inFigure 1.2, the field lines are shown on the surface of an imaginary sphere concentric with, but at a

distance from the particle. For those functions where the electric field has no radial component they are

called Transverse Electric or TE modes, and for those with no magnetic radial components they arecalled Transverse Magnetic or TM modes. The magnetic field can also be represented by these patterns

(Bohren C., 1983), since:

( ) ( )

Incident Field

Scattering Fields

Internal fields

Figure 1.1. Scattering by a spherical particle (Bohren C.,

1

2


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Figure 1.2. Electric field patterns of the first two coefficient of the Mie Scattering functions (Bohren C., 1983)

The coefficients for the scattering (and ) and internal fields (and ) are given bellow:

Where , and , , "" and "" are the refractive index of the particle and thesurrounding medium respectively, "" is the wavevector in the surrounding medium, and " is theparticle radius. Note that the denominators of and are identical so when the denominator to zero,which leads to the following equation: Means that and will dominate, and the electric field will be mostly represented by . Similarlythe electric field will be mostly represented by , for those frequencies which makes thedenominator of

and

,- which are identical-, zero, leading to:

Frequencies for which one of these two last equations is exactly satisfied are complex, but there are

some frequencies whose imaginary part is small compared with the real one, and could be considered

real. Note that for the two lowest resonant modes, and the sphere exhibit electric and magneticdipoles respectively.

Transverse Magnetic

(no radial H component)Transverse Electric

(no radial E component)


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2.

Lewin's Model

If the particle is smaller than the wavelength in the surrounding medium, , theRiccati-Bessel functions take the following form:

For the functionwellkeep the general form of the first order Riccati -Bessel functions:

In this case, only the first dipole terms will dominate:

( )

( )

Where we consider the fact that: ; and . Reorganizing theexpression for the coefficients and , we derive to the following:

Where:

At some frequencies, dominates over and the scattered electric field takes the following form: [ ]

Where: .This equation has a dipole form (Jackson, 1999, p. 441):

[ ]

Where is the electric polarizability.Equating both equations, the electric polarizability can be deduced in terms of the Mie coefficient :


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The electric polarizability can be related to the effective permittivity by using the Classius-Mossotti

equation (Jackson, 1999): By replacing the expression for the polarizability, the effective permittivity can be obtained in terms Mie

coefficient : Finally taking the expression for , we obtain an expression for the effective permittivity.

Similarly, at frequencies when dominates over we can obtain an expression for the effectivepermeability:

Where , , and is the filling fraction.The same result was obtained by Lewins (1946) but using a different procedure.

3.

Classius-Mossoti

The Classius-Mossoti equation relates the external electric field acting upon a material and the local field

acting on a molecule. We define a sphere, called the Lorentz sphere, separating the contribution from

closest molecules from those that are far away. This sphere is microscopically large to consider the

inside media as a continuous but macroscopically small in order to accommodate the discrete nature of

the medium very close to the molecules (Cai & Shalaev, 2010). The medium surrounding the molecules

inside the Lorentz sphere is the vacuum.

We will use the same model to develop an expression for a homogeneous media comprising a group of

small particles embedded in a host medium, where the particles can be treated as atoms according the

definition of metamaterials. Unlike the classical model, the medium surrounding the particles inside the

Lorentz sphere is the host medium.

In dense media, the polarization of neighboring particles gives rise to an electric field on the surfaceof the Lorentz sphere as a result of the electric charges distributed around the surface. The local electricfield acting on any particle, is given by the contribution of the macroscopic field and the fieldacting on the surface of the Lorentz sphere: 1


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There is also a contribution from closest particles located inside the Lorentz sphere. This term is difficult

to determine and depends strongly on the crystal structure. However this term vanishes for highly

symmetric crystal like a simple cubic array, or for dipoles randomly distributed with uncorrelated

positions (Jackson, 1999).

Figure 3.1 The Lorentz sphere concept for calculating the local field (Cai & Shalaev, 2010)

The charge density induced on the surface of the sphere is given by . Then the resulting fieldat the center, which is parallel to , has a magnitude:

The polarization is related to the local field acting on every particle by:

Where is the total number of dipoles and is the polarizability. Since there is a medium insidethe Lorentz sphere, the electric field acting in the effective medium is equivalent to the sum of the

polarization and the electric field acting on the host inside the sphere:

Combining the equations 1, 2 and 3 we obtain to the following expression:

By using the relation 4 we obtain the Classius-Mossoti equation: This relates the electric polarizability with the permittivity of the effective medium and the host.

4

3

2

Host media

Inclusions


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dielectric metamaterials

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