Polarizability dyadics of small bianisotropic spheres

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Polarizability dyadics of small bianisotropic spheres

Akhlesh Lakhtakia

To cite this version:

Short Communication

Polarizability dyadics

of small

_{bianisotropic}

_spheres

Akhlesh Lakhtakia

Department

of

_Engineering

Science and

_Mechanics,

The

_Pennsylvania

State

_{University, University}

Park,

PA

16802,

U.S.A.

(Received

23 April

1990,

accepted 6 A ugust

1990)

Abstract. 2014 The

low-frequency scattering

response of a

_{homogeneous sphere}

made of a

_general

linear

_{bianisotropic}

material is

_quantified

in terms

_{of polarizability}

_{dyadics. The}

_{polarizability dyadics}

are then used to

_generalize

the Maxwell-Garnet

model

for a

_composite

obtained

_by

_randomly

sus-pending

small

_{bianisotropic spheres}

in an

_isotropic

achiral host medium.

Classification

Physics A bstracts

41.10

1. Introduction.

Treatment of

_{scattering by homogeneous isotropic spheres}

can be traced back to Mie

_[1]

and

Debye

[2] ;

the

_resulting

formulation has

_enjoyed

enormous

_popularity,

and has been extended to

bi-isotropic

spheres

as well

_[3].

But a treatment of

_comparable

_simplicity,

_elegance

and

_generality

for

_{homogeneous anisotropic spheres}

is still elusive.

_Analytic

solutions

_requiring

intensive

compu-tation have become available for

_{scattering by radially}

uniaxial dielectric

_spheres

_[4,5].

In recent

years,

a

_{semi-microscopic}

numerical

_approach

due to Purcell and

_{Pennypacker [6]}

has been used for

_anisotropic

_spheres

_[7].

For

_electrically

small

_{anisotropic spheres,}

however,

the most common

procedure

used is to

_compute

the

_scattering

characteristics of an

_{"averaged" isotropic sphere}

_[7-9].

It is well-known that the field scattered

_by

_any

scatterer

_(in

free

_space)

can be

_decomposed

into

multipoles [10] .

In

_particular,

at low

_enough

_frequencies,

the

_scattering

_response

of a

homoge-neous dielectric

_sphere

is

_isomorphic

with the radiation characteristics of a

_point

electric

_dipole;

hence,

a small dielectric

_sphere

_may

be

_adequately

characterized

_by

an electric

_{polarizability [9] .}

This

_fact,

_along

with the Clausius-Mosotti relation

_{[10] ,}

was utilized

_by

Maxwell-Garnet

_[11]

to fashion a

_theory

for the

_{macroscopic properties}

of a

_composite

medium constructed

_{by randomly}

dispersing

small

_spherical

inclusions in a dielectric host material. The

_{resulting approach}

has been

considerably augmented

for different cases

[12,13],

and it has its

_competitors

in the

_Bruggeman

model

_[14]

and its variants

_[e.g.,

_15],

as well as in the more

_{rigorous multiple scattering}

theo-ries

_[16-18].

The

_{elegant simplicity}

of this

_approach

has lead to its extensive

_usage,

_{nevertheless,}

despite

its

_many

limitations

_{[12,13 .}

2236

The

_objectives

of this

_paper

are twofold.

First,

the

_scattering

_response

of a

_general,

homoge-neous, small

_{bianisotropic}

_{[19, 20]}

_sphere

will be obtained.

_Second,

the Maxwell-Gamet model will be

_generalized

in order to estimate the effective

_properties

of a

_composite

constructed

_by

randomly

suspending

small

_{bianisotropic spheres}

in a

_{homogeneous, isotropic,}

achiral

_[21]

host

medium;

without loss of

_generality,

the host medium will be taken to be free

_space.

_Coupled

volume

_{integral equations}

will be used in section 2 to obtain the

_{polarizability}

_dyadics

of a small

bianisotropic

sphere suspended

in free

_space.

_{Parenthetically,}

it should be noted that these

po-larizability dyadics

can be used in

_extending

the

_{Purcell-Pennypacker}

_technique

_[6]

for

_arbitrary

scatterers

_{[e.g., 7, 22].}

_Following

the

_developments

of section

2,

the constitutive

_parameters

of the Maxwell-Garnet model of the

_composite

will be estimated in section 3.

2. Polarizabilities of a small

_{bianisotropic sphere.}

Let a

_spherical

_region

of radius a be

_{occupied by a general,}

linear,

_homogeneous

medium

specified

by

the constitutive

_equations

be embedded in free

_space

_{(go, uo).}

The

_sphere

is assumed to be centered at the

_origin

r =

0, e

is the relative

_{permittivity dyadic, y}

is the relative

_permeability

_dyadic,

while a

_{and 8}

_represent

the

_{magnetoelectric dyadics;}

no restrictions have been

_placed

as of now on these

_dyadics,

and a

medium described

_{by (1)}

is called

_{bianisotropic [19, 20] .}

Bianisotropic

media occur

_readily

in nature. Materials

_exhibiting

the dielectric

_Faraday

effect abound as

_crystals

_{[23] .}

Ferrites and

_{plasmas [19]}

exhibit the

_magnetic

Faraday

effect

1

Natural

_optically

active materials

are well-known to

_organic

and

_physical

chemists

[21, 24] .

Even.

a

_simply

moving, isotropic

dielectric scatterer

_appears

to be

_{bianisotropic}

to a

_stationary

observer

_{[25] .}

Thus,

a material characterised

_{by (1)}

is the most

_general,

linear,

non-diffusive

_{electromagnetic}

substance.

_Physically

realizable forms of the constitutive tensors have been discussed at

_{length by}

Post

_[19]

within the framework of Lorentz covariance

_{[26] .}

In the absence of

_any

_impressed

_sources, Maxwell curl

_equations

satisfied

_by

time-harmonic

[e-iú1t]

electromagnetic

fields

_everywhere

can be

_{expressed concisely}

as

In which I is the

_idempotent

_Thus,

the influence of the

_{bianisotropic}

scatterer is

_being

treated as

that due to certain volume distributions of the electric and the

_magnetic

current densities in free

space.

The solution of

₍₂₎

is

well-known,

and is

_given

for all r

_by

_[27]

In these

_equations,

the free

_space

Green’s

_dyadic

Go

(r, r’)

is defined as

where

is the free

_space

scalar Green’s

_{function, ko}

=

ú.JV(éoJ.lo)

is the free

_space

wavenumber,

and

Einc(r)

and

_Hinc(r)

_represent

the field incident on the scatterer. The

_{specific properties}

of the solution

₍₅₎

have been discussed in detail

_by

Jones

_[27],

to which work the interested reader is referred to for further details.

It is assumed here that

_{k4 a « 1; hence,}

the field inside V can be

_easily

estimated

_{by making}

a

long-wavelength

approximation.

First,

the

_equations

₍₅₎

can be transformed to

_[28]

in which P.V denotes the

_principal

value,

while the

_{singularities}

at r = r’ have been

_clearly

identi-fied.

_{Parenthetically,}

it is to be noted here that

_{(7a, b) considerably generalize}

and extend Jones’ formulation

_[27].

_Next,

the

_P.V{.}

are set to zero since the scatterer is

_electrically

small,

and one

gets

It must be remarked here that the

_spheres

considered here have radü small

_compared

to the

free-space

wavelength

at a

_{given frequency}

w. From available literature

_concerning

_isotropic

di-electric

_spheres,

it

_appears

that the

_{approximation}

_{(8a, b)}

_may

be valid for

koa

0.1 or there-abouts

_{[29, 30].}

2238

in which Ur is a unit vector

_parallel

_{to r,} while

is an electric

_dipole

moment, and

is a

_magnetic

_dipole

moment.

_{Consequently,}

in the

_{low-frequency}

_régime,

the scattered fields at-tributable to the

_{bianisotropic}

scatterer can be

_thought

of as due to the combination of an electric

dipole

and a

_magnetic

_dipole.

Next,

it is assumed that the

_{electromagnetic}

field induced inside the small

_sphere

is

_spatially

constant;

i.e.,

E(r)

=

E(0)

and

H(r)

=

H(0)

for all r

e V

_{Solving (8)}

and

_{(10) together}

_yields

in which the four

_{polarizability dyadics}

for the small

_{bianisotropic sphere}

are

_{given by}

In thèse

_equations,

_B-1

is the

_dyadic

inverse of

B,

etc.; while

and

It is assumed here and hereafter that all

_dyadic

inverses exist

_{[19 (Chaps.}

6 and

_8)]

or can be

satisfactorily

taken into account.

Expressions

(12a-d)

constitute the most

_general

result for the

_{polarizability-representation}

of

a small

_{homogeneous sphere}

made of the most

_general,

_linear,

non-diffusive material and

em-bedded in free

_space.

Since

_they

hold for

_fully

_{bianisotropic}

_spheres,

these results subsume not

only

the traditional result for

_isotropic

dielectric

_spheres

_[10]

but also that for chiral

_spheres

3. Effective

_properties

of a

_composite.

Finally,

a discrete random medium made

_up

of identical

_{bianisotropic spheres}

distributed in a free

space

is considered. The

spheres

are

_randomly

distributed with N

_being

the number

_density

_per

unit volume. It is also assumed in this section that the volumetric

_proportion

of the chiral material is

small,

and it is

_expected

that the

_analysis

will hold for

_{N(4x /3)a3}

0.1 or so

_[12].

_Finally,

it is also assumed here that all

_spheres

have identical constitutive tensors within a

_global

coordinate

system.

If the

_{resulting composite}

medium is to be viewed as

_being

_{effectively homogeneous,}

its constitu-tive

_parameters

must be of the form

The

_concept

of flux

_densities,

D and

_B,

_implies

a

_polarization

field P =

(D - so E)

and a

magne-tization field M =

(B - lioh).

The

polarization

field P is defined as the electric

_dipole

moment

per

unit

volume,

while the

_{magnetization}

field M is the

_{magnetic dipole}

moment

_per

unit

volume;

thus,

The

_equivalent

_{moments, p and} m, of a

_{single sphere}

are

_proportional

to the local

_(Lorentz)

electric and

_magnetic

fields

_exciting

it

_[10].

Therefore,

from

_(11),

must be used in

_(15);

here,

the

_subscript

"L" stands for Lorentz.

_Assuming

that

_spheres

are

_weakly

anisotropic,

the usual

_[13,32]

_prescription

for the Lorentz field can be

_{followed; Le.,}

2240

It follows from

_(14),

as well as from the definitions of P and

_M,

that the constitutive

_parameters

of the effective medium can now be estimated as

When

_(12a-d)

are substituted into

_(19a-d),

the

_resulting

values of the constitutive

_parameters

of the

_{bianisotropic composite}

₍₁₄₎

are too cumbersome for inclusion here.

_{Nevertheless,}

_they

constitute an extensive

_{generalization}

of the Maxwell-Garnet formula

_given

for

_composites

made of

_isotropic

dielectric

_{spheres suspended}

in an

_isotropic

dielectric host medium.

Before

_concluding,

it

_may

be of use to consider the

_limiting

case

N( 41r /3)a3

« 0.1. In such a

dilute

_composite,

the Lorentz fields

_{{EL, HL}}

can be

_{simply replaced}

_by

{E, H}

in

(16),

because

Acknowledgements.

This work was

_supported

in

_part

_by

a Research Initiation Grant from the

_Pennsylvania

State

Uni-versity.

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