Light scattering by non-spherical plasma-particles

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Light scattering by non-spherical plasma-particles

V. Karathanos, A. Modinos, N. Stefanou

To cite this version:

V. Karathanos, A. Modinos, N. Stefanou. Light scattering by non-spherical plasma-particles. Journal

de Physique I, EDP Sciences, 1992, 2 (7), pp.1279-1286. �10.1051/jp1:1992210�. �jpa-00246621�

Classification

Physics

Abstracts

41.10 42.20

Light scattering by non-spherical plasma-particles

V. Karathanos

('),

A. Modinos

(2)

^and N. Stefanou

(Ii

(~) ^Solid State Section,

University

of Athens,

Panepistimioupolis

New

Buildings,

GR-157 71 Athens, Greece

(~)

Department

of

Physics,

National Technical

University

of Athens,

Zografou Campus,

GR- l57 73 Athens, Greece

(Received 5 November 1991, accepted in

final form

18 March 1992)

Abstract. -A

point matching

method is

applied

to the problem of

light

scattering

by

_non-

spherical

bodies. The

electromagnetic

field is

expanded

into _a series of

spherical

_waves which include all

multipole

terms which contribute

significantly

_to the wavefield. We examine, in particular, the absorbance of

light by

a

plasma particle

and demonstrate the existence of various peaks associated with

dipole

and

quadrupole

_resonances of the induced

charge

distribution.

I. Introduction.

The

problem

^of

scattering

of

electromagnetic

_waves

by

isolated

objects

has

long

been

investigated

in its different

aspects.

^These

investigations

are motivated

by

_a

pletllora

of diverse

applications ranging

from radar

meteorology

to

biological

sciences. When the size of the

panicle

is much smaller than the

wavelength

of the incident radiation A _one

speaks

of

Rayleigh scattering

which

explains

the blue colour of the

sky [I].

In the short

wavelength

limit the

phenomena

related to

light scattering

_are described

by

^the

Huygens-Kirchhoff

diffraction

theory [2]

or even, as A

-0, by geometrical optics. However,

the intermediate

region

between these two

regimes

has defied _a unified _treatment.

A

rigorous

solution of the

problem

of

scattering

of a

plane

monochromatic

electromagnetic

wave

by

a

homogeneous sphere

of

arbitrary

radius situated in _a

homogeneous

medium _was

given early

in this

century by

Mie

[3]

and

Debye [4]

in terms of classical

electromagnetic theory

and since then it has been used _to

great advantage

in the

study

of various

phenomena.

The

problem

of

scattering

of

electromagnetic

_waves

by non-spherical objects

has also been examined

by

several authors.

Analytically rigorous

solutions to this

problem

have

generally

been based _on the

separation

^of variables

technique.

This is

only

effective for bodies whose

boundary

surface is described

conveniently

in _one of the coordinate

systems

^{for which the}

vector Helmoltz

equation

is

separable [5].

Because of the

analytical complexity

^{of the overall}

boundary

value

problem,

a

large

number of

approximate

methods have been

developed

for

dealing

with the

scattering problem.

In the

perturbation expansion approach [6, 7]

the solution is

expanded

into powers of the deviation of the

shape

of the

non-spherical object

1280 JOURNAL DE

PHYSIQUE

I N° 7

from _a

perfect sphere.

In the

point matching

method

[8]

the

spherical

_wave

components

^{of the} scattered

electromagneIic

field and Ihe

corresponding scattering

Iransition T-matrix _are determined

by applying

^the

boundary

conditions for the wavefield _{at a} number of selected

points

_on the surface of the _scatterer. The extended

boundary

condition method is _an

integral

equation

method

leading

to a matrix formulation of

scattering.

It _was first

developed by

Waterman

[9]

and

subsequently

used

by

a number of authors in its

original [10]

or iterative form it

].

The method of moments

[12]

and the unimoment method

[13]

have also been used to great

advantage by

_many

investigators

to solve the

problem

of

scattering by

_non

spherical

bodies

through integral equation

formulations. In the so-called _«

coupled dipole

method _» of Purcell and

Pennypacker [14]

_an

object

of

arbitrary shape

and

electromagnetic properties

is

replaced by

_a lattice array of

elementary dipoles.

A self-consistent solution _to the wavefields inside and outside the _scatterer is

sought by

an iterative

procedure.

Further progress has

recently

been made in the

study

of

scattering by arbitrarily shaped

scatterers in

conjunction

with

inhomogeneity [15, 16].

In the last few years it has been discovered that films

consisting

of small metallic

particles

embedded in _a dielectric host material have

optical properties

that

might

be useful in _a

variety

of

technological applications,

_e,g, as

coatings

for solar energy absorbers

[17].

Calculations

relating

to the

optical properties

of

inhomogeneous

systems

consisting

of

non-overlapping

metallic

particles

in _a dielectric host have been

published by

_a number of authors

(see [18, 19]

and references

therein).

In most of these calculations the

particles

_are assumed _to be

spheres

or

spheroids

^and the

scattering

of

light

is evaluated in the

dipole approximation (long wavelength limit). Using

^the

point matching

method _we calculate in this paper the absorbance of

light by

isolated

plasma-particles

of

arbitrary

but smooth

shape, including

all

multipole

terms which contribute

significantly

_to the wavefield. Our method

is, therefore, applicable

to the

scattering

of

light

of

arbitrary wavelength.

In section 2 _we summarise the formalism. In section 3 _we examine the convergence of _our method and present ^{results for selected} cases.

We have

recently developed

_a method which describes the

scattering

of

electromagnetic

waves

by

a

periodic monolayer

of

spherical particles

_on a substrate

[20, 21].

In this method the

scattering properties

of the individual

particle

enters

through

the

corresponding (diagonal

for

a

sphere)

T-matrix. The extension of the method _to the _case of

non-spherical particles

is

straightforward.

For this purpose one needs to know the

non-diagonal

T~matrix which describes the

scattering

of

light by

_a

single non-spherical particle

as evaluated here. We shall present ^results on the

scattering

of

light by

_a two-dimensional _array of

non-spherical particles

in _a

forthcoming publication.

2. Method of calculation.

In _a medium characterised

by

a

complex frequency-dependent permittivity E(w

the electric field

E(r, t)

₌ Re

[il(r) _{exp(- iwt)]}

_{and its associated}

_magnetic

_field

can be

expanded

into

spherical

waves as follows

il _{(r )}

=

( _{jj a(~}

V _{A zI}

(kr

_xi~

(f)

₊

at

_zi

(kr

_xi~

(f

t=J m=-t k

i~(r

= El

~o)~'~ f _{jj a(~}

zi

(kr )

_xi~

(f) at

_V

A zI

(kr )

_xi~

(f) ( ii

f i m f

~

We _assume that the

magnetic permeability equals

that of _vacuum ~o. For

magnetic

materials

~o ^must be

replaced by

_~ in the relevant formulae. k

=

(~o e)~'~

_w is the _wave number. The

functions _zi

(kr )

may be any linear combination of the

spherical

Bessel function

ji(kr

and the

spherical

Hankel function

h/ (kr).

These

quantities

as well _as

xi~(f)

_are ^{defined in} reference

[20].

The coefficients

af)~~

in

equation (I)

_are constants to be determined.

A

plane electromagnetic

_wave described

by

it(r)

=

Eo(k) e'~'~, #(r)

=

S

V _A

it(r) (2)

where

Eo~k)

_m

Eo(k) fi specifies

the

magnitude Eo

and the

polarization fi

of the electric

field,

has _a

corresponding ^spherical

wave

expansion given by equation (I)

with

zi(kr)

=

ji(kr)

and

a()~~=a/~f~~~ Explicit expressions

for

a/~f~~~

are obtained

by substituting equations (2)

into

equations (I)

and

expanding exp(ik r)

^into

spherical

waves.

When the

electromagnetic

_wave described

by equations (2)

is scattered

by

_a

particle

of

permittivity e~(w

embedded in _a medium of

permittivity E(w )

it

gives

rise _{to a} total wavefield

(E°~~,

H°~~) ^{outside the}

particle, composed by

the incident and scattered _wave. The

spherical

_wave

expansion

for the

electromagnetic

field of the scattered _wave is

given by equations (I)

with

zi(kr)

=

hf (kr)

and

a()~~

= af~~~~~.

The

electromagnetic

^{field inside} the

particle

(E~~, H~~) ^{is also}

given by equations (I)

with

E = EM,

zi(kr)

=

ji(kM r), a()~~

=

a©~~~

The coefficients

af~~~~~, af~~~

_are determined

by

the

requirement

that the

tangential

components of the electric and the

magnetic

field must be continuous at each

point

_on the surface of the

particle.

A

point

_r on an

arbitrary

but smooth surface is

generally given by

r =

f(9, ~)

?

(3)

where

f

is _a smooth function of the

angular

_components

9,

_q~ of _r in _a

spherical

coordinate system with unit vectors

f, 4,

_{i~. We define}

a local

orthogonal

coordinate system on the tangent

plane

at _r

by

the unit _vectors

The

boundary

conditions for the

electromagnetic

field

i1°~~(r kg

=

il~~(r) _kg

,

i1°~~

(r) k~

=

il'~ _{(r) k~}

11°~~

(r kg

₌ ^ii~~

(r)

_kg

,

11°~~

(r)

k

~ ⁼

ii~~(r ) k~

,

(5)

for every r on the

surface,

lead to _a

system

^of

infinitely

_many linear

equations. By truncating

the

spherical

_wave

expansions

of the scattered

electromagnetic

field _as well _as of (E~~, H~~) to

i~~~

^and

applying

the

boundary

conditions

(5)

at

N~

₌

(i~~~

+

1)~

l selected

points

_on the surface of the

particle

the size of the system ^{is reduced} to

4[(f~~~ +1)~

II- However,

it may be necessary in _{some cases to use a} number of mesh

points larger

than

(i~~ +1)~

_{l in order to}

_represent

_{the surface}

accurately. Then,

the

boundary

conditions

(5)

lead to _a number of linear

equations larger

than the number of coefficients _to be

determined. This is

equivalent

to _a best fit

problem

which _can be

solved,

for

instance, by

_a

least squares method.

The energy absorbed per unit time

by

the

particle

is

given by

^the

negative integral

of the

Poynting

vector over the surface of the

particle.

We denote the average of this

quantity

_{over a}

1282 JOURNAL DE

PHYSIQUE

I N° 7

period

T

=

2 gr/w

by

W. When the

particle

is embedded in _{a non}

absorbing

medium _we obtain

~

lw

f

W

= ~ ~~~ x Re

~ ~ [af~f* af~~

+

(af~~[

^~ +

a/#* _af~~

₊

_{af~~[ ~]}

2 ~°

(J~

0 ^~

)

f _I

m f

(6)

For any

given

scatterer with dimensions smaller than _or

comparable

to the

wavelength

of the incident radiation

only

_a finite number of the

scattering

coefficients

af~~~~~ corresponding

to

I

=

1, 2,

..., i~~~ are

significant.

The number of components to be retained in

equation (6)

to obtain convergence, I,e, i~~~, increases with tile size of the

panicle.

In the _case of

scattering by

_a

spherical particle

tile infinite many

boundary-condition equations separate

^and one obtains

atmE~H> =

Tf~H~ailf~H~ (7)

Explicit expressions

for the T-matrix _are

given

in reference

[20].

If the

bounding

surface is

non-spherical,

the different

I-components

are

coupled

and _we ohtain

w f'

+ E

~ ~

_j~mEE

^~0E

~ ~mEH

~0H j

~im ~ im;i'm' i'm' im:i'm' i'm'

f'= _{i m'=} f'

al

^{H ~}

~'

~

f~j

~.

~p ^~~~~

^i'm'

^~l'$,

⁺

^Tfi

^~,~

^~0H

~'~'

^j (8)

When the

particle

has

cylindrical

_symmetry around the z-axis

~f (

9, _{q~ m}

f (9 ))

symmetry

requires

that

Tlllfi,fl~~

=

TIII(I>§l~>

8mm

(9)

Moreover,

from

equations (I)

and

(5)

and

using symmetry properties

of

spherical

harmonics

one can

easily

show that

T(~m;I>-m

=

T($;I>m, Tf~m;I,-m

=

T©I,m T(im;I,-m

=

T($;I.m, Tiim;I,-m

=

T©;I,m. (lo)

If in addition to

cylindrical _symmetry

the

plane

_z

= 0 is _a

symmetry plane (f(gr

9

=

f(9 ))

the system ^{of the}

boundary-condition equations

_can be further reduced.

Indeed,

one can then show that the

only nonvanishing

elements of the submatrices

T~~

_and

T~~

occur for

[I I'[

=

0, 2, 4,

and of the submatrices

T~~

_and

T~~

_for

[I I'[

=

1, 3, 5,

3. Results and discussion.

We used _our method _to calculate the absorbance of _a

plane electromagnetic

_wave

by smoothly shaped plasma particles

in _vacuum. We _assume that the

optical

_response of the

particles

is

described

by

the Drude dielectric function

~2

~~°'~

w

(w (

ilr)

^~~~~

We have taken _:

hw~

= 6.93 eV and fir

=

0,158 eV

which,

it is

assumed,

_are

appropriate

values for silver

particles [22].

In the

applications presented

here _we deal with

particles having cylindrical

_symmetry

and, therefore,

we can

exploit

the symmetry ^relations

given

in the

preceding

section. In this _case the surface is

conveniently

defined

by

_a mesh of

uniformly

distributed

points

_{on a} meridian

plane (the

_q~

=

0

plane

for

simplicity). However,

calculations _were also carried _out for different sets of mesh

points (not equally spaced)

and _we found that the scattered wavefield is

relatively

insensitive to the choice of

angles

eat which the

boundary

conditions _are

applied.

A number of

points leading

to a number of

equations

twice the number of unknown coefficients is sufficient _to obtain

good

convergence in all the _cases examined.

We examined the accuracy of _our method

by calculating

the absorbance of _a

sphere

of

radius S

=

200h

_centred

50h

_{away from the}

_origin

of coordinates. Within the

present

formalism the T-matrix is

non~diagonal.

We obtained convergence for

i~~=9

_and

i~~~ =

3. The results for the absorbance agree with the results of the Mie

theory (the

absorbance does not

depend

on the

position

of the

sphere)

^with an accuracy of

10~~.

Next _we considered _a

nearly spherical particle

^described

by f(

9, _q~

)

=

300 ₊ 60 _cos 9

(h ₎

and calculated the absorbance _{as a} function of

frequency

for various

angles

^of incidence and

polarization

directions. The results _are not, ^{in this} case,

significantly

different from those calculated for _a

sphere

of

equal

^{volume. This is} understandable since the

shape

under

consideration is to a first

approximation

a shifted

sphere.

Finally,

and in order _to demonstrate the effects of _{a more}

pronounced

deviation from

sphericity,

_we consider _an

ellipsoid

with _a

major

axis b

=

664h

_and

a minor axis

a =

570

h,

_the

_major

_axis

_being

_{the axis of revolution}

z. The accuracy of _our

results,

obtained with cutoff values

i~~

=

7 and i~~~ =

3,

is better than 10-3

In

figure

I _we show the absorbance _versus

frequency

of

light

incident

perpendicular

_to the

major

axis of the

ellipsoid

and electric field

polarization parallel

and

perpendicular respectively

_to this

axis,

and compare it with the absorbance

by

a

sphere

of

equal

volume

(S

=

300

h _).

_All

curves exhibit _two

peaks.

The low energy

peak corresponds

to the excitation

of the

dipole (I

= I

) plasma

mode. The 3-fold

degeneracy

of this mode in the case of _a

sphere

is

split

in the _case of the

ellipsoid giving

rise _{to two}

peaks,

the lower in energy

corresponding

to oscillations of the induced

charge parallel

to the

major

axis

(excited by

a field

polarized along

this

axis)

and the

higher-energy

_one

corresponding

to oscillations

perpendicular

to the

major

axis

(excited by

_a field

perpendicular

to this

axis).

Calculations for _a

sphere

with diameter

equal

to the

major

axis of the

ellipsoid

show _a

dipole peak

at 3.46

eV,

whereas for _a

sphere

with diameter

equal

to the minor axis this

peak

_appears at 3.58 eV. The shift of the

peak position

relative to that of the

equal-volume sphere (3.54 eV)

is _{not as}

large

as the shift observed when the incident

light

is

polarized parallel

to the

major (3.31eV)

and minor

(3.66 eV)

axis of the

ellipsoid.

The second

peak

^of each _curve in

figure

I is associated with

excitation of _a

quadrupole (I

=

2) plasma

mode. The 5-fold

degeneracy

of this mode in the

case of _a

sphere

is

split

in the _case of the

ellipsoid

into _{one non}

degenerate ((3 z~ r~)-like)

and two 2-fold

degenerate (xz-

and

x~-like)

_modes, excited

by

the

appropriate

_component of the electric field

gradient.

We

analyse

the

quadrupole

_resonances further in what follows.

In

figure

2 _we

present

^the same results _as in

figure

I for the

ellipsoid

but in addition _we show the absorbance for

light

incident

parallel

to the

z-axis,

which is

independent

of the

polarization

direction for symmetry reasons.

Finally,

in

figure

3 _we show the absorbance _versus

frequency

for three different

angles

^of

incidence : 9

=

0°, 45°,

90° with respect ^{to the}

major

axis of the

ellipsoid.

The electric field vector oscillates in the

plane

defined

by

the

propagation

direction and the z-axis

(Fig. 3a)

and

1284 JOURNAL DE

PHYSIQUE

I N° 7

~ /~

i, j'

/

[

/

I i

/ I

'

if ^{' < '}

' Ii

ii W /

j

j I I

j (

,'

/ ',

J ' _I

, _. ' / '

, ',

, / '

/.

l' '~ ~ " i' ' ~

l' _'~

",

~ ~

~°~

~~~

~ ~

hw

eV

Fig.

I.

Fig. ^2.

Fig.

I.- Absorbance _versus

frequency

for

light

incident

perpendicular

_to the

major

axis of _an

ellipsoidal plasma-particle

(a

= 570

h,

_b

= 664

h)

electric field

polarization parallel

to the

major

axis (- -) _; electric field polarization

perpendicular

to the major axis (.. ). The full _curve shows the

absorbance

by

_a

sphere

of the _same volume.

Fig. 2.- Absorbance _versus

frequency

for light incident

perpendicular

_to the

major

axis of _an

ellipsoidal plasma-panicle

(a

=

570

h,

_b

=

664

h)

: electric field

polarization parallel

to the

major

axis (- -)_; electric field

polarization perpendicular

to the

major

axis (.. ). Incidence parallel to the

major

^axis ( ).

~ a

3~

b

~ ~

hw

/

eV

Fig.

3. Absorbance _versus frequency for

light

incident _at

angles

₌ 0°, 45° and 90° with respect to the

major

axis of _an

ellipsoidal plasma-particle

(a

=

570h,

_b

=

664h).

The electric field vector oscillates in the plane defined

by

the

propagation

direction and the major axis

(Fig.

3a) and

perpendicular

to this

plane (Fig.

3b).

perpendicular

to this

plane (Fig. 3b).

In the former _case

(Fig. 3a)

the low energy

peaks correspond

to a

parallel dipole

mode _resonance

(9

=

90°)

and to a

perpendicular

_one

(9

=

0°

).

For 9

=

45° both the

parallel

and the

perpendicular dipole

modes _are excited

by

the

corresponding component

^{of the electric field. The}

high

_energy

peaks correspond

to an

xz4ike

quadrupole

mode _resonance

(9

=

0° and 9

=

90°)

whereas for 9 ₌ 45°

quadrupole

resonances of the 3

z~- r~

_and

x~ _type

are excited. In the latter _case

(Fig. 3b) only

the

perpendicular dipole

mode is

excited, irrespective

of the

angle

of

incidence,

since the electric field oscillates

always perpendicular

to the

major

axis of the

ellipsoid. Conceming

the

quadrupole modes,

the 9

=

0 incidence excites the xz-like mode and the 9 =90° the

x~-type.

For 9

=

45° both the _xz and the

x~

_modes

are excited

by

the

corresponding

components of the electric field

gradients.

Conclusion.

We have shown that the

scattering

of

light by

_a

non-spherical body

_can be calculated

accurately

and

efficiently using

_a

point matching

method. We have shown that the structure

(peaks)

in the absorbance _curve of _a

non-spherical plasma-particle

_may differ

significantly

from that of _a

spherical particle

and demonstrated the relation of the absorbance

peaks

to

oscillations of the induced surface

charge

in _resonance with the

dipole

and

quadrupole

terms of the wavefield.

References

[1] Lord RAYLEIGH, Philos. Mag. ^XLI (1871) 274, 447 XLVII (1899) 375.

[2] BORN M. and WOLF E.,

Principles

of

Optics (Oxford

_: Pergamon Press, 1975) _p. 370.

[3] MIE G., Ann.

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