Research Article

Nanoscience Journal

Volume 1(1): 1-4

ScienceText

Estimation of Electromagnetic Property of Thin Film with Mie Scattering TheoryArun K. Saha*Department of Chemistry and Forensic Science, Albany State University, Albany, Georgia, USA

AbstractDielectric thin films are grown layer by layer in spin coating process. Each layer can be considered as a two dimensional plane composed of nano particles of spherical shapearranged periodically in a host medium with infinite extent in x- and y- direction but with finite extent in z- direction. Well known “Mie scattering theory” is applied to calculate the scattered fields from all individual spherical dielectric inclusions. Effective material permittivity is calculated from the total scattered field at a certain point. Calculated effective permittivity is then compared with a special case in Lewin’s model and a numerical value as obtained from commercial HFSS simulation software.

IntroductionAn artificial dielectric material is considered to have spherical dielectric particles in a

host medium arranged periodically in all three dimensions. A plane electromagnetic wave incidence is considered on one side of the artificial material. Total field at a certain point inside the artificial material is calculated as the summation of all scattered fields from all individual neighboring particles plus the incident field at the point of investigation. From this calculated total field at that point, effective material parameter is determined after some mathematical manipulations. Calculated material parameter is then verified with ANSYS HFSS simulation software.

Background Study: Revist of Mie ScatteringLet’s consider that spherical dielectric particles, with relative dielectric constants ɛp, μp and

radius a, are embedded periodically into a host medium with relative dielectric constant ɛh, μh as shown in Figure 1. The loaded medium is assumed to have thickness t in z direction but extended infinitely (from –∞ to +∞) in x and y direction.

Correspondence to: Arun K. Saha, Department of Chemistry and Forensic Science, Albany State University, Albany, Georgia, USA; E-mail: Arun.saha@asurams.edu

Received:

Accepted:

Published:

Keywords: artificial material; effective permittivity; spherical dielectric; mie scattering

Figure 1. Electromagnetic wave incidence on artificial material composed of spherical dielectric objects arranged periodically in a host medium

Saha AK (2018) Estimation of Electromagnetic Property of Thin Film with Mie Scattering Theory

Volume 1(1): 2-4

By interchanging electric and magnetic component, y component of scattered magnetic field is

( )

( )0

0

22

20

3

0

2

2

re

zzEjk

ky

zHaH

hhjkr

heffp

heffp

p

hhh

effp

heffp

ys

εµ

εε

εεε

εµµµ

µµ

−

∂∂

+

−

−

+

∂

∂

+

−=

(7)

Excitation Field Calculation Let’s suppose the electromagnetic field that excite the particle

located at point (x0, y0, z0) is ( )0zE and ( )0zH . Both ( )0zH and ( )0zH are the sum of incident field and mutual field. Mutual field

at any particle is, in fact, the summation of all scattered fields from remaining particles. If the lattice constant and thickness of the artificial material is assumed s and t respectively, then positions of the remaining particles can be expressed as (ls, ms, ns), where l and m varies from -∞ to +∞ and n varies from 0 to t/s. Therefore, distance of the point (x0, y0, z0) from all other remaining points can be calculated as

( ) ( ) ( )202

02

00 znsymsxlsR −+−+−= (8)

Now, total electric field at the point (x0, y0, z0) is

( )

( )

( )

( )

( )00

2/

2/

220

23

2/

2/

2/

2/

00

/

0

220

23

0

0

0

0

0

0

0

0

0

0

2

2

2

2

ˆ

Re

znsHjk

kx

nsEa

Re

znsHjk

kx

nsEa

eEzE

hh

hh

hh

jkR

heffp

heffp

p

sz

szhh

heffp

heffpsy

sy

sx

sx

jkR

heffp

heffp

p

stn

nhh

heffp

heffpm

m

l

l

jkzi

εµ

εµ

εµ

µµ

µµµ

εµεε

εε

µµ

µµµ

εµεε

εε

−

+

−

+

−

+

−

−

+=

=

+∞=

−∞=

+∞=

−∞=

−

∂∂

+

−−

+

∂

∂

+

−

−

∂∂

+

−−

+

∂

∂

+

−

+=

∑∑∑

∑∑∑

(9)

Assuming ls=α, ms=β, ns=γ and considering the material under investigation is infinite in x and y direction, the above equation takes the following form

( )

( )

( )

( )

( ) γβαµµ

µµγµ

εµεε

εεγ

γβαµµ

µµγµ

εµεε

εεγ

εµ

εµ

εµ

dddR

ez

Hjk

kx

E

dddR

ez

Hjk

kx

Es

eEzE

hh

s

s

s

s

s

s

hh

hh

jkR

heffp

heffp

p

z

zhh

heffp

heffp

y

y

x

x

jkR

heffp

heffp

p

thh

heffp

heffp

jkzi

00

220

2

00

0

220

2

3

0

0

20

20

20

20

20

20

0

0

2

2

2

2

1

ˆ

−

+

−

+

−

+

−

−

∞+

∞−

∞+

∞−

−

∂∂

+

−−

+

∂

∂

+

−

−

∂∂

+

−−

+

∂

∂

+

−

+=

∫∫∫

∫∫∫

(10)

A plane electromagnetic wave, with electric field polarized in x direction, is traveling in the host medium (ɛh, μh) in the direction of z. Electric and magnetic component of such wave can be expressed as

hhjkzixi eEE εµ−= ˆ (1)

hhhhhh jkz

ijkz

iyi eEeHH εµµεεµ −− −== ˆˆ (2)

Here k=propagation constant of the host material. When this incident wave falls on the artificial material under consideration at the plane z=0, some part of the wave will be reflected and some will be transmitted through the material. Reflected will be a plane wave at a sufficiently long distance from the boundary and transmitted wave will vary from point to point as the wave travels. Any individual particle is, therefore, under the influence of two electromagnetic fields – one is incident field and other one is mutual field which is the summation of all scattered fields by all neighboring particles.

Now let’s consider a spherical particle located at point (x0, y0, z0). The problem of plane wave scattering by a dielectric sphere (with radius much less than the wavelength) is solved by G. Mie [1]. After some manipulation, the x component of that scattered electric field at a certain point can be written as [2].

++

∂

∂= − )()()(

32 2

2

210

2/32hhhhxs k

xjbzEkE εµεµ

010

0)(

re

zazHk

hhjkrh

εµµ

-

∂∂ (3)

Here E(z0) is the electric field at point (x0, y0, z0), r0 is the distance from (x0, y0, z0) to the point of investigation and a1 and b1 are magnetic and electric coefficients or Mie coefficients which are given by following two equations

effph

effph

pp

effph

effph

hh

akjb

akja

εε

εεεµ

µµ

µµεµ

+

-=

+

-=

2)(

32

2)(

32

32/321

32/321

(4)

Here effp

ffp εµ , are the effective permeability and permittivity of

the particleand related to actual material permeability and permittivity

pp εµ , of which the particles are composed. effp

ffp εµ , and pp εµ ,

are related according to the following equations

)(

)(

cossin)1(

)cos(sin2)(2

θεε

θµµ

θθθθ

θθθθ

εµθ

F

F

F

ka

peffp

pffp

pp

=

=

+−

−=

=

(5)

Now substitution of 1a and 1b in equation 3 yields

( )

( )0

0

22

20

3

0

2

2

re

zzHjk

kx

zEaE

hhjkr

heffp

heffp

p

hhh

effp

heffp

xs

εµ

εµ

µµµ

εµεε

εε

−

∂∂

+

−

−

+

∂

∂

+

−=

(6)

Saha AK (2018) Estimation of Electromagnetic Property of Thin Film with Mie Scattering Theory

Volume 1(1): 3-4

After performing first and second integration with respect to dα and dβ,

( )

( )

( )

( ) γγµµ

µµγµ

εε

εεγεµ

εµπ

εε

εεπ

εµγ

εµ

deHjk

Eksa

kj

zEsa

eEzE

hh

hh

zjk

heffp

heffp

t

heffp

heffp

hhhh

heffp

heffp

jkzi

0

0

2

2

2

234

ˆ

1

0

23

3

03

30

--

-

∂∂

+

-

++

-

-+

-

+=

∫ (11)

Now let’s write the above equation in a compact form assuming

3

3

3

4

factorfillingand

2

1

2

1

s

af

b

R

Q

hp

heffp

heffp

heffp

heffp

π

εµ

µµ

µµ

εε

εε

=

=

+

-=

+

-=

( ) ( )

( ) ( ) γγ

γµγπ γ deHR

jkEQbk

kbsaj

zEQfeEzE

bzjkht

bjkzi

0

0

0

22

3

3

00

2

ˆ

−−

−

∂∂

+

−+=

∫ (12)

Similarly the equation for H(z0) can be calculated as

( ) ( )

( ) ( ) γγ

γεγπ γ deEQ

jkHRbk

kbsaj

zHRfeHzH

bzjkht

bjkzi

0

0

0

22

3

3

00

2

ˆ

−−

−

∂∂

+

−+=

∫ (13)

Now these two simultaneous equations are to be solved assuming a general solution

( )( ) 0

0

0

0zj

zj

BezH

AezEβ

β

−

−

=

= (14)

Here β=propagation factor of artificial material and µεβ k= . μ, ɛ are relative permeability and permittivity of the artificial material under investigation.

( )

( ) 0

1

00

00

00

00

00

=

−

+−

−

−+

−

+−

−

−−

−

−

−−+−

−−

−−+−

−−

−−

zjkbzkbttj

jkbzzj

zjkbzkbttj

jkbzzj

jkbzi

zj

eekb

BD

eekb

BD

eekb

AC

eekb

AC

eEAeQf

ββ

β

ββ

β

β

β

β

β

β

(15)

and

( )

( ) 0

1

00

00

00

00

00

=

−

+−

−

−+

−

+−

−

−−

−

−

−−+−

−−

−−+−

−−

−−

zjkbzkbttj

jkbzzj

zjkbzkbttj

jkbzzj

jkbzi

zj

eekb

NA

eekb

NA

eekb

MB

eekb

MB

eHBeQf

ββ

β

ββ

β

β

β

β

β

β

(16)

3

3

3

3

3

3

3

3

2,2

2,2

Qs

kaN

Rs

kbaM

Rs

akD

Qs

kbaC

h

h

πεπ

µππ

==

==

Now simultaneous equations (15) and (16) are ready to be solved. Assuming the artificial material is nonmagnetic (μ=1.0), these equations are function of A, B and β. To solve these two equations, let’s write those in the following forms –

( ) ( )( ) ( ) 0,,,,

0,,,,=+=+

ββββ

BAjGBAGBAjFBAF

IR

IR (17)

To solve A, B and β, only the following 3 equations will be necessary-

( )( )( ) 0,,

0,,0,,

===

βββ

BAGBAFBAF

R

I

R (18)

Newton’s method is used to solve these 3 simultaneous equations. If the initial guess of iaA = , ibB = and ic=β corresponding updated values are uuu cba ,, , then

( )( )( )

×

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

-

=

-

βββ

β

β

β

,,,,,,

1

BAGBAFBAF

GB

GA

G

FBF

AF

FB

FA

F

cba

cba

R

I

R

RRR

III

RRR

i

i

i

u

u

u

(19)

A MATLAB program is developed to implement this iteration process until ( )β,, BAFR + ( )β,, BAFI + ( )β,, BAFI ≤10-7

Result The above theoretical calculation to estimate relative permittivity of

the composite is verified by HFSS simulation and Lewin’s model [3]. In the calculation, host medium is considered as air, spherical inclusions are considered as nonmagnetic material with relative permittivity of 3. Signal frequency, radius of sphere and lattice constant are assumed 1 GHz, 5 mm and 10 mm respectively as suitable parameters to be used in HFSS simulation. The best algorithm [4] to extract material property is used in the simulation. Calculated value of relative permittivity of composite is shown in Figure 2 where it is seen that relative permittivity

Saha AK (2018) Estimation of Electromagnetic Property of Thin Film with Mie Scattering Theory

Volume 1(1): 4-4

value settles down to a value of 2.0 when thickness is too big or tends to infinite. HFSS simulation and Lewin’s model are based on composite of infinite size. So, results from these two methods are also very close to each other as shown in Figure 3.

ConclusionA three dimensional artificial material is assumed to be made

of multiple layers, where each layer is composed of nonmagnetic spherical dielectric balls arranged in two dimensional (x, y) arrays. Assuming radius of each ball = 5 mm, lattice constant = 10 mm, and relative dielectric constant of ball material = 3 estimated relative permittivity with respect to thickness or number of layers shows some unstable values when number of layers are lessand settles down for large number of layers as per expectation. Relative permittivity of the proposed composite as obtained from HFSS Simulation result and Lewin’s model are very close to each other as both methods are based on infinite thickness of material.

Acknowledgment The author acknowledges the support from the Department of

Chemistry and Forensic Science, Albany State University, Albany, Georgia 31705.

References1. Gustav Mie, “BeiträgezurOptiktrüberMedien, speziellkolloidalerMetallösungen”,

Annalen der Physik, vol. .330, no. 3, pp. 377-445, 1908.

2. J. A. Stratton, “Electromagnetic Theory”, New York: McGraw-Hill, 1941.

3. L. Lewin, “Radio Science Paper”, vol. 94, part III, pp. 65-68, Jan 1947.

4. James Baker-Jarvis, Eric J. Vanzura and William A. Kissick, “Improved Technique for Determining Complex Permittivity with the Transmission/Reflection Method,” IEEE Transaction on Microwave Theory and Techniques, vol. 38, no. 8, pp. 1096-1103, August 1990.

Figure 2. Relative permittivity of composite material as obtained.

Figure 3. Comparative analysis of relative permittivity of the proposed material as obtained by the theoretical calculation, Lewin’s estimation and HFSS simulation whenmaterial thickness tends to infinity.

This work is partially supported by Department of Chemistry & Forensic Science, Albany State University, Albany GA31705.