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: [email protected]
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.