Resonant Degenerate Crystal Made of Spheres Located in
Magnetodielectric Medium
Anatoliy I. Kozar
Kharkov National University of Radio Electronics, Kharkov, Ukraine
Abstract In the paper, a computational method is proposed to analyze phenomena occurring when electromagnetic waves
are scattered by a finite resonant crystal with the cubic lattice having small resonant magnetodielectric spheres in its nodes.
The method is based on the second kind Fredholm integral equations. Investigated are properties of a resonant degenerate
crystal located in magnetodielecrtic environment with positive and negative values of permittivity 0 and permeability 0 . It
is shown that media with positive and negative values of permittivity and permeability differently influence on scattering
properties of the degenerate resonant crystal. The effect of degeneration canceling, when the permittivity and permeability
jump, is found such that coincidence of the lattice resonance and the internal crystal sphere resonance is canceled, and the
resonance curves split. Graphic dependences given in the paper numerically define a magnitude of occurring splitting. The
influence of an internal resonant crystal cavity size on the scattered field structure in the Fresnel zone is graphically estimated.
Keywords Magnetodielectric, Integral Equation, Crystal
second type Fredholm integral electromagnetic equations[2,
3, 4].
1. Introduction
Our attention was attracted by work[1] wherein unusual
properties of substances with simultaneously negative values
of permittivity  and permeability μ were considered. It was
of interest to study scattering properties of a finite discrete
resonant crystal made of magnetodielectric spheres, if the
crystal is located in external media with simultaneously
either positive or negative permittivity  and permeability μ.
Investigated in this paper are crystal properties when a
degenerate resonance is excited in it (st+ m +e) in the case
the structural (st) (lattice) crystal resonance coincides with
the internal, also coincided, magnetic (μ) and electric ()
resonances of spheres. This is a study of the case equivalent
to roentgen crystals optics, with a λ  1 ; a λg ~1;
d , h, l λ ~1 where a is the sphere radius;
, g are the
lengths of waves scattered outside and inside spheres; d , h, l
are the orthogonal lattice constants. We investigated the
me
multimode structure (d, h, l~ 2 λr ) of the resonant (r) field
inside and outside the cubic lattice in the Fresnel and
Fraunhofer zones for external media with the permittivity
and permeability  0   0  1 .
An effect of canceling the resonant crystal degeneracy is
considered. The solution is obtained on the basis of the
* Corresponding author:
fizika@kture.kharkov.ua (Anatoliy I. Kozar)
Published online at http://journal.sapub.org/ijea
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved
2. Main Part
Let us find the field scattered over the known internal field
re r
 r ,t  and magnetic
of scatterers through the electric П
r r
П m  r , t  Hertz potentials of the spatial lattice


 П  rr ,t   ik  , П  rr ,t  , (1)
r
r
 П  rr ,t   ik , П  rr ,t  .
r
Еscatt    k 2 0 0
r
H scatt    k 2 0 0
re
rm
0
m
e
0
The lattice Hertz potentials are presented as a
superposition of Hertz potentials for separate lattice spheres.
The electric Hertz potential of the lattice looks like
N
r r
1
П e  r , t    3  sin k1ac  k1ac cos k1ac  х
c 1 k1
  cef
r
r
eik1rc
х
 1 Ec0( p , s ,t ) (r ', t )
.
rc
 0

(2)
Expression (2) describes the scattered field at arbitrary
distance rc from the sphere centers to observation points
r0 r
outside the spheres. Here Ec (',r )t is the induced internal
field of the interacting spheres. The field can be found by the
algebraic system of inhomogeneous equations[3], N is the
number of spheres. The field scattered by the system of
ur
r
spheres Е scatt (,r )t
is found by (1), (2) as
me
N
r
r
1
Escatt  r , t    3  sin k1a  k1a cos k1a  x
c 1 k1
) r r
 cef

  cef
x 
 1 Lc Ec0 (r ')  ik 0 
 1 x
  0

 0

) r0 r
i t  k1rc 
x Pc H c  r ' e
,
where
)
Lc
)
Pc
and
 xxc 
ik1
where
1 2 3( x  xc)0
k1 
rc
rc5
 x, y, z  are
2
 rc2
2
 0yc 

 xc 

0 

 zc
0
 0xc
 rc2 ( 2 x 
) xc 0
 k1
rc3
2
the coordinates of observation point;
 xc0 , yc0 , zc0  are the coordinates of spheres centers.
Expression (3) describes the scattered field consisting of
propagating and damped spatial harmonics inside and
outside the lattice in the Fresnel and Fraunhofer zones.
The entire field at an arbitrary medium point outside the
spheres is defined as
E( ,r )t 
( ,E)0 r t( , )Escatt r t ,
(4)
where E(0 ,r )t is the undisturbed field of the scattered
wave.
Expressions (3) and (4) are analyzed numerically for the
resonant degenerate cubic crystal. The analysis results are
given in Figs. 1, 2, 3. The number of spheres here is N = 64000;
their radii are а = 0.5 cm; their permittivity and permeability are
me
r
2λ
Fig. 1 b).
 0  0  +1
(curve 2 in Fig. 1 b). Here resonances of
lattices and spheres do not coincide.
Figs. 2 and 3 show modules of fields (4) and (3) versus
changes in coordinates x, y and z in the direction of scattered
wave propagation (Fig. 1 b), inside and outside the
degenerate crystal in media with  0  0  +1 (Fig. 2) and
 0  0  –1 (Fig. 3).

,
    9.75; the lattice constants are d  h  l 
14.82 cm (Fig. 1 a) and
 0  0  –1 (curve 1,
Considered for the same crystals are also the similar
dependences for the cases when the medium parameters
change and become  0  0  –1 (curve 2 in Fig. 1 a) and
are the functional matrices
 xyc
3( x  xc)0
rc4
and in the medium with

 0
 xzc 


 yyc  yzc  ; Pˆc    0zc


 yc
 zyc  zzc 

)
Element  xxc of matrix Lc looks like
  xxc

Lˆc    yxc

  zxc
(3)
d  h  l  2 λr  12.52 cm (Fig. 1 b).
Figs. 1 а and 1 b show the fields (4) and (3) modules
versus scattered wavelength l for the resonant degenerate
crystals in the medium with  0  0  +1 (curve 1, Fig. 1 a)

|E|
Figs. 2 с and 3 с show the damped component of the
multimode scattered field in the Fresnel zone; curve 1 is
given for crystals with internal cavities[5, 6].
Figs. 2 d and 3 d show modules of fields (4) and (3) versus
coordinate z for the crystal side area along the y-axis (Fig. 1 b)
in the Fraunhofer zone in media with  0  0  +1 (Fig.
2 d) and
 0  0  –1 (Fig. 3 d).
It follows from the numerical analysis that the resonant
degenerate cubic crystal in media with simultaneously either
positive or negative permittivity and permeability
 0  0  1 has different electromagnetic properties. In a
medium with  0  0  –1 the crystal is characterized by
the pronounced reflecting properties. It can play a role of
reflecting mirror and resonator with a strong internal field.
By forming nodes and antinodes on crystal faces, one can
control the scattering crystal properties. In a medium with
 0  0  +1, a phenomenon of scattered wave resonant
propagation is inherent in the crystal. This can be used when
creating non-reflecting devices.

E0
Escatt
x
4,95

H0
4,95
2(m+e)
z
1(st+m+e)
1(st+m+e)
1,65
y
2(m+e)
1,65
, сm
, сm
7
6
a
8
6,5
5,6
b
Figure 1. Dispersion dependences of entire (a) and scattered (b) fields in media with
 0  0  1
7,4
|E|
|E|
5,175
5,175
1,725
0,575
y, сm
x, сm
-6000
0
6000
-5000
5000
0
a
b
|E|
|E|
6,3
2,7
1
1,5
2,1
z, сm
z, сm
-825
750
-4200
2325
c
Figure 2. The entire field in the medium with
 0  0  +1
0
4200
d
Escatt
Escatt
5,265
5,265
1,755
1,755
y, сm
x, сm
-6000
0
-6000
6000
0
a
6000
b
Escatt
Escatt
6,75
1,89
1
2,25
0,63
z, см
-2000
-1000
0
400
z, см
-4500
c
Figure 3. The field scattered in a medium with
 0  0  –1.
0
4500
d
When varying the environment parameters  0 and 0 , an effect of canceling the degeneracy in the resonance degenerate
crystal, which entails violation of the crystal resonances (st+ m +e) superposition, can occur. As an example, this effect is
shown in Fig.4 where splitting of resonant curves as a consequence of this effect (Fig.1 b) and occurrence of noncoincident
resonances (st), (m), (e) (Fig. 4 a) and (e), (m) (Fig. 4 b) can be seen. In the case of splitting resonance curve 1 (st+ m +e),
medium
parameters have the values:  0  1 ; 0  1.71 ; whereas in the case of curve 1 (m + e) they take the values:  0  1 ;
0  1.71 .
After the degeneracy is cancelled, the structural resonance (st) of curve 1 (st+ m +e) (Fig. 4 а) is deposed to the domain of
wavelength
st
λ
r  8.2 , and therefore is not presented in Fig. 4a, as it is beyond the figure limits.
Escatt
Escatt
1(st+m+e)
4,95
4,950
1(m+e)
(e)
(m)
(e)
(m)
1,65
1,65
5,84
6,56
6,2
, сm
7
7,5
a
8
, сm
b
Figure 4. The influence of environment on cancellation of the resonant cubic crystal degeneracy
3. Conclusions
[3]
A.I. Kozar 2004, Electromagnetic Wave Scattering with
Special Spatial Lattices of Magnetodielectric Spheres // J.
Telecommunication and Radio Engineering. – New York,
N.Y. (USA): Begell House Inc. Vol. 61, No.9. – P. 734-749.
[4]
A.I. Kozar 2005, Structural Function Development for
Electromagnetic Interactions in the System of Multiple
Resonant Magnetodielectric Spheres // J. Telecommunication
and Radio Engineering. – New York, N.Y. (USA): Begell
House Inc. Vol. 63, No.7. – P. 589-605).
[5]
A.I. Kozar 2008, The effect of defects on scattering properties of
the resonant magneto-dielectric spherical crystal / A.I. Kozar //
Microwave and Telecommunication Technology: IEEE 18th
International Crimean Conference, September 8-12, 2008. –
Sevastopol, Crimea, Ukraine. Vol. 2. – P. 560-561.
[6]
A.I. Kozar 2007, Resonant degenerate crystal made of
magnetodielectric spheres with defect / A.I. Kozar // Opto-,
nanoelectronics, nanotechnology and Microsystems: IХ Int.
conf. September, 24-30, 2007: abstracts. – Ulyanovsk,
Ulyanovsk State University, Russia, – P. 201. (in Russian).
The resonant degenerate crystal located in a
magnetodielectric medium with varying parameters  0 and
0 can be used for creation of resonant structures with
abnormal electrodynamic properties.
REFERENCES
[1]
[2]
V. Vesalago 1968, The electrodynamics of substances with
simultaneously negative values of  and  // Uspekhi
Sovetskoy Fiziki. V.10, No.4. – P. 509-514. (in Russian)
N.A. Khyzhnyak 1958, The Green function of Maxwell’s
equations for inhomogeneous media // J. technical physics.
V. 28, №7. – C. 1592 – 1609. (in Russian)