New type of eigen oscillations based on evanescent waves
In this paper we represent the results of investigations of the electromagnetic waves in a layered dielectric that is limited from one side by a metal plate. There are electromagnetic oscillations, which evanesce in the direction of periodicity and propagate along the perpendicular direction, so the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | New type of eigen oscillations based on evanescent waves / M.I. Ayzatsky // Вопросы атомной науки и техники. — 2001. — № 1. — С. 89-91. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859576288472203264 |
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| author | Ayzatsky, M.I. |
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| citation_txt | New type of eigen oscillations based on evanescent waves / M.I. Ayzatsky // Вопросы атомной науки и техники. — 2001. — № 1. — С. 89-91. — Бібліогр.: 3 назв. — англ. |
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| description | In this paper we represent the results of investigations of the electromagnetic waves in a layered dielectric that is limited from one side by a metal plate. There are electromagnetic oscillations, which evanesce in the direction of periodicity and propagate along the perpendicular direction, so they can be treated as surface waves. Existence of such waves gives possibility to create resonators filled by a layered dielectric with new type of eigen oscillations based on these evanescent waves.
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| first_indexed | 2025-11-27T02:10:13Z |
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T H E O R Y A N D T E C H N I C S O F P A R T I C L E A C C E L E R A T I O N
NEW TYPE OF EIGEN OSCILLATIONS
BASED ON EVANESCENT WAVES
M.I. Ayzatsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
In this paper we represent the results of investigations of the electromagnetic waves in a layered dielectric that is
limited from one side by a metal plate. There are electromagnetic oscillations, which evanesce in the direction of
periodicity and propagate along the perpendicular direction, so they can be treated as surface waves. Existence of
such waves gives possibility to create resonators filled by a layered dielectric with new type of eigen oscillations
based on these evanescent waves
PACS: 84.40.Cb
INTRODUCTION
It is well known [1], that in 1-D periodic mediums
there are two different in basis electromagnetic eigen-
oscillations supported by medium without external
currents and charges. In the certain frequency intervals
(passbands) the electromagnetic oscillations represent
wave process, which carry a constant energy (in the case
of absence of absorption) in direct or opposite direction.
Between the passbands the electromagnetic oscillations
have a structure that is distinct from previous case. In
these frequency intervals electromagnetic oscillations
transfer no energy in the direction of periodicity and
have decreasing (increasing) dependence on the
coordinate. These frequency intervals are called
forbidden bands (stopbands). Results of our
investigations of the properties of the electromagnetic
oscillations in the stopbands of a layered structure are
represented. Particularly, we have shown that under
some conditions the eigen-functions, which describe the
field distributions, can have the zero values at definite
planes, which are perpendicular to the direction of
periodicity. Placing metal sheets in this points we can
essentially change performances of the waves reflected
from finite number of dielectric layers lying on a metal
plate and create dielectric-metal resonance systems,
which shall have oscillations with increasing or
decreasing field distributions along the coordinate z.
These phenomena can be treated in the terms of the so-
called “defect modes” [2] as the “surface defect levels”
that arise as the result of existence of the interface
between the medium with a band structure and the
medium that can be considered as an infinite potential
wall. In this paper we represent the results of
investigations of the evanescent in one direction and
propagating in perpendicular one electromagnetic waves
in the case when a layered structure is limited from one
side by metal (or in the case when periodicity is
reflected symmetrically relatively some plane).
Existence of such waves gives possibility to create
resonators filled by a layered dielectric with new type of
eigen oscillations based on these evanescent waves.
1. MULTI-LAYER DIELECTRIC
Let us consider properties of eigen electromagnetic
oscillations in a layered dielectric, which represents
periodically repeating along the axes z a set of layers
with thickness 1d and 2d and with permittivities 1ε end
2ε . In transversal directions (x, y) the layers are not
limited. Dependence on time and the transversal
coordinate x we shall suppose as exp{ { )}xi k x t− ω .
Dependence of the transversal components of
electromagnetic field on the longitudinal coordinate z
can be found from the Maxwell equations and for
arbitrary selected two adjoining layers (i=1,2) (from
which we shall conduct numbering periods (s=0))
tangential electric field components can be written as:
, ,(0) ( ) ( )
,
i z i zik z ik z
i i iE E e E e−+ −= +τ ,
, ,(0) ( ) ( )
,
0
z i i zik z ik zi
i i i
qH E e E e
Z
−+ − = − τ ,
where 2 2 2
, / ( / )i z i xk c sqrt k c= × −ω ε ω , ( )
iE ± - constants
and 2 2 2/ ( / )i i i xq sqrt k c= −ε ε ω - for s-polarization and
2 2 2( / )i i xq sqrt k c= − −ε ω -for p-polarization
As the considered system is periodic along the axes
z with period D=d1+d2, field components within the
period with number s can be determined by the
expression:
, ,( ) ( )( ) ( ) ( )
,
i z i zik z sD ik z sDs s
i i iE E e E e− − −+ − = + τ ρ
where ρ - some complex number. Similarly, one can
write the expression for ( )
,
s
iHτ . It is easy to show that at
fixed frequency the boundary conditions for the field
components are fulfilled only for two values of
parameter ρ - 1ρ and 2 11/=ρ ρ .:
2
1,2 1Q Q= ± −ρ
where Q is determined by the expression
2 2
1 1, 1 2, 2 2 1, 1 2, 2
2 2
1 2
cos( ) cos( )z z z zk d k d k d k d
Q
+ − −
=
−
α α
α α
,
and 1 2 1 2 2 1(1 / ) / 2 , (1 / ) / 2= + = −α ε ε α ε ε .
Inside the passbans 1 2 1= =ρ ρ and inside the
stopbans 1 1<ρ and 2 1>ρ (see, Fig. 1, where the
dependence of 1( )abs ρ on dimensionless frequency
/(2 ) /D c DΩ = =ω π λ is depicted for the case when
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 89-91.
89
/ 2 / 0x x xk D D= = =κ π λ and 1 1( ) 2n sqrt ′= =ε ,
2 2( ) 1.38n sqrt ′= =ε , 1 2 0′′ ′′= =ε ε , 1 2 1 2/( )d n n n D= + ,
2 1 1 2/( )d n n n D= + . If / 2 / 0x x xk D D= = >κ π λ , there
is the gap for frequencies 00 < Ω < Ω . This gap we shall
call the natural or zero stop band.
Fig. 1
Let us consider a structure of an electrical field in
some stop band. An example can be seen in Fig. 2
where the dependencies of the modulus of Eτ on the
longitudinal coordinate /z D=ξ are shown within one
period in the middle of the first stopband ( 0.31Ω = ,
see, Fig. 1).
From Fig. 2 it follows that both eigen non-
propagating oscillations (d-decreasing - 1 1<ρ , i-
increasing - 2 1>ρ ) of layered dielectric have an
interesting feature - in the certain planes, perpendicular
to the z axes, the tangential component of an electrical
field equals to zero [3]. And for two fundamental
solutions these planes do not coincide. Putting in these
places metal planes, we can effectively control the field
distribution
Fig. 2
Indeed, a general solution of electrodynamic
problem with a finite layered structure is represented as
the sum of two fundamental (partial) solutions
( ) ( )1 ,1 2 ,2( )E z C E z C E z= +τ τ τ . If the metal plate is
located at z z∗= , such condition must be fulfilled
( )1 ,1 2 ,2( ) 0C E z C E z∗ ∗+ =τ τ .
From this condition it follows that if ,1( ) 0E z∗ =τ (
,2 ( ) 0E z∗ =τ ), then С2=0 (С1=0). Thus, choosing some
z∗ , we can create the field distribution that appropriate
only to one fundamental solution, i.e. either pure
increasing or decreasing one.
Transversal planes with zero value of the tangential
component of electric field exist in all gaps under some
conditions, but for the zero gap they locate near the
edge of the gap and when 0x →κ the frequency width
of this interval tends to zero.
In section 2 we represent the results of exploring of
the surface waves that can exist in the forbidden zones
in the semi-limited layered dielectric, which ends by a
metal plate.
Using the described circumstances, we can, at first,
essentially change performances of waves reflected
from finite number of dielectric layers, lying on a metal
plate. Secondly, we can create metal-dielectric
resonance systems with eigen-oscillations that is base
on evanescent (non-propagating) waves and have
increasing (decreasing) field distributions along the
coordinate z. Results of these investigations is given in
section 3.
2. SURFACE WAVES IN THE HALF-
LIMITED LAYERED DIELECTRIC
Assume that the front surface of the metal plate has
the coordinate z=zm. The layered dielectric is spaced so
that a layer with the higher refractive index starts at the
coordinate z=0 (see Fig. 3).
Fig. 3
Changing the position of the front surface of the
metal plate (0<zm<D), we can study the different
“surface levels” of the electromagnetic oscillations. We
shall consider the case when optical length of each layer
equals a quarter of some reference wavelength ∗λ -
/(4 )i id n∗= λ . For ideal metal we have a boundary
condition at z=zm that coincide with (1.6). If ,1( )E zτ
describes the decreasing solution then we have to
suppose that C2=0 and try to find such frequency Ω and
transversal wavenumber xκ inside a forbidden band
when
,1( , , ) 0x mE z zΩ = =τ κ .
In Fig. 4 (p-polarization) and Fig. 5 (s-polarization)
we represent the results of solving this dispersion
equation (symbols {} - zm=0, the layered medium
starts with more dense layer - n1=2; and symbols {+} -
zm=d1, layered medium starts with a less dense layer -
n2=1.38) . White color marks forbidden bands, grey
color – passbands.
We can see that for p-polarization there are surface
waves both for zm=0 and zm=d1, but surface waves with
s-polarization exist only when zm=0. The surface wave
in the zero forbidden band exists only for p-polarization
when 0<zm<d1/2 and (d1+d2/2)<zm<D. In the first
forbidden band for p-polarization waves can travel
91
90
along the x-axis with phase velocities less or greater
than that of light (0<zm<d1/2 and (d1+d2/2)<zm<D) and
they are slow waves when d1/2<zm<(d1+d2/2).
For s-polarization surface waves in the zero
forbidden band can not exist. In the first forbidden band
waves travel along the x-axis with phase velocities less
or greater than that of light only when (0<zm<d1/2 and
(d1+d2/2)<zm<D). So, surface waves with s-polarization
do not exist when d1/2<zm<(d1+d2/2).
Our preliminary simulations have shown that studied
surface waves can exist in the layered structures with
finite number of layers limited from one side by metal.
Fig. 4
Fig. 5
3. NEW TYPE OF EIGEN OSCILLATIONS
IN RESONATORS FILLED WITH A
LAYERED DIELECTRIC
As the considered above system is periodic, then the
condition (1.7) is fulfilled not only for mz z= but also
for the set of points with longitudinal coordinates
, 1,2mz z D s s= + × = K . So, we can put an additional
metal sheet in some point from this set and thus we get
the waveguide with the non-uniform transverse
distribution. All results obtained in section 2 can be
used for describing waves between such two metal
sheets. We must note that in the case of two metal
sheets there are two types of waves. The first type we
shall obtain if we look at the system as beginning from
one sheet (for example, zm,1=0), and the second type - if
we look at the system as beginning from another sheet
(if zm,1=0 then zm,2=d1).
We can fix the value of the transversal wavenumber
xκ by restricting the layered dielectric along the
transversal direction by the metal cylinder with radius b
and in such way we obtain a closed cavity (see, Fig. 6).
For example, in the case of E-wave
0/ 2 /( 2 )x x pk D D b= =κ π λ π , where 0 pλ are the roots
of the Bessel function J0(x).
From Fig. 4 it follows that for different values of xκ
we may have either a decreasing (along the coordinate
z) eigen oscillation ( xκ <0.5 ) or an increasing ( xκ
>0.5 ) one.
Fig. 6
Using SUPERFISH code we simulated the
properties of the resonator depicted on the Fig. 6 (n1=2,
n2=1.38, D=2 cm, d1 = n2/(n1+n2)D = 0.816 cm, d2=D-
d1). For b=4 cm ( xκ =0.191) the value of eigen
frequency equals Ω=0.3275 and from Eq.(1.7) we
obtained Ω=0.3274. For this value of xκ the simulated
longitudinal distribution of the electric field on the axis
of resonator (r=0) is a decreasing one (see, Fig. 7-1).
For b=1 cm ( xκ =0.7655) the value of eigen frequency
equals Ω=0.5905 and from Eq.(1.7) we obtained Ω
=0.591. In this case the simulated longitudinal
distribution of the electric field is an increasing one
(see, Fig.7, curve 2).
Fig. 7
4. CONCLUSION
It was usually assumed that the resonator based on
smooth waveguide has the eigen oscillations that are
formed by interference of two waves which propagate in
different directions and have equal amplitudes. These
patterns are usually called standing waves. We have
shown that the eigen oscillations of a resonator can be
base on the evanescent (non-propagating) waves. In
some cases we need only one eigen wave to compose
the eigen oscillation of a closed cavity.
REFERENCES
1. L. Brillouin, M. Parodi Propagation des ondes dans
les milieux periodiques, 1956.
2. E. Yablonovish. Inhibited Spontaneous Emission in
Solid-State Physics and Electronics// Phys. Rev.
Lett. 1987, v.58, N.20, p. 2059-2062.
3. M.I. Ayzatsky. Electromagnetic oscillations in
periodic mediums and waveguides outside the
90
passband // Problems of Atomic Science and
Technology, Kharkov, NSC KIPT, 1999, N.3, p. 6-8.
91
92
THEORY AND TECHNICS OF PARTICLE ACCELERATION
NEW TYPE OF EIGEN OSCILLATIONS
BASED ON EVANESCENT WAVES
INTRODUCTION
1. MULTI-LAYER DIELECTRIC
2. SURFACE WAVES IN THE HALF-LIMITED LAYERED DIELECTRIC
3. NEW TYPE OF EIGEN OSCILLATIONS IN RESONATORS FILLED WITH A LAYERED DIELECTRIC
4. CONCLUSION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-78519 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-27T02:10:13Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ayzatsky, M.I. 2015-03-18T17:27:39Z 2015-03-18T17:27:39Z 2001 New type of eigen oscillations based on evanescent waves / M.I. Ayzatsky // Вопросы атомной науки и техники. — 2001. — № 1. — С. 89-91. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 84.40.Cb https://nasplib.isofts.kiev.ua/handle/123456789/78519 In this paper we represent the results of investigations of the electromagnetic waves in a layered dielectric that is limited from one side by a metal plate. There are electromagnetic oscillations, which evanesce in the direction of periodicity and propagate along the perpendicular direction, so they can be treated as surface waves. Existence of such waves gives possibility to create resonators filled by a layered dielectric with new type of eigen oscillations based on these evanescent waves. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration New type of eigen oscillations based on evanescent waves Новый тип собственных колебаний на основе исчезающих волн Article published earlier |
| spellingShingle | New type of eigen oscillations based on evanescent waves Ayzatsky, M.I. Theory and technics of particle acceleration |
| title | New type of eigen oscillations based on evanescent waves |
| title_alt | Новый тип собственных колебаний на основе исчезающих волн |
| title_full | New type of eigen oscillations based on evanescent waves |
| title_fullStr | New type of eigen oscillations based on evanescent waves |
| title_full_unstemmed | New type of eigen oscillations based on evanescent waves |
| title_short | New type of eigen oscillations based on evanescent waves |
| title_sort | new type of eigen oscillations based on evanescent waves |
| topic | Theory and technics of particle acceleration |
| topic_facet | Theory and technics of particle acceleration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78519 |
| work_keys_str_mv | AT ayzatskymi newtypeofeigenoscillationsbasedonevanescentwaves AT ayzatskymi novyitipsobstvennyhkolebaniinaosnoveisčezaûŝihvoln |