Self-focusing of radiation in non-uniform mediums
The possibility of self-focusing in propagating the rays in the non-uniform mediums is shown. This self-focusing is similar to a Veksler-Mac-Millan’s autophasing in the theory of accelerators. The self-focusing of the rays are able to weaken essentially an effect of random non-uniformities and to in...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2005 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79805 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Self-focusing of radiation in non-uniform mediums / A.V. Buts, V.A. Buts, G.I. Churyumov // Вопросы атомной науки и техники. — 2005. — № 2. — С. 146-148. — Бібліогр.: 3 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860152750769176576 |
|---|---|
| author | Buts, A.V. Buts, V.A. Churyumov, G.I. |
| author_facet | Buts, A.V. Buts, V.A. Churyumov, G.I. |
| citation_txt | Self-focusing of radiation in non-uniform mediums / A.V. Buts, V.A. Buts, G.I. Churyumov // Вопросы атомной науки и техники. — 2005. — № 2. — С. 146-148. — Бібліогр.: 3 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The possibility of self-focusing in propagating the rays in the non-uniform mediums is shown. This self-focusing is similar to a Veksler-Mac-Millan’s autophasing in the theory of accelerators. The self-focusing of the rays are able to weaken essentially an effect of random non-uniformities and to increase the threshold of development of stochastic instability.
Показана можливість автофазування променів, що поширюються в неоднорідних середовищах. Це автофазування аналогічне автофазуванню Векслера-Мак-Міллана в теорії прискорювачів. Автофазування променів істотно послабляє вплив випадкових неоднорідностей і збільшує поріг розвитку стохастичної нестійкості.
Показана возможность автофазировки лучей, распространяющихся в неоднородных средах. Эта автофазировка аналогична автофазировке Векслера-Мак-Миллана в теории ускорителей. Автофазировка лучей существенно ослабляет влияние случайных неоднородностей и увеличивает порог развития стохастической неустойчивости.
|
| first_indexed | 2025-12-07T17:52:06Z |
| format | Article |
| fulltext |
SELF-FOCUSING OF RADIATION IN NON-UNIFORM MEDIUMS
A.V. Buts 1, V.A. Buts 2, G.I.Churyumov 1
1Kharkov National University of Radio Electronics, 61166, Kharkov, e-mail: buts1974@front.ru;
2NSC Kharkov Institute of Physics and Technology,
61108, Kharkov, Ukraine, e-mail: vbuts@kipt.kharkov.ua
The possibility of self-focusing in propagating the rays in the non-uniform mediums is shown. This self-focusing is
similar to a Veksler-Mac-Millan’s autophasing in the theory of accelerators. The self-focusing of the rays are able to
weaken essentially an effect of random non-uniformities and to increase the threshold of development of stochastic
instability.
PACS: 42.15.-i
INTRODUCTION
In many cases in the non-uniform mediums like
laboratory plasma, ionosphere, ocean, or fiber
communication lines, the dimensions of the non-
uniformities are able to exceed considerably a wavelength
which propagates in these mediums. In order to analyze
the propagation of the waves in this case it is necessary to
apply an approximation of geometrical optics. Taking into
account that the geometrical optics so concerns to the
wave optics, as the classical mechanics concerns to the
quantum mechanics we are able to expect that many
important features of classical dynamics of charged
particles in electromagnetic fields can develop in the
dynamics of the rays too. In particular, there is a
phenomenon of the autophasing of accelerated particles in
physics of charged particles. This phenomenon was
discovered by Veksler and Mac-Millan and is a
fundamental one in the theory of the accelerators. We can
hope that analogical phenomenon will take place also in
propagating electromagnetic rays in the non-uniform
mediums, for example, in the non-uniform plasma. It is
necessary to note that a phenomenon of the self-focusing
of the rays was studding in [1].
PARAXIMAL APPROXIMATION
The simplest we can see analogy between dynamics of
charged particles in the external electromagnetic fields
and dynamics of the electromagnetic rays in the non-
uniform dielectric mediums if the paraxial approximation
for describing dynamics of the rays will be used:
2
1
0 12
( , )d r n r zn n
dz r
∂= ∇ =
∂
, (1)
where 0 1( , ) ( ) ( , )n r z n r n r z= + ; 0 ( )n r - independent
from longitudinal co-ordinate z - component of the
refraction coefficient; 1( , )n r z - slowly changing along
axis z component. As seen from (1), we have confined
the simplest case when dynamics of the rays is considered
at the plane ( ,r z ). For comparison let us consider the
non-relativistic motion equation of charged particle in
potential ( , )V r t
2
2 ( , )d rm F V r t
dt
= = −∇ . (2)
By comparing the equation (1) with motion equation of a
material point (2), we can see that these equations are
completely analogical. The analogua of the potential V−
in the geometrical optics is the coefficient refraction 1n ,
and analogue of mass is 0n . If we will choose the
potential as
( )2 4( , ) / 2 / 4 sinV r t r r r tα β ε= ⋅ + ⋅ − ⋅ ⋅ Ω ⋅ ,
we will obtain the equation that will be similar to the
equation of the Duffing’s oscillator which is acted an
external periodical force having amplitude ε and
frequency Ω on. Before the dynamics of the Duffing
oscillator that is subjected both the similar external effect
and parametrical perturbation action have been
investigated in [2]. In particular, the condition of
appearance of moving particles stochastic instability has
been defined in this paper. It has been shown that
dynamics of the rays in propagating in the medium having
the refraction coefficient (4) should be similar dynamics
of the charged particles.
SELF-FOCUSING OF THE RAYS
The paraxial approximation in the geometrical optics
corresponds to the non-relativistic moving of charged
particles. It means that the generalized pulse of the rays
can not become by sufficiently major. In order to describe
dynamics of the rays having great values of the pulse we
should refuse from the paraxial approximation. In this
case it is necessary to use more common equations of the
geometric optics which are similar to the relativistic
motion equations of charged particles. These equations
may be written
0Hp
x
∂= −
∂
& 0Hx
p
∂=
∂
& , (3)
where 2 2
0 ( , , ) ( , )H x p z n x z p= − − is hamiltonian,
,x p are the generalized coordinate and pulse
correspondently. Let us consider the first case when the
rays propagate in the medium. The parameters of the
medium depend only on transverse coordinate x (
( )n n x= ). From the equations (5) we can get the
following equations for a generalized coordinate x .
0
2
0 0 0
( ) ( ) ( )p Hp n x n xx F x
H H H x
⋅ ∂= − + = ⋅ =
∂
&&
&& . (4)
The equation (4) is one of the non-linear pendulum. In
particular, if the refraction coefficient of the medium will
be given as
2 2 2 2
0 / xn n ch
a
µ= + , (5)
146 Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 146-148
mailto:vbuts@kipt.kharkov.ua
mailto:buts1974@front.ru
the qualitative view of dynamics of the rays will be
similar to a view that is presented in Fig. 1.
Fig. 1
In Fig. 1 the trajectories of 20 rays which were entered
into the medium with different input characteristics are
shown. A value
x
a
Ψ = is called by a phase of a ray
relatively of non-uniformity. As this takes place, a value
a defines transversal dimensions of a canal of the
medium. Let us assume that typical transversal
dimensions of the canal of the medium depend on
longitudinal coordinate z , e.g. ( )a a z= . In this case a
phase of a ray will be a function not only x but also z ,
e.g. ( , )x zΨ = Ψ . Taking into account an analogy with
dynamics of the charged particle in our case the self-
focusing of the rays will take place. In order to define the
conditions of appearance of the self-focusing it is
necessary to consider the equation that describes changing
a phase of a ray with coordinate z :
2 0a a x
a a a
Ψ + Ψ + Ψ − =
& && &&&& & (6)
2 2
0 0 0
n n p n nx
H x H H z
∂ ∂= ⋅ + ⋅ ⋅
∂ ∂
&& . .
A synchronous phase we can define using the following
condition: ( ) / ( )s x z a z constαΨ = = = . The small
deviations of the phase from the synchronous phase
sϕ = Ψ − Ψ satisfy an equation of the linear pendulum
with attenuation:
0
s
a F
a ψ
ϕ ϕ ϕ
ϕ
∂+ ⋅ + ⋅ = ∂
&
&& & , (7)
where ( )
( ) ( )
x aF
a z a z
Ψ = − + Ψ
&& &&
.
In particular it follows from equation (7) that if the
transversal dimensions of the dielectric waveguide will
increase ( 0a >& ) the phases of all rays will aim at a phase
of a synchronous ray. In this case the self-focusing will
take place. Hence the condition of the self-focusing in the
considered model will be simple condition of growth of a
typical transversal dimension of a canal in the medium.
Fig. 2 and 3 show the generalized coordinates and
generalized pulse vs the longitudinal coordinate z for
the rays used before in Fig. 1. The difference is to change
the value ( )a z in the last case take place in according to
the following law: 0a a z= + . The characteristics of the
refraction coefficient have been chosen the following:
0 00.8; 3; 1a nµ= = = . It is seen from Figs. 2 and 3, that
in growing ( )a z dynamics of the rays are organized.
Fig.2
Fig. 3
A value of the generalized impulse decreases, e.g. the
angle of incidence of the rays to the axis z decreases.
The absolute values of deviations of the rays are grown.
INFLUENCE OF FLUCTUATION UPON
DYNAMICS OF RAYS
There are fluctuations in the real mediums always. In
order to take into account the fluctuations the refraction
coefficient can be written as:
( )2 2
0 1 , ( )n n n x z q z= + + 1q << . (8)
It is to be noted that the fluctuations are by random
functions, which have zero mean values, and these
functions are delta-correlated:
( )( ) ( )q z q z D z zδ′ ′⋅ = ⋅ − .
By assuming that the fluctuations are not great the
generalized impulse and Hamiltonian may be presented as
a sum of perturbed and unperturbed parts:
0p p p= + % 0 1( , , ) ( , , ) ( , , )H x p z H x p z H x p z≈ + ,
where 2 2
0 0 1H n n p= − + − 1
02
qH
H
=
Then for perturbed generalized impulse the following
equation can be written as
0 01
2 2
0 0
1 1
2 2
H dpHdp q q
dz x x dzH H
∂∂
= − = − =
∂ ∂
%
. (9)
From equation (9) it is seen that a mean value of p% equals
zero, e.g. 0p =% .
147
0 25 50 75 100
5
2.5
0
2.5
5
Fig.2
2.577
2.577−
U 1〈 〉
1000 U 0〈 〉
0 25 50 75 100
4
2
0
2
4
Fig.3
2.201
2.201−
U 2〈 〉
1000 U 0〈 〉
0 25 50 75 100
2
1
0
1
2
Fig.1
1.002
1.002−
U1〈 〉
1000 U0〈 〉
It is necessary to note that the unperturbed value of the
impulse has regular periodical variations along of axis z
with a typical period equal β . In addition the value of
unperturbed impulse attenuates slowly according to the
law ( )exp zδ− ⋅ . By using these considerations from
equation (9) can get the following expression for mean
square of p% :
( ) ( )22
02
4
0
1 exp( 2
24
p z
p D
H
β δ
δ
⋅ ∆ − −
= ⋅ ⋅% . (10)
If self-focusing of the rays is absent then the value
0δ = . In this case an expression (10) is transformed in
the famous law of a diffusion:
( ) 22
02
4
04
p
p D z
H
β ⋅ ∆
= ⋅% . (11)
As appears from expression (11) the scattering of
generalized impulse grows with increasing the distance.
Eventually, the rays will abandon the canal.
As may be seen from expression (10), the presence of
self-focusing can essentially confine an influence of the
fluctuation in propagating of the waves.
STABILIZATION OF STOCHASTIC
INSTABILITY
If parameters of the medium are constant along of axis
z , hamiltonian will depend on canonical variables (
0 ( , )H H x p= ). In this case the new canonical variables
which include an action and angle ,I Θ can be entered.
With these new canonical variables initial hamiltonian
will be a function only of one action. Let us assume that
there have small periodical perturbations. In this case
hamiltonian can be written as
0 ( ) ( , , )H H I V I zε= + Θ , (12)
where
,
, 0
1 ( )exp . .
2
z
m s
m s
V V I im is dz k c
= Θ + ℵ +
∑ ∫ .
The difference of given perturbation from well-known
perturbation is to take place a dependence of the
parameter ℵ from coordinate z . In spite of hamiltonian
(21) allows getting the following equations set that
describe dynamics in the vicinity of one cyclotron
resonance:
, sinm s sI mVε= ⋅ ⋅ Φ& ,
( ),( ) cos /s m sI V Iω εΘ = + Φ ∂ ∂& , ( )m I sωΦ = + ℵ& . (13)
It is known, that when the non-linear resonances overlap
the stochastic instability develops. If the parameter ℵ
does not depend on z then by using (13) a condition of
the stochastic instability can be found easily (see, for
example, [2]). In our case dynamics of a small deviations
from stationary points may be found easily by using:
cos( ) 0n
αϕ ϕ β ϕ
λ
+ − ⋅ Φ ⋅ =&& & . (14)
Here is nϕ = Φ − Φ , nΦ is stationary phase. In getting
(14) we assumed that the parameter ℵ depend on z in
according to the following law:
( )2 / zπ λ αℵ = + , where /s mλ π ω= − .
As will be seen from (14) we have obtained an damping
oscillator. In common case the presence of the attenuation
allows increasing the threshold of appearance of the
stochastic instability.
REFERENCES
1. V.A. Buts, V.A. Chatskaya. Dynamic focusing of
rays in wave-guiding mediums // JTF. 1995. v. 65,
N4, p. 195-198.
2. A.V. Buts, G.I. Churyumov. Regular and chaotic
dynamic of Duffing oscillator // Electromagnetic
waves & electron systems. 2003, v.8.
3. А. Lihtenberg, М. Liberman. Regular and stochastic
dynamic. М.: “Мir”. 1984, p.528.
САМОФОКУСИРОВКА ИЗЛУЧЕНИЯ В НЕОДНОРОДНЫХ СРЕДАХ
А.В. Буц, В.А. Буц, Г.И. Чурюмов
Показана возможность автофазировки лучей, распространяющихся в неоднородных средах. Эта
автофазировка аналогична автофазировке Векслера-Мак-Миллана в теории ускорителей. Автофазировка лучей
существенно ослабляет влияние случайных неоднородностей и увеличивает порог развития стохастической
неустойчивости.
САМОФОКУСУВАННЯ ВИПРОМІНЮВАННЯ В НЕОДНОРІДНИХ СЕРЕДОВИЩАХ
А.В. Буц, В.А. Буц, Г.І. Чурюмов
Показана можливість автофазування променів, що поширюються в неоднорідних середовищах. Це
автофазування аналогічне автофазуванню Векслера-Мак-Міллана в теорії прискорювачів. Автофазування
променів істотно послабляє вплив випадкових неоднорідностей і збільшує поріг розвитку стохастичної
нестійкості.
148
|
| id | nasplib_isofts_kiev_ua-123456789-79805 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:52:06Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, A.V. Buts, V.A. Churyumov, G.I. 2015-04-04T20:07:50Z 2015-04-04T20:07:50Z 2005 Self-focusing of radiation in non-uniform mediums / A.V. Buts, V.A. Buts, G.I. Churyumov // Вопросы атомной науки и техники. — 2005. — № 2. — С. 146-148. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 42.15.-i https://nasplib.isofts.kiev.ua/handle/123456789/79805 The possibility of self-focusing in propagating the rays in the non-uniform mediums is shown. This self-focusing is similar to a Veksler-Mac-Millan’s autophasing in the theory of accelerators. The self-focusing of the rays are able to weaken essentially an effect of random non-uniformities and to increase the threshold of development of stochastic instability. Показана можливість автофазування променів, що поширюються в неоднорідних середовищах. Це автофазування аналогічне автофазуванню Векслера-Мак-Міллана в теорії прискорювачів. Автофазування променів істотно послабляє вплив випадкових неоднорідностей і збільшує поріг розвитку стохастичної нестійкості. Показана возможность автофазировки лучей, распространяющихся в неоднородных средах. Эта автофазировка аналогична автофазировке Векслера-Мак-Миллана в теории ускорителей. Автофазировка лучей существенно ослабляет влияние случайных неоднородностей и увеличивает порог развития стохастической неустойчивости. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Self-focusing of radiation in non-uniform mediums Самофокусування випромінювання в неоднорідних середовищах Самофокусировка излучения в неоднородных средах Article published earlier |
| spellingShingle | Self-focusing of radiation in non-uniform mediums Buts, A.V. Buts, V.A. Churyumov, G.I. Plasma electronics |
| title | Self-focusing of radiation in non-uniform mediums |
| title_alt | Самофокусування випромінювання в неоднорідних середовищах Самофокусировка излучения в неоднородных средах |
| title_full | Self-focusing of radiation in non-uniform mediums |
| title_fullStr | Self-focusing of radiation in non-uniform mediums |
| title_full_unstemmed | Self-focusing of radiation in non-uniform mediums |
| title_short | Self-focusing of radiation in non-uniform mediums |
| title_sort | self-focusing of radiation in non-uniform mediums |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79805 |
| work_keys_str_mv | AT butsav selffocusingofradiationinnonuniformmediums AT butsva selffocusingofradiationinnonuniformmediums AT churyumovgi selffocusingofradiationinnonuniformmediums AT butsav samofokusuvannâvipromínûvannâvneodnorídnihseredoviŝah AT butsva samofokusuvannâvipromínûvannâvneodnorídnihseredoviŝah AT churyumovgi samofokusuvannâvipromínûvannâvneodnorídnihseredoviŝah AT butsav samofokusirovkaizlučeniâvneodnorodnyhsredah AT butsva samofokusirovkaizlučeniâvneodnorodnyhsredah AT churyumovgi samofokusirovkaizlučeniâvneodnorodnyhsredah |