Peculiar properties of systems under secondary resonances with an external perturbation
This report is about the dynamics of the Hamiltonian systems with two degrees of freedom, in the presence of a low-frequency disturbance. It is proved that a low-frequency disturbance induces a stochastic instability. Besides, a low-frequency disturbance can stimulate the development of a parametric...
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nasplib_isofts_kiev_ua-123456789-800442025-02-09T17:31:39Z Peculiar properties of systems under secondary resonances with an external perturbation Особенности динамики систем при вторичных резонансах с внешним возмущением Buts, V.A. Kinetic theory This report is about the dynamics of the Hamiltonian systems with two degrees of freedom, in the presence of a low-frequency disturbance. It is proved that a low-frequency disturbance induces a stochastic instability. Besides, a low-frequency disturbance can stimulate the development of a parametric instability. This instability is a new mechanism of transformation of low-frequency oscillation energy to a energy of high-frequency oscillation. As examples of the parametric instability development in distributed systems, we have examined the possibility of the parametric amplification of the x-ray radiation emission in crystals and the possibility of the parametric amplification of the waves propagating in gyrotropic media. The author thanks A.S. Bakai for useful discussions. The work was supported by STCU (project №855). 2001 Article Peculiar properties of systems under secondary resonances with an external perturbation / V.A. Buts // Вопросы атомной науки и техники. — 2001. — № 6. — С. 329-333. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 05.45.+b; 41.20.Jb; 41.50.+h https://nasplib.isofts.kiev.ua/handle/123456789/80044 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Kinetic theory Kinetic theory Buts, V.A. Peculiar properties of systems under secondary resonances with an external perturbation Вопросы атомной науки и техники |
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This report is about the dynamics of the Hamiltonian systems with two degrees of freedom, in the presence of a low-frequency disturbance. It is proved that a low-frequency disturbance induces a stochastic instability. Besides, a low-frequency disturbance can stimulate the development of a parametric instability. This instability is a new mechanism of transformation of low-frequency oscillation energy to a energy of high-frequency oscillation. As examples of the parametric instability development in distributed systems, we have examined the possibility of the parametric amplification of the x-ray radiation emission in crystals and the possibility of the parametric amplification of the waves propagating in gyrotropic media. |
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Buts, V.A. |
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Buts, V.A. |
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Buts, V.A. |
| title |
Peculiar properties of systems under secondary resonances with an external perturbation |
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Peculiar properties of systems under secondary resonances with an external perturbation |
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Peculiar properties of systems under secondary resonances with an external perturbation |
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Peculiar properties of systems under secondary resonances with an external perturbation |
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Peculiar properties of systems under secondary resonances with an external perturbation |
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peculiar properties of systems under secondary resonances with an external perturbation |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Kinetic theory |
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https://nasplib.isofts.kiev.ua/handle/123456789/80044 |
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Peculiar properties of systems under secondary resonances with an external perturbation / V.A. Buts // Вопросы атомной науки и техники. — 2001. — № 6. — С. 329-333. — Бібліогр.: 2 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butsva peculiarpropertiesofsystemsundersecondaryresonanceswithanexternalperturbation AT butsva osobennostidinamikisistemprivtoričnyhrezonansahsvnešnimvozmuŝeniem |
| first_indexed |
2025-11-28T18:22:47Z |
| last_indexed |
2025-11-28T18:22:47Z |
| _version_ |
1850059444149288960 |
| fulltext |
PECULIAR PROPERTIES OF SYSTEMS UNDER SECONDARY
RESONANCES WITH AN EXTERNAL PERTURBATION
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: abuts@kipt.kharkov.ua
This report is about the dynamics of the Hamiltonian systems with two degrees of freedom, in the presence of a
low-frequency disturbance. It is proved that a low-frequency disturbance induces a stochastic instability. Besides, a
low-frequency disturbance can stimulate the development of a parametric instability. This instability is a new
mechanism of transformation of low-frequency oscillation energy to a energy of high-frequency oscillation. As
examples of the parametric instability development in distributed systems, we have examined the possibility of the
parametric amplification of the x-ray radiation emission in crystals and the possibility of the parametric
amplification of the waves propagating in gyrotropic media.
PACS: 05.45.+b; 41.20.Jb; 41.50.+h
INTRODUCTION
As it is known, the secondary resonance’s can
essentially influence dynamics of Hamiltonian systems
only if the disturbance amplitude could be large enough
(see, for example, [1]). Really, when the disturbance is
small ( ε < < 1 ), the scale of the effects conditioned by
the secondary resonance’s also small, being
proportional to ( )1 1/ / !ε (e.g., the secondary nonlinear
resonance width). Below it is to be demonstrated that if
there exists an external low-frequency disturbance, the
influence of the secondary resonance’s locked with it
could be much more essential.
1. STATEMENT OF THE PROBLEM AND
GENERAL EQUATIONS
Let us investigate dynamics of the following system
with two degree of freedom, characterized by the
Hamiltonian
H H J H J t= + ⋅0 1( ) ( , , )
ε θ (1)
We now regard the disturbance as a periodic one over
the angular variable. We also suppose that it can be
presented as a series:
( )∑ θ⋅=
mk
mk nitJhH
,
,1 exp),(
.
Here
n k m⋅ = +θ θ θ1 2 .
Let us consider that the resonance relations
s s1 1 2 2⋅ = ⋅ω ω are justified for the undisturbed systems
(here sk denotes integer numbers; ω ∂ ∂k kH J= 0 / ).
According to the resonance theory of disturbances, one
can apply the following course-of-value function in
order to make a transition to new canonical variables:
( )W I s s I I( , )
θ θ θ θ= − +1 1 2 2 1 2 2 . (2)
It is worth emphasizing that the course-of-value
function is explicitly independent of time. Making use
of this function, one gets the following expression for
the new Hamiltonian:
H I H I h I tk m
k m
( , ) ( ) ( , ),
,
Ψ = + ⋅ ∑0 ε
⋅ + ⋅ +
⋅
exp i k
s
k
s
s
m
1
1
2
1
2Ψ Ψ (3)
Here
I , Ψ denote new canonical variables. The new
angular variable Ψ 1 1 1 2 2= −s sθ θ is changing slowly,
while the angular variable Ψ 2 2= θ keeps on changing
quickly. Let us average the Hamiltonian (3) over the fast
angular variable. We also suppose that the coefficients
in the expansion of the disturbance are characterized by
the following symmetry: h hk m k m− −=, , i.e., the
disturbance is real. Thus, the averaged Hamiltonian
takes the form:
H H I h ms msm
= + ⋅ −∑0
1 2
( )
,
ε ( )⋅ ⋅2 1cos mΨ (4)
2. DYNAMICS OF NONDEGENERATE
SYSTEMS
Let us suppose that
I0 designates the values of
actions, under which the resonance condition is
precisely true. Limiting ourselves just to the dominant
terms in (4), we get the following simple expression for
the Hamiltonian:
( )H H
I I
I h s s I t=
⋅ ⋅ + ⋅ − ⋅
∂
∂
ε
2
0
1
2
0
2
1 2
0
1
2
2 1
∆ Ψ, ( , ) cos . (5)
Here it is taken into account that I 2 is an integral.
Hence ∆ I I I= −1 10 . The Hamiltonian (5) is the
standard one with the time dependent coefficients.
Dynamics of the system, characterized by the
Hamiltonian (5), is describable by the following
equation of the mathematical pendulum:
sinΨ Ψ1
2
1 0+ ⋅ =ω B . (6)
Here ω ε
∂
∂B h s s I t
H
I I
2
1 2
0
2
0
1
2
0
2= − ⋅ − ⋅
, ( , )
.
For simplicity, let us choose the temporal dependence of
the disturbance coefficients in the form
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 329-333. 73
( )h h I t h h s s= + ⋅ ≡ −( ) cos ; ,
0
1 2
1 µ Ω . In this
case (6) may be presented as:
( ) ( )[ ] sin sin sinΨ Ψ Ψ Ω Ψ Ω+ = + + −µ τ τ
2 1 1 (7)
with Ψ Ψ Ω Ω≡ = =1 1; ; /τ ω ωB Bt .
Dynamics of the system, describable by Eq. (7), is
characterized by three nonlinear resonances. The half-
width of the primary one is equal to 2 ( ∆ Ψ = 2 ),
whereas the half-widths of the two other make 2 µ .
The distance between these nonlinear resonances is Ω 1 .
Consequently, if the condition ( )Ω 1 2 1< + µ is true,
the system dynamics becomes chaotic. In particular, this
condition indicates the following: if the frequency of the
external low-frequency disturbance is on the order of
the bound system beat frequency or smaller than the
bounce frequency, the disturbed system dynamics is
always chaotic, notwithstanding the value of the
disturbance amplitude. Surely, one should keep in mind
that the chaotic motion characteristics (e.g., the
diffusion coefficient) essentially depend on the
disturbance amplitude.
It is worth mentioning another important specificity
of dynamics of the disturbed system small oscillations.
As one can see, equation (6) describes dynamics of the
linear pendulum when the oscillation amplitudes are
small. If the disturbance frequency is so that the
condition for the parametric resonance is satisfied
Ω = 2ω B , the external low-frequency disturbance
induces the parametric instability. In this case, the
oscillation amplitudes exponentially increase with the
increment ~ µ .
3. DYNAMICS OF DEGENERATED
SYSTEMS
Above we have investigated the nondegenerated
case, i.e. the second derivative with respect to the
undisturbed Hamiltonian is regarded as nonzero. This
condition corresponds to the undisturbed frequency
dependence on the magnitude of the action. We now
examine dynamics of the degenerate system. For such
system ( ) ( )∂ ∂2
0
2
0
0H I
I
/ = . As it is known, changes
in both the magnitude of the action and the angular
variable are equally small under the influence of a small
disturbance. Let us expand the expression for the
Hamiltonian (4) in the close vicinity to the points that
correspond to the stationary ones in the absence of the
explicit temporal dependence. The expansion is made
not only in the action variable but in the angular
variable as well. The Hamiltonian derived corresponds
to the Hamiltonian of the linear pendulum. It is handy to
write the equation of motion of this pendulum as:
( ) ; ( )ϕ ϕ= ⋅ = ⋅A t j j B t . (8)
Here ϕ ≡ −Ψ Ψ 0 is a small deviation of the angular
variable from the stationary state; j I I≡ −1 10 is a small
deviation of the action variable from the value that
corresponds to the precise resonance;
A h I h s s I= + ⋅ −
∂ ∂ ∂ ∂2
00 1
2 2
1 2
1
22 , / ;
B h s s= −2
1 2, .
In particular, Eq. (8) indicates the following: if the ratio
of the coefficients A and B are time-independent, the
system (8) is characterized by an additional integral:
ϕ 2 2− ⋅ =c j const . In this case equation (8) is
completely integrable. There do not develop any
parametric instability. If the ratio of A to B is time-
dependent, than the system (8) is equivalent to the
ordinary differential equation of the second order with
the coefficients that vary in time:
( )[ ] /z AB A A A A z+ − + + ⋅ =2 2 0
2
, (9)
here z A= Ψ / .
As equation (9) indicates, there can exist the
parametric instability under certain values of the
disturbance parameters.
4. COUPLED LINEAR OSCILLATORS
Linear systems make an important example of
degenerate systems. For instance, let us examine a
system that consists of two coupled linear oscillators.
The system Hamiltonian has the form:
[ ]
[ ]
[ ]
H a p b q
t p t q
c t p p d t q q
k k k k
k
k k k k
k
= ⋅ ⋅ + ⋅∑ +
+ ⋅ ⋅ + ⋅∑ +
+ ⋅ ⋅ ⋅ + ⋅ ⋅
=
=
1
2
2
2 2
1
2
2 2
1
2
1 2 1 2
ε α β
ε
( ) ( )
( ) ( ) .
(10)
In (10) the first sum describes the system of two
independent oscillators. The second sum describes their
small disturbance. The third addendum describes the
small coupling between the oscillators. It is easy to
show that in the system of the ordinary differential
equations of the (8)-type, which corresponds to the
Hamiltonian (10), the ratio of the coefficients A to B is
independent of time. Thus, the system of coupled linear
oscillators, describable by the Hamiltonian (10),
acquires intricate temporal dynamics under the
influence of the time-dependent disturbance. However,
the conditions for the parametric instability are
unrealizable under the action of an external low-
frequency disturbance. Whether always it so? Below
we shall see, that for non-stationary systems with two
degree of freedoms with imperfect connections it not so.
Besides in the distributed systems also it is possible to
realize the conditions for development of parametrical
instability. Let's illustrate the formulated statement on
the elementary example. This example can be a system
of two connected identical oscillators, the connection
between which depends on time and is unmutual. Set of
equations which describes dynamics of such oscillators
is possible to write as:
( )x x t x1 1 1 2+ = ⋅µ
74
( )x x t x2 2 2 1+ = ⋅µ . (11)
If (µ1/µ2) ≠ 1 the system (11) cannot have Hamiltonian
(10) (Hamiltonian for a system (11) in general case can
be written in an extended phase space). The reason of
absence of a Hamiltonian consists that the virtual work
of connections is not equal to zero, i.e. the connection in
this case are imperfect. By a physical example of a
system, which can be circumscribed by a set of
equations (11), is dynamics of fields of two connected
identical resonators. The connections between these
resonators are different. Such connections can be
realized, for example, with the help of channels, which
have gyrotropic insertions. Let coupling coefficients
look like µ α β γi i i t= + ⋅ ⋅cos( ) , where α βi i, -
constants. Than, if condition of parametrical instability
γ = 2Ω ( Ω = ⋅α α1 2 2/ ) is fulfilled, system (11) has
solution a a t1 2~ ~ exp( )Φ , where ( )Φ = ⋅β α1 12 .
5. SYSTEMS WITH AN INFINITE NUMBER
OF THE DEGREES OF FREEDOM
Thus, in the linear system with two degrees of
freedom does not exist the mechanism for the
parametric pumping of the low-frequency oscillation
energy to the energy of high-frequency motions. For
obtaining the energy pumping in this system, an element
of nonlinearity is required. At the same time, for the
most interesting cases (e.g., the stimulation of the X-ray
radiation emission), the magnitudes of nonlinearity are
vanishingly small. Seemingly, the systems with
nonideal constraints possess the necessary properties.
However, these systems are rather rare. The system with
a large number of the degrees of freedom must possess
the necessary properties. Therefore, it is of practical
interest to determine the conditions for this parametric
amplification in the system with a large number of the
degrees of freedom. Below we will demonstrate that
there does really take place the amplification of this
kind in distributed systems.
To prove the existence of this possibility, let us
examine a medium where occurs interaction between
two wave processes. For instance, one can talk about the
three-wave interaction in the approximation of a
constant value of the amplitude of one of the waves (the
pumping wave amplitude). Besides, we also suppose
that the medium parameters are slowly changing in
space and in time. Specific examples will be given
below. The equations that describe evolution of the
wave amplitudes may be presented as:
α
∂
∂
µ
∂
∂ τ0
0
0
0
0 0
0
1
1
2 2
A
z
A
i
Q A
q
A+ = +
α
∂
∂
µ
∂
∂ τ1
1
1
1
1 1
0
0
1
2 2
A
z
A
i
Q A
q
A+ = +
. (12)
Here A i denote the slowly-varying amplitudes of the
waves that interact with one another; q i designate the
constants that determine the wave coupling;
Q Q zi i= ( , )τ prescribed functions, slowly varying in
time and space. Description of a rather large number of
physical processes is reducible to the system (12).
First, let us determine the most general conditions
under which the system (12) can possess parametrically-
increasing solutions. For this purpose, let us rewrite the
system (12) in partial derivatives in the form of the
system of characteristic equations:
dA
dz i
Q A
q
A
d
dz
0
0
0 0
0
1
0
0
1
2 2
= ⋅ +
=
α
τ µ
α
; ;
dA
dz i
Q A
q
A d
dz
1
1
1 1
1
0
1
1
1
2 2
= ⋅ +
=
α
τ µ
α
; . (13)
In (13) the first pair of equations is equivalent to the
first equation in the system (12), whereas the second
pair is equivalent to the second equation in equation
(12). As it is easy to see, if the functions Q0 and Q1
coincide and if the characteristics are equal
( )µ α µ α0 0 1 1= , the system (13) is completely
integrable. As well as in the system with two degrees of
freedom, in Eq.(13) does not exist any parametric
instability. In this case, the presence of low-frequency
disturbances results just in a certain complication in
dynamics of the amplitudes A i . It is also evident that
even if one of this conditions (either the equality of the
characteristics or coincidence of the functions Q0 and
Q1 ) is not fulfilled, generally speaking, the system
equation (12) (or 13)) can possess the solution that
describes the parametric amplification.
Let us now proceed to a more detail determination of
the conditions for the parametric amplification. Let us
suppose that the coefficients Q i are periodic functions
of the time τ and coordinate z . We choose this
dependence in the form Q zi i= ⋅ −ε τcos( )Κ . Hence,
one can look for the solution to the system (12) in the
following form:
( )A a ini i n= ⋅∑ , exp τ . (14)
In order to find the Fourier amplitudes a i n, , we
substitute the solution (14) into (12). Thus, one gets the
following infinite coupled system of ordinary
differential equations of the first order:
( ) ( )[ ]
α
∂
∂
µ
ε
0
0
0 0
0
1
0
0 1 0 1
4
4
a
z
in a
q
i
a
i
a i z a i z
n
n n
n n
,
, ,
, ,exp exp
+ ⋅ = +
+ ⋅ + ⋅ −+ −Κ Κ
( ) ( )[ ]
α
∂
∂
µ
ε
1
1
1 1
1
0
1
1 1 1 1
4
4
a
z
in a
q
i
a
i
a i z a i z
n
n n
n n
,
, ,
, ,exp exp
+ ⋅ = +
+ ⋅ + ⋅ −+ −Κ Κ .
(15)
This system useful to rewrite as
′ ′ + =v v Fn n n nλ ε2
0 . (16)
Here
( )v a iR zn n n= ⋅0 2, exp / ,
75
B
q q
nn =
⋅
−
1 0
0 1
2 0 1
0 116 α α
µ µ
α α ,
R nn = +
µ
α
µ
α
0
0
1
1
,
λ n n nB R2 2 4= + / ,
( ) ( )[ ]D = + ⋅ ⋅1 1 0 0 1ε α ε α ,
( )
( )
( )
( )
F
i
v i z iR z
C i DR v
i z iR z
C i DR
D
v i z iR z
v i z iR z
n
n n
n n n
n
n n
n n
n n
=
⋅ − ⋅
⋅ −
+ ⋅
⋅ − − ⋅
⋅ −
+
+
′ ⋅ − +
′ ⋅ − −
+ +
+
+ −
−
−
−
+ +
− −
1
4
2
2
2
2
2
2
0
1 1
1 1
1
1
1 1
1 1
α
exp /
exp /
exp /
exp /
Κ
Κ
Κ
Κ
,
( )
C i n
n
n
± = ± + +
±
Κ
µ
α
ε α
ε α α
µ1
1
1 0
0 1 0
0
1
.
The system (16) describes an infinite system of linear
coupled oscillators. Their coupling is conditioned by the
presence of a small ( ε i < < 1 ) slow and periodic change
in the parameters of the medium where takes place the
wave interaction. We now look for the solution to the
system (16) in the following form:
( )
( )
v w z i z
z i z V z
n n n
n n
= ⋅ +
+ ⋅ − +
( ) exp
( ) exp ( )
λ
ς λ ε 0
. (17)
Let us substitute (17) into system equation (16),
imposing the condition of the absence of driving
resonant forces in the equation for the function V z( ) . If
the resonance conditions
( )Κ = − − −+ +λ λn n n nR R1 1
1
2
(18)
are satisfied, than w n are joint with w n+ 1 and ς n are
joint with ς n+ 1 . The equations for each of the
amplitudes w n or ς n are independent of the
amplitudes characterized by other numbers. For
instance, to determine the function w n , one can derive
the following ordinary differential equation of the
second order with constant coefficients:
′′ + ⋅ =w G wn n 0 . (19)
Here
( )
( )
( )
G
C i D R
C i D R
n n
n n n
n n n
≡
− ⋅
⋅ ⋅
⋅
⋅ − ⋅ −
⋅
⋅ − ⋅ −
+
+
+ +
+
−
1
64
2
2
2
2
0
2
0
2
1
1 1
1
ε
α λ λ
λ
λ
.
As equation (19) indicates, when the system parameters
are so that G<0, than the functions w n will
exponentially increase. The field Fourier amplitudes
a i n, will be increased too.
Let us suppose that another resonance condition
( )Κ = + − −+ +λ λn n n nR R1 1
1
2
(20)
is fulfilled. Then the condition of the absence of the
resonant driving terms on the right of the equation for
V(z) lead to the joint between the functions w n and
ς n+ 1 . As well as in the previous resonance case, for
determining the function w n , one can derive the
ordinary differential equation of the second order,
analogous with (19):
′′ + ⋅ =w G wn n1 0 . (21)
Here
( )
( )
( )
G
C i D R
C i D R
n n
n n n
n n n
1
0
2
0
2
1
1 1
1
1
64
2
2
2
2
≡
+ ⋅
⋅ ⋅
⋅
⋅ − ⋅ +
⋅
⋅ − ⋅ −
+
+
+ +
+
−
ε
α λ λ
λ
λ
.
As well as in the previous resonance case if G1 0<
there takes place amplification of the radiation emitted.
6. PARAMETRIC AMPLIFICATION
OF THE X-RAYS IN CRYSTAL
Our goal is to demonstrate that the process of scattering
of X-rays by the perfect crystals is reducible to the case
described above when the crystal susceptibility is
modulated by a low-frequency disturbance.
KRYSTAL EXTINCTION
LENTH
MINUS FIRST
ORDER OF
DIFFRACTION
FELLING WAVE
Fig. 1.
We now suppose that the crystal is located in the
lower half-space z>0 (see Fig. 1) and its susceptibility is
describable by the formula:
χ χ κ ε= + ⋅ + ⋅ ⋅ −0 q r z tcos( ) cos( )
Κ Ω (22)
Here
κ denotes the vector of the inverse grate of the
crystal; q ~ χ ε0 > > - designates the degree of the
spatially-temporal periodic inhomogeneity, induced by
an external source; Κ < < κ .
For describing the process of scattering of X-rays by
this crystal, let us limit ourselves to the framework of
the two-wave dynamic theory of diffraction. In this
case, the field in the crystal can be presented as
76
E A r t ik r i tj j
j
= ⋅ − +
=
∑ ( , ) exp( )ω
0
1
(23)
Here
k k1 0= + κ ; k k c1
2
0
2 2 2= = ω .
Now let us consider the incident radiation to be
completely unlimited in the direction transverse to z -
axis, i.e. we neglect the effects, conditioned by bounded
crystal and incident radiation in the transverse direction.
Besides, we suppose that Ω < < ω . Thus, changes in
amplitudes of the interacting waves depend only on time
and z-coordinate. Maxwell's equations readily yield the
averaged equations that describe dynamics of changes
in these amplitudes. In the dimensionless variables
τ ω= = =Ω t z k z z c; 0 , these equations can be written
in the following form:
α ∂
∂
µ ∂
∂ τ
α ∂
∂
µ ∂
∂ τ
δ
0
0 0
0 1
1
1 1
1 0
1
2 2
1
2
2
2
A
z
A
i
Q A q A
A
z
A
i
Q A q A
+ = ⋅ +
+ = + +
( ),
[( ) ],
(24)
here
µ
ω
≡ Ω ~ ;q
Q z≡ + ⋅ −χ ε τ0 cos( )Κ ;
α θi
i z
i
k
k
≡ =
0
cos ;
k k ko
2
1
2
0
22− = ⋅δ .
If µ ε= = 0 , the system equation (24) takes the form of
the well-known one (e.g., see [2]), which possesses
pendular solutions. We also suppose that the Bragg
resonance conditions are precisely satisfied i.e., there is
now Bregg detuning ( δ = 0 ). Besides, the wave
numbers are chosen so that it corresponds to the wave
propagation not in vacuum (as in (23)) but in the
medium characterized by the susceptibility χ 0 , that is,
( )k k c1
2
0
2 2 2
01= = ⋅ −ω χ . In this case, the system
(24) completely coincides with the system (12), where
one must equate: µ µ µ0 1= = , Q Q Q0 1= = ;
q q q0 1= = .
7. WAVE PROPAGATION IN GYROTROPIC
MEDIA
As it is known, in generally case, E and H − waves
are coupled in boundless gyrotropic media. The total
electromagnetic wave field in these media possesses all
components of the electric and magnetic fields. We now
examine a gyrotropic half-space, on the boundary of
which, for instance, the E -wave field ( E x , H y , E z ) is
determinate. As the wave is propagating deep into the
gyrotropic medium, the E -wave field energy
completely transfers into the H -wave energy ( H x , E y
, H z ) (see Fig. 2). Looking like rotation of the plane of
the wave polarization, this energy transformation
represents the Faraday effect. Thus, the Faraday effect
can be presented as a result of dynamics of two coupled
waves. It is worth mentioning that the resonance of the
interaction between E and H -waves plays the part of
the primary resonance in this model. In addition, if the
components of the tensor of the dielectric or magnetic
susceptibility are periodic functions of the coordinate
and time, in this system secondary resonances can
effectively indicate themselves. Moreover, it is possible
to show, that dynamics of amplitudes Å and Í waves is
described by a set of equations (12) and one can use
external low-frequency wave for amplification high-
frequency waves.
E-wave
H-wave E H E
Fig. 2
ACKNOWLEDGMENTS
The author thanks A.S. Bakai for useful discussions.
The work was supported by STCU (project №855).
REFERENCES
1. AJ. Lichtenberg, I.A. Liebermanю Regular and
Stochastic Motion, N.Y.: “Springer-Verlag”, 1983,
528 p.
2. S.A. Àhmanov, Yu.A. Dyakov, A.S. Chirkin.
Introduction to Statistic Radiophysic and Optic.
Moskov: “Science”, 1981, 640 p.
77
V.A. Buts
INTRODUCTION
OF THE X-RAYS IN CRYSTAL
7. WAVE PROPAGATION IN GYROTROPIC MEDIA
REFERENCES
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