Evolution of oscillator spectrum in periodic potential
Spectral power of the oscillator radiation, which moves in periodically inhomogeneous potential, is investigated analytically and numerically. Spectrum of the nonrelativistic oscillator can have a maximum at high numbers of harmonics of basic frequency. Amplitudes of potential at which the motion of...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Evolution of oscillator spectrum in periodic potential / V.A. Buts, A.M. Yegorov, V.I. Marekha, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2007. — № 1. — С. 107-109. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-1105342017-01-05T03:04:35Z Evolution of oscillator spectrum in periodic potential Buts, V.A. Yegorov, A.M. Marekha, V.I. Tolstoluzhsky, A.P. Plasma electronics Spectral power of the oscillator radiation, which moves in periodically inhomogeneous potential, is investigated analytically and numerically. Spectrum of the nonrelativistic oscillator can have a maximum at high numbers of harmonics of basic frequency. Amplitudes of potential at which the motion of oscillator become irregular are found. In relativistic case a small value of potential practically not influence on character of spectrum. The dependence of high-frequency range of spectrum from value of potential inhomogeneity period is investigated. With decreasing of inhomogeneity period the spectrum maximum is shifting into short-wave range. In linear approximation the dispersion equation for oscillation excitation by ensemble of oscillators at frequency, which corresponds to the maximum of radiation spectrum of single oscillator is found. Аналітично й чисельно досліджена спектральна потужність випромінювання осцилятора, що рухається в періодично-неоднорідному потенціалі. Показано, що спектр нерелятивістського осцилятора може мати максимум на високих номерах гармонік основної частоти. Знайдено амплітуди потенціалу, при яких рух осцилятора стає нерегулярним. В релятивістському випадку мала величина потенціалу практично не впливає на характер спектра. Досліджено залежність високочастотної області спектра від величини періоду неоднорідності потенціалу. Зі зменшенням періоду неоднорідності максимум спектра пересувається в короткохвильову область. У лінійному наближенні отримане дисперсійне рівняння збудження коливань ансамблем осциляторів на частоті, що відповідає максимуму спектра випромінювання індивідуального осцилятора. Аналитически и численно исследована спектральная мощность излучения осциллятора, который движется в периодически-неоднородном потенциале. Показано, что спектр нерелятивистского осциллятора может иметь максимум на высоких номерах гармоник основной частоты. Найдены амплитуды потенциала, при которых движение осциллятора становится нерегулярным. Показано, что в релятивистском случае малая величина потенциала практически не влияет на характер спектра. Показано, что с уменьшением периода неоднородности максимум спектра передвигается в коротковолновую область. В линейном приближении получено дисперсионное уравнение возбуждения колебаний ансамблем осцилляторов на частоте, соответствующей максимуму спектра излучения индивидуального осциллятора. 2007 Article Evolution of oscillator spectrum in periodic potential / V.A. Buts, A.M. Yegorov, V.I. Marekha, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2007. — № 1. — С. 107-109. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 05.45.-a, 41.60.-m, 02.60.Cb, 29.27.Bd http://dspace.nbuv.gov.ua/handle/123456789/110534 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics |
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Plasma electronics Plasma electronics Buts, V.A. Yegorov, A.M. Marekha, V.I. Tolstoluzhsky, A.P. Evolution of oscillator spectrum in periodic potential Вопросы атомной науки и техники |
| description |
Spectral power of the oscillator radiation, which moves in periodically inhomogeneous potential, is investigated analytically and numerically. Spectrum of the nonrelativistic oscillator can have a maximum at high numbers of harmonics of basic frequency. Amplitudes of potential at which the motion of oscillator become irregular are found. In relativistic case a small value of potential practically not influence on character of spectrum. The dependence of high-frequency range of spectrum from value of potential inhomogeneity period is investigated. With decreasing of inhomogeneity period the spectrum maximum is shifting into short-wave range. In linear approximation the dispersion equation for oscillation excitation by ensemble of oscillators at frequency, which corresponds to the maximum of radiation spectrum of single oscillator is found. |
| format |
Article |
| author |
Buts, V.A. Yegorov, A.M. Marekha, V.I. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Yegorov, A.M. Marekha, V.I. Tolstoluzhsky, A.P. |
| author_sort |
Buts, V.A. |
| title |
Evolution of oscillator spectrum in periodic potential |
| title_short |
Evolution of oscillator spectrum in periodic potential |
| title_full |
Evolution of oscillator spectrum in periodic potential |
| title_fullStr |
Evolution of oscillator spectrum in periodic potential |
| title_full_unstemmed |
Evolution of oscillator spectrum in periodic potential |
| title_sort |
evolution of oscillator spectrum in periodic potential |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Plasma electronics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110534 |
| citation_txt |
Evolution of oscillator spectrum in periodic potential / V.A. Buts, A.M. Yegorov, V.I. Marekha, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2007. — № 1. — С. 107-109. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butsva evolutionofoscillatorspectruminperiodicpotential AT yegorovam evolutionofoscillatorspectruminperiodicpotential AT marekhavi evolutionofoscillatorspectruminperiodicpotential AT tolstoluzhskyap evolutionofoscillatorspectruminperiodicpotential |
| first_indexed |
2025-07-08T00:42:56Z |
| last_indexed |
2025-07-08T00:42:56Z |
| _version_ |
1837037390225473536 |
| fulltext |
PLASMA ELECTRONICS
Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 107-109 107
EVOLUTION OF OSCILLATOR SPECTRUM IN PERIODIC POTENTIAL
V.A. Buts, A.M. Yegorov, V.I. Marekha, A.P. Tolstoluzhsky
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
e-mail: tolstoluzhsky@kipt.kharkov.ua
Spectral power of the oscillator radiation, which moves in periodically inhomogeneous potential, is investigated
analytically and numerically. Spectrum of the nonrelativistic oscillator can have a maximum at high numbers of
harmonics of basic frequency. Amplitudes of potential at which the motion of oscillator become irregular are found. In
relativistic case a small value of potential practically not influence on character of spectrum. The dependence of high-
frequency range of spectrum from value of potential inhomogeneity period is investigated. With decreasing of
inhomogeneity period the spectrum maximum is shifting into short-wave range. In linear approximation the dispersion
equation for oscillation excitation by ensemble of oscillators at frequency, which corresponds to the maximum of
radiation spectrum of single oscillator is found.
PACS: 05.45.-a, 41.60.-m, 02.60.Cb, 29.27.Bd
1. INTRODUCTION
In the most of nonlinear Hamilton systems, which
describe the dynamics of particles, it is possible to sort
regions of phase space where trajectories have regular
character and where they are stochastic. If we throw away
stochastic trajectories (i.e. trajectories which lies near
separatrix) then all the rest of trajectories will be periodic. In
our case regular trajectories will be periodic. Particles, which
will have a chaotic dynamics, will radiate random fields.
We restrict ourselves with particles, which have a
regular dynamics because only those ones can radiate
intensive coherent radiation.
We'll specify and realize such conditions, fulfillment
of which leads to minimum of particles with stochastic
dynamics. Therefore in further we'll orient, first of all, by
particles with regular dynamics. The influences of
stochastic particles at this stage of analysis we'll be
neglect, although we will make estimate their number and
minimize it.
2. RADIATION OF PARTICLE WHICH
MOVES IN PERIODIC POTENTIAL
We’ll describe the mechanism of high numbers
harmonics generation with nonrelativistic oscillators. Let
a charged particle moves in time-periodic electric field
( )( ) sinext extE t E tω= ⋅ ⋅ and in field of periodic
potential ( )0( ) cosU z U g zκ= + ⋅ ⋅ . For simplicity we’ll
consider that motion occur only along z-axis. In general
the equation of electron motion will be write:
U
dP eE eE
dt
= −
r
r r
, 2, / 1dr V V P P
dt
= = +
r
r r r r
, (1)
where UE U= −∇
rr
, e - electron charge.
As far as we, first of all, are interested in particle motion
along z axis, so we’ll get dimensionless form of equation
(1) for given component
( ) ( )2 1
0
2
sin sin ,
,
1
d p
d
d p
d p
κζ ε τ
τ
ζ β
τ
− + Ω ⋅ = ⋅ Ω ⋅
= =
+
(2)
where /zp p mc= , zζ κ= , c tτ κ= ⋅ , 2 2
0 /eg mcΩ = ,
2/ 2exteE d mcε π= , /zV cβ = , /ext dκ λΩ = , 2 / dκ π= .
When intensities of the fields are small enough it is
possible to consider nonrelativistic motion of particles:
( ) ( )2 1
0 sin sin κζ ζ ε τ
⋅⋅
−+ Ω ⋅ = ⋅ Ω ⋅ . (3)
Moreover, let us consider case E gκ>> . While passing
into moving coordinate frame ( )2 1sinκ κζ ξ ε τ−= − ⋅Ω ⋅ Ω ⋅
equation (3) takes form
( ) ( )( )2 1
0 sinn
n
J n κξ µ ξ τ
∞⋅⋅
−
=−∞
= −Ω ⋅ ⋅ − ⋅Ω ⋅∑ , (4)
where ( )nJ µ - Bessel function, 2
κµ ε= ⋅Ω .
Equation (4) describe changing of “particle” phaseξ , at
which many of waves acts on. Amplitudes of those waves
( )2
0 nJ µΩ ⋅ are increasing with growing of harmonic’s
number and in region ~n µ have a local maximum.
Amplitudes of harmonics with number n µ> are
exponentially decrease [1].
( ) ( ) ( ) ( )1/ 3 1/ 3 42 / 2 / (1/ 2 )nJ n Ai z n z e ςµ π −
: : .
Thus, it is possible to expect, that radiation field will
contain harmonics with frequencies up to extnω .
Radiation intensity into space angle unit dο with
frequency extnω ω= is equal to [2]:
2 2
02n n
cdI H R dο
π
=
r
, (5)
where H i k Aω ω
=
r rr
, and Fourier component of vector
potential defined by
( )0
0 0
exp( ) 1 ( ) exp ( )
T
n ext
ikRA e v t in t kr t dt
cR T
ω = − ∫
rr r r , (6)
where 2 / extT π ω= , ( )r tr , ( )v tr - particle’s radius-vector
and velocity, k
r
- wave vector, 0R - distance to point of
observation. In the common case, it isn't seemed possible
to get the analytical dependencies of spectral density from
parameters of external fields.
3. RADIATION SPECTRUM OF PARTICLE
MOVING IN PERIODIC POTENTIAL
The investigation of spectral characteristic of fields
radiated by charged particle moving in external electrical
field and in the field of potential, was carried out by the
numerical solution of equations (3) and substitution of its
solutions into (5) and (6).
mailto:tolstoluzhsky@kipt.kharkov.ua
108
For a case of nonrelativistic motion the amplitude of
an external electrical field is equal to Eext=104 V/cm, for
relativistic - Eext=106 V/cm. Frequency of the external
electromagnetic field was fixed.
Investigation was carried out for two value of
potential period d=0.0025· ext and d=0.00125· ext with
ext=10cm. Value of potential amplitude varied
within ( ) 10 0.125 extg E κ −= − . Initial conditions for
particles were equal to ( ) 00ζ τ ζ= = ; ( )0 0ζ τ = =& . For
that the right-hand of (3) had been presented as:
( )1cos κε τ−⋅ Ω ⋅ .
Calculation accuracy was controlled with the help of
motion integrals
( ) ( )( ) ( ) ( )
2
2 1
0 0
0
cos cos cos
2 kI d
τβ
ζ ζ ε β τ τ τ−= − Ω ⋅ − + ⋅ Ω∫ ,
( ) ( ) ( ) ( )
2
2 1
0 0 0
sin ( ) cos
2r k
pI p d p d
τ τ
τ ζ τ τ ε τ τ τ−= + Ω + Ω∫ ∫ ,
their absolute values was less than 1210I −< .
In absence of the periodic potential influence
2
0 0.0Ω = the equation (3) has a simple analytical solution.
Motion of the particle is periodic with frequency extω ω= ,
and spectrum of its speed and spectrum of the radiation
field are linear.
Presence of space-periodic potential with amplitude
of 2
0 0.025εΩ = ( 0.0025 extd λ= ⋅ ) (Fig.1) qualitatively
changes the picture of charged particle motion and
radiation. At phase plane the trajectory is not strictly
periodic curve (Fig 1,a), because particle motion is
determined by acting, both external electrical field, and
periodic potential. In velocity spectrum (Fig. 1,b) the base
frequency dominates. Components on its harmonics occur
under acting of the periodic potential.
a b
c d
Fig.1. a) Phase space; b) Spectrum of velocity;
c) Influence of potential on particle trajectory;
d) Spectrum of radiated field
For the analysis of influence of the periodic potential
on particle motion it is convenient to introduce
variable ( ) ( ) sin( )p κβ τ β τ ε τ= − Ω . It is visible, that
under acting of the potential the particle performs high-
frequency oscillations (Fig. 1,c), which results to
appearance of harmonics in the spectrum of velocity. At
that in power spectrum of radiated field (Fig.1,d) also
appear the components on harmonics of external field
with relative maximum on harmonic number n=11, that in
order of magnitude is in very good accord with position
of relative maximum n ~ 2 12kµ ε −= ⋅Ω ≈ .
With growing of amplitude of periodic potential
2
0 0.075 εΩ = ⋅ ( 0.0025 extd λ= ⋅ ) (Fig.2) occurs the
growing of harmonics amplitudes in spectrum of velocity
and its enriching at intermediate frequencies. Amplitudes
of high-frequency oscillations grow up at the influence of
periodic potential. Amplitudes of all harmonics in
spectrum of radiated field are growing up too. Kind of the
spectrum practically hasn't changes.
a b
c d
Fig.2. a) Phase space; b) Spectrum of velocity;
c) Influence of potential on particle trajectory;
d) Spectrum of radiated field
For potential amplitude 2
0 0.125 εΩ = ⋅
( 0.0025 extd λ= ⋅ ) (Fig.3) occur qualitative changes of
phase plane – the particle motion isn’t localized in limited
region of space and represented series of oscillations near
locally stabled state. Spectrum of particle velocity and,
consequently, spectrum of radiated field lot enriching at
all of intermediate frequencies.
a b
c d
Fig.3. a) Phase space; b) Spectrum of velocity;
c) Influence of potential on particle trajectory;
d) Spectrum of radiated field
Further growing of potential amplitude leads to
appearance of non-regular motion of particle hence the
spectrum of radiated field also becomes non-regular.
For a potential with period 31.25 10 extd λ−= ⋅ ⋅ the
number of harmonics, both in the spectrum of velocity
and radiated field, are proportional to parameter µ .
a b
c d
Fig.4. a) Phase space; b) Spectrum of velocity;
c) Influence of potential on particle trajectory;
d) Spectrum of radiated field
109
At amplitudes of potential 2
0 0.07 εΩ = ⋅ (Fig.4) the
motion of particle is quasiregular. Spectrum of velocity
and spectrum of radiated field have line structure. The
local maximum of spectrum fall on harmonics with
number n =23, that is in very good accord with
value 2 25kµ ε −= ⋅Ω ≈ .
a b
c d
Fig.5. a) Phase space; b) Spectrum of velocity; c) Particle
velocity as time function; d) Spectrum of radiated field
For the case of relativistic motion the amplitude
Eext=106 V/cm ( 2
0 0.15εΩ = , d=0.0025· ext) typical is
motion of particle with almost constant velocity, close to
velocity of light (Fig. 5), practically during all half
period of the external electrical field. Spectrum of
velocity has line nature. Spectrum of radiation has a
maximum on the harmonics with frequency ωmax=γ3 / 2
that completely corresponds to analytical result [3].
4. DISPERSION EQUATION
For obtaining of dispersion equation it is necessary to
solve self-consistent systems, which implies of the
Maxwell equations for field and equations of charged
particles’ motion in exited fields. In linear approximation
we choose a field of such type:
0Re exp( ( ) exp( )E i kx z i tε κ ω= + −
r r .
Executing necessary transformations, we’ll obtain a
set of linear algebraic equations. The equality to zero of a
determinant of this algebraic system represents a
dispersion equation. We shall keep only resonant
members extnω ω≈ . In these conditions dispersion
equation takes on enough simple form:
22 2 22 2
0
2 2 2 2
( )11 1 0,
4 ( )
b b b n
ext ext
Jk c
n n
ω ω ω ω µ
ω ω ω ω ω ωω
⋅ − − − − = −
(9)
with increment ( )1/ 32 20
0
3Im ( )
4 n bJω
δ µ ω ω
ω
= .
CONCLUSIONS
The presence of space-periodic potential even small
amplitude leads to generation of high-number harmonics
of radiated field. At that, local maximum in spectrum of
radiation lays at high-number harmonics of external
electric field. Number maxn of harmonic, at which local
maximum of radiation lays, is in accord with analytical
results very well. Radiation frequency in its maximum
max max extnω ω= is sufficiently higher than external field
frequency ( max 1n >> ). Intensity of harmonic radiation at
local maximum is high enough.
While, that for the case of nonrelativistic oscillator
discussed in [3] the intensity of n -th harmonic radiation
is proportional 2nβ , so intensity of harmonic radiation
with numbers equal to appropriate numbers of harmonics
at local maximum (in our case max~ 11n n ≈ ) will be
negligible small.
REFERENCES
1. M. Abranovitz, A. Stegun. Handbook of
mathematical functions. Pergamon Press, 1968,
p.1054.
2. L.D.Landau, I.M.Lifshits. Field theory. Moscow,
1971, p.424.
3. . Sokolov, I. . Ternov. Relativistic electron.
Moscow, 1974, p. 392.
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