High harmonics oscillator radiation in a periodic structure
We discuss spectral and angular characteristics of radiation by a harmonic oscillator placed in a field of a crystal lattice. The main attention is paid to high harmonics which can not be described in a dipole approximation. Certain possibilities of X-rays generation using this effect are discusse...
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2006 |
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| Format: | Article |
| Language: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Cite this: | High harmonics oscillator radiation in a periodic structure / M.A. Gorbunov, A.N. Lebedev // Вопросы атомной науки и техники. — 2006. — № 2. — С. 140-141. — Бібліогр.: 1 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859901047376445440 |
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| author | Gorbunov, M.A. Lebedev, A.N. |
| author_facet | Gorbunov, M.A. Lebedev, A.N. |
| citation_txt | High harmonics oscillator radiation in a periodic structure / M.A. Gorbunov, A.N. Lebedev // Вопросы атомной науки и техники. — 2006. — № 2. — С. 140-141. — Бібліогр.: 1 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | We discuss spectral and angular characteristics of radiation by a harmonic oscillator placed in a field of a crystal lattice.
The main attention is paid to high harmonics which can not be described in a dipole approximation. Certain possibilities
of X-rays generation using this effect are discussed as well.
Рассмотрены спектрально-угловые характеристики излучения гармонического осциллятора, находящегося в поле кристаллической решётки. Основное внимание уделяется высокочастотным гармоникам, не удовлетворяющим условию дипольности. Обсуждается возможность использования этого эффекта для генерации рентгеновского излучения.
Розглянуто спектрально-кутові характеристики випромінювання гармонійного осцилятора, що перебуває в полі
кристалічної ґратки. Основна увага приділяється високочастотним гармонікам, що не задовольняють умові дипольності.
Обговорюється можливість використання цього ефекту для генерації рентгенівського випромінювання.
|
| first_indexed | 2025-12-07T15:57:52Z |
| format | Article |
| fulltext |
HIGH HARMONICS OSCILLATOR RADIATION IN A PERIODIC STRUC-
TURE
M.A. Gorbunov, A.N. Lebedev
P.N. Lebedev Physical Institute, Moscow, Russia
E-mail: lebedev@sci.lebedev.ru
We discuss spectral and angular characteristics of radiation by a harmonic oscillator placed in a field of a crystal lat-
tice. The main attention is paid to high harmonics which can not be described in a dipole approximation. Certain possi-
bilities of X-rays generation using this effect are discussed as well.
PACS: 41.60.Cr
Intense monochromatic radiation of high frequency can
find a lot of scientific and technical applications. In the ma-
jority, installations for generation of high quality short-
wave radiation are rather costly, e.g. free electron laser, un-
dulator devices and so on. The point of this paper is a dis-
cussion of the method in this field, suggested in [1].
The method implies electrons moving under influence
of a harmonics electric filed in a field of a periodic struc-
ture where high harmonics of the diving field can be gen-
erated, especially under the strong Doppler effect.
The electron motion can be driven by a laser or by a
HF oscillator. Inhomogeneity of the environment can be
natural as in the case of a crystal or produced with the
help of various grids and lattices.
The basic opportunity of high frequency generation
will be estimated here using an elementary model, follow-
ing that of the paper [1].
The electron motion will be considered in the field of
the wave periodical in time tEtE Ω= cos)( . In a peri-
odic structure created, for example, by a crystal lattice, the
motion can be one-dimensional, as it is supposed to be be-
low. In principle, the presence of a large magnetic field
could provide that. We suppose that the wave comes along
the normal of a crystalline plane. As it concerns the peri-
odic structure we shall suppose that its potential varies
along the particle motion as xUxU κρ cos)( 0 += .
The general layout is presented in Fig.1.
Fig.1. The general layout
We do not consider here the transverse electron mo-
tion although a transverse acceleration generally provides
larger radiation power than a longitudinal one.
Quantitative characteristics of radiation power can be
obtained, of course, with detail of the method realization
only. In particular, spatial distribution of radiators is of
importance, including influence of skin effect, the possi-
bility of induced effects and so on. For these reasons we
shall restrict ourselves by spectral-angular distribution of
single particle radiation only.
In its essence the described picture is a modification of
undulator radiation or of Smith-Purcell effect. The latter
means that a high-power electron beam has to propagate
along a short-period system. The main problem then is the
damage of the structure by the beam.
Similar effect can be obtained with electrons oscillat-
ing under action of an external wave. In this case the elec-
tron velocity is a function of time. Note that only those
parts of the trajectory where the velocity is almost equal to
that of light (relativistic case) can provide the spatial co-
herency from various parts of the lattice. This is essential
for getting a monochromatic radiation.
Fig.2. An oscillating velocity in case of a strong electric
field of a wave
The relative power of various harmonics can be direct-
ly obtained from the harmonics of the vector potential.
The latter can be presented as an integral along the parti-
cle trajectory
0
0
k rikR i teA e e dr
cR
ω
ω
ω
ж ц
−з чз чи ш= т
rr
uur r . (1)
It is convenient in the one-dimension case to present
the particle motion in a rather unusual way – i.e. as a de-
pendence of time upon the particle coordinate. Then, the
equation of motion of an electron in the periodic electric
field and in the field of the periodic structure
2
cos sin
1
x
d eEc t x
dt mc mcx
c
ρ κ κ= Ω +
ж ц− з ч
и ш
&
& ,
will take the form
( ) 3
2
2
2 2
cos sin
1
c t eE t x
mc mcc t
ρ κ κ
ўў
= − Ω −
ў −
.
___________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.140-141.140
Solving this equation with substitution into the integral
(1) yields spectral-angular characteristics of radiation
within the frames of the accepted model.
We solve Eq.(1) by perturbation methods supposing
the field of the periodic structure is smaller than the driv-
ing field eE< <κ ρ . Then the electron passes many peri-
ods of the lattice during one period of the external wave.
This approach still gives a rather complicated con-
struction of the vector potential. However, for very high
harmonic numbers the stationary phase method can be
used for evaluations. The physical basis of the method is
the fact that an extensive wave-particle energy exchange
takes place at the moments when the phase slippage ve-
locity is minimal. Then the argument of the exponent can
be expanded in the vicinity of these points. The result can
be writhed as
( ) ,
11
1
1
2
2
222
22
4
224
2
242
2
22
2
222
2
32
2
2
0
2
2
−−
−
+⋅
⋅
−
==
α
λ κ
α
κλ
ακ
ρ
λ
πωω
bb
mc
eE
bbb
mc
eEn
b
mc
eEcmc
n
R
eAP
where ( ) 22 cos 1nb λ κθ −= − ; 2
1 eE
mc
λ
α
= − a wave
strength parameter being a measure of the particle maxi-
mal energy.
The zeroes of the denominator determine the resonant
harmonics
2 21 1 1 1
nλ κ λ κ
α α
Ј Ј
+ + + −
. (2)
For integral numbers satisfying the inequality (2) one can
find an angle θ at which the radiation intensity is maximal
−+=
ncr
λ καθ 21arccos .
The dependence of the radiated power upon the angle
θ is presented in the Fig.3.
The inequality (2) means that the maximal harmonic
number is equal to
22
1 1
n
mc
eE
λ κ
λ
=
ж ц+ −з ч
и ш
The relative radiation power versus the harmonic num-
ber at the angle optimal for the 500-th harmonic is pre-
sented in Fig.4 for the same parameters as in Fig.3. One
can see that for a fixed angle only one harmonic is effec-
tively radiated.
Fig.3. Dependencies of harmonics power on angle θ
for the electric field strength of E=108V/cm, the pumping
wave length λ =10-2см, the structure period 2π/κ =10 μ
(the wave strength parameter =2)
0 50 100 150 200 250 300 350 400 450 500 550 600
0
5
10
15
20
25
30
P1 E λ, θ 500( ), κ, n,( )
nFig.4. The relative radiation power versus the harmonic
number at the angle optimal for the 500-th harmonic
The main results can be formulated as follows:
• The radiation spectrum consists of sharp lines.
• A spectral coherency is possible for large field
strengths. So, the spatial coherency of bunched elec-
trons is available in principle.
• The maximal harmonic number increases with the
field strength increase.
• The absolute value of intensity requires more detailed
model of the phenomenon.
REFERENCES
1. V.A. Buts, E.A. Kornilov. High number harmonic ex-
citation by oscillators in periodic media and in peri-
odic potential // Problems of Atomic Science and
Technology. Series: Plasma Electronics and New
Methods of Acceleration. 2003, №4(3), 114-118.
СПЕКТРАЛЬНО-УГЛОВЫЕ ХАРАКТЕРИСТИКИ ИЗЛУЧЕНИЯ ОСЦИЛЛЯТОРА В ПОЛЕ КРИСТАЛЛИЧЕ-
СКОЙ РЕШЁТКИ
М.А. Горбунов, А.Н. Лебедев
Рассмотрены спектрально-угловые характеристики излучения гармонического осциллятора, находящегося в поле кри-
сталлической решётки. Основное внимание уделяется высокочастотным гармоникам, не удовлетворяющим условию диполь-
ности. Обсуждается возможность использования этого эффекта для генерации рентгеновского излучения.
СПЕКТРАЛЬНО-КУТОВІ ХАРАКТЕРИСТИКИ ВИПРОМІНЮВАННЯ ОСЦИЛЯТОРА
В ПОЛІ КРИСТАЛИЧНОЇ ҐРАТКИ
М.А. Горбунов, А.М. Лебедєв
Розглянуто спектрально-кутові характеристики випромінювання гармонійного осцилятора, що перебуває в полі
кристалічної ґратки. Основна увага приділяється високочастотним гармонікам, що не задовольняють умові дипольності.
Обговорюється можливість використання цього ефекту для генерації рентгенівського випромінювання.
132
References
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| id | nasplib_isofts_kiev_ua-123456789-78880 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:57:52Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Gorbunov, M.A. Lebedev, A.N. 2015-03-22T09:39:42Z 2015-03-22T09:39:42Z 2006 High harmonics oscillator radiation in a periodic structure / M.A. Gorbunov, A.N. Lebedev // Вопросы атомной науки и техники. — 2006. — № 2. — С. 140-141. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS: 41.60.Cr https://nasplib.isofts.kiev.ua/handle/123456789/78880 We discuss spectral and angular characteristics of radiation by a harmonic oscillator placed in a field of a crystal lattice. The main attention is paid to high harmonics which can not be described in a dipole approximation. Certain possibilities of X-rays generation using this effect are discussed as well. Рассмотрены спектрально-угловые характеристики излучения гармонического осциллятора, находящегося в поле кристаллической решётки. Основное внимание уделяется высокочастотным гармоникам, не удовлетворяющим условию дипольности. Обсуждается возможность использования этого эффекта для генерации рентгеновского излучения. Розглянуто спектрально-кутові характеристики випромінювання гармонійного осцилятора, що перебуває в полі кристалічної ґратки. Основна увага приділяється високочастотним гармонікам, що не задовольняють умові дипольності. Обговорюється можливість використання цього ефекту для генерації рентгенівського випромінювання. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Линейные ускорители заряженных частиц High harmonics oscillator radiation in a periodic structure Спектрально-угловые характеристики излучения осциллятора в поле кристаллической решётки Спектрально-кутові характеристики випромінювання осцилятора в полі кристаличної ґратки Article published earlier |
| spellingShingle | High harmonics oscillator radiation in a periodic structure Gorbunov, M.A. Lebedev, A.N. Линейные ускорители заряженных частиц |
| title | High harmonics oscillator radiation in a periodic structure |
| title_alt | Спектрально-угловые характеристики излучения осциллятора в поле кристаллической решётки Спектрально-кутові характеристики випромінювання осцилятора в полі кристаличної ґратки |
| title_full | High harmonics oscillator radiation in a periodic structure |
| title_fullStr | High harmonics oscillator radiation in a periodic structure |
| title_full_unstemmed | High harmonics oscillator radiation in a periodic structure |
| title_short | High harmonics oscillator radiation in a periodic structure |
| title_sort | high harmonics oscillator radiation in a periodic structure |
| topic | Линейные ускорители заряженных частиц |
| topic_facet | Линейные ускорители заряженных частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78880 |
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