On the nature of sources of pulsating radiation in weakly inverted media
The conditions for the occurrence of a periodic sequence of pulses of coherent radiation in a weakly inverted two-level medium are investigated. It is studied the dependence of the pulse period and amplitude on the inversion pumping levelandradiation losses. It is shown that an increase in the size...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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irk-123456789-1122452017-01-19T03:03:02Z On the nature of sources of pulsating radiation in weakly inverted media Kirichok, A.V. Kuklin, V.M. Zagorodny, A.G. Теоретические проблемы физики плазмы The conditions for the occurrence of a periodic sequence of pulses of coherent radiation in a weakly inverted two-level medium are investigated. It is studied the dependence of the pulse period and amplitude on the inversion pumping levelandradiation losses. It is shown that an increase in the size of the radiating system leads to the growth of the total radiation intensity and the pulse repetition period. This dependence is consistent qualitatively with the observedcharacteristics of the cosmic sources of pulsed radiation. Розглянуто умови виникнення періодичної послідовності імпульсів когерентного випромінювання в слабоінвертованому дворівневому середовищі. Досліджено залежність періоду і амплітуди виникаючих імпульсів від рівня накачки інверсії та радіаційних витрат. Показано, що зі збільшенням розмірів випромінюючої системи зростає інтегральна інтенсивність випромінювання і збільшується період генерації імпульсів. Така залежність якісно збігається з характеристиками космічних джерел пульсуючого випромінювання. Рассмотрены условия возникновения периодической последовательности импульсов когерентного излучения в слабоинвертированной двухуровневой среде. Изучена зависимость периода и амплитуды возникающих импульсов от уровня накачки инверсии и радиационных потерь. Показано, что с увеличением размеров излучающей системы растет интегральная интенсивность излучения и увеличивается период генерации импульсов. Такая зависимость качественно совпадает с наблюдаемыми характеристиками космических источников пульсирующего излучения. 2015 Article On the nature of sources of pulsating radiation in weakly inverted media / A.V. Kirichok, V.M. Kuklin, A.G. Zagorodny // Вопросы атомной науки и техники. — 2015. — № 4. — С. 9-11. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 98.65.Fz; 98.80.Bp http://dspace.nbuv.gov.ua/handle/123456789/112245 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Теоретические проблемы физики плазмы Теоретические проблемы физики плазмы |
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Теоретические проблемы физики плазмы Теоретические проблемы физики плазмы Kirichok, A.V. Kuklin, V.M. Zagorodny, A.G. On the nature of sources of pulsating radiation in weakly inverted media Вопросы атомной науки и техники |
| description |
The conditions for the occurrence of a periodic sequence of pulses of coherent radiation in a weakly inverted two-level medium are investigated. It is studied the dependence of the pulse period and amplitude on the inversion pumping levelandradiation losses. It is shown that an increase in the size of the radiating system leads to the growth of the total radiation intensity and the pulse repetition period. This dependence is consistent qualitatively with the observedcharacteristics of the cosmic sources of pulsed radiation. |
| format |
Article |
| author |
Kirichok, A.V. Kuklin, V.M. Zagorodny, A.G. |
| author_facet |
Kirichok, A.V. Kuklin, V.M. Zagorodny, A.G. |
| author_sort |
Kirichok, A.V. |
| title |
On the nature of sources of pulsating radiation in weakly inverted media |
| title_short |
On the nature of sources of pulsating radiation in weakly inverted media |
| title_full |
On the nature of sources of pulsating radiation in weakly inverted media |
| title_fullStr |
On the nature of sources of pulsating radiation in weakly inverted media |
| title_full_unstemmed |
On the nature of sources of pulsating radiation in weakly inverted media |
| title_sort |
on the nature of sources of pulsating radiation in weakly inverted media |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Теоретические проблемы физики плазмы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112245 |
| citation_txt |
On the nature of sources of pulsating radiation in weakly inverted media / A.V. Kirichok, V.M. Kuklin, A.G. Zagorodny // Вопросы атомной науки и техники. — 2015. — № 4. — С. 9-11. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T03:35:30Z |
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2025-07-08T03:35:30Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 9
ON THE NATURE OF SOURCES OF PULSATING RADIATION
IN WEAKLY INVERTED MEDIA
A.V. Kirichok*, V.M. Kuklin*, A.G. Zagorodny**
*V.N. Karazin Kharkiv National University, Institute for High Technologies, Kharkov,
Ukraine;
**Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
E-mail: kuklinvm1@google.com
The conditions for the occurrence of a periodic sequence of pulses of coherent radiation in a weakly inverted
two-level medium are investigated. It is studied the dependence of the pulse period and amplitude on the inversion
pumping levelandradiation losses. It is shown that an increase in the size of the radiating system leads to the growth
of the total radiation intensity and the pulse repetition period. This dependence is consistent qualitatively with the
observedcharacteristics of the cosmic sources of pulsed radiation.
PACS: 98.65.Fz; 98.80.Bp
INTRODUCTION
As known, a two-level system demonstrates the pos-
sibility of generation of both spontaneous and induced
(stimulated) emission when the initial population inver-
sion is sufficiently large [1]. Usually, the term sponta-
neous emission denotes the emission of oscillator (or
other emitter) which not forced by external field of the
same frequency. By induced or simulated emission is
usually meant the emission produced because of an ex-
ternal field action on the emitting source at the radiation
frequency [2 - 4].
In the earlier paper [5], the authors have discovered
a threshold of induced radiation at a certain critical val-
ue of population inversion. The specific feature of this
threshold is that it follows from the condition that the
initial value of the population inversion is equal to the
square root of the total number of states. Above this
threshold, the number of photons begins to grow expo-
nentially with time. It was shown that the threshold cor-
responds to the case when the intensity of the spontane-
ous and stimulated coherent radiation become equal. In
further work [6], it was demonstrated that the pulse of
stimulated radiation with a characteristic profile is
formed when the initial population inversion slightly
exceeds the threshold. The leading edge of the pulse is
very sharp due to the exponential growth of the field,
and the trailing edge is rather broadened. Further
threshold overriding with growing of the population
inversion leads to the growth of the ratio of the pulse
trailing-edge time to the leading-edge time.
If there is exist a recovery mechanism for the popu-
lation inversion, the pulsating mode of stimulated emis-
sion generation becomes possible. The integral radiation
intensity at this may be increased several times. This
approach can be used for analysis of the cosmic radia-
tion that might help explain a great variety of pulsating
radiation sources in space. In present work, we investi-
gate the characteristics of the periodic pulse generation
depending on the initial inversion, the pumping level
and the absorption rate.
1. THE MODELS FOR DESCRIPTION
OF A TWO-LEVEL SYSTEM
A two-level system with transition frequency
2 1 12ε ε ω− = can be described by following set of
equations:
2 21 21 2 12 1/ ( )k kn t u w N n w N n∂ ∂ = − + ⋅ ⋅ + ⋅ ⋅ , (1)
1 12 1 21 21 2/ ( )k kn t w N n u w N n∂ ∂ = − ⋅ ⋅ + + ⋅ ⋅ ,
where the sum of level populations 1 2n n N+ = remains
constant, 21 2u n is the rate of change in the number den-
sity of atoms due to spontaneous emission. The rates of
change in level population due to stimulated emission
and absorption are 21 2kw N n and 12 1kw N n correspond-
ingly. The number of quanta kN on the transition fre-
quency kω is governed by the equation
21 21 2 12 1( ) ( )k
k k
N
u w N n w N n
t
∂
= + ⋅ ⋅ − ⋅ ⋅
∂
. (2)
The energy losses in active media are caused mainly
by radiation outcome from a resonator. These radiative
losses can be calculated by imposing the correct bound-
ary conditions on the field. Thus, they can be estimated
in rather common form with the following parameter:
2 2
1 [ ( , )]
4
1 (| | | | ) ,
8
S V
kE Hds
k
E H dv
ω ωε ωδ
π ω
π
∂ ∂
= × ×
∂∂
× +
∫∫ ∫∫∫
(3)
i. e. as the ratio of the energy flow outside the system
should be divided by the total field energy within the
system. It is important, that the characteristic size of the
system L should be much less than the characteristic
time of field variation 2 2 1~| | | |( / )E E tτ −∂ ∂
multiplied
by the group velocity of oscillations | / |kω∂ ∂
. In this
case, the radiative losses can be replaces by distributed
losses within the volume. The threshold of instability
leading to exponential growth of the stimulated emis-
sion in this case is defined by condition 0 1THµ µ> (see,
for example [6], where
1 21/TH wµ δ= . (4)
Equations (1)-(2) can be rewritten with considera-
tion of the energy losses δ and steady pumping rate I ,
caused for example by permanent heating of the system:
2 2/ kn n N Iτ µ∂ ∂ = − − ⋅ + , (5)
2/ 2 2 kn N Iµ τ µ∂ ∂ = − − ⋅ + , (6)
2/ ,k k kN n N Nτ µ δ∂ ∂ = + ⋅ − ⋅ (7)
where 21w tτ = ⋅ , 21 21 12u w w= = .
mailto:kuklinvm1@google.com
ISSN 1562-6016. ВАНТ. 2015. №4(98) 10
It can be assumed, at least qualitatively, that the
terms in r.h.s. of Eq. (1)-(2)proportional to Nk corre-
spond to the coherent processes, as well as the photons,
which number Nk is incorporated in these terms, will be
assumed coherent. With these general principles in
mind, we expand the total number of photons into two
components ( ) ( )incoh coh
k k kN N N= + , where ( )incoh
kN is
the number of quanta, corresponding to the spontaneous
emission, and ( )coh
kN is the number of quanta, corre-
sponding to the stimulated emission. Then Eqs. (1)-(2)
can be rewritten as follows [6]:
( )
2 2/ ,coh
kn n N Iτ µ∂ ∂ = − − ⋅ + (8)
( )
2/ 2 2 2 ,coh
kn N Iµ τ µ∂ ∂ = − − ⋅ + (9)
( ) ( )
2/ ,incoh incoh
k kN n Nτ δ∂ ∂ = − ⋅ (10)
( ) ( ) ( )/ ,coh coh coh
k k kN N Nτ µ δ∂ ∂ = ⋅ − ⋅ (11)
where 1 2N n n= + is a total number of emitters and
2 ( ) / 2n N µ= + .
It was shown in the paper [5] that the scenario of the
process changes, if the initial value of the inversion µ0
is more or less than the threshold value:
1/2
2 2TH Nµ = . (12)
The suppression of the exponential growth of the
photon number, when 1/2
0 2 2TH Nµ µ< = demonstrates
not only the changes in scenario of the process, but
suggests that the stimulated emission is suppressed by
preferential growth of the spontaneous emission.
Let discuss the reasons why it makes sense to use a
qualitative system of equations (8)-(11) near the thresh-
old (12).
Within the framework of the classical description,
the total intensity of the spontaneous emission of an
ensemble of particles-oscillators, whose phases are dis-
tributed randomly and uniformly, can be found as a sum
of individual intensities produced by each particle-
oscillator being in an excited state. As for the stimulated
emission, the radiation field strength is so great that
synchronizes the phase both of the emitting and absorb-
ing oscillators. Thus the sign of the population inversion
2 1n nµ = − determines is the stimulated field will in-
crease or decrease. Note, that the characteristic time of
this process is inversely proportional to µ. However, if
the coherent field is absent, the oscillators in the excited
state will emit only spontaneously, because their phases
are not synchronized. The absorption of the spontaneous
field by the unexcited particles-oscillators can be ig-
nored since they are placed in a random rapidly alternat-
ing field, which averaged effect is negligible.
In the quantum case, the traditional model (5)-(7) in-
cludes the term kNµ ⋅ which is responsible for the
stimulated processes of excitation and absorption. But it
has no physical meaning below the threshold (12), since
in this case there is no an intense stimulated field, which
is able to synchronize the emission of many particles. In
this case Eq. (9) takes the form
/ 2 .N Iµ τ µ∂ ∂ = − − + (13)
In the steady state 2st THI Nµ µ= − < the intensity
of the radiation source will be determined only by the
spontaneous emission ( ) / 2incoh
kN Nδ ⋅ . The term N in
Eq. (13) determines the effect of the spontaneous emis-
sion on the inversion. But when the threshold (12) is
exceeded µ > 2THµ , the term kNµ ⋅ in r.h.s of Eqs. (5)-
(7) and (8)-(11) plays an important role, providing an
effect of the stimulated processes.
Near the threshold (12), it is reasonable to use name-
ly a qualitative system of equations (8)-(11).Then the
steady state of the spontaneous emission is determined
by the value ( ) 2incoh
kN Nδ ⋅ , but the energy flow of the
stimulated emission is equal to ( )incoh
kNδ ⋅ . In order to
describe the behavior of a two-level system in presence
of the radiation losses and continuous external pumping
and neglecting the small values of the order of 1
0µ
− ,
Eqs. (9)-(11) can be rewritten in a convenient form:
0 0/ 2 2cT N I∂Μ ∂ = − − Μ ⋅Ν + , (14)
0/ / 2inc incT N θ∂Ν ∂ = − ⋅Ν , (15)
/c c cT θ∂Ν ∂ = Μ ⋅Ν − ⋅Ν , (16)
where ( )
0/incoh
inc kN µΝ = , ( )
0/coh
c kN µΝ = , 0/µ µΜ = ,
0/µ µΜ = , 21 0 0T w tµ µ τ= ⋅ ⋅ = ⋅ , 2
0 0/I I µ= , and the
only convenient for analysis free element
is 2
0 0/N N µ= . Let specify the initial values as follows:
( 0) 1TΜ = = ,
4
0 0( 0) / 3 10 /inc incT µ µΝ = = Ν = ⋅ ;
4
0 0( 0) / 3 10 /c cT µ µΝ = = Ν = ⋅ ;
4
1 0 0( 0) / 3 10 /kT µ µΝ = = Ν = ⋅ .
The energy losses are taken into account by the pa-
rameter 0/θ δ µ= , where δ is defined by Eq. (3).
In the case, when 2THµ µ> and (0)M θ> , the re-
laxation oscillations appear in the system resulting in a
stationary state 0 0( ) / 2cst I N θΝ = − , stM θ= . The
total radiation flow outside the system in assumed terms
is equal to
0 0 0 0( ) / 2 / 2 / 2cst incst I N N Iθ θ⋅Ν + ⋅Ν = − + = .
Note, that in presence of an external mechanism,
which provides an exceedance of the inversion over its
stationary value stM θ= , the Eq. (14) can be supple-
mented by the driving term
0 0/ 2 2cT N I∂Μ ∂ = ΓΜ − − Μ ⋅Ν + . (17)
The Eqs. (16), (17) are similar to so-called Statz-
DeMars equations [7], which describe the relaxation
oscillations in a two-level media in the presence of the
pumpandenergy losses. The only difference in equation
(17) is the first term in r.h.s. that provides the mainte-
nance of the population inversion. Namely this term
changes the characteristics of pulse generation from
relaxation to periodic.
The Eqs. (15)-(17) have a solution in a form of peri-
odical sequence of coherent pulses (Figure) against a
background of the mean radiation flow
0( ) / 2cst incst Iθ θ θ⋅Ν + ⋅Ν = Γ + . (18)
Pulse repetition rate is θ ⋅Γ . The integral radiation
intensity on the pulse peak can exceed the background
value in several times.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 11
The repetition pulse train, resulting as a solution
of Eqs. (16), (17) for 0.4Γ = and 0.3θ =
It should be noted that the radiation losses of the
field energy θ in open systems is defined as the ratio of
the energy flux from the object to the energy in its vol-
ume, and therefore this parameter decreases with in-
crease of the radius of the system R as /c R , where c
is the speed of light. This means that an increase in
size R reduces the losses θ, which in turn, as shown in
[6], provides a higher intensity of the stimulated emis-
sion. That is, at the same parameters of the system, the
larger objects should generate more intense pulses but
with less repetition rate.
REFERENCES
1. A. Einstein. Quantentheorie der Strahlung // Mittei-
lungend. Phys. Ges. Zurich. 1916, № 18; Phys. Zs.
1917, № 18, р. 121.
2. C.Η. Τоwnes. Production of Coherent Radiation by
Atoms and Molecules // IEEE Spectrum. 1965, iss. 2
(2), p. 30.
3. G. Birnbaum. Optical masers. New York and Lon-
don: Academic Press, 1964.
4. N. Blotmbergen. Nonlinear Optics. A Lecture Note /
W.A. Benjamin, Inc. New York-Amsterdam, 1965.
5. V.M. Kuklin, A.G. Zagorodny. To realization condi-
tion of maser radiation // XIV Khariton's Topical
Scientific Readings "High-Power Pulsed Electro-
physics" March 12-16, 2012, Sarov, Russia.
6. A.V. Kirichok, V.M. Kuklin, A.V. Mischin,
A.V. Pryjmak, A.G. Zagorodny. On the formation of
pulses of coherent radiationin weakly inverted media
// PACS. Series “Plasma Electronics and New
Methods of Acceleration”. 2013, № 4 (86), iss. 8,
p. 267-271.
7. C.L. Statz, G. DeMars // Quantum Electronics.
N.Y.: Columbia Univ. Press, 1960, p. 530.
Article received 12.05.2015
О ПРИРОДЕ ИСТОЧНИКОВ ПУЛЬСИРУЮЩЕГО ИЗЛУЧЕНИЯ
В СЛАБОИНВЕРСНЫХ СРЕДАХ
А.В. Киричок, В.М. Куклин, А.Г. Загородний
Рассмотрены условия возникновения периодической последовательности импульсов когерентного излу-
чения в слабоинвертированной двухуровневой среде. Изучена зависимость периода и амплитуды возника-
ющих импульсов от уровня накачки инверсии и радиационных потерь. Показано, что с увеличением разме-
ров излучающей системы растет интегральная интенсивность излучения и увеличивается период генерации
импульсов. Такая зависимость качественно совпадает с наблюдаемыми характеристиками космических ис-
точников пульсирующего излучения.
ПРО ПРИРОДУ ДЖЕРЕЛ ПУЛЬСУЮЧОГО ВИПРОМІНЮВАННЯ
В СЛАБОІНВЕРСНИХ СЕРЕДОВИЩАХ
О.В. Киричок, В.М. Куклін, О.Г. Загородній
Розглянуто умови виникнення періодичної послідовності імпульсів когерентного випромінювання в сла-
боінвертованому дворівневому середовищі. Досліджено залежність періоду і амплітуди виникаючих
імпульсів від рівня накачки інверсії та радіаційних витрат. Показано, що зі збільшенням розмірів випромі-
нюючої системи зростає інтегральна інтенсивність випромінювання і збільшується період генерації імпуль-
сів. Така залежність якісно збігається з характеристиками космічних джерел пульсуючого випромінювання.
ON THE NATURE OF SOURCES OF PULSATING RADIATION IN WEAKLY INVERTED MEDIA
О ПРИРОДЕ ИСТОЧНИКОВ ПУЛЬСИРУЮЩЕГО ИЗЛУЧЕНИЯ В СЛАБОИНВЕРСНЫХ СРЕДАХ
Про ПРИРОДУ джерел ПУЛЬСУЮЧОГО випромінювання в СЛАБОІНВЕРСНИХ СЕРЕДОВИЩАХ
|