On the formation of pulses of coherent radiation in weakly inverted media
A change in the character of maser generation in a two-level system is found when the initial population inversion exceeds some threshold value equal to the square root of the total number of atoms. Above this threshold, the number of photons begins to grow exponentially with time and the pulse with...
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| Zitieren: | On the formation of pulses of coherent radiation in weakly inverted media / A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 267-271. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1121592017-01-18T03:03:58Z On the formation of pulses of coherent radiation in weakly inverted media Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryjmak, A.V. Zagorodny, A.G. Нелинейные процессы в плазменных средах A change in the character of maser generation in a two-level system is found when the initial population inversion exceeds some threshold value equal to the square root of the total number of atoms. Above this threshold, the number of photons begins to grow exponentially with time and the pulse with short leading edge and broadened trailing edge is generated. In this work, we attempt to explain the nature of this threshold. Coherent pulse duration, estimated by its half-width, increases significantly with increasing inversion, if all other parameters are fixed and the absorption is neglected. The inclusion of the energy loss of photons leads to the fact that the duration of coherent pulse is almost constant with increasing inversion, at least well away from the threshold. Виявлено зміну характеру процесу генерації випромінювання в дворівневій системі при перевищенні початкової інверсії заселеності величини, що дорівнює кореню квадратному з повного числа станів. При перевищенні цього порога число квантів починає рости з часом за експонентою. Зроблена спроба пояснити природу цього порога: при його перевищенні виникає генерація когерентного випромінювання у вигляді імпульсів з коротким переднім фронтом і протяжним заднім фронтом. Якщо всі параметри, окрім інверсії, зафіксувати, то з подальшим ростом інверсії при відсутності поглинання тривалість когерентного імпульсу, оцінена за його напівшириною, помітно збільшується. Урахування втрат енергії квантів призводить до того, що тривалість когерентного імпульсу практично не змінюється при зростанні інверсії, принаймні, досить далеко від порога. Обнаружено изменение характера процесса генерации излучения в двухуровневой системе при превышении начальной инверсии заселенностей величины, равной корню квадратному из полного числа состояний. При превышении этого порога число квантов начинает расти экспоненциально со временем. Сделана попытка пояснить природу этого порога: при его превышении возникает генерация когерентного излучения в виде импульсов с коротким передним фронтом и протяженным задним фронтом. Если все параметры, кроме инверсии, зафиксировать, то с ростом инверсии в отсутствие поглощения длительность когерентного импульса, оцененная по его полуширине, заметно увеличивается. Учет потерь энергии квантов приводит к тому, что длительность когерентного импульса практически не изменяется при росте инверсии, по крайней мере, достаточно далеко от порога. 2013 Article On the formation of pulses of coherent radiation in weakly inverted media / A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 267-271. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 42.50.Fx http://dspace.nbuv.gov.ua/handle/123456789/112159 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
| spellingShingle |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryjmak, A.V. Zagorodny, A.G. On the formation of pulses of coherent radiation in weakly inverted media Вопросы атомной науки и техники |
| description |
A change in the character of maser generation in a two-level system is found when the initial population inversion exceeds some threshold value equal to the square root of the total number of atoms. Above this threshold, the number of photons begins to grow exponentially with time and the pulse with short leading edge and broadened trailing edge is generated. In this work, we attempt to explain the nature of this threshold. Coherent pulse duration, estimated by its half-width, increases significantly with increasing inversion, if all other parameters are fixed and the absorption is neglected. The inclusion of the energy loss of photons leads to the fact that the duration of coherent pulse is almost constant with increasing inversion, at least well away from the threshold. |
| format |
Article |
| author |
Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryjmak, A.V. Zagorodny, A.G. |
| author_facet |
Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryjmak, A.V. Zagorodny, A.G. |
| author_sort |
Kirichok, A.V. |
| title |
On the formation of pulses of coherent radiation in weakly inverted media |
| title_short |
On the formation of pulses of coherent radiation in weakly inverted media |
| title_full |
On the formation of pulses of coherent radiation in weakly inverted media |
| title_fullStr |
On the formation of pulses of coherent radiation in weakly inverted media |
| title_full_unstemmed |
On the formation of pulses of coherent radiation in weakly inverted media |
| title_sort |
on the formation of pulses of coherent radiation in weakly inverted media |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Нелинейные процессы в плазменных средах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112159 |
| citation_txt |
On the formation of pulses of coherent radiation in weakly inverted media / A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 267-271. — Бібліогр.: 11 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT kirichokav ontheformationofpulsesofcoherentradiationinweaklyinvertedmedia AT kuklinvm ontheformationofpulsesofcoherentradiationinweaklyinvertedmedia AT mischinav ontheformationofpulsesofcoherentradiationinweaklyinvertedmedia AT pryjmakav ontheformationofpulsesofcoherentradiationinweaklyinvertedmedia AT zagorodnyag ontheformationofpulsesofcoherentradiationinweaklyinvertedmedia |
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2025-07-08T03:28:46Z |
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2025-07-08T03:28:46Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 267
ON THE FORMATION OF PULSES OF COHERENT RADIATION
IN WEAKLY INVERTED MEDIA
A.V. Kirichok*, V.M. Kuklin*, A.V. Mischin*, A.V. Pryjmak*, A.G. Zagorodny**
*Kharkov National University, Institute for High Technologies, Kharkov, Ukraine;
**Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
E-mail: kuklinvm1@rambler.ru
A change in the character of maser generation in a two-level system is found when the initial population inver-
sion exceeds some threshold value equal to the square root of the total number of atoms. Above this threshold, the
number of photons begins to grow exponentially with time and the pulse with short leading edge and broadened
trailing edge is generated. In this work, we attempt to explain the nature of this threshold. Coherent pulse duration,
estimated by its half-width, increases significantly with increasing inversion, if all other parameters are fixed and the
absorption is neglected. The inclusion of the energy loss of photons leads to the fact that the duration of coherent
pulse is almost constant with increasing inversion, at least well away from the threshold.
PACS: 42.50.Fx
INTRODUCTION
Description of physical phenomena based on the
systems of partial differential equations, derived from
the observations and experimental facts, often conceals
from an investigator some essential features, especially
in those cases, when the researchers do not expect to
find anomalies and qualitative changes in the dynamics
of systems in given range of variables and parameters.
Namely such a case of unusual behavior of a two-level
quantum system was found in attempting to separate a
coherent component from the total radiation flow.
In the beginning of the past century, A. Einstein has
proposed the model of two-level system, which has
demonstrated the possibility of generation of both spon-
taneous and induced (stimulated) emission when the
initial population inversion is sufficiently large [1].
Usually, the term spontaneous emission denotes the
emission of oscillator (or other emitter) which not
forced by external field of the same frequency. As for
other influences on the characteristics of the spontane-
ous emission, there is nothing to say definitely. Al-
though the dynamics of spontaneous processes usually
shows a steady recurrence and invariance, there is evi-
dent [2] that the characteristics of the spontaneous proc-
esses can vary with change of environment. By induced
or simulated emission is usually meant the emission
produced because of an external field action on the
emitting source at the radiation frequency.
There were difficulties in the quantum description
with interpretation of the stimulated emission as coher-
ent, where in contrast to the classical case it was impos-
sible to say anything about the phases of the fields emit-
ted by individual atoms and molecules. However, C.
Townes believed that "… the energy delivered by the
molecular systems has the same field distribution and
frequency as the stimulating radiation and hence a con-
stant (possibly zero) phase difference" [3].
If we assume, relying upon the results of the studies
of fluctuation correlations in the laser radiation [4], that
a stimulated emission has a high proportion of the co-
herent component, one can find a threshold of coherent
radiation at a certain critical value of population inver-
sion [5]. 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. On the other hand, the change in
the nature of the process near the threshold is evident,
even without making any other assumptions. Above this
threshold, the number of photons begins to grow expo-
nentially with time. Herewith, below the threshold there
no exponential growth.
It is known that at low levels of spontaneous com-
ponent and far above the maser generation threshold the
number of photons growths exponentially and the radia-
tion is largely a coherent [6, 7]. The meaningful indica-
tor of the collective character of stimulated emission is
the so-called photon degeneracy, which is defined as the
average photon number contained in a single mode of
optical field (see, for example [8]). For the incoherent
light, this parameter does not exceed unity, but for even
the simplest He-Ne maser it reaches the value of 1210 as
was shown in the early works (see [6]).
It is of interest to go further and analyze the conse-
quences of consideration of the spontaneous emission as
a random process (at least, in a homogeneous medium)
and induced process as a coherent process. It is clear
that the separation of total radiation into two category:
the stimulated – coherent and spontaneous – random or
incoherent will be idealized simplification. However,
such separation may explain, at least qualitatively, the
nature of the radiation emitted by two-level quantum
system near to exposed threshold.
Another indirect proof of the existence of such a
threshold is the following observation. The intensity of
the spontaneous emission, which is non-synchronized
(randomly distributed) over oscillators phases is known
to be proportional to their number. The intensity of the
coherent stimulated emission is proportional in turn to
the square of the number of oscillators. It is easy to see
that the exposed threshold corresponds to the case when
the intensity of spontaneous and stimulated coherent
radiation become equal.
In [5] we have shown that under these conditions the
pulse of coherent radiation with a characteristic profile
is formed when the initial population inversion slightly
exceeds the threshold. The leading edge of the pulse due
to the exponential growth of the field is very sharp due
to the exponential growth of the field, and the trailing
edge is rather broadened. Further overriding of the thre-
shold, that is growing of the initial population inversion,
ISSN 1562-6016. ВАНТ. 2013. №4(86) 268
results in the ratio of the trailing edge duration to the
leading edge duration becomes greater. At large times
the incoherent radiation dominates.
Because very small value of the initial population
inversion can provide generation of pulses of coherent
radiation, it is of interest to determine the shape of these
pulses for different values of the initial population in-
version levels and when the field energy absorption
should be taken into account. These pulses can be eas-
ily detected in experiments. In addition, after experi-
mental validation of this model, it will be possible to
use these approaches for analysis of the cosmic radia-
tion that might help explain such abundance of coherent
radiation sources in space.
In this paper, we study the characteristics of the
pulses of coherent radiation as a function of the initial
inversion and absorption level in the system. The dy-
namics of the emission process in the simplified model is
compared with the dynamics of change in the number of
quanta in the traditional model, where the separation into
coherent and incoherent components is not carried out.
1. TRADITIONAL DESCRIPTION OF
TWO-LEVEL SYSTEM
Following to A. Einstein [1], a two-level system
with transition frequency 2 1 12ε ε ω− = h can be de-
scribed 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 corre-
spodingly. The number of quanta kN on the transition
frequency 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 losses of energy in active media are caused
mainly by radiation outcome from a resonator. These
radiative losses can be calculated by imposing the cor-
rect boundary 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
∂ ∂
= × ×
∂∂
× +
∫∫ ∫∫∫
r
r r
r
r r
(3)
i.e. as the ratio of the energy flow passing through the
resonator mirrors to the total field energy within resona-
tor. It is important, that the characteristic size of the
resonator L should be much less than the characteristic
time of field variation 2 2 1~| | | |( / )E E tτ −∂ ∂
r r
multiplied
by the group velocity of oscillations | / |kω∂ ∂
r
. In this
case the radiative losses through the mirrors can be re-
places by distributed losses whithin the resonator vol-
ume. The threshold of instability leading to exponentiol
growth of coherent emission 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 in the form
2 2/ kn n Nτ μ∂ ∂ = − − ⋅ , (5)
2/ 2 2 kn Nμ τ μ∂ ∂ = − − ⋅ , (6)
2/ ,k kN n Nτ μ∂ ∂ = + ⋅ (7)
where 21w tτ = ⋅ , 21 21 12u w w= = . Since the purpose of
this work is to find the threshold of the initial population
inversion, which starts the exponential growth of the
number of emitted quanta, we will restrict our consid-
eration by the case 2 1 1 2,n n n nμ = − << . It follows from
Eqs. (5) - (7) that 0 0 0( ) / 2 ( ) / 2k kN N μ μ μ μ= + − ≈ − ,
and at large times 2 0/ 2 ( ) / 2st st stn N μ μ μ≈ = − ⋅ − ,
where 0 ( 0)μ μ τ= = , 0 ( 0)k kN N τ= = . Hence, we find
the stationary value of the inversion
2
0 0( / 2) ( / 2)st Nμ μ μ= − + . (8)
Two cases are of interest. When the initial popula-
tion inversion is sufficiently large 2
0( / 2) Nμ >> , it
rapidly decreases to its steady-state value
1 0( / )st Nμ μ μ→ = − with 1 0| |stμ μ<< .The number of
quanta at this growths exponentially and asymptotically
tends to a stationary level 1 0 / 2k kstN N μ→ = . It is ob-
viously that in this case the stimulated emission domi-
nates (the second terms in r.h.s. of Eqs. (5) - (7)).
The second case of interest corresponds to relatively
small initial inversion 2
0( / 2) Nμ << . Here, μ tends to
its stationary value 1/2( )st Nμ μ→ = − , where 0| |stμ μ> ,
and the number of quanta reaches the limit
1/2
2k kstN N N→ = .
If the spontaneous emission only dominated (the
first terms on the r.h.s. of Eqs.(5) - (7)), the characteris-
tic time to reach the steady-state number of photons will
be of the order of 1
0 0/m Nτ τ μ μ −Δ = > in the first
case and 1
01/ Nτ μ −Δ < in the second case,
where 1
0μ
− is the characteristic time of exponential
growth of the number of photons in the first case. This
means that the exponential growth of the number of
photons in the second case is suppressed and the role of
the second terms in r.h.s. of Eqs. (5) - (6) comes to sta-
bilize the number of particles and the inversion level
due to the absorption process.
Thus, it is clear that the scenario of the process
changes, if the initial value of the inversion μ0 is more
or less thana threshold value [5]:
1/2
2 2TH Nμ = . (9)
The suppression of the exponential growth of the
number of photons when 1/ 2
0 2 2TH Nμ μ< = demon-
strates not only the changes in scenario of the process,
but it suggests that the stimulated emissionis suppressed
by preferential growth of spontaneous emission. Indeed,
the first term in r.h.s. of Eqs. (5) - (7), which is respon-
sible for the spontaneous emission, reduces in version to
zero in a very short time 21/ THτ μ< , thus excluding the
possibility of exponential growth of the number of pho-
tons, which is characteristic for the induced processes.
ISSN 1562-6016. ВАНТ. 2013. №4(86) 269
It is useful, at least qualitatively, to examine the na-
ture of changes in emission characteristics of an in-
verted system near the threshold μTH2. It should be ex-
pected also other specific features in the radiation na-
ture, including the formation of a short pulse of coherent
radiation against the background of incoherent field [5].
2. QUALITATIVE MODEL
OF TWO-LEVEL SYSTEM
First of all, in order to understand the further, it
should be remembered that the oscillator emits under
the action of an external coherent field with the same
frequency and phase as the stimulating field, that is, the
external radiation and radiation of the oscillator stimu-
lated by it occur to be coherent [6, 7]. Moreover, the
greater intensity of the coherent component of the ex-
ternal field, the more energy the oscillator loses per unit
time by radiation. On the other side, the spontaneous
emission is the process independent of the external ra-
diation field and incoherent, at least for a uniform dis-
tribution of emitters.
Neglecting the stage when the number of photons is
saturated, we can at least qualitatively assume 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= + and rewrite Eqs. (2)-
(3) as follows [11]
( ) ( )
2 12 1 21 21 2/ ( ) ,coh coh
k kn t w N n u w N n∂ ∂ = + ⋅ ⋅ − + ⋅ ⋅ (10)
( ) ( )
1 12 1 21 21 2/ ( ) ,coh coh
k kn t w N n u w N n∂ ∂ = − ⋅ ⋅ + + ⋅ ⋅ (11)
( )
21 2/ ,incoh
kN t u n∂ ∂ = ⋅ (12)
( )
21 2 12 1/ .coh
k k kN t w N n w N n∂ ∂ = ⋅ ⋅ − ⋅ ⋅ (13)
Assuming 21 21 12u w w= = and 2 ( ) / 2n N μ= + , we obtain
( )
2 2/ coh
kn n Nτ μ∂ ∂ = − − ⋅ , (14)
( )
2/ 2 2 coh
kn Nμ τ μ∂ ∂ = − − ⋅ , (15)
( )
2/incoh
kN nτ∂ ∂ = (16)
( ) ( )/ ,coh coh
k kN Nτ μ∂ ∂ = ⋅ (17)
where 1 2N n n= + is a total number of emitters.
Let compare the dynamics of the processes de-
scribed by Eqs. (14) - (17) and by Eqs. (5) - (7). In order
to do this, we represent themas shown in Table:
The modelling set of equations with separation of
quanta into coherent and incoherent sorts
0/ 2 cT N∂Μ ∂ = − − Μ ⋅Ν , (18)
0/inc incT N θ∂Ν ∂ = − ⋅Ν , (19)
/c c cT θ∂Ν ∂ = Μ ⋅Ν − ⋅Ν . (20)
Traditional set of equation
1 21 1 1/ 2 2T n∂Μ ∂ = − − Μ ⋅Ν , (21)
1 21 1 1 1/ T n θ∂Ν ∂ = +Μ ⋅Ν − ⋅Ν . (22)
where ( )
0/incoh
inc kN μΝ = , ( )
0/coh
c kN μΝ = , 0/μ μΜ = ,
1 0/μ μΜ=Μ = 21 0 0T w tμ μ τ= ⋅ ⋅ = ⋅ 1 0/kN μΝ = . The only
free parameter convenient for the analysis is
2
0 0/N N μ= . For correct comparison, we assume that the
total number of real states is 12
1 2 10N n n= + = , and the
threshold inversion is 6
0 10th Nμ = = . Transition to a
unified time scale will be carried out as follows
0T τ μ= ⋅ , where T is time for each case. Let choose
the following initial values
1( 0) ( 0) 1T TΜ = = Μ = = ,
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 radiation losses are taking into account by the
term 0/θ δ μ= , where δ is defined in (3).
Fig. 1 demonstrates a change in dynamics of the
process with increase in the starting population inver-
sion (9) simulated by Eqs. (21) - (22), where
0 (30...0.01)N ⊂ .
The attention should be given to a change in the rate
of emitted quanta with crossing of the threshold (9). For
greater values of the initial inversion, the stimulated
emission begins to prevail and the regime of exponential
growth in the number of quanta becomes more pro-
nounced.
In the absence of radiative losses, the simulation of
Eqs. (18) - (20) shows that after the coherent pulse
drops, the spontaneous emission continues to increase.
Within framework of the traditional model (21) - (22),
absorption restricts the growth of the number of quanta
and radiation intensity tends to a stationary level.
Fig. 1. Evolution of the value 1 1ln( )dN N dT
for different 2
0 1 2 2 1( ) / ( )N n n n n= + − : 1) 0 30N = ;
2) 0 10N = ;3) 0 5N = ;4) 0 2N = ; 5) 0 1N = ;
6) 0 0.5N = ; 7) 0 0.2N = ; 8) 0 0.1N = ; 9) 0 0.03N =
However, comparing the dynamics of the processes
it can be understood that after the amplitude of the co-
herent pulse decreases, the spontaneous emission be-
cames dominant. That is, attimes exceeding the duration
of the coherent pulse the incoherent radiation prevails.
The absorption of photons suppresses the generation,
so we choose relatively lowlevel of energy loss, that is
52 10δ = ⋅ and 54 10δ = ⋅ . The generation process in
this case keeps the same features, but the absorption
limits the lifetime of the generation and the differences
between two models areless pronounced.
ISSN 1562-6016. ВАНТ. 2013. №4(86) 270
Fig. 2. Evolution of M1 and N1 (dot line), M (dash line),
Nc and Ninc (solid and dash-dot line correspondingly)
in lossless case ( 0θ = ) and N0 =N/μ2
0 =0.05
Fig. 3. Evolution of M1 and N1 (dot line), M (dash line),
Nc and Ninc (solid and dash-dot line correspondingly)
for lossless case ( 0θ = ) and 2
0 0/ 0.01N N μ= =
Fig. 4. Evolution of M1 and N1 (dot line), M (dash line),
Nc and Ninc (solid and dash-dot line correspondingly)
with absorption ( 52 10δ = ⋅ ,θ =δ/μ0=0.045) and
2
0 0/ 0.05N N μ= =
Fig. 5. Evolution of M1 and N1 (dot line), M (dash line),
Nc and Ninc (solid and dash-dot line correspondingly)
with absorption (δ = 4⋅105, θ =δ/μ0=0.04)
and 2
0 0/ 0.01N N μ= =
Now, let discuss the quantitative characteristics of
the coherent pulse. Figs. 6 and 7 demonstrate the shape
of the coherent pulse in lossless case and in presence of
absorption for different initial value of the population
inversion.
Fig. 6. Evolution of coherent pulse shape in absence of
absorption (θ =0) for different values of initial popula-
tion inversion 1) 0μ = 62 10⋅ ; 2) 62 10⋅ ; 3) 610 10⋅ ;
4) 620 10⋅ ; 5) 650 10⋅ ; 6) 710 ;7) 72 10⋅ ;
8) 72 10⋅ ; 9) 710 10⋅
Note the fact that in the case of a fixed finite level of
loss, the shape and duration of the coherentpulsedoes
not change even when the population inversion level
increases significantly. Thus, the formation of the lead-
ing edge of the pulse is determined by the initial inver-
sion level, the duration of its trailing edge is determined
mostly by the rate of radiative loss.
Fig. 7. Evolution of coherent pulse shape in presence of
absorption ( 54 10δ = ⋅ ) for different values of initial
population inversion 1) 0μ = 62 10⋅ ; 2) 62 10⋅ ;
3) 610 10⋅ ; 4) 620 10⋅ ; 5) 650 10⋅ ; 6) 710 ;
7) 72 10⋅ ; 8) 72 10⋅ ; 9) 710 10⋅
CONCLUSIONS
The threshold of coherent emission generation, dis-
cussed in this paper, corresponds to the case when the
intensity of spontaneous and stimulated coherent radia-
tion become equal. The stimulated emission in this case
can be considered as completely coherent or as a set of
narrow wave packets of coherent radiation. When the
initial population inversion crosses the threshold (9), the
ISSN 1562-6016. ВАНТ. 2013. №4(86) 271
process of generation undergoes qualitative changes.
The excess of the threshold (9) leads to an exponential
growth in the number of quanta. If we make the as-
sumption that the stimulated emission is mainly coher-
ent, the nature of this threshold can be explained as fol-
lows: generation of coherent radiation begins only after
crossing of this threshold. In this work, we have tried to
develop a qualitative model of this process.
It follows from results of numerical simulation that
the number of coherent quanta tends to μ0/2 with in-
crease of the initial population inversion in agree with
the theory of super radiance [9 - 11]. If we fix all pa-
rameters except the inversion, the duration of the coher-
ent pulse estimated by its half-width significantly in-
creases with increasing initial inversion in the absence
of absorption. The foregoing estimates of the character-
istic times of the process are confirmed by numerical
calculations. For relatively small inversion levels
0N Nμ<< << the coherent emission is always presents
as a rather short pulse with duration of τ ∼ (μ0/N). At
large times τ > (μ0/N) the incoherent radiation domi-
nates. Since the model (18) - (20) doesn’t take into ac-
count absorption of the incoherent radiation, it becomes
inapplicable after this time. It is important to note that
the time when the total number of photons reaches the
steady state in the model (21) - (22) after exceeding the
threshold (9) is comparable with the time when the
number of spontaneous photons achieves the same val-
ues Δτ ∼ τm =μ0/N in the model (18) - (20).
If the absorption is taken into account, even a small,
the coherent pulse duration remains almost unchanged
with an increase in the population inversion, at least far
enough above the threshold. The ratio of the pulse-
trailing edge duration to the pulse-leading edge duration
(the latter, by the way, is inversely proportional to the
initial inversion) is growing with increasing of the initial
inversion. The duplication of absorption reduces the
pulse duration by half. In an absorbing medium and
when with significant overriding of the threshold (9),
the difference between the traditional model and our
qualitative description become insignificant.
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Article received 16.04.2013.
О ВОЗМОЖНОСТИ ФОРМИРОВАНИЯ ИМПУЛЬСОВ КОГЕРЕНТНОГО ИЗЛУЧЕНИЯ
В СЛАБОИНВЕРСНЫХ СРЕДАХ
А.В. Киричок, В.М. Куклин, А.В. Мишин, А.В. Приймак, А.Г. Загородний
Обнаружено изменение характера процесса генерации излучения в двухуровневой системе при превышении началь-
ной инверсии заселенностей величины, равной корню квадратному из полного числа состояний. При превышении этого
порога число квантов начинает расти экспоненциально со временем. Сделана попытка пояснить природу этого порога:
при его превышении возникает генерация когерентного излучения в виде импульсов с коротким передним фронтом и
протяженным задним фронтом. Если все параметры, кроме инверсии, зафиксировать, то с ростом инверсии в отсутствие
поглощения длительность когерентного импульса, оцененная по его полуширине, заметно увеличивается. Учет потерь
энергии квантов приводит к тому, что длительность когерентного импульса практически не изменяется при росте инвер-
сии, по крайней мере, достаточно далеко от порога.
ПРО МОЖЛИВІСТЬ ФОРМУВАННЯ ІМПУЛЬСІВ КОГЕРЕНТНОГО ВИПРОМІНЮВАННЯ
В СЛАБОІНВЕРСНИХ СЕРЕДОВИЩАХ
О.В. Киричок, В.М. Куклін, О.В. Мішин, О.В. Приймак, О.Г. Загородній
Виявлено зміну характеру процесу генерації випромінювання в дворівневій системі при перевищенні початкової ін-
версії заселеності величини, що дорівнює кореню квадратному з повного числа станів. При перевищенні цього порога
число квантів починає рости з часом за експонентою. Зроблена спроба пояснити природу цього порога: при його пере-
вищенні виникає генерація когерентного випромінювання у вигляді імпульсів з коротким переднім фронтом і протяж-
ним заднім фронтом. Якщо всі параметри, окрім інверсії, зафіксувати, то з подальшим ростом інверсії при відсутності
поглинання тривалість когерентного імпульсу, оцінена за його напівшириною, помітно збільшується. Урахування втрат
енергії квантів призводить до того, що тривалість когерентного імпульсу практично не змінюється при зростанні інвер-
сії, принаймні, досить далеко від порога.
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