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ОСНОВАН В МАРТЕ 1873 ГОДА 
ТОМ 165, ВЫПУСК 4 
ВЫХОДИТ 12 РАЗ В ГОД 
АПРЕЛЬ 2024 
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СОДЕРЖ АНИЕ
А ТО М Ы , М О Л Е К У Л Ы , О П Т И К А
Excitation of Wannier-Stark states in a chain of coupled optical resonators with linear gain and nonlinear
losses ............................................................................................................ Verbitskiy A., Yulin A. 
455
Атомистический анализ рекомбинационной десорбции водорода с поверхности вольфрама ............
........................ Дегтяренко Н. Н., Гришаков К. С., Писарев А. А., Гаспарян Ю. М. 
470
Диффузное рентгеновское рассеяние на пленке 1-додеканола на границе н-гексан-вода .................
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486
Особенности излучения смеси молекулярных газов .................Ж иляев Д . А., Смирнов Б. М. 
494
Я Д Р А , Ч А С Т И Ц Ы , П О Л Я , Г Р А В И Т А Ц И Я  И А С Т Р О Ф И ЗИ К А
Нелокальные гравитационные теории и изображения теней черных дыр .............................................
........................................ Алексеев С.О., Байдерин А.А., Немтинова А.В., Зенин О.И. 
508
Т В Е Р Д Ы Е  Т Е Л А  И Ж И Д К О С Т И
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527
П О Р Я Д О К , Б Е С П О Р Я Д О К  И Ф А ЗО В Ы Е  П Е РЕ Х О Д Ы  
В К О Н Д Е Н С И Р О В А Н Н Ы Х  С Р Е Д А Х
Ferromagnetic response of thin NiE flakes up to room temperatures ..........................................................
..O rlova N. N., Avakyants A. A., Tim onina A. V., Kolesnikov N. N., Deviatov E. V. 
536
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Исследование взаимосвязи топологического фазового перехода, аксионо-подобного состояния и магнитоэлектрического эффекта в антиферромагнитном топологическом изоляторе MnBi2 Te4 ....
.. Шикин А. М., Естюнина Т. П., Ерыженков А. В., Зайцев Н. Л., Тарасов А. В. 
544
О решении электростатических задач методом собственных функций ................Балагуров Б. Я. 
558
Э Л Е К Т Р О Н Н Ы Е  С В О Й С Т В А  Т В Е Р Д Ы Х  Т Е Л
Сильно нелинейный эффект Холла в макроскопически неоднородной двумерной системе ..............
................... Шуплецов А. В., Нунупаров М. С., Приходько К. Е., Кунцевич А. Ю. 
572
С Т А Т И С Т И Ч Е С К А Я  И Н Е Л И Н Е Й Н А Я  Ф И ЗИ К А ,
Ф И ЗИ К А  «М Я Г К О Й » М А Т Е РИ И
Генерация плоской стационарной ударной волны при предельно высокой передаче давления твердому веществу от малоплотного поглотителя излучения тераваттного лазерного импульса ............
............................................................................................................................................ Белов И. А.,
Бельков С. А., Бондаренко С. В., Вергунова Г. А., Воронин А. Ю., Гаранин С. Г.,
Головкин С. Ю., Гуськов С. Ю., Демченко Н. Н., Деркач В. Н., Змитренко Н. В., 
Илюшечкина А. В., Кравченко А. Г., Кузина А. А., Кузьмин И. В., Кучугов П. А.,
Мюсова А. Е., Рогачев В. Г., Рукавишников А. Н., Соломатина Е. Ю., Стародубцев К. В., Стародубцев П. В., Чугров И. А., Шаров О. О., Яхин Р. А. 
581
Термодинамический критерий нейтральной устойчивости ударных волн в гидродинамике и его следствия .............................................................................................................................Конюхов А. В. 
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ЖЭТФ, 2024, том 165, вып.4, стр. 455-469
©  2024
EXCITATION OF W A N N IE R  -  ST A R K  ST A T E S IN A CHAIN OF 
C O U P L E D  O PT IC A L  R E S O N A T O R S  W IT H  LINEAR GAIN A N D
N O N L IN E A R  LO SSES
A. Verbitskiy*, A. Yulin
School of Physics] mnd, Engineering, IT M O  University 
197101, St. Petersburg, Russia
Received July 14, 2023, 
revised version December 28, 2023 
Accepted for publication January 10, 2024
In this paper, we theoretically study the nonlinear dynamics of Wannier-Stark states in a dissipative system 
of interacting optical resonators whose resonant frequencies depend linearly on their number. We show that 
negative losses in some resonators can switch the system to a lasing regime with Wannier-Stark states acting 
as working modes. Our extensive numerical simulations show that single-frequency stationary regimes can exist 
in such a system as well as multi-frequency ones. In the latter case, Bloch oscillations can appear in the system. 
We investigate selective excitation of Wannier-Stark states enabled by an appropriate dissipation profile. A 
simple perturbation theory describing the quasi-linear regimes is developed and compared with the numerical 
results.
DOI: 10.31857/S0044451024040011
The advantage of optical systems over solid-state 
ones is that optical experiments for observation of the 
aforementioned effects are more feasible. Therefore, 
theoretical prediction of optical WSLs and BOs [31-42] 
was quickly followed by experimental demonstrations.
1. IN T R O D U C T IO N
Wannier-Stark ladders (WSLs) continue to be of 
great interest to scientists in different areas of physics, 
such as solid-state physics, condensed matter, and 
quantum magnets [1-3]. The WSL effect consists in 
the presence of equidistant lines in the spectrum, which 
correspond to the eigenmodes of the system (Wannier- 
Stark states) [4,5]. Beating between these states in time 
may result in periodic motion, i.e. Bloch oscillations 
(BOs) [6- 8].
BOs were first predicted in solid-state physics. 
However, their experimental observation in solids is 
quite challenging, and it took many years to confirm 
the effect experimentally [9]. BOs turned out to be a 
very common phenomenon, and they were found in a 
large variety of physical systems such as atomic systems 
[10-15], lasers [16], coupled LC circuits [17], mechanical systems [18-21], and plasmonic [22-27] or exciton- 
polariton systems [28-30].
E-mail: alexey.verbitskiy@metalab.ifmo.ru
One of the first experimental observations of an optical WSL was reported in [43]. Here, the Wannier- 
Stark (WS) states are realized using a chirped Moire 
grating. Another evidence of the existence of WSLs is 
presented in the work [44], where a spatial tilt of the 
minibands occurs due to a linear gradient of the optical thickness of the superlattice layers. In addition, 
WS states were observed in photonic lattices formed in 
a photorefractive material using relatively strong light. 
For this purpose, the formed lattices were irradiated 
with laser beams with a WS profile retrieved from a 
hologram [45]. Moreover, strongly localized WS states 
in a curved photonic lattice were obtained due to a 
large gradient of the refractive index and weak interaction between the waveguides [46]. BOs were also experimentally detected in the optical range, see, for example 
[47,48]. In these works, under the influence of temperature or by changing the waveguides’
 width, a linear gradient of the refractive index was created in a waveguide 
array, which led to periodic spatial oscillations of the 
light beam due to Bragg and total internal reflections
455

A. Verbitskiy, A. Yulin
ЖЭТФ, том 165, вып.4, 2024
at the opposite edges of the array. In addition, BOs 
were observed in porous silicon structures, in which, to 
incline the optical band authors used a cavity whose 
width increased linearly across the structure [49,50]. 
Besides, by bending a waveguide array, BOs can also 
be achieved in optical systems [51-53]. Another important system supporting BOs are parity-time synthetic 
photonic lattices [54,55]. A comprehensive review of 
the research on BOs and related phenomena is provided 
in [56].
to achieve lasing in these resonators. These are serious 
issues, but the advances of modern technologies allow 
us to expect that systems with the required parameters could be manufactured in near future. In particular, new materials such as perovskites demonstrate 
fascinating properties in laser devices [60], including 
very high linear gain. Dielectric resonators with a high 
quality factor based on bound states in the continuum 
(BICs), which have been actively developed recently 
[64, 65], are another promising platform for creating 
active microresonators with WS modes. The third possible platform for experimental observation of BOs is 
polariton lasers based on micropillars [66]. Considering these prospects, theoretical investigation of optical 
systems supporting WS states and BOs is of great physical interest, and the respective theoretical findings can 
boost further experimental activity in this direction.
The presence of dissipation, pump, and nonlinear effects in optical systems (for example, arrays of interacting nonlinear optical cavities) calls for generalization of 
the BOs theory to nonlinear dissipative systems. Let us 
note that optical systems such as microlaser arrays are 
promising sources of coherent radiation [57-62]. Thus, 
the study of these systems is not only of fundamental, 
but also of practical interest.
Active systems of microcavities with WS states and 
BOs have not yet been realized experimentally. However, lurrently, other optical and waveguide systems 
are actively studied in practice, in which interesting 
phenomena from the physics of WS states are also observed [21,30,63]. These advances in not only theoretical but also experimental demonstration of BOs in different optical systems stimulate further research of this 
effect in photonics and, in particular, in laser systems.
In this paper, we aim to study the nonlinear dynamics of WS states in one-dimensional systems of coupled optical yavities, in which each of the resonators 
supports only one mode defined by the material and 
geometry of the resonator. The described system is 
schematically shown in Fig. 1. To obtain a Bloch-type 
system, we introduce linear dependence of the cavities’
 
frequency on their index (i. e., number). A similar system driven by a train of coherent laser pulses is considered in [67], where the resonant excitation of WS modes 
and chaotic BOs were demonstrated. The present paper is focused on the dynamics of WS states in micro- 
laser arrays with population inversion created either by 
optical or electric pump.
Experimental implementation of microlaser systems 
with WS modes is indeed not an easy task. First, 
the manufacturing of such systems requires cutting- 
edge precise technologies that are not readily available. 
Second, the quality factors of typical optical microresonators are low, and thus, high linear gain is required
Below, we consider in detail different regimes of WS 
lasers, their switching from single-frequency to multi456

ЖЭТФ, том 165, вып.4, 2024
Excitation of Wannier-Stark...
125 m-1, which is determined by the effect of gain saturation and selection of the appropriate absorber. For 
the sake of mathematical convenience, we normalize the 
coefficients of the equation (1) by the strength of the 
coupling between the neighboring resonators a, and as 
a consequence, we obtain normalized time t, a = 1, 
(j, = 0.2, and (3=1. We choose 7 = 0.01 as an appropriate value for linear losses.
frequency regimes, and the appearance of BOs. To 
explain the behaviour of such systems near the lasing threshold, we develop a perturbation theory. We 
consider this work as a proof of concept rather than 
a discussion of the optimal experimental system, and, 
therefore, we choose the simplest lasing cavity model. 
We should acknowledge that for a real experiment, the 
scheme and, consequently, the theoretical model might 
require elaboration.
To describe the dynamics of light in microresonators, we use a well-known discrete model for slowly 
varying complex amplitudes Un(t) of the modes of individual resonators [68-79]:
idtUn +  a(Un+1 +  Un—
i) +  (mUn +  
+
+*Aj
 \u n 
y u „  =  o, 
(i)
The paper is structured as follows. For a systematic 
study of the problem, we start with the simplest case, 
in which only one resonator is pumped (Section 2 of 
the paper). In Section 3, we show that simultaneous 
excitation of several resonators makes the system’s dynamics richer, giving rise to multi-frequency regimes, 
including self-sustained BOs. In Section 4, mode selection is considered. We show that the efficiency of 
mode excitation depends on the pump profile, and by 
controlling the pump shape, we can extend the range 
of intensities where the single-frequency regime takes 
place. The main findings of the work are briefly discussed in the Conclusion.
2. SY ST E M S E X C IT E D  B Y  L IN E A R  G A IN  IN  
ONLY O N E R E SO N A T O R
where n is the index enumerating the resonators, a is 
the coupling strength between the resonators, p, accounts for the dependence of the resonant frequency 
on the resonator index, and j n and (3
n are the linear 
and nonlinear losses, respectively. Both j n and /3n can 
differ for different resonators. Let us note that here we 
consider a simple, but physically meaningful case: we 
assume that the nonlinear effects change the effective 
losses, but not the resonant frequencies of the individual resonators. We acknowledge that nonlinear correction of the resonant frequencies can be of importance, 
but it requires a special consideration, which will be 
done elsewhere.
We start with a simple case where 7,,. is negative 
in only one resonator with n = 0, and in all other resonators, 7„ is a positive constant. This means that we 
have a linear amplification in the resonator n = 0, and 
the other resonators have linear losses.
A sufficiently strong incoherent pump can not only 
change the linear losses, but also make them negative. 
Thus, such a pump can transform an individual cavity into a laser. However, as we consider a system of 
resonators, we need to calculate the effective gain of 
the supermodes of the system rather then the effective 
gain of individual resonators. For a rough estimate, we 
can consider the stationary states as a balance between 
the effective gain and effective nonlinear losses calculated for the WS state. Importantly, nonlinear losses 
might be present only in the pumped cavities, or in all 
the resonators. Further, we will show that in these two 
cases, the WS modes’
 dynamics is different.
We choose the linear losses to be j n = 7 for n y
A
 0 
and 70 = 7 —
 o, where a is the pump amplitude, and 
study the dynamics of the system numerically. Our numerical simulations reveal that only the trivial solution 
Un = 0 is possible as long as the linear gain a is lower 
than the lasing threshold, which depends on the parameters of the system 7 and (i. If the gain exceeds the 
threshold, the eigenmodes emerge in the system. If the 
dissipative and nonlinear terms are small, then these 
emerging modes can be very accurately approximated 
by the WS states, which are known analytically for the 
equation (1) in the conservative limit j n =
■
 0, [33]. 
The eigenvalues of the WS states form an equidistant 
spectrum ivm
 = /.mi with eigenfunctions
1
 I'// m
 
'If!' 'I
I
/ [
 
]
 :
V
 A
1
 J
where the index m enumerates the eigenstates. We use 
WS states normalized so that
V l F 2
 
= 1.
/  v 
n—
m
The parameters of a coupled waveguide array differ depending on their experimental implementations. 
We use typical data from work [47]: a = 125 m-1, 
(j, = 25 m-1, and 7 «  0.5 dB/cm, suitable for demonstrating the discussed effects. However, the value of 
linear losses 7 in this work is significantly higher than 
we need. In practice, this circumstance can be overcome by using high-Q BIC-based systems [64,65]. We 
also assume the nonlinear parameter /3 to be equal to
457

A. Verbitskiy, A. Yulin
ЖЭТФ, том 165, вып.4, 2024
Fig. 2. (Color online) a— Effective linear gains —
Гт  of the Wannier-Stark states with the fastest-growing amplitudes (m = ±8) 
vs. the pump amplitude a: obtained by numerical simulation (red circles), the perturbation method (dash-dotted blue line), and 
the eigenvalues (dashed green line), b
 — Effective linear gains —
 Гт of Wannier-Stark states with m = ±8 (red circles), m = ±9 
(brown circles), and m = ±2 (magenta circles) vs. the pump amplitude a, obtained by numerical simulation. The solid lines are
guides for eyes. The used parameters are: fi = 0.2, T =  0.01
Tm = 'y -a W lm. 
(4)
If the dissipation is so low that it does not affect 
the spatial structure of the eigenstates, a simple perturbation theory can be developed. The quantity
e =
 J
2 \
u
-\2
П
(energy of the field in the system) is conserved if j n = 0 
and /3
n = 0. If у,, and /3
n are nonzero, but small, the 
field in the system can be found in the form
The intensity distributions of the WS states are symmetric and have two main maxima, located symmetrically with respect to the center of the mode. Therefore, if the system is excited by linear gain only in one 
resonator, then there are two modes with the fastest- 
growing amplitudes and the same increment. For the 
parameters used in the numerical simulations, the indexes of such modes are mmax = ± 8.
= Am(t)Wn- m exp(ifimt),
where Am(t) is the time-dependent complex amplitude 
of the m-th WS state. Substituting this into (1), multiplying by Wn- m, and calculating the sum over n, we 
obtain ordinary differential equations for Am:
&tAm 
I''m
 -I'm
 
Ь
З
П
!
 | ,1,,, | ~
 ,17
I
( 2 )
where
г т  =  E
 7«w„2_ m, 
в т =
Now let us compare the results of the perturbation 
theory with those of the direct numerical simulations 
of the master equation (1). It is natural to introduce 
the effective linear gain of a mode as —
Гт . Figure 2 a 
shows the effective linear gains extracted from the numerical simulations and calculated by formula (4) as 
functions of the pump amplitude a. One can see that 
in the vicinity of the lasing threshold, where the dissipative terms can be considered as small corrections, 
the results of the perturbation theory are in very good 
agreement with the numerical simulations.
n 
n
are the effective linear and nonlinear losses for the m-th 
mode.
The complex frequencies of the modes can also be 
found by analyzing the linearized equation for the amplitudes Un:
idtUn +  a(Un+1 +  Un—
i) +  i~mUn +  iynUn =  0. 
(5)
For a purely dissipative nonlinearity (i.e., affecting 
only the effective losses, but not the resonant frequency 
of the cavities), the equations (2) can be re-formulated 
as a set of equations for the intensities Im
 = |Д „|2:
Then, by choosing a solution in the form
dtlm —
 2(—
Г 
m/ m
 —
 BmI;n). 
(3)
Un(t) = Vn exp(iujt Jk
For our choice of j n 
7 aSon (% is the Ivronecker 
we obtain an eigenvalue problem: 
symbol), the sum in the expression for the effective linear losses Гт  can be easily calculated analytically: 
ivVn = a(Vn+\ + F)j- i) + iinVn + bfaVn. 
(6)
:
458

ЖЭТФ, том 165, вып.4, 2024
Excitation of Wannier-Stark...
grow. The numerical simulations show that if there are 
nonlinear losses only in the pumped resonator, a singlefrequency stationary state is formed as a WS state with 
m = 8 or m = —
 8. The probability of the formation of 
each of the states is 1/2. The formation of the stationary states is illustrated in Fig. 4 a, b, d and e.
The real part of to is the eigenmode frequency, the imaginary part is its dissipation rate, and the eigenvector Vn 
describes the structure of the eigenmode. If there are 
no dissipative terms, the eigenstates are the conservative WS states discussed above. The solution of the 
spectral problem allows us to find the exact solutions 
for the eigenstates in the dissipative case. We solved 
the spectral problem numerically to confirm that the 
dissipative terms do not significantly affect the structure of the eigenmodes.
If nonlinear losses are distributed evenly in the system, there are different regimes of stationary states formation. The excitation thresholds, of course, remain 
the same, but the stationary state forming from a weak 
noise varies periodically in time. Very close to the excitation threshold, the stationary state can be considered 
as a superposition of the WS states with m = 8 and 
m = —
8;
 consequently, the stationary state contains 
temporal harmonics with frequencies equal to the WS 
states’
 eigenfrequencies. The formation of such a state 
is illustrated in Fig. 4 c,f.
Comparing effective linear gains of different WS 
states can also be useful. The numerically found —
 
Tm
 
for the six modes with the fastest-growing amplitudes 
are shown in Fig. 2 &
 as a function of the pump amplitude a. One can see that for our parameters, the modes 
with the fastest-growing amplitudes and the lowest lasing threshold are the modes with m = ± 8;
 the second 
and the third fastest-growing modes have the indexes 
m = ±9 and m = ±2, respectively.
The intensity of the stationary states Im
 formed in 
the system can be easily found from (3):
To explain such a behaviour of the system, we expand the perturbation theory described above by writing the equations for the amplitudes A± of two interacting modes m = ±m with the highest effective linear 
gains. Thus, we seek the held in the form
U„ i= A+Wn-m exp(imt) + A -W n+lb exp(-im t).
Substituting this ansatz into (1) and projecting the 
equation on the eigenstates, we obtain equations for 
A±. These equations can be reduced to equations for 
the intensities I± in a similar way to (3):
dtI+ = —
2(Г +  BI+ + £/_)/+, 
(8)
dtI -  =  —
2(Г +  B I- +  % ) / _ ,  
(9)
where
S = 2] T /3nW^_Aif/ +
Л
щ
 
г = Г±А.
n
Figure 3 shows the dependencies of the stationary intensities of three pairs of WS states with the highest 
effective linear gains on the pump a for two cases: (a) 
when the nonlinear losses are nonzero only in the excited resonator with n = 0: /3
q
 = /3 and (&
) for spatially 
uniform nonlinear losses: [
3
n = /3. The stationary intensities can be higher for the modes with lower effective linear gains, see Fig. 3 a. The possible reason is, 
if the nonlinear losses are nonzero only in the excited 
resonator, the modes with the highest effective linear 
gains have the highest nonlinear losses, and their ratio (7) is lower than that of the modes with the lower 
effective linear gains.
As we derived these equations, we assumed that the difference between the eigenfrequencies of these states is 
large, and we can safely neglect the quickly oscillating 
terms.
Let us analyse the fixed points of the dynamical 
system (8)-(9). For Г > 0, there is only a trivial solution I± = 0. For negative losses (and, correspondingly, 
positive gain), there are four solutions:
Our numerical simulations reveal that for small 
pump intensities, only one pair of the WS states with 
the highest effective linear gain is dynamically stable. 
The dependencies of the stationary intensities of the 
WS states extracted from the numerical simulations 
are shown in Fig. 3. The perturbation theory and the 
numerical simulations are in good agreement for low 
pump intensities.
I± = 0;
-Г
/_
1+ —
 o,
1 Г ’
-Г
1- = 0,
1+
1 Г ’
-Г
I± =
It can be interesting to study the dynamics when 
the initial conditions have the form of low-intensity 
noise. As we mentioned above, the modes are formed 
when the pump exceeds a certain threshold. We choose 
the pump exceeding only the threshold for the modes 
with the largest increment. Thus, for the parameters 
we chose, only the amplitudes of the modes m = ±8
B + B'
459

A. Verbitskiy, A. Yulin
ЖЭТФ, том 165, вып.4, 2024
nonlinear losses 
nonlinear losses
Fig. 3. (Color online) Stationary intensities Im of different Wannier-Stark states vs. the pump amplitude a, obtained with the 
perturbation method (dash-dotted lines) and by numerical simulation (circles) for (a) nonzero nonlinear losses only in the excited 
resonator with n =  0, i.e., /Зо =  /3, and (&) spatially uniform nonlinear losses, i.e., /3
,,. =  /3. The used parameters are: fi =  0.2,
7 =  0.01, /3 =  1
nonlinear losses 
nonlinear losses
nonlinear losses 
in all resonators
8
6
4
2
n
0
-20 
0 
20 
П
-20 
0 
20 
П
Fig. 4. (Color online) Stationary states \Un\
 in the form of WS states with m =  —
8 (a), m =  8 (&), m =  —
8(c) and m =  8 
(time-averaged field); figs, d, e and /show  the respective evolutions of the field module |t/IJ(f)|. The pump amplitude slightly 
exceeds the excitation threshold. Nonlinear losses are nonzero only in the excited resonator with n =  0, i.e., /Зо =  /3, for a, b, d, 
and e; nonlinear losses are spatially uniform, i.e., /3,, =  /3, for c and /. The blue circles correspond to the resonators, the solid 
blue lines are guides for eyes, and the dashed red lines correspond to the pumped resonator with n =  0. The used parameters
are: ц =  0.2, 7 =  0.01, a = 0.1, /3 =  1
for Г <0, Ai is negative for В < В and positive otherwise. Therefore, this state can be either a stable node 
for В < В or a saddle for В > B. The last state
-Г
We can directly explore the stability of these states 
by writing linearized equations for small perturbations 
of the intensities I± and finding the eigenvalues governing the evolution of the perturbations. The trivial state is, of course, always unstable Л17 = —
2Г. 
The second and the third states have the eigenvalues 
Ai = —
 
2Г(1 —B/B) and A
2
 = 2Г; A
2
 is always negative
I±
B + B
460

ЖЭТФ, том 165, вып.4, 2024
Excitation of Wannier-Stark...
has the eigenvalues
For /4
 = 0.2 used in our direct modelling, the coefficients are В = 0.07 and В = 0.04. Therefore, in this 
case, there is only one stable stationary state
Л
2Г
B + B (в ± в у
-Г
I±
B + B
From this, we can conclude that this state is stable (a 
stable node) for В > В or unstable (saddle) for В < B.
Thus, the stability analysis tells us that if В > В, 
then for the system (8)—
(9), there is only one stable 
stationary state,
-Г
B + B
Thus, we can expect that in this case, the final state 
consists of two WS states of the same intensity, oscillating with different frequencies. 
This perfectly 
agrees with the results of our numerical simulations, 
see Fig. 4 c. In addition, for p, > 0.6, there are regions 
where В > В, see Fig. 5 a. Hence, in these bands, there 
should be two stable states:
For В < B, there are two stable states:
1+
0,
-Г
-Г
~B*:
/ + = 0, 
/_
I T ’
and
and
/+ = 1 Г ’
 
=
C  = T  
7_=C.
instead of the previously observed single state. This is 
confirmed by numerical calculations.
Now let us estimate the values В and B. When only 
/?o 
0, then
n
and
B = 2 Y JPnWl_1
-J¥l+lb= 2W i.
П
We would like to note that the developed perturbation theory not only gives a qualitative explanation of 
the observed effect, but also allows determining the intensities of the two-component states with a good precision. The intensity I±(t) dependencies extracted from 
numerical simulations overlap with those calculated by 
formulas (8)-(9). For low linear gain a, the simulated 
and calculated results are in good agreement.
This means that В = 2B, and, as our stability analysis 
shows, in this case, the stable stationary states are
3. L A SIN G  W IT H  L IN E A R  G A IN  IN  SE V E R A L  
R E SO N A T O R S
1+
0,
-Г
I T ’
and
In numerical simulations, only a stable state can be 
observed as a stationary state, which explains why for 
the chosen [
3
n, we see the formation of either one or the 
other WS state.
To increase the radiation power, it seems reasonable to introduce linear gain in several resonators. Let 
us first consider nonlinear losses present only in the 
pumped resonators. If the gain is uniformly distributed 
in the pumped resonators, we expect lasing to begin at 
lower pump amplitudes for a larger number of pumped 
resonators. Thus, Fig. 6 a shows the total energy E of 
the single-mode stationary state as a function of the 
pump amplitude a for different numbers of pumped 
neighbouring resonators M. In this figure, the stationary energy values E obtained via the perturbation 
method (solid line) and by numerical simulations (circles) are in good agreement for different M .
When /3
n = /3, the ratio between В and В can be 
different. The coefficient В depends on the overlap of 
the intensity distributions of the states Wn±m, and this 
overlap decreases with increasing width of the WS state 
defined as
H =  l'Z ,W * { n -n c)*,
У
 n
The single-frequency state is the only possible solution within the pump range athi < a <
; ath
%
$
 where athi 
is the excitation threshold for the pair of WS modes 
with the fastest-growing amplitudes, and ath2 is the 
excitation threshold of the second fastest-growing pair. 
The simulations show that if the number of the pumped
where nc is the center of the WS state. Figure 5 shows 
the dependencies of В and В on the width of the states 
H (a state width H is determined by /4).
461