Reflection of resonant light from a plane surface of an ensemble of motionless point scatters: Quantum microscopic approach

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by  A. S. Kuraptsev

Reflection of resonant light from a plane surface of an ensemble of
motionless point scatters: quantum microscopic approach
A.S. Kuraptsev1 and I.M. Sokolov1,2
1Peter the Great St.Petersburg Polytechnic University, 195251, St.-Petersburg, Russia
2Institute of Analytical Instrumentation, Russian Academy of Sciences, 198103, St.-Petersburg, Russia
(Dated: January 15, 2016)
6
1 On the basis of general theoretical results developed previously in [JETP 112, 246 (2011)], we
0 analyzethereflectionofquasi-resonantlight from aplanesurfaceofdenseanddisorderedensemble
2 of motionless point scatters. Angle distribution of the scattered light is calculated both for s-
and p- polarizations of the probe radiation. The ratio between coherent and incoherent (diffuse)
n
components of scattered light is calculated. We analyze the contributions of scatters located at
a
differentdistancesfrom thesurfaceanddetermineonthisbackgroundthethicknessofsurfacelayer
J
responsible for reflected beam generation. The inhomogeneity of dipole-dipole interaction near the
4
surface isdiscussed. Westudyalso dependenceof totalreflectedlight powerontheincidenceangle
1
and compare the results of microscopic approach with predictions of Fresnel reflection theory. The

calculations are performed for different densities of scatters and different frequencies of a probe
]
h radiation.
p
PACSnumbers:
-
m
o
I. INTRODUCTION Collectiveeffects causedensity-dependentshifts of
t
a atomic transition as well as distortion of spectral
. line shape[28],[29]. The realpartofdielectricper-
s The vast majority of experimental optical de-
c mittivityofdenseatomicensemblecanbenegative
tectionmethods arebasedonanalysisofradiation
i
s scatteredfrominvestigatedmedium. Amongthese in some spectral area [30]-[32].
y
methods ones based on measurements of coherent Modificationofopticalcharacteristicscausedby
h
component of scattered radiation have a range of dipole-dipoleinteractionmanifestsitselfdifferently
p
[ advantages. Reflection of light from resonant me- for spatial areas inside the medium and near its
dia have even formed a special area of optics (see surface. The atomslocatedinthe subsurfacelayer
1 [1]-[16]andreferencestherein). Amongagreatva- interactpredominantlywithatomssituatedatone
v
rietyofresonantmediathedisorderedensemblesof side of them, inside the ensemble. The surface
3
7 point-like scatters which motion can be neglected layerisgenerated. Itswidthdependsondensityof
5 takeaspecialplace. Thephysicalmodelofmotion- the sample and is about 1.5-2 inverse wave num-
3 less scatterers is commonly used for description of bers[33]. Thissubsurfaceregionsignificantlyinflu-
0 interaction between impurity centers in solids and ences both the incoherent scattering and coherent
1. electromagnetic radiation. It also can be used for reflection.
0 description of cold atomic ensembles prepared in The goal of this paper is to study the reflection
6 special atomic trap. of quasi-resonant light from the plane boundary
1 Dense ensembles in which the average inter- betweenvacuumanddenseensembleofpointscat-
:
v atomic distance and the mean free path of photon ters. We analyze the influences of the features of
i are comparable with the wave length of resonant dipole-dipole interaction caused by the subsurface
X
radiationhaveattractedaspecialinterestrecently. spatial inhomogeneity on the properties of reflec-
r It is connected with both exciting physical prop- tion.
a
erties of such systems as its widespread practical Inhomogeneityoftheopticalpropertiesandspa-
applicationinquantummetrology,frequencystan- tial disorder of atoms in the ensemble restrict the
dardizationandquantuminformationscience[17]- classicaldescriptionusing Fresnelequations which
[25]. requirethemeanfreepathofphotonandthewave-
The interaction of resonant light with dense en- length of probe radiation much greater than the
semble has a range of important features which average interatomic distance. In this paper we
are usually neglected in case of dilute ensembles. usetheconsequentquantummicroscopicapproach
If the average interatomic distance is comparable [26]. This approach allows us to obtain both co-
with resonant wavelength the atoms can not be herentandincoherent(diffuse)componentofscat-
considered as independent scatters of electromag- tered light. Note that nearly all previous appli-
netic waves(hereafterwewillassociatepointscat- cations of the method developed in [26] were de-
ters with atoms for brevity). In this case we deal voted to analysis of incoherent scattering of the
with so-called cooperative scattering [26],[27]. In- light from random media, particularly for study
teratomicdipole-dipoleinteractionsignificantlyin- of multiple recurrent scattering. This analysis re-
fluencesontheopticalcharacteristicsofamedium. quires calculation of averageintensity of scattered
2
light. In the present paper for the first time we of the weak probe radiation [36]. Nonresonant
use this approach for description of coherent mir- states with two excited atoms and one photon
ror scattering from random media. We calculate ψemaemb′ are necessary for a correct description of
mean electric field of the light scattered by disor- the dipole-dipole interaction at short interatomic
dered media with sharp boundary as a sum of in- distances.
dividualcontributionsofallatoms. Suchapproach Theamplitudeofstateψg′ doesnotchangedur-
allowsustoanalyzethepartialcontributionsofthe ing the evolution of the system, because transi-
layersofthe mediumlocatedatdifferentdistances tions to this state from other states taken into
from the surface and analyze the properties of re- account are impossible. The transition from ψg′
flected wave depending on density of the atoms, to any other state is also impossible. The total
frequency of the probe light, its polarization and set ofequations for the other quantum amplitudes
on angle of incidence. bema , bg and bemaemb′ is infinite because of the infi-
nite number of field modes. From this infinite set
ofequationwecanpickoutthefinite subsetof3N
II. BASIC ASSUMPTIONS AND
algebraic equations for the Fourier component of
APPROACH
amplitudes of one-fold atomic excitation bema . Its
formal solution can we written as follows:
The calculationofresonantreflectioninthis pa-
pqueranwtuilml baeppmraodaechondevtheleopbeadsisinof[2a6].micTrhosiscoappic- bema (ω)=bX,m′Remaemb′(ω)b0emb′(ω) (2.1)
proachis basedonthe non-stationarySchrodinger
equation for the wave function of the joint sys- The matrix Remaemb′ is the resolvent of considered
tem consisting of N motionless atoms and elec- system projected on the one-fold atomic excited
tromagnetic field. All atoms are assumed to be states. This matrix describes the multiple pho-
identical and have a ground state J = 0 sepa- ton exchange among atoms and it depends both
rated by the frequency ω from an excited J = 1 on the spatial location of atoms and on the fre-
0
state. The excited state has the Zeeman structure quency of probe light (see [26], [32]). The vector
so there are three sublevels for each atom which b0em′(ω)describestheinteractionofatomswithex-
b
differ by the value of angular momentum projec- ternal radiation which is is assumed to be a plane
tion m = −1,0,1. Such structure of atomic levels monochromatic wave. This vector depends on the
allows us to describe the effects connected with direction of probe light and its polarization.
vector nature of electromagnetic field in case of
dtwenos-leevmeledsicuamlarcomroredcetllyd.oeNsonteotthhaatvethtehisstaanddvaarnd- b0emb′(ω)=−demb′;gbe/~E0exp(ik0rb) (2.2)
t[3a4g]e,.[3F5o]r. detailed comparison of these models see Inthisequationdemb′;gb isdipolematrixeleme′ntfor
transitionfromthegroundgtotheexcitedm state
The Hamiltonianofthe joined system H canbe of atom b, E is an amplitude of probe radiation,
0
presentedasa sum oftwooperatorsH0+V. Here k0 and e are its wave vector and unit polarization
H0isthesumoftheHamiltoniansofthefreeatoms vector, rb is the radius-vectorof the atom b.
and the free field, V is the operator of their inter- Microscopic approach allows us to consider
action. Thewavefunctionisfoundasanexpansion atomic ensemble with arbitrary shape and spatial
in a set of eigenstates {|li} of the operator H0. distribution of atomic density. In this paper we
The key simplification of the approach is in re- will consider sample in the form of a rectangular
strictionofthetotalnumberofstates|litakeninto parallelepiped with random but uniform on aver-
account. We consider only the vacuum state ψg′ agespatialdistributionofatoms. Theedgelengths
(all atoms are in the ground state and there is no of the parallelepipedarel , l andl . The quanti-
x y z
photon), the one-fold excited atomic states ψema zationaxisz is directedperpendicular to the front
(one atom is excited and there is no photon), the surface of an ensemble, the axis x along the pro-
one-foldexcitedstatesoffieldsubsystemψg (there jection of the wave vector of the probe light on
isonephotonandtherearenoexcitedatoms),and this surface. The angle between the direction of
the nonresonant states ψemaemb′ with two excited probe light propagation and the axis z is θ0. The
atomsandonephotoninthe fieldsubsystem. The polarization of light is assumed to be linear. We
complex index em used here contains information will analyze two types of linear polarization: par-
a
about both the number of excited atom a and the alleltotheplaneofincidence(p-polarization)and
Zeeman sublevel which is populated. perpendicular to this plane (s- polarization). The
In the rotatingwaveapproximationit is enough polarization vectors corresponding to these polar-
tSotattaeksewiintthoouatcceoxucnittatoinolnybtohtehsitnataetsomψeicma aannddfiψelgd. iuznaittiocnyscleicpvaencdtoerssec−an1,bee0,parnesdenet+e1duisnintghestbaansdisarodf
subsystemψg′ allowus to describe coherentstates relations [37]. The advantage of this basis is that
3
the dipolemomentprojectionsaregivenbysimple
equations deb−1;gbe−1 = de0b;gbe0 = de+b1;gbe+1 =
3~c3γ /4ω3 (γ is the natural linewidth).
0 0 0
pDescribing the polarization properties of scat-
tered light we will use the frame of reference con-
nected with the direction of scattered light (with
′
z axis alongthe wavevector)It is connected with
typical arrangement of polarization measurement
when polarization analyzers are oriented in accor-
dance with the direction of detected radiation. So
long as our paper is devoted mainly on the inves-
tigation of coherent scattering we will further re-
strictourselvestothecasewhenbothincidentand
scatteredwaveshavethesametypeofpolarization
(s- or p-).
Numerical calculation of the amplitudes bema (ω)
onthebasisof(2.1)-(2.2)allowsustoobtainampli-
tudesofotherquantumstatesbg andbemaemb′. Thus
we can obtain the wave function and any physi-
cal observable, particularly electric field strength
of scattered light and light intensity (see [26] for
detail). It gives us opportunity to compare the
results obtained in the framework of quantum mi-
croscopic approachwith Fresnel equations.
Theangulardistributionofscatteredlightpower
contains a speckle component because of light in-
terference from a big number of randomly dis-
tributed point scatters. In experiments the radi- Figure1: (color online)Angledistribution ofthescat-
ation averaged over an area of photodetector and tered light power. (a) ϕsc = 0, (b) θsc = π−θ0. 1
s-polarization, 2p-polarization, 3s-coherentcompo-
integrated over definite time interval is measured.
nent, 4 p-coherent component
Therefore in our calculations we perform multiple
averagingofresultsoverrandomspatialconfigura-
tions of atoms by Monte-Carlo method. To ana-
lyze the coherent component of scattered light we ◦
incidence is θ =17.5 . As the computational dif-
0
averagethe electric fieldstrengthandthenwe cal-
ficulty increasesrapidlywith the number ofatoms
culate intensity of this component. To calculate
the atomic density is chosen not very big n=0.05
total light intensity we average light intensity it-
which corresponds to the mean free path of pho-
self. Note also that this averaging allows taking
ton l = 1.63. However, as it was shown in [32]
ph
into accountpartlythe residualthermalmotionin
the collective effects causedby dipole-dipole inter-
cold atomic ensembles.
action play a significant role for such density.
Under considered conditions the reflection of
light takes place not only from the front surface
III. RESULTS AND DISCUSSION
of the ensemble but from the side surfaces as well.
To eliminate the influence of reflection from side
A. Angular distribution of scattered light surfaces we take into accountonly secondaryradi-
ation from atoms located sufficiently far from the
We start with analysis of angular distribution sides, approximately 60% of the total number of
of light scattered by optically dense plate. Fig- atoms. Inexperimentsucheliminationcanbeper-
ure 1 shows the power of light scattered in a unit formed by means of a diaphragm.
spherical angle as a function of both polar angle Fig. 1(a) shows that the maximum of the scat-
[Fig. 1(a)] and azimuthal angle [Fig. 1(b)]. The tered light power obtained in the frame of micro-
calculation is performed for atomic ensemble with scopic approach corresponds to a well known re-
l = 110, l = 55, l = 6.53 (hereafter in this pa- flection law (θ =π−θ , ϕ =0). Besides ofthe
x y z sc 0 sc
per we use the inverse wave number of resonant main peak we observe weak satellite peaks caused
light k−1 as a unit of length, k−1 = λ /2π). The by diffraction from the rectangular front surface
0 0 0
probe radiation is assumed to be exactly resonant andasmallcontributioncausedbydiffuse scatter-
with free atom transition, its frequency detuning ing. The height and width of the main peaks in
is equal to zero ∆ = ω − ω = 0, the angle of the Fig. 1(a) and Fig. 1(b) are determined by the
0
4
sizes l and l respectively. As we have l > l
x y x y
themainpeakofθ-distribution(Fig. 1(a))ismore
narrow than one of ϕ-distribution (Fig. 1(b)).
For comparison we performed similar calcula-
tions for different sizes of atomic ensemble. The
height of the main peak increased with size but
its width decreased so that the total intensity of
reflected light was proportional to the area l l .
x y
InFig.1weincludedtheresultsofcalculationof
total scattered light power and its coherent com-
ponent. The ratio of coherent component to total
powerexceeds0.85evenforrelativelysmallatomic
density n=0.05. Itconfirms the factthatcooper-
ative effects play a significant role for this density.
Note, that there are several physically different
cooperativeeffectswhichcantakeplaceunderlight
interactionwith dense and cold atomic ensembles.
Such phenomena as super-radiance, lasing in dis-
ordered media and Anderson localization attract
great attention recently. Physical effect studied in
thepresentworkhasabitdifferentnaturethanall
phenomena mentioned above. In our case collec-
tive effect does not assume multiple scattering. It
is determined by interference of secondary radia-
tion emitted by different atoms located in subsur-
face layer of the cloud. Similar interference causes
coherent Rayleigh forward scattering which cross
sectionisproportionaltosquarednumberofatoms
in all ensemble. In our case intensity of reflected
light is proportional to squared amplitude of elec- Figure 2: (color online) a) Angle distribution of light
tric field and consequently to squared number of scatteredbysubsurfacelayers;s-polarization;ϕsc =0;
atoms in the mentioned subsurface layer. In the depthofthelayer: 1-0.5l ,2-l ,3-2l ,4-7l ,b)
ph ph ph ph
next subsection we will analyze formation of re- MaximumofthefunctionP(θsc,ϕsc)dependingonthe
flected beam in more details. depth of the layer near the front surface; s- polariza-
tion;Forbothfiguresn=0.05,∆=0,θ0 =17.5◦,lx =
102,ly =51,lz =11.42
B. Microscopic analysis of reflected beam
generation
wavesscatteredbydifferentatomsbecomesimpor-
Letusanalyzenowhowtheatomslocatedatdif- tant and the destructive influence of the interfer-
ferent distances from the surface influense on the ence has to be taken into account. It looks like an
reflectedsignalgeneration. Ourapproachallowsus interference of light in transparentthin films. But
tocalculatethereflectedsignaltakingintoaccount in our case we deal with resonant atomic ensem-
only finite size layer near the front surface. Fig- bles and the absorption is very important. Under
ure2ashowscorrespondingresultsfor threelayers considered conditions the mean free path of pho-
withthedepthd=0.5l ,l ,2l (l isthemean ton is approximately equal to quarter of the reso-
ph ph ph ph
free path of photon obtained in [31]) as well as for nant light wavelength. Atoms located sufficiently
thewholeensemblewithdepth 7l . Incaseofthe far from the front surface do not influence on the
ph
first and the second layer the power of reflected coherent scattering, i.e. on reflection. It can be
light exceeds the total reflected signal. For the seen in the Fig. 2b. We observe saturation in de-
third layer the result is less than one for a whole pendence of reflected light power on the depth of
ensemble. Such behavior can be explained by the layer. The curve in Fig.2b flattens out at depth of
interferenceoftheelectromagneticwavesscattered thelayersgreaterthan(3.5÷4)l . Forbiggerden-
ph
by different atoms. When we consider relatively sity the mean free path of photon is smaller and
thin layers the phase increment on its thickness the saturation is observed for smaller depth.
is small so the interference is constructive and the Note, however that attenuation of the curve
powerofreflectedsignalincreaseswiththedepthof shown in the Fig.2b is essentially slower than it
layer. Ifthicknessofthelayerbecomescomparable can be expected if we suggest wave damping tak-
with wavelength the dephasing of electromagnetic ingintoaccountitspropagationintwodirections–
5
toward scattered atom and from them outside the
medium. Fig.2bshowstotalcontributionofatoms
located inside layer with thickness d. From this
contribution we can calculate partial contribution
ofatoms locatedin thin layersituatedatthe arbi-
trary depth l. Corresponding analysis shows that
this partial contribution decreases approximately
exponentiallyexp(−αl/cosθ )(thedeviationfrom
0
exponential dependence connects with mentioned
above boundary effects). Index of exponential at-
tenuation α is close to inverse mean free path of
photon in the considered medium α ≈ 1/l . So
ph
this index is two times smaller as compared with
its expected value if we consider light propagation
in two directions. This discrepancy connects with Figure 3: (color online) Dependence of reflectivity
on the angle of incidence; 1,2 s- polarization, 3,4 p-
the fact that inside the atomic ensemble there is
polarization; microscopic approach 1,3; Fresnel equa-
no coherent wave propagating in "backward" di-
tions 2,4; the parameters of ensemble are the same as
rection (of course, if the medium is semi-infinite
in Fig. 1
or scattering from far edge can be neglected). Re-
flected coherentlight beam exists only outside the
atomicensembleandthisbeamisaresultofcollec-
tivescatteringbyallatomsoftheensemble. Fig. 2 Figure 3 demonstrates that for the atomic den-
demonstrates that the determinative contribution sity n = 0.05 we have a good agreement between
to the reflectionsignalis givenby subsurfacelayer quantum microscopic approach and Fresnel equa-
with the depth comparable with wavelength. tions for p- polarization. However for s- polar-
izations there is noticeable quantitative disagree-
mentbetweenthesetwoapproaches. Thesituation
C. Comparison with Fresnel equations changes more dramatically for bigger densities.
Figure 4 shows the angular dependence of re-
Inthissubsectionweconsiderthedependenceof flectivity for the atomic density n = 0.5. Reso-
the reflection coefficient on the angle of incidence nant dipole-dipole interaction is so strong for this
and show that under considered conditions such density that it causes negative real part of the
dependence cannot be described by Fresnel equa- dielectric permittivity in some spectral area [30]-
tions. [32]. For example, for the probe light with detun-
Comparing our results with Fresnel theory we ing ∆ = γ the dielectric permittivity is equal to
0
have to take into account that the angular size of ε = −0.125+1.542i. The mean free path of pho-
reflectedlightconeinourcaseisfiniteevenforthe ton corresponding to these parameters l =0.55.
ph
plane incident wave because of finite sizes of the Curves in the Fig. 4.a are calculated for this case.
frontsurfaceoftheatomicsample. Forthisreason For comparison we add the plot corresponding to
it is naturally to determine the reflectivity R as the negative detuning ∆=−γ (ε=1.80+1.40i).
0
the ratio of the light power in the main maximum The mean free path of photon for this detuning
of angular distribution P (see Fig.1) to the total l = 1.03. Note that the dielectric permittivity
ph
power of light scattered in all directions P0. was calculated in [31] for spatially homogeneous
P can be obtained as an integral of the angular (on average) atomic ensembles by the analysis of
distribution of scattered light P(θsc,ϕsc) over the lightpropagationsufficientlyfarfromtheirbound-
reflected light cone Ωc. aries.
FromtheFig. 4itisclearlyseenthatresultsob-
P = P(θ ,ϕ )dΩ (3.1) tainedintheframeofmicroscopicapproachdiffers
Z sc sc
essentiallyfrompredictionsofFresnelequationsfor
Ωc
both polarizationchannelsaswellasforbothcon-
P0 can be obtained using the optical theorem. sidered detunings ∆=γ0 and ∆=−γ0.
Figure 3 shows the dependence of reflectivity In our opinion there are two main reasons for
on the angle of incidence. Two couples of curves such discrepancy. Firstof allFresnel equationsre-
are shown. The first one is calculated on the ba- quirethataveragedinteratomicseparationsshould
sis of microscopic approach and the second cou- be much less than light wavelength and photon
ple of curves are obtained from Fresnel equations. mean free path in consideredmedium. In our case
Dielectric permittivity required for corresponding itisnotso. Bothwavelengthandmeanfreepathof
calculation were calculated by the method de- photon are comparable with inrteratomic separa-
scribed in [30],[31]. tions. The secondimportantpeculiarityofconsid-
6
IV. CONCLUSION
In this paper we analyze the reflection of quasi-
resonant light from a plane surface of dense and
disordered ensemble of motionless point scatters
like impurity centers in solid. The calculation is
performed on the basis of quantum microscopic
approach. Solving the nonstationary Schrodinger
equation for the joint system consisting of atoms
and a weak electromagnetic field we calculate an-
gularandpolarizationcharacteristicsoflightscat-
tered by an ensemble in the form of a rectangular
parallelepiped with big optical depth. The ratio
between coherent and incoherent (diffuse) compo-
nents of scattered light is also studied.
Microscopic approach allows us to analyze the
influence of scatters located at different distances
from the surface. This analysis shows that main
contribution into reflected light comes from the
surface layerwhichdepth is determinedby several
mean free path of photon in considered medium.
It proves that the inhomogeneity of dipole-dipole
interaction near the surface essentially influences
on the coherent reflection.
Figure 4: (color online) Reflection coefficient depend- We studied the dependence of total reflected
ing on the angle of incidence, n = 0.5; ∆ = γ0 (a); light power on the incidence angle for both s- and
∆ = −γ0 (b); 1,2 s- polarization, 3,4 p-polarization; p- polarizations. The calculations are performed
microscopic approach 1,3; Fresnel equations 2,4
for different densities of scatters and different fre-
quencies of a probe radiation. The reflection coef-
ficient obtained in the framework of quantum mi-
croscopicapproachiscomparedwithFresnelequa-
ered physical conditions is essential role of bound-
tions. Itisshownthatadisagreementbetweentwo
ary effects. As we showed in the previous subsec-
approaches increases with atomic density. This
tion under resonant reflection the main contribu-
discrepancyisexplainedbysubsurfaceviolationin
tion into reflected signal is given by the surface
the spatialhomogeneityofthe mediumandbythe
layer which depth is about several photon mean
fact that for resonant light the mean free path of
free path. But just in this spatial domain the
photoniscomparablewiththeaverageinteratomic
inhomogeneity in optical properties of a medium
distance. It is shown that an important parame-
caused by the features of resonant dipole-dipole
ter here is k l . A disagreementbetween two ap-
0 ph
interaction is very essential [33]. Atoms located
proachesincreaseswithdecreaseofthevaluek l .
0 ph
in the surface layer responsible for reflected sig-
nalareindifferentphysicalconditionascompared We expect that observed disagreement between
with atoms located inside the medium. quantum microscopic approach and Fresnel equa-
tions (in case of resonant light) will be especially
At the end of this section note that the impor-
important for the case of light reflectionfrom thin
tantfeatureofquantummicroscopicapproachused
films. If the thickness of the film is comparable
in this work is that the resolvent matrix (2.1) is
with the resonant wavelength we can consider the
determined numerically so it does not allow us to
whole volume of the medium as the subsurface
consider atomic ensemble with verybig number of
area. The approachemployed in present work can
atoms. The calculations described in this paper
besuccessfullyusedforthiscaseevenforfilmswith
were made for 2000 - 7000 atoms. In this regard,
inhomogeneous spatial distribution of atomic den-
to make sure that observed results are not caused
sity. Furthermore,ourapproachallowstodescribe
by small size of the cloudwe repeated for compar-
the light scattering by nanoclusters with a small
isonour calculations for different sizes ofthe front
number of atoms.
surfaceofatomic ensemble. We increasedthe area
of the front surface from 1.2 to 1.6 times and the In our opinion, microscopic analysis of resonant
difference in the reflection coefficient was at the reflection performed here will be useful for further
level of computational error caused by statistical improvement of optical detection methods based
error of Monte-Carlo averagingmainly. on coherent scattering of resonant light.
7
V. ACKNOWLEDGEMENTS (State Assignment3.1446.2014K).A.S.K. appreci-
ates also financialsupport fromthe RFBR (Grant
WeacknowledgefinancialsupportfromtheRus- No.14-02-31422), the Council for Grants of the
sian Foundation for Basic Research (Grant No. President of the Russian Federation and the non-
RFBR-15-02-01013)and from the Ministry of Ed- profit foundation "Dynasty".
ucation and Science of the Russian Federation
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