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Laser Physics, Vol. 14, No. 5, 2004, pp. 685 ­ 691.
Original Text Copyright © 2004 by Astro, Ltd. Copyright © 2004 by MAIK "Nauka / Interperiodica" (Russia).

PHOTONIC CRYSTALS

Nonlinear Magnetooptics in Magnetophotonic Crystals and Microcavities
O. A. Aktsipetrov1, T. V. Dolgova1, A. A. Fedyanin1, R.V. Kapra1, T. V. Murzina1, K. Nishimura2, H. Uchida2, and M. Inoue2
1

Department of Physics, M.V. Lomonosov Moscow State University, Moscow, 119992 Russia 2 Toyohashi University of Technology, Toyohashi, 441-8580 Japan e-mail: fedyanin@shg.ru
Received October 24, 2003

Abstract--In this article, recently published and new results of nonlinear magnetooptical studies of magnetic photonic band-gap materials--magnetophotonic crystals (MPCs) and microcavities--are surveyed. Nonlinearoptical magnetic Kerr effect in MPC with a half-wavelength-thick Bi-substituted yttrium­iron­garnet layer sandwiched between two dielectric Bragg reflectors is studied in transversal, longitudinal, and polar configurations of the dc magnetic field application when the fundamental radiation is in the resonance with the allowed microcavity mode located in the photonic band gap. Localization of the resonant fundamental radiation in the garnet spacer enhances the absolute values of both nonmagnetic (crystallographic) and magnetization-induced components of second-harmonic generation manifold. Specific symmetry properties of magnetization-induced second-harmonic generation (MSHG) allow the observation of magnetization-induced variations in the intensity and relative phase of the second-harmonic (SH) wave in transversal configuration. Magnetization-induced variations in the SH intensity with a contrast up to 0.7 and relative SH phase shifts up to 180° are observed upon changing the dc magnetic field direction. They are odd in magnetization due to the interference between magnetization-induced and nonmagnetic components of the SH field that manifest the internal homodyne effect in MSHG. The longitudinal nonlinear-optical magnetic Kerr effect reveals itself in rotation of the polarization plane of the reflected SH wave, which reaches values up to 250°/µm. The polarization rotation of the SH wave in the polar configuration is found to be approximately 20°/µm for fundamental radiation with a wavelength of 860 nm.

1. INTRODUCTION Photonic crystals and microcavities have been a subject of intensive studies over recent years [1]. Tremendous progress in microfabrication techniques of these photonic band-gap (PBG) materials brought about the observation of new phenomena in contemporary optics related to the interaction between the radiation field and matter. These studies also generate a vast range of technological applications of photonic crystals in microelectronics and optical and microwave communications. Magnetic materials are prospective components of photonic crystals since they open up the creation of magnetophotonic crystals (MPCs). These gyrotropic PBG materials with broken time-reversal symmetry yield a mechanism for molding the flow of light that is flexible under external control impacts, such as a dc magnetic field. For this reason, they can find widespread use as optical isolators, optical switchers, magnetic-field sensors, spatial light modulators, and as new materials for magnetooptical imaging and detection. MPCs with a single magnetic layer squeezed between two high-finesse Bragg reflectors have been designed recently [2]. Such MPCs act as microcavities and have a resonant optical transition (microcavity mode) located in the PBG. Spatial localization of the optical wave resonant to the microcavity mode leads to the enhancement of the Faraday effect observed in

MPCs with magnetic garnet [2], Co­ferrite [3], or Co­ Sm­O granular film [4] spacers. One of the prospective applications of photonic crystals is for enhancing nonlinear-optical effects, such as second-harmonic generation (SHG). For example, the anomalously small group velocity [5] of optical waves allows one to fulfill effectively the phase-matching conditions for SHG in the case where the fundamental or second-harmonic (SH) wave is tuned near the PBG edge. The SHG enhancement in one-dimensional (1D) photonic crystals--distributed Bragg reflectors-- due to the SHG phase matching was proposed in [6] and later observed in Bragg reflectors formed from different semiconductors and dielectrics in [7­10]. Another mechanism of SHG enhancement is realized in photonic-crystal microcavities possessing the microcavity (allowed) mode located in the photonic band gap. The fundamental field is strongly localized in the microcavity spacer if the wavelength and wave vector are close to the mode. This brings about the enhancement of the nonlinear-optical response of microcavities that was recently observed in microcavities with chromophore [11], polymeric [12], or semiconductor [13, 14] spacers. However, nonlinear magnetooptical studies of magnetophotonic crystals were initiated quite recently, as the nonlinear magnetooptical Kerr effect (NOMOKE) of MPCs has been observed in second[15, 16] and third-harmonic [17] generation. Nonlinear

685


686

AKTSIPETROV et al.

magnetooptical diffraction in magnetic films with an ordered periodical domain structure, which can be treated also as MPCs, was observed [18] and described [19] recently. In this paper, recently published and new results of transversal, longitudinal, and polar nonlinear magnetooptical Kerr effect studies in magnetophotonic microcavities (MMCs) formed from dielectric Bragg reflectors and a magnetic Bi-substituted yttrium­iron­garnet (Bi:YIG) spacer are presented. Magnetization-induced variations in the SH intensity and rotations of the SH wave polarization and the relative SH phase shifts are observed upon wavelength (angular) resonance of the fundamental radiation with a microcavity mode. The paper is organized as follows. Section 2 presents a brief phenomenological description of magnetization-induced second-harmonic generation in magnetic films and an analysis of the symmetry properties of the quadratic susceptibility tensor of a Bi:YIG spacer of MMCs. The details of the MMC sample fabrication and experimental setup are given in Section 3. Section 4 is devoted to the observation of the nonlinear magnetooptical Kerr effect in MPCs. The studies are concluded in Section 5. 2. BACKGROUND SHG arises from quadratic nonlinear polarization (2) P 2 at the double frequency of the fundamental radiation E, which is given in the electric-dipole approximation as follows: P
(2) 2

27], and nanoparticles [28]. In the electric-dipole approximation, MSHG arises from regions with broken space-inversion symmetry and MPCs have to be fabricated from noncentrosymmetric transparent magnetic materials. Bi-substituted yttrium­iron­garnet is an excellent material for this purpose due to the high magnetooptical response, small saturating fields, and transparency in the red and IR spectral regions [29, 30]. Bisubstitution is of crucial importance since bismuth atoms break initial cubic lattice symmetry and make garnet films macroscopically noncentrosymmetric. This has been directly probed by SHG studies of Bi:YIG films of different Bi content both in epitaxial films with different crystallographic orientation and polycrystalline layers [31­33]. Polycrystalline Bi:YIG film can be considered as an isotropic film in its plane and anisotropic along the normal. The corresponding symmetry group is m. Nonzero elements of (2, 0) and (2, 1) (pseudo)tensors are given in the Cartesian frame, with the z axis being the film normal and xz being the plane of incidence. (2, 0) has three nonequivalent elements: zzz,
zxx

=

zyy

,



xxz

=

yyz

.

(3)

(2, 1) has six nonequivalent elements [34]:
xzzY

= ­ = ­ = ­

yzzX

, ,

zxzY

= ­ = ­

zyzX

, , , (4)

yxzZ

xyzZ xxyX

,

yyyX xyyY

xxxY yxxX

yxyY

= ­

=

(2)

: E E ,

(1)

where (2) is the dipole quadratic susceptibility tensor. In magnetic materials with a simultaneous breaking of space-inversion and time-reversal symmetries, (2) becomes a function of the magnetization vector M; thus, the nonmagnetic (crystallographic) and magneti(2) zation-induced electric-dipole contributions to P 2 coexist, leading to magnetization-induced second-harmonic generation (MSHG) [20]. Function (2)(M) can be expanded into a series over M: (M) =
(2) ( 2, 0 )

where the capital subscript corresponds to the M component. Since in transparent materials (2, 0) is a real tensor but (2, 1) is an imaginary tensor, interference between nonmagnetic and magnetization-induced components of the total SH field requires absorption. Bi:YIG films have an absorption band above 550­ 500 nm depending on the Bi content [29]. This allows the high contrast multiple interference at the fundamental radiation in the red and IR spectral regions and the strong localization at the microcavity mode combined with the interference between nonmagnetic and magnetization-induced SHG components in the absorption band. 3. SAMPLES AND SETUP MMCs are grown on a glass substrate by RF sputtering of the corresponding targets in an Ar+ atmosphere with a sputtering pressure of 6 mTorr. Bragg reflectors of MMC are formed from five repeats of MC /4-thick SiO2 and Ta2O5 layers, where MC denotes the PBG center at normal incidence. The cavity spacer is a Bisubstituted yttrium­iron­garnet layer, Bi1.0Y2.5Fe5Ox , with an optical thickness of MC /2. The MMC samples have MC 900 nm and MC 1115 nm, which correspond to a Bi:YIG spacer thickness of approximately 195 and 245 nm, respectively. After fabrication of the
LASER PHYSICS Vol. 14 No. 5 2004

+

( 2, 1 )

M + ....

(2)

The nonmagnetic (crystallographic) contribution in the SH field is governed by tensor (2, 0), and the magnetization-induced SHG component is related to the pseudotensor (2, 1). Typically, magnetization-induced changes in the parameters of the SH radiation, such as its amplitude (intensity), polarization, and relative phase, are several orders of magnitude larger than in the magnetooptical Kerr effect [21]. This makes MSHG sensitive to the surface/interface magnetism that has been demonstrated in studies of ultrathin films [22, 23] and crystal surfaces [24, 25] in UHV conditions, planar nanostructures [26,


NONLINEAR MAGNETOOPTICS IN MAGNETOPHOTONIC CRYSTALS

687

SH intensity, arb. units

bottom Bragg reflector and the Bi:YIG spacer, the sample is annealed in air at 700°C for 20 min for residual oxidation and crystallization of garnet. Figure 1 shows a field-emission scanning electron microscope image of the MPC sample cleavage. The image demonstrates the high quality of the interfaces and the proper reflectors periodicity. The sample structure is clearly seen. MMCs are characterized by the linear transmission (solid circles in Fig. 2) and Faraday rotation (open circles in Fig. 2) spectra. The small transmission observed in the spectral interval from approximately 750 to 1000 nm corresponds to the photonic band gap of the MPC. The PBG spectral width and the drop of the transmission coefficient inside the PBG are determined by the number of periods and refractive index difference in the SiO2/Ta2O5 Bragg reflectors. The sharp peak in transmittance at 900 nm is attributed to the microcavity mode and shows the high quality factor of MPC, Q 75. The small red shift of its spectral position from the center of PBG indicates that the optical thickness of the Bi:YIG layer is slightly larger than a halfwavelength. The spectrum of the Faraday rotation angle of the linear polarized wave also has a peak at the microcavity mode where rotation is enhanced to approximately 1.5°. This corresponds to an effective value of 7.7°/µm, that is, approximately 50 times larger than the Faraday rotation angle for the single Bi:YIG films at these wavelengths. The output of an optical parametric generator directed at an angle of incidence = 30° is used for wavelength-domain SHG spectroscopy of the MMC sample with MC 900 nm. The pulse duration is 2 ns, the energy is below 5 mJ/pulse, and the fundamental wavelength is tunable from 730 to 1050 nm. Wavevector-domain SHG spectroscopy of the MMC sample with MC 1115 nm is performed by changing the angle of incidence of the 10 ns-YAG:Nd3+ laser output at 1064 nm with an energy of below 10 mJ/pulse. The SH radiation reflected from the MPC is selected by a series of glass filters (BG39) and detected by a photomultiplier tube and a boxcar. The polarization of the fundamental radiation is controlled by a Glan prism polarizer and varied by a Fresnel rhombus. The polarization of the SH radiation is controlled by a Glan prism analyzer. A saturating dc magnetic field with a strength up to 2 kOe is applied tangentially to the MMC for the longitudinal and transversal NOMOKE or normal to it for the polar one. 4. RESULTS AND DISCUSSION 4.1. Resonant SHG in Garnet MMCs Figure 3 presents the SHG spectra measured for different polarization combinations when the fundamental radiation is tuned across the microcavity mode. The spectra have peaks in the interval from 850 to 880 nm separated by 10 nm for s- and p-polarized fundamental radiation.
LASER PHYSICS Vol. 14 No. 5 2004

100 nm NONE COMPC 15CkV â 30 000 WD 7.0 mm

Fig. 1. Field-emission scanning electron microscope image of the MMC sample.

1.0 0 Transmittance ­ 0.4 0.6 ­ 0.8 0.4 0.2 ­ 1.2 Faraday rotation, deg 0.8

­ 1.6 0 600 700 800 900 1000 1100 1200 Wavelength, nm
Fig. 2. Spectra of the transmittance (filled circles) and the Faraday rotation angle (open circles) measured in a Bi:YIGbased MMC MC 900 nm at normal incidence.

SH wavelength, nm 400 410 420 430 440 450 460 470 12 10 8 6 4 2 0 800 820 840 860 880 900 920 940 Fundamental wavelength, nm
Fig. 3. The SHG spectra of garnet MMC measured in p-in, p-out and s-in, p-out polarization combinations (open and filled circles, respectively).


688 SH magnetic contrast 1.0 0.8 0.6 0.4 0.2 0 10 SH intensity, arb. units. 8 6 4 2 0 (b) (a)

AKTSIPETROV et al. SHG magnetic contrast 1.0 0.8 0.6 0.4 0.2 0 12 10 8 6 4 2 0 20 22 24 26 28 30 32 34 Angle of incidence, deg
Fig. 5. Intensity effects in MSHG: (a) the spectrum of the SHG magnetic contrast in the angular vicinity of the microcavity mode and (b) transversal NOMOKE measured in the p-in, p-out polarization combination for opposite directions of the magnetic field (solid and open circles, respectively).

(a)

(b)

854 856 858 860 862 864 866 868 Fundamental wavelength, nm

Fig. 4. Intensity effects in MSHG: (a) the spectrum of the SHG magnetic contrast in the spectral vicinity of the microcavity mode and (b) transversal NOMOKE measured in the p-in, p-out polarization combination for opposite directions of the magnetic field (solid and open circles, respectively).

The peaks correlate with the microcavity mode, which is spectrally shifted to a shorter fundamental wavelength for oblique angles of incidence and is ­2 observed at = MC(1 ­ n YIG sin2)1/2 depending on the refractive index nYIG of the Bi:YIG spacer. The SH intensity I2 at the microcavity mode is enhanced by at least a factor of 103 in comparison with that outside the PBG, where fundamental wave propagation is allowed. No significant increase in I2 is observed in the spectral region of the PBG edge. The Q factor of the SHG resonances, Q2 = 0 /, where 0 denotes the resonant fundamental wavelength and is the full width at half-maximum, is 160 ± 5, which is approximately twice as large as Q. In the transversal configuration of the dc magnetic field application and in the s-in, s-out and p-in, s-out polarization combinations, the SH intensity is found to be negligible. This confirms the symmetry considerations for nonzero elements of the (2, 0) and (2, 1) (pseudo)tensors, in which both magnetization-induced and nonmagnetic SHG contributions are expected to be zero. The SH intensity is found to be negligible also at the PBG edge in the s-in, p-out and p-in, p-out polarization combinations, where phase-matched SHG from layers forming mirrors is expected [8, 10]. Thus, the SiO2/Ta2O5 Bragg reflectors are considered as linear media. The primary mechanism of the SHG enhancement in garnet MMCs is the spatial localization in the

spacer of the fundamental radiation resonant with the microcavity mode. The spectral splitting of the SHG peaks for the s- and p-polarized fundamental wave is associated with the splitting of the microcavity modes for these polarization states, which can be a manifestation of the stress-induced anisotropy of the Bi:YIG layer and nonequal diagonal elements of the permittivity tensor, zz() and yy(). 4.2. Transversal NOMOKE Effect in MMCs Figure 4b shows the magnetization-induced SH intensity as a function of the fundamental wavelength measured in the spectral (angular) vicinity of the microcavity mode for two opposite directions of the dc magnetic field. The magnetic field is applied in the transversal configuration as M = (0, MY , 0). The SH intensity is enhanced as the fundamental wave is in resonance with the microcavity mode. The ratio of the intensities for the two directions of the magnetic field is almost 2. Figure 4a shows the spectral dependence of the magnetic contrast in the SH intensity, = (I+ ­ I-)/(I+ + I­), where + and ­ denote the direction of the field. reaches values of 0.3 and appears to be almost spectrally independent. Changing the magnetic field direction varies only the SH intensity, and no spectral shifts of SHG resonances are observed. Figure 5b shows the magnetization-induced SH intensity as a function of the angle of incidence measured
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SH intensity, arb. units


NONLINEAR MAGNETOOPTICS IN MAGNETOPHOTONIC CRYSTALS

689

in the angular vicinity of the microcavity mode in transversal configuration. Changing the magnetic field direction varies I2 approximately by a factor of 4. The spectrum of the SHG magnetic contrast shown in Fig. 5a is angular-independent and achieves values of 0.65. Magnetization-induced changes of the relative phase of the SH wave are observed using SHG interferometry [35]. The SHG interference patterns are obtained by translation along the laser beam of the SHG reference sample varying the distance l between the reference and the MMC sample. The SHG reference sample is a 30-nm-thick indium tin oxide film deposited on a fused quartz plate. The total SH intensity, I2(l, M), is produced by the coherent sum of the SH waves from the reference, E E2(M):
r 2

180 Magnetoinduced SH phase shift, deg

120
SH intensity, arb. units

9 6
175°

60

3

0

160 240 80 Relative distance, nm

, and the MMC sample,
2

0

25

26 27 28 29 Angle of incidence, deg

30

cr I 2 ( l, M ) = ----- E 2 ( l ) + E 2 ( M ) 8 =I
r 2

(5)

+ I 2 (M) + 2 I

r 2 I 2 (M

) cos (2 kl + rs(M)) ,

where k = 2n/ with n = n2 ­ n describing air dispersion, rs is the phase difference between the reference and sample SH waves, and < 1 is a phenomenological parameter accounting for both spatial and temporal coherences of the laser pulses. Changing the magnetic field direction shifts the SHG interference patterns by almost half of a period (inset in Fig. 6), which indicates a shift of the relative SH phase at approximately 180°. The SH phase shifts measured in the angular vicinity of the mode are shown in Fig. 6. Their values are slightly smaller than 180° and almost constant in . In p-in, p-out polarization combination, the nonNM magnetic (crystallographic) SH field E 2 , which is induced by zzz , zxx , and xxz elements of the (2, 0) tensor, interferes with the magnetization-induced SH field, M E 2 exp(i M), generated by xzzY , zxzY , and xxxY elements of the (2, 1) pseudotensor. I2(± MY) contains the cross-term ± 2 E 2 E 2 cos M , which changes the sign upon changing the magnetic field direction. This term leads to variations in I2, which are linear in M, and
NM M

Fig. 6. Phase effects in MSHG. The magnetoinduced shift of the relative SH phase measured in the transversal NOMOKE configuration in the angular vicinity of the microcavity mode. Inset: row SHG interference patterns for opposite directions of the magnetic field measured at = 28°.

ments, where the magnetization-induced shift of the relative SH phase is close to 180°. The spectral dependence of M is attributed to the Bi:YIG absorption at the SH wavelength. Absorption is vital for the SHG cross term, since in transparent materials (2, 0) is a real tensor NM but (2, 1) is an imaginary tensor and fields E 2 and E 2 do not interfere if they are generated at the same point in space. For small refraction angle in the Bi:YIG layer, the ratio between E estimated as E For 0.15.
NM 2 NM 2 M

and E

M 2

can be (6)

/E

M 2



xxxY

MY /(2

zxx

tan ) .

0.65 and M

0, it gives the ratio xxxY MY /zxx

depends on the relative phase M between E E
M 2

NM 2

and

. The constant value of in the vicinity of the
NM 2

microcavity mode indicates that the SH fields E
M 2

and

are enhanced similarly due to the fundamental E field localization. The large value of in angular MSHG spectroscopy is attributed to the small M value providing better interference between E 2 and E 2 . The fact that M takes values close to 0° or to 180° for opposite directions of M is seen also in phase measureLASER PHYSICS Vol. 14 No. 5 2004
NM M

4.3. Longitudinal NOMOKE Effect in MMCs Dependences of the SH intensity on the angle of rotation of the analyzer axis, , measured for opposite directions of the magnetic field at the microcavity mode are shown in Fig. 7. The longitudinal NOMOKE manifests itself in the magnetization-induced rotation of the SH wave polarization, which reaches values of 50°. The rotation angles increase as the angle of incidence decreases. In the longitudinal NOMOKE configuration, the nonmagnetic and magnetization-induced SH fields are NM polarized orthogonally, E 2 being p-polarized and E 2 s-polarized, respectively. The magnetizationinduced effects appear only in the rotation of polarizaM


690 (a) 90 120

AKTSIPETROV et al. 60 30 (b) 120 150 90 60 4 2
48°

10 SH intensity, arb. units

5

0 180

180

38°

00 2

5

250 240 270 300

330 210 240 270 300

330 4

10

Fig. 7. Polarization effects in MSHG. SH polarization diagrams measured for two opposite directions (open and solid circles) of the magnetic field applied in the longitudinal configuration. The angle of incidence of s-polarized fundamental radiation is 15° (panel a) and 30° (panel b). The zeroth value of the analyzer angle corresponds to the p-polarized SH wave. Curves are the fit to the intensity of the linear polarized wave.

tion of the total SH light. The SH intensity depends on the analyzer angle as follows: I2( ) E
NM 2

cos + E

M 2

exp ( i M ) sin ,

2

sors. For longitudinal NOMOKE and the s-polarized fundamental radiation, can be expressed in the following form: tan ( /2 ) F ( )
yyyX

(7)

M X/(

zyy

sin ) ,

SH intensity, arb. units

150

30

(8)

where the phase shift M describes the degree of ellipticity of the SH field. Since the SH polarization diagrams reach a value of I2 = 0, the SH wave is considered to be linear polarized with M 0. The angle of rotation of the SH field polarization upon changing the magnetic field direction is estimated as M NM 2arctan( E 2 / E 2 ) and depends on the ratio between the corresponding elements of the (2, 0) and (2, 1) ten9 SH wave polarization rotation, deg 8 7 6 5 4 3 2 1 0 880 884 888 892 896 900 Fundamental wavelength, nm

where F() is a weak function of the angle of incidence . According to Eq. (8), increases with decreasing . For = 48°, it gives the ratio yyyX MX /zyy 0.1, which is close to the value of the xxxY MY /zxx ratio estimated in transversal NOMOKE. 4.4. Polar NOMOKE Effect in MMCs The spectrum of the polar NOMOKE is shown in Fig. 8. Tuning across the microcavity mode leads to a gradual increase in from 1° to 7°. In the polar NOMOKE configuration, the magnetoinduced rotation of the SH wave polarization is observed due to the generation of the s-polarized magnetization-induced SH field by the yxzZ element and Faraday rotation of the SH and fundamental waves. Tuning through the microcavity mode of the ppolarized wave greatly enhances the Faraday rotation of the fundamental radiation due to multiple reflection in the Bi:YIG spacer [2] and leads to the appearance of the s-polarized component of the fundamental radiation, which allows the generation of the s-polarized SH wave in the mix-in, s-out polarization combination. The best condition for this magnetization-induced polarization plane rotation via the nonmagnetic yyz element is achieved as the s and p modes overlap, which leads to a spectral dependence of . For small angles of incidence, the value can be estimated as [15]
yxzZ

Fig. 8. Polarization effects in MSHG. Spectrum of the SH wave polarization rotation upon changing the magnetic field direction in the polar NOMOKE configuration.

MZ/

zxx

+ /2 + 2 ,
Vol. 14 No. 5 2004

(9)

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NONLINEAR MAGNETOOPTICS IN MAGNETOPHOTONIC CRYSTALS

691

where and 2 are the linear (Faraday) rotation angles of fundamental and SH waves, respectively. 5. CONCLUSIONS In conclusion, magnetization-induced second-harmonic generation is observed in magnetophotonic microcavities with a spacer formed from bismuthdoped yttrium­iron­garnet. Localization of the fundamental radiation, resonant with the microcavity mode, in the garnet spacer enhances the absolute values of both nonmagnetic (crystallographic) and magnetization-induced SHG in reflection from the Bi:YIG microcavity manifold. This allowed the first observation of transversal and longitudinal nonlinear magnetooptical Kerr effects in magnetic garnet films. Transversal NOMOKE reveals itself in the magnetization-induced variations of the SH intensity with a magnetic contrast up to 0.65 and in the large, close to 180°, magnetization-induced shift of the relative SH phase. Large, up to 50°, magnetization-induced rotations of the SH wave polarization plane are observed for the longitudinal NOMOKE. Multiple reflection interference of the resonant fundamental radiation, which additively enhances the Faraday rotation of fundamental wave polarization, results in enhancement of the SH wave polarization rotation in the polar NOMOKE configuration. ACKNOWLEDGMENTS The work was supported by a Grant-in-Aid from the Ministry of Education, Science, Culture, and Sport of Japan (grant nos. 14205045 and 14655119), the Russian Foundation for Basic Research, and the Presidential Grant for Leading Russian Science Schools. REFERENCES
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