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Influence of magnetic field on nonlinear magneto-optical diffraction on two-dimensional hexagonal magnetic bubble lattice Nataliya N. Dadoenkova, Igor L. Lyubchanskii, Maxim I. Lyubchanskii, and Evgeniy A. Shapovalov Donetsk Physical and Technical Institute, National Academy of Sciences of Ukraine, 72, R. Luxemburg str., 83114, Donetsk, Ukraine
Andrey E. Zabolotin Department of Physics, Donetsk National University, 24, Universitetskaya str., 83055, Donetsk, Ukraine
Theo Rasing Institute for Molecules and Materials, Radboud University Nijmegen, 6525 ED Nijmegen, The Netherlands Received May 10, 2004; accepted August 5, 2004 The nonlinear (at the second-harmonic frequency of the incident light) optical diffraction by a two-dimensional lattice of magnetic bubbles (cylindrical magnetic domains) is theoretically studied. Because the periods of these structures are comparable with the wavelengths of the fundamental and the second-harmonic radiation, diffraction at the second-harmonic frequency can be expected. We investigate the influence of an external magnetic field on the nonlinear magneto-optical diffraction. © 2005 Optical Society of America OCIS codes: 050.1940, 160.3820, 190.1900.
1. INTRODUCTION During past few years the nonlinear magneto-optical diffraction (NMOD) (at the second harmonic of the incident light frequency) by one- and two- dimensional magnetic domain structures has been investigated theoretically.1–4 First experimental observation of the NMOD from onedimensional periodic structures was reported by Lazarenko et al.5 It is well known that parameters of magnetic domain structures are very sensitive to an external magnetic field.6 As a result, the intensities of the nonlinearly diffracted light should be very sensitive to the magnetic field variation. The main goal of this paper is the theoretical investigation of the influence of an external magnetic field on the NMOD from a two-dimensional (2D) magnetic bubble lattice (MBL). The article is organized as follows. In Section 2 we present the polarization analysis of the diffracted secondharmonic generation (SHG) light. Section 3 is devoted to the NMOD theory. In Section 4 we present the numerical results. The main conclusions are summarized in Section 5.
2. GENERAL RELATIONSHIPS Let us consider a thin magnetic film of thickness h with a hexagonal MBL, which is located in the xy plane, with the z axis perpendicular to the surface. The external magnetic field is applied along the z axis. The magnetization inside the film and inside the magnetic bubbles is ori0740-3224/2005/010215-05$15.00
ented along the z axis and in opposite directions, respectively (see Fig. 1). The magnetization distribution in this magnetic film can be presented as follows7: m Z 共 r⬜兲 ⫽
M Z 共 r⬜兲 Ms
⫽1⫺2
兺 共 R k
k
⫺ 兩 r⬜ ⫺ rk 兩 兲 , (1)
where M s is a saturation magnetization, r ⫽ (r⬜ , z), r⬜ is the radius vector in the xy plane, (r) is a Heaviside function, and R k and rk are the kth magnetic bubble radius and the radius vector of the magnetic bubble, respectively. The summation in Eq. (1) is assumed to be over the entire MBL. The electric field of an electromagnetic wave at the second-harmonic frequency of the incident light can be determined as a solution of the wave equation with a nonlinear source term on the right-hand side8: ⵜ 2 E共 r, t 兲 ⫹
n 2 2 E共 r, t 兲 c2
t2
4 ⫽⫺ PNL共 r, t 兲 , c2
(2)
where c is the velocity of light in vacuum, n is the refractive index at frequency and PNL is the nonlinear polarization vector, which can be presented in the dipole approximation8: P iNL共 2 兲 ⫽ ijk 共 ⫺2 : , 兲 E j 共 兲 E k 共 兲 ,
(3)
where E() is the electric field of the incident light at frequency and ijk is the quadratic nonlinear optical sus© 2005 Optical Society of America
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Dadoenkova et al.
冕
A E i 共 2 , q兲 ⫽ ⫺ V
V
drP iNL共 2 , r兲 exp共 iqr兲 ,
(9)
where A ⫽ ⫺i /cn and V is the interaction volume. Substituting Eqs. (7) into Eq. (9), after integration we obtain the following results2:
冉
冊 冋
s;ss E s共 2 兲 ⫽ E p共 2 兲 p;ss
s; pp p; pp
册冉 冊
E 2s 共 兲 , E p2 共 兲
(10)
where
s;ss ⫽ 0, Fig. 1. Schematic image of nonlinear diffraction on magnetic bubble lattice.
ceptibility (NOS) tensor. For a magnetic medium the latter can be expanded on terms of the magnetization unit vector m ⫽ M/M s as9 0兲 m兲 ijk ⫽ 共ijk ⫹ 共ijk ,
(0) ijk
(4)
(m) ijk
where and are the magnetization independent and linear in magnetization parts of the quadratic NOS tensor9: m兲 m,1兲 共ijk ⫽ 共ijkL mL .
(5)
For a (001) magnetic film that is characterized by C 4v (0) symmetry, the NOS tensors in Eqs. (4) and (5), ijk and ( m,1) ijkL , are characterized by the following nonzero components8,9: 0兲 0兲 0兲 0兲 共xzx ⫽ 共yzy ⫽ 共xxz ⫽ 共yyz , 0兲 0兲 共zxx ⫽ 共zyy ,
0兲 共zzz ,
m,1兲 m,1兲 共xyzZ ⫽ ⫺ 共yxzZ .
(6)
Taking into account the nonzero components of the NOS tensors from Eqs. (6), we obtain the following components of the nonlinear polarization:
m,1兲 ⫹ 2 共xyzZ m Z 共 r兲 E y 共 兲 E z 共 兲 ,
0兲 ⫹ 2 共yyz E y共 兲 E z共 兲 ,
r兲 ⫽
⫹
(11)
In Eqs. (11) the Fourier component m Z (q) of the unit magnetization vector m(q) is calculated over the interaction volume V: m Z 共 q兲 ⫽
1 V
冕 冕 h
dz
0
S
m 共 r⬜兲 exp共 iqr兲 dr⬜ ,
(12)
where S is the laser spot area (see Fig. 1). As follows from Eqs. (10)–(12), the square of the magnetization vector Fourier component is proportional to the relative intensity of the diffracted SHG light: 兩 m Z 共 q兲 兩 ⬃
共 I 2s 兲 2
2
共 I p 兲 2 兩 s; pp 兩 2
.
(13)
As one can see from Eqs. (10) and (12), only one of four main possible geometries, namely, p( ) → s(2 ), allows for a NMOD, as in this case the SHG signal is determined completely by the m Z component of the magnetization. In other possible geometries, namely s( ) → p(2 ) and p( ) → p(2 ), the SHG is produced purely by the nonmagnetic part of the NOS tensor.2 The geometry s( ) → s(2 ) reveals no SHG signal.
兺 l
(7)
Within the slowly varying amplitude approximation the wave equation [Eq. (2)] for the second-harmonic electric field can be written as8
2 2ik 2 ,i ⵜl E l 共 2 , r兲 ⫽ ⫺ P iNL共 2 , r兲 exp共 iqr兲 , c2
0兲 ⫹ 共xxz sin 2 cos 2 .
m,1兲 共ijkL ⫽
0兲 2 共zyy Ey共兲
0兲 2 ⫹ 共zzz Ez 共 兲.
0兲 0兲 p; pp ⫽ 共 共zxx cos2 ⫹ 共zzz sin2 兲 sin 2
As the magnetic bubbles are ordered in two directions (along the x and y axes), the magnetization-induced NOS ( m,1) tensor ijk in Eq. (5) can be presented as follows:
共 m,1兲 P NL y 共 2 , r 兲 ⫽ 2 yxzZ m Z 共 r 兲 E x 共 兲 E z 共 兲
0兲 2 共zxx Ex共兲
0兲 p;ss ⫽ 共zyy 共 sin 2 兲 ⫺1 ,
3. NONLINEAR MAGNETO-OPTICAL DIFFRACTION
共0兲 P NL x 共 2 , r 兲 ⫽ 2 xzx E x 共 兲 E z 共 兲
P zNL共 2 ,
m,1兲 s; pp ⫽ 共yxz m Z 共 q兲 sin cos ,
(8)
where q ⫽ 2k ⫺ k2 is the wave-vector mismatch and k and k2 are the wave vectors of light at the fundamental and second-harmonic frequencies, respectively. After integration of Eq. (8) we obtain the electric field at the second-harmonic frequency
共 m,1兲 ijkL 共 l 兲 exp共 iQlr⬜ 兲 ,
(14)
where Ql is a reciprocal vector of the 2D magnetic superstructure and l is a 2D vector that is determined via two integer numbers, l 1 and l 2 . For the hexagonal MBL Ql can be presented as
冉
Ql ⫽ l 1 ⫺
1
冊
l 2 ex ⫹
冑3
(15) l 2 ey , 2 2 where ex and ey are the unit vectors directed along the x and y axes. The direction in which one can observe diffracted second-harmonic light can be determined from the nonlinear Bragg law for the three-wave interaction k2 ⫽ 2k ⫹ Ql. (16) For a hexagonal MBL film, we obtain from Eq. (16)
Dadoenkova et al.
共 l ,l 2 兲
sin 2 1
⫽
n n 2 ⫹
Vol. 22, No. 1 / January 2005 / J. Opt. Soc. Am. B
sin 2 D
冋冉
l1 ⫺
1 2
冊
l 2 cos ⫹
冑3 2
册
l 2 sin , (17)
( l ,l ) 2 1 2
where D is the period of the MBL, is the angle of nonlinear diffraction, is the incidence angle, 2 and n 2 are the wavelength and the refractive index at the frequency 2, respectively, and is the azimuthal angle (the angle between the crystallographic direction X and the optical plane). The number of diffraction orders for MBL in yttrium– iron garnet (YIG) were estimated in our previous paper.2
4. NUMERICAL RESULTS It is known that one can govern the period of MBL and magnetic bubble radiuses10 by changing the value of the external magnetic field, directed perpendicularly to the
Fig. 2.
217
magnetic film. The detailed analysis of the magnetic bubble radius and the hexagonal MBL period behavior as a function of the magnetic field is given in Ref. 10. Following the procedure described in Ref. 10 we have minimized the total energy density for the hexagonal MBL structure with respect to two dimensionless variables: the MBL diameter 2R/h and the period of MBL D/h. Subsequently, the period of MBL and the magnetic bubble radius as a functions of the magnetic field, obtained in Ref. 10, were numerically reproduced. For the numerical estimations we used the dimensionless parameter w /4 M s 2 h ⫽ 0.25 ( w is the Bloch-wall surface energy density). In Fig. 2(a) the dependence of the normalized magnetic bubble radius 2R/h on normalized magnetic field H/4 M s is depicted. The magnetic bubble radius monotonically decays with the magnetic field increase. In Fig. 2(b) the normalized period D/h of the hexagonal MBL is shown to vary slowly with the normalized magnetic field amplitude in the range 0 ⬍ H 0 /4 M s ⬍ 0.22 and to increase abruptly for values H 0 /4 M s ⬎ 0.22. These data were used for the further numerical calculations of the
Dependencies of (a) the normalized magnetic bubble radius and (b) the MBL period on the normalized external magnetic field.
Fig. 3. Dependence of 兩 m Z 兩 2 on the normalized wave vector qh for the different values of the external magnetic field H/4 M s . In the legends magnetic bubble radii R and MBL periods are given in the units of the film thickness h. The solid curve corresponds to the magnetic bubbles’ collapse and transition to the uniform magnetization state. The calculations carried out for the incidence angles are (a) ⫽ 30°, (b) ⫽ 60°.
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In Fig. 4 one can see the azimuthal dependencies of the diffracted SHG light for different values of the external magnetic field and the MBL lattice parameters. The calculations are performed for the incidence angle ⫽ 45°, and the laser spot radius is R L ⫽ 20h. The pronounced sixfold axis in the Figs. 4(a)–4(d) reflects the symmetry of the hexagonal MBL. We calculated the dependence of the angle of the NMOD [Eq. (17)] as a function of the period of the MBL for the first NMOD orders l 1 ⫽ l 2 ⫽ 1 (see Fig. 5), using the wavelength ⫽ 0.775 m (Ti:sapphire laser) and the refractive indices of YIG n ⫽ 2.309 and n 2 ⫽ 2.970.11
5. CONCLUSIONS
Fig. 4. Azimuthal dependencies of the first-order diffracted SHG light for different values of the external magnetic field and MBL parameters: (a) H 0 /4 M s ⫽ 0, D/h ⫽ 5.08, 2R/h ⫽ 3.72; (b) H 0 /4 M s ⫽ 0.18, D/h ⫽ 4.98, 2R/h ⫽ 2.46; (c) H 0 /4 M s ⫽ 0.25, D/h ⫽ 6.92, 2R/h ⫽ 2.17; (d) H 0 /4 M s ⫽ 0.28, D/h ⫽ 16.90, 2R/h ⫽ 2.11. The laser spot radius is R L ⫽ 20h.
In conclusion, we demonstrated the sensitivity of the NMOD from a 2D hexagonal MBL on the applied magnetic field, calculating the relative intensity and the azimuthal dependencies of the diffracted SHG light for different values of the external magnetic field and the NMOD diffraction angle as a function of the MBL period, governed by the magnetic field. In our opinion, the best candidates for the NMOD observation are thin films of impurity-doped YIG, as these materials combine all the necessary conditions for NMOD. First, in such samples the stable 2D hexagonal MBL can be realized under certain conditions.6 Second, YIG films are widely used in integrated magneto-optics.12 Third, SHG has been observed in the uniformly magnetized YIG films13–15 as well as in the YIG films with magnetic domains.14,16
ACKNOWLEDGMENTS This research was performed with the partial support of the International Association for Cooperation with Scientists from the former Soviet Union (INTAS) under Grant 03-51-3784. I. L. Lyubchanskii
[email protected].
can
be
reached
at
REFERENCES 1.
2. Fig. 5. Angle of NMOD as a function of the period of MBL for the first order of the nonlinear diffraction (l 1 ⫽ l 2 ⫽ 1). 3.
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4.
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