Magneto-optical enhancement through gyrotropic ... - OSA Publishing

Magneto-optical enhancement through gyrotropic gratings Y. H. Lu,1 M. H. Cho,1+ J. B. Kim,1 G. J. Lee,1 Y. P. Lee,1∗ and J. Y. Rhee2 1 Quantum

2 BK21

Photonic Science Research Center and BK21 Program Division of Advanced Research and Education in Physics, Hanyang University, Seoul 133-791, Korea

Physics Research Division and Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea + Co-corresponding

author: [email protected]

∗ Corresponding

author: [email protected]

Abstract: Diffracted magneto-optical (MO) effects are numerically investigated for one-dimensional lossy gyrotropic gratings in the zeroth and the first orders for the polar magnetization by utilizing the rigorous coupled-wave approach implemented as an Airy-like internal-reflection series. The simulated Kerr spectra agree well with the experimental ones. The dependence of the MO Kerr enhancement on the grating depth in the first-order diffraction, compared with that in the zeroth one, is illustrated, and the diffracted MO Faraday effect is theoretically investigated as well. Such a MO enhancement through the gyrotropic gratings is superior to the conventional MO devices and magneto-photonic crystals. The potential applications are also suggested. © 2008 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (050.1960) Diffraction theory; (050.5298) Photonic crystals; (160.3820) Magneto-optical materials.

References and links 1. M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768– 5770 (1999). 2. E. Takeda, N. Todoroki, Y. Kitamoto, M. Abe, M. Inoue, T. Fujii, and K. Arai, “Faraday effect enhancement in Co-ferrite layer incorporated into one-dimensional photonic crystal working as a Fabry-P´erot resonator,” J. Appl. Phys. 87, 6782–6784 (2000). 3. M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Technol. Lett. 12, 1171–1173 (2000). 4. M. J. Steel, M. Levy, and R. M. Osgood, “Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects,” J. Lightwave Technol. 18, 1289–1296 (2000). 5. M. J. Steel, M. Levy, and R. M. Osgood, “Photonic bandgaps with defects and the enhancement of Faraday rotation,” J. Lightwave Technol. 18, 1297–1308 (2000). 6. I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, “Magnetic photonic crystals,” J. Phys. D: Appl. Phys. 36, R277–R287 (2003). 7. M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys. 39, R151–R161 (2006). 8. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003). 9. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Institute of Physics Publishing, London, 1997).

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Received 2 Jan 2008; revised 24 Mar 2008; accepted 29 Mar 2008; published 3 Apr 2008

14 April 2008 / Vol. 16, No. 8 / OPTICS EXPRESS 5378

10. S. Fan, M. F. Yanik, Z. Wang, S. Sandhu, and M. L. Povinelli, “Advances in theory of photonic crystals,” J. Lightwave Technol. 24, 4493–4501 (2006). 11. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981). 12. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996). 13. K. Watanabe, “Study of the differential theory of lamellar gratings made of highly conducting materials,” J. Opt. Soc. Am. A 23, 69–72 (2006). 14. M. Grimsditch and P. Vavassori, “The diffracted magneto-optical Kerr effect: what does it tell you?” J. Phys.: Condens. Matter 16, R275–R294 (2004). 15. A. Westphalen, A. Schumann, A. Remhof, H. Zabel, T. Last, and U. Kunze, “Magnetization reversal of equilateral Fe triangles,” Phys. Rev. B 74, 104417 (2006). 16. J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006). 17. F. Jonsson and C. Flytzanis, “Nonlinear magneto-optical Bragg gratings,” Phys. Rev. Lett. 96, 063902 (2006). 18. J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of a magnetic grating,” J. Appl. Phys. 101, 09C518 (2007). 19. R. Antos, J. Mistik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first-diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005). 20. R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evaluation of the quality of Permalloy gratings by diffracted magneto-optical spectroscopy,” Opt. Express 13, 4651–4656 (2005). 21. R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006). 22. H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue, “Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003). 23. G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003). 24. P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999). 25. D. F. Edwards, “Silicon (Si)” in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic, New York, 1998); H. R. Philipp, “Silicon Dioxide (SiO2 ) (Glass),” ibid. 26. E. Hecht, Optics (Addison-Wesley, New York, 1998). 27. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A: Pure Appl. Opt. 1, 646–653 (1999).

1. Introduction It is well known that a magneto-optical (MO) enhancement can be realized in magneto-photonic crystals (MPCs) [1, 2, 3, 4, 5], which are very promising for applications in modern photonics. In contrast to the conventional photonic crystals, MPCs possess an additional degree of freedom, spin, so that their optical and MO properties can be tuned with applied magnetic field [6, 7]. In such structures, the time reversal symmetry is broken down and entails nonreciprocity in the system [6, 8, 9]. As a result, they are expected to be a good candidate to provide sufficient signal isolation and suppress parasitic reflections between devices in on-chip large-scale integration. More importantly, one of the advantages of MPCs over the conventional MO devices is that MPCs need much less propagation distance to occupy a large footprint because the weakness of MO effects [10] is overcome by light localization in the vicinity of the defects with a consequent increase in the mean optical-path length of the emitted light [4]. The gyrotropic gratings and the MPCs are different in terms of the geometry for applications, as well as the theoretical model. For example, light generally propagates along the periodic direction of alternating magnetic and nonmagnetic layers in MPCs. For gyrotropic gratings, light is illuminated upon the patterned surface, and more stress is laid on the diffraction efficiency [11, 12, 13], the magnetization reversal process [14, 15] and so on. In fact, one definition in#91178 - $15.00 USD

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cludes another, and vice versa [7, 16, 17, 18], because both of them make use of periodic magnetic structures to modulate the MO effects. Antos et al. [19, 20] have observed a MO enhancement experimentally at the first-order diffraction from gyrotropic gratings. However, unlike Kim et al. [16, 18], they did not focus on the significance of the observation. Anyhow, it was supposed that gyrotropic gratings share some of the aforementioned advantages of MPCs. It would be of great significance if the MO enhancement is also able to be obtained through gyrotropic gratings. A huge decrease in thickness compared with the conventional MPCs, from the micrometer level down to the nanometer one, benefits the miniaturization and the integration of MO devices considerably. In this paper we focus on the demonstration of the geometrical effects on the diffracted MO enhancement utilizing a rigorous theoretical model to achieve trustful and predictable analyses. The influences of the depth of grooves on the diffracted MO enhancement are systematically investigated in order to explore the possibility of miniaturizing and integrating the MO devices. Furthermore, the diffracted MO Faraday spectra are also presented theoretically. 2. Theoretical approach For the calculation of diffracted MO effects, the rigorous coupled-wave approach (RCWA) implemented as an Airy-like internal-reflection series (AIRS) [21] was employed. Unlike isotropic gratings in Ref. [21], lossy anisotropic gratings with multilayered structures for a polar MO configuration are taken into account, as shown in Fig. 1.

Fig. 1. (Color online) Diffracted MO effects from a one-dimensional multilayered gyrotropic grating for the polar magnetization.

The governing equation is the time-harmonic Maxwell’s equations, for convenience, in which the space coordinates with the free-space wavenumber k 0 are transformed to dimensionless ones by setting r¯ = k0 r: ˜ ∇ × E = −iH, (1) ˜ = i[ε (y)]E, ∇×H ¯ (2) ˜ = cμ0 H. c and μ0 represent light velocity in vacuum and magnetic permeability in where H vacuum, respectively. The components of the wave vector in periodic media are defined as q= #91178 - $15.00 USD

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λ , d

(3)



qn = sin θi + nq,

(4)

where λ , d, and θ i denote the wavelength of the incident beam, the period of the grating, and the incident angle, respectively. Using the Floquet theorem and assuming only propagation in the direction of the increasing z coordinate, the propagation eigenmodes of electromagnetic fields are expressed in terms of pseudo-Fourier series: +∞

∑

Ek (y, ¯ z¯) =

fk,n exp[−i(qn y¯ + s¯z)],

(5)

gk,n exp[−i(qn y¯ + s¯z)],

(6)

n=−∞ +∞

¯ z¯) = H˜ k (y,

∑

n=−∞

where k denotes x or y. The permittivity is a periodic function of only one lateral coordinate y with periodicity d: [εw ], 0 ≤ y ≤ w, [ε (y)] = (7) [εb ], w ≤ y ≤ d, where [εw ] and [εb ] are the permittivity in wires with the width of w and spaces between them, respectively. For the wires made of magnetic materials, [ε w ] is a 3 × 3 tensor with off-diagonal elements, whose positions are dependent on the direction of magnetization with respect to the incident plane. In the case of polar magnetization, the permittivity is written as below: ⎞ ⎛ εxx εxy 0 [εw ] = ⎝ εyx εyy 0 ⎠ . (8) 0 0 εzz The components of the y-dependent permittivity can be expressed in terms of Fourier series: +∞

∑

ε (y) ¯ αβ =

n=−∞

εαβ ,n exp(−inqy), ¯

(9)

where α and β represent any of x, y and z. Substituting Eqs. (5), (6) and (9) into Maxwell’s equations, a system composed of ordinary differential equations are obtained as d2 [εxy ] fx [εxx ] − q2 fx = , (10) − 2 fy fy (1 − q[εzz ]−1 q)[εyx ] (1 − q[εzz ]−1 q)[εyy ] dz d dz

gy −gx

=

[εxx ] − q2 [εyx ]

[εxy ] [εyy ]

fx fy

,

(11)

where [εαβ ] is a Toeplitz matrix consisting of Fourier coefficients ε αβ ,n and the tangential column vectors are denoted to f x , fy , gx , and gy composed of f k and gk in Eqs. (5) and (6). Here q is a diagonal matrix in the form of ⎛ ⎞ .. 0 0 0 0 ⎟ ⎜ . ⎜ 0 q−1 0 0 0 ⎟ ⎜ ⎟ 0 q0 0 0 ⎟ q=⎜ (12) ⎜ 0 ⎟. ⎜ 0 ⎟ 0 0 q 0 1 ⎝ ⎠ .. . 0 0 0 0

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Eventually, matching the boundary conditions at each interface and considering multiple reflections with AIRS, the diffracted MO Kerr rotation can be calculated [22] for the nth order as follows, 1 2Re(χ (n) ) (n) θK = tan−1 ( ), (13) 2 1 − |χ (n)|2 r r / fx,n . The procedure of modwhere χ (n) is estimated by the ratio of reflection amplitudes f y,n eling diffracted MO Faraday effect is almost identical to the case of the diffracted MO Kerr r / f r with that of transmission effect except for replacing the ratio of reflection amplitudes f y,n x,n t /ft . amplitudes f y,n x,n In the actual calculations, the infinite sums of the pseudo-Fourier and Fourier series are remax placed with ∑+n n=−nmax in Eqs. (3), (4) and (7). Thus, the truncation order n max is chosen carefully to guarantee a sufficient convergence. In the analysis, the grating period and the width of gyrotropic rod are 910 and 700 nm, respectively. The thicknesses of Cr 2 O3 capping layer, Ni 81 Fe19 permalloy layer, and SiO 2 oxide layer on the substrates are 2, 12, and 3 nm, respectively. All the wavelength-dependent optical constants are taken from the literature [23, 24, 25] considering absorption.

3. Results and discussion Figure 2 presents the theoretical MO spectra, together with the experimental ones of Ref. [19], for the zeroth- and the first-order diffractions. As can be seen in Fig. 2, the overall agreement between the experiment and the theory is good except for some small discrepancies at low photon energies, which is attributed to the free-electron Drude components of the optical constant being sensitive to the effect of the reduced dimensionality in the nanoscale systems [19, 21]. Comparing Fig. 2(a) with Fig. 2(b), the magnitude of Kerr rotation in the zeroth order is one order smaller than that of the first order at most of the energy range. It must, however, be noted that the two Kerr spectra were measured at different incident angles. 0.4

0.11 Theory Ref. 19

0.3

Kerr rotation θ(−1) (deg.)

0.09 0.08

Ks

Kerr rotation θ(0) (deg.) Ks

0.1

0.07 0.06 0.05 0.04

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 (b)

(a) 0.03 1

2

3

4

Photon energy (eV)

5

−0.5 1

2

3

4

5

Photon energy (eV)

Fig. 2. (Color online) Theoretical and experimental Kerr rotation spectra in the 0th (a) and the -1st (b) order diffractions for s-polarized incidence. Incident angle was 7◦ for the 0th order. For the -1st order diffraction, the angle between incident and reflected beams was fixed to 20◦ . A truncation order of nmax = 10 was sufficient enough for the good convergence.

In order to investigate the MO enhancement in gyrotropic gratings more accurately, the incident angle and wavelength were fixed to 7 ◦ and 632.8 nm, that of a He-Ne laser. The diffracted

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0.1

(deg.)

0.16

(−1)

0.14

Kerr rotation θs

Kerr rotation θ(0) (deg.) s

0.18

0.12 0.10 0.08 0.06 0.04

100

200

300

400

0.0 −0.1 −0.2 −0.3 −0.4

500

100

Grating depth (nm)

200

300

400

500

Grating depth (nm)

Fig. 3. Simulated Kerr rotation as a function of grating depth. Incident angle was set to 7◦ for both (a) the 0th and (b) the -1st orders. The truncation order nmax was 50 because of the increase in the grating depth. 9 8 7 6

M

5 4 3 2 1 0

50

100

150

200

250

300

350

400

450

500

Grating depth (nm)

Fig. 4. Magnification M as a function of grating depth.

MO enhancement is defined as the magnification M as below: (−1)

M =| (−1)

θKs

(0)

θKs

|,

(14)

(0)

where θKs and θKs are the Kerr rotation of the first- and the zeroth-order diffraction with s-polarized incidence, respectively. It is possible to observe the MO responses in the zerothand the first-order diffraction according to the grating equation [26], for both transmission and reflection, as below: (15) sin θD = sin θi + N λ /d, where θD and N represent the diffraction angle and order, respectively. In the following calculations, the effects of the capping and the oxide layers are excluded to investigate the diffracted MO effects from pure gyrotropic gratings. The simulated Kerr rotation is shown in Fig. 3 as a function of the grating depth varying from 10 to 500 nm. The magnitudes of the Kerr rotation in two diffraction orders vary in opposite ways with the increase in the depth even though #91178 - $15.00 USD

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a fluctuation is found in the depth greater than 100 nm. Obviously, we can suppose that the diffracted MO enhancement through gyrotropic gratings comes out only when the grating is thin compared with the grating period and the width of gyrotropic rod, as presented in Fig. 4, where M monotonously decreases down to zero with depths. It is thought that the exaltation of the internal diffraction-edge effects in thick gratings suppresses the MO effect in high orders. The maximum Kerr rotation is even up to ∼36 ◦/μ m for the 10 nm grating in the first-order diffraction. The thickness is reduced from the micrometer level down to the nanometer level compared with conventional MPCs to achieve such a huge MO effect. Also in the form of the single layer with the same thickness the estimated Kerr rotation is merely ∼5 ◦ /μ m by the propagation-matrix method [27]. As a result, the realization of the diffracted MO enhancement in thin gyrotropic gratings is of great advantage to the the miniaturization and integration of devices. In addition to the diffracted MO Kerr effect, the spectra of the diffracted MO Faraday effect is theoretically evaluated for the first time, as shown in Fig. 5. The MO enhancement can also be realized with the Faraday effect as well as the Kerr effect and the magnification is very close to the one illustrated in Ref. [16] experimentally. 0.3

Faraday rotation θ(−1) (deg.) F

(0)

Faraday rotation θF (deg.)

0.15 0.12 0.09 0.06 0.03 0.00 −0.03 (a) 2

3

4

Photon energy (eV)

5

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4

(b) 2

3

4

5

Photon energy (eV)

Fig. 5. Simulated spectra of Faraday rotation in (a) the 0th and (b) the -1st orders. The grating depth was set to 12 nm and the truncation order nmax was equal to 10.

4. Conclusions The RCWA implemented as AIRS was utilized to model the diffracted MO effects in onedimensional gyrotropic gratings. The theoretical and the experimental diffracted MO spectra indicate that there exists a consistency in the zeroth- and the first-order diffractions for a polar magnetization. For the diffracted MO Kerr effect, the calculation illustrates that the maximum of the MO Kerr rotation is even up to ∼8 times larger than that of the 10 nm single layer, which is favorable to miniaturize and integrate MO devices. Moreover, the MO enhancement is also demonstrated theoretically in the diffracted MO Faraday effect for the first time. It is believed that the absolute magnitudes and the MO magnification can be improved further by elaborately selecting the grating structures and the gyrotropic materials. Acknowledgments The authors would like to express their gratitude to Dr. R. Antos for valuable discussions and suggestions. This work was supported by MOST (Ministry of Science and Technology) and KOSEF (Korea Science and Engineering Foundation) through the Quantum Photonic Science Research Center at Hanyang University, Korea. #91178 - $15.00 USD

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