356 OPTICS LETTERS / Vol. 34, No. 3 / February 1, 2009

Voltage-controlled Kerr effect inmagnetophotonic crystal

Hai-xia Da,1,2,4 Zi-qiang Huang,3 and Z. Y. Li1,*1Department of Physics and Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou, 215006, China

2Current address: School of Chemical and Biomedical Engineering, Nanyang Technological University,50 Nanyang Avenue, 639798, Singapore

3School of Opto-Electronic Information, University of Electronic Science and Technology of China,Chengdu, 610054, China

4E-mail address: hxda2005@yahoo.com.cn*Corresponding author: zyli@suda.edu.cn

Received September 10, 2008; revised November 2, 2008; accepted December 3, 2008;posted December 22, 2008 (Doc. ID 101290); published January 29, 2009

A magneto-optical tunable device based on a one-dimensional magnetophotonic crystal infiltrated with anematic liquid crystal has been proposed. By applying a tunable electric field the voltage-induced reorien-tation of the director results in an alternating magneto-optical effect. The transfer matrix method was per-formed to verify the controllable magneto-optical properties. The present results may be useful for futureapplication of liquid crystal-based tunable magneto-optical devices. © 2009 Optical Society of America

OCIS codes: 160.3710, 230.3810.

The ability to rotate the polarization state of light bymagnetic materials is one of the most fundamentalphenomena of electrodynamics, for example, the Kerreffect. It was discovered by John Kerr in 1875 and isnow widely used in integrated optics and magneto-optical devices for magnetic domain imaging, map-ping of hysteresis loops, and high-density recording[1–3].

Magnetic materials with large magneto-optical re-sponses are always the attractive ones used in mag-netophotonic crystals (MPCs). The recent explosiveinterest in MPCs is driven by an opportunity for theenhancement of magneto-optical effects, wherecavity-based periodic and aperiodic structures are re-quired to achieve such an enhancement [4–13]. Tun-able magneto-optical devices have important applica-tions in optical switches and displays. So far, thereare few reports on the tunable MPCs. Magnetic con-ducting spheres and semiconductor quantum wellshave been discussed as possible candidates forachieving artificial tunable MPCs [14,15]. It wasknown that nematic liquid crystals (NLCs) are goodchoices for tunable photonic crystals owing to theirunique sensitivity. However, few reports concerningthe tunable magneto-optical effect based on NLCshave been presented in the literature. Therefore, itwould be interesting to investigate the opportunity ofcreating an electrically controlled Kerr effect in aMPC infiltrated with the liquid crystal. Furthermore,from the application-oriented perspective it would bevery desirable if a tunable magneto-optical effectcould be achieved by controlling the applied electricfield, thus making such an effect a potential proposi-tion.

In this Letter we first present a rigorous theoreti-cal demonstration that the tunable Kerr effect can beachieved in MPCs based on NLCs by performing a4�4 faster Berreman matrix [16,17]. Although sucha tunable magneto-optical effect is expected to beseen with a variety of patterns containing NLCs, we

investigate a multilayered structure infiltrated with

0146-9592/09/030356-3/$15.00 ©

NLCs. The key enabling ingredient is the existence ofa NLC that is an outstanding material with easilycontrollable optical properties via external param-eters such as electric field or temperature. Owing tothe intrinsic anisotropic properties of NLCs, a rigor-ous anisotropic treatment of NLCs is essential to con-sider the NLC director’s spatially inhomogeneousproperty upon an applied external voltage. We stud-ied Kerr rotation angle and reflection of the structureat normal incidence. Because of the voltage-inducedreorientation of the director, the NLC-based multi-layered structure can exhibit different Kerr effectsunder different applied voltages.

We consider a one-dimensional MPC infiltratedwith the defect of the NLC, as shown schematicallyin Fig. 1. The gray and white regions are magneticmaterials, yttrium–iron garnet (YIG), and nonmag-netic materials, gadolinium–gallium garnet (GGG)with permittivity �2, respectively. The magnetizationvector is oriented along the z axis in the YIG layers,which are characterized by antisymmetrical permit-tivity and have the nonzero components, �xx=�yy=�zz=�1, �xy=−�yx= ig, where g is the gyrotropy of themagnetic layer. The thicknesses of magnetic and non-magnetic materials are d1 and d2, respectively. TheNLC is characterized by a long-range uniaxial orien-tational order of rodlike anisotropic molecules along

Fig. 1. Schematic view of the magnetophotonic crystal.

2009 Optical Society of America

February 1, 2009 / Vol. 34, No. 3 / OPTICS LETTERS 357

a common director, whose dielectric tensor is given by[18]

�LC = �n0

2 + �� cos2 � 0 �� cos � sin �

0 n02 0

�� cos � sin � 0 n02 + �� sin2 �

� , �1�

where � is the tilt angle between the principal axis ofthe NLC molecule and the x axis. �� is the optical di-electric anisotropy given as no

2−no2; no and ne are or-

dinary and extraordinary refractive indices, respec-tively. Upon an applied external voltage, thedistribution of the NLC director exhibits a spatiallyinhomogeneous property. The distribution of � isgiven in the NLC slab by the Ossen–Frank elastictheory when an electric field is applied along the zaxis. The relations between � and the voltage V ap-plied to the NLC slab can be achieved according tothe following equations [19]:

�k1 cos2 � + k3 sin2 ��� dz

d��−2

= ��0E2�sin2 �m − sin2 ��, �2�

V

Vc=

2

��

0

�2�1 + k�2 sin2 �

1 − �2 sin2 �d�, �3�

with �=sin �m and k= �k3−k1� /k1. The parameters k1and k3 are the Frank elastic constants for splay andbend modes of the nematic director distortion, respec-tively. �m is the maximum tilt angle in the NLC slab;��0 is the dielectric anisotropy at zero frequency.Note that the direction of the NLC molecules is af-fected by the external voltage only when the appliedvoltage V exceeds a threshold value Vc�=��k1 /��0�.

Based on the Maxwell equations, for a mono-chrome light propagating in the medium, we have[16]

d��z�

dz=

i

c��z���z�, �4�

where ��z� is the field vector provided as ��z�= �Ex ,Hy ,Ey ,−Hx�t, and ��z� is the Berreman matrix,which is the function of permittivity, permeability,and the incident wave vector. The subscript t denotesthe transpose operator. In the frame of the transfermatrix method, the field vectors are connected with��z�=T�z ,z0���z0�, where T�z ,z0� is the transfer ma-trix, which can be provided with the Cauchy andHamilton theorem [16]

T�z,z0� = 0I + 1� + 2�2 + 3�3, �5�

with

0 = − i=1

4

�j�k�l

fi

� � �,

ij ik il

1 = i=1

4

��j�k + �j�l + �k�l�fi

�ij�ik�il,

2 = − i=1

4

��j + �k + �l�fi

�ij�ik�il,

3 = i=1

4 fi

�ij�ik�il,

where �ij=�i−�j, fi=ei�/c��i�z−z0�, i , j ,k , l=1,2,3,4.Owing to the continuity of boundary conditions, theelectric fields of the reflection light are related tothose of the incident light by [20]

�Erp

Ers� = �rpp rps

rsp rss��Ei

p

Eis� . �6�

Now �p= p+ i�p=rps /rpp, �s= s+ i�s=rsp /rss, where p � s� and �p ��s� are the Kerr rotation angle and theellipticity for the p�s� polarized wave, respectively[20]. The reflection spectrum is also presented toclarify the physical explanations.

For the one-dimensional multilayered structure�GGG YIG�n�YIG GGG�mNLC�YIG GGG�n, at thenear-IR wavelength �=1.55 �m, YIG and GGG arecharacterized by �1=5.5+ i0.0025, g= �1− i0.15��10−2, and �2=3.709, respectively [21]. The thick-nesses are d1=0.466D and d2=0.534D, respectively.k1=7.4 pN, k3=10.2 pN, ��0=1.43�10−10 F/m, ne=1.75, n0=1.54 for a typical NLC (5CB) [22], dLC= 20

6 D. The repetition numbers are n=16 and m=29.Detailed studies on the influence of the magnetic ma-terials loss on the tunable Kerr effect will be carriedout in our future works.

Figure 2(a) corresponds to the distribution of �with different voltages V=0, 2Vc, 5Vc by solid,dashed, and dotted curves, respectively, for thep-polarized wave. The anisotropic character of theNLC can be considered exactly by utilizing the ob-tained distribution of �. The calculated results for pas functions of normalized frequency D / �2�c� withdifferent voltages are plotted in Fig. 2(b). The solid,dashed, and dotted curves correspond to p with V=0, 2Vc, 5Vc, respectively. It can be seen clearly thatthe spectrum of p moves at different voltages, whichis a direct manifestation of the voltage-controlledKerr effect. The frequency dependence of p in thepresent case exhibits the following features. First,the positions of p peaks are dependent on the ap-plied voltage owing to the reorientation of the direc-tor. It is seen that the positions of peaks move towardhigh frequencies with the applied voltages. This isbecause of the NLC’s high sensitivity to the appliedvoltage. When the applied voltage is very small, thedistribution of the director aligns in the x directionowing to the surface anchoring. With the increase ofapplied voltage, the energy of the electric field in-creases gradually. When the applied voltage is biggerthan the threshold value, the function of the electric

field plays a leading role. In such a case, the align-

358 OPTICS LETTERS / Vol. 34, No. 3 / February 1, 2009

ment of the director along the electric field can be ob-served. The reorientation of the NLC’s director willlead to the change of its permittivity, which directlyaffects the Kerr rotation angles. Second, the ampli-tudes of p depend strongly on the applied voltage;i.e., the heights of p peaks increase considerablywith the increase of V. For example, the maximum of p at V=5Vc � p max=−30° � is approximately threetimes larger than that of p � p max=−10° � at V=0,which describes the large difference in the indices ofrefraction between the right and left circularly polar-ized light. This can be understood according to thecircular birefringence, in which different refractiveindices due to different localization wavelengths forleft- and right-circularly polarized lights leads to thelarge rotation of the reflected light [23]. In addition,the positions of p peaks agree with those of the dipsin the reflection spectrum exactly. Taking V=5Vc asan example, we calculated p and the correspondingreflection spectrum, which are exhibited in Figs. 2(c)and 2(d). Such a phenomenon is directly related tothe edges of photonic bandgaps, where the abnormaldispersion relation leads to strong photonic localiza-tion. As a result, p can be enhanced at these specificfrequencies in the present structure [24]. Owing tothe degenerate properties of p- and s-polarized wavesat normal incident light, the conclusions are true forthe s-polarized wave.

In summary, we demonstrated theoretically that

Fig. 2. (Color online) (a) � versus the distance at V=0,2Vc ,5Vc, respectively; (b) p versus the normalized fre-quency under V=0,2Vc ,5Vc at the normal incident angle,are shown by solid, dashed, and dotted curves, respectively.(c) The dependence of p and (d) the reflectance on the nor-malized frequency at V=5Vc

the tunability of the multilayered structures infil-

trated with NLCs can be realized owing to the reori-entation of the molecules of the NLCs. Our resultsshow that the peaks of Kerr rotation angles move to-ward the high frequencies with the the applied volt-age. The results imply a general strategy for con-structing a MPC supporting tunable magneto-opticaleffect. The large sensitivity of NLCs to the appliedvoltage make it possible for the present results to bemore easily distinguished compared with the othermethods.

This work was supported by the National NaturalScience Foundation of China (NSFC) (10774107,10547128, 90201030) and by the Jiangsu provincialinnovation project under grant ZY320607.

References

1. S. Y. Yang, Y. P. Chiu, B. Y. Jeang, H. E. Horng, C.-Y.Hong, and H. C. Yang, Appl. Phys. Lett. 79, 2372(2001).

2. H.-E. Horng, C.-Y. Hong, S. L. Lee, C. H. Ho, S. Y.Yang, and H. C. Yang, J. Appl. Phys. 88, 5904 (2000).

3. C.-Y. Hong, J. Appl. Phys. 85, 5962 (1999).4. H. Kato, T. Matsushita, A. Takayama, M. Egawa, K.

Nishimura, and M. Inoue, J. Appl. Phys. 93, 3906(2003).

5. S. Sakaguchi and N. Sugimoto, Opt. Commun. 162, 64(1999).

6. M. Inoue, K. Arai, T. Fujii, and M. Abe, J. Appl. Phys.83, 6768 (1998).

7. M. Mansuripur, The Principles of Magneto-OpticalRecording (Cambridge U. Press, 1995).

8. M. Levy, J. Appl. Phys. 99, 073104 (2006).9. M. Levy and R. Li, Appl. Phys. Lett. 89, 121113 (2006).

10. V. I. Belotelov, V. A. Kotov, and A. K. Zvezdin, PhaseTransitions 79, 1135 (2006).

11. V. I. Belotelov and A. K. Zvezdin, J. Opt. Soc. Am. B 22,286 (2005).

12. S. I. Khartsev and A. M. Grishin, Appl. Phys. Lett. 87,122504 (2005).

13. S. Kahl and A. M. Grishin, Appl. Phys. Lett. 84, 1438(2004).

14. M. Golosovsky, Y. Neve-Oz, and D. Davidov, Phys. Rev.B 71, 195105 (2005).

15. J.-T. Liu, and K. Chang, Appl. Phys. Lett. 90, 061114(2007).

16. H. Wohler, G. Haas, M. Fritsch, and D. A. Mlynski, J.Opt. Soc. Am. A 5, 1554 (1988).

17. H. X. Da, C. Xu, and Z. Y. Li, Eur. Phys. J. B 45, 347(2005).

18. I.-C. Khoo, Liquid Crystals, 1st ed. (Wiley, 1995), p. 21.19. L. Liebert, Liquid Crystals, 1st ed. (Academic, 1978), p.

81.20. C.-Y. You and S.-C. Shin, Appl. Phys. Lett. 70, 2595

(1997).21. A. Zvezdin and V. Kotov, Modern Magneto-Optics and

Magneto-Optical Materials (IOP, 1997).22. D. W. Berreman, J. Opt. Soc. Am. 63, 1374 (1973).23. M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P.

B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T.Murzina, and A. Granovsky, J. Phys. D 39, R151(2006).

24. A. G. Zhdanov, A. A. Fedyanin, O. A. Aktsipetrov, D.Kobayashi, H. Uchida, and M. Inoue, J. Magn. Magn.

Mater. 300, e253 (2006).