May 15, 2005 / Vol. 30, No. 10 / OPTICS LETTERS
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Faraday rotation in a two-dimensional photonic crystal with a magneto-optic defect Amir A. Jalali and Ari T. Friberg Department of Microelectronics and Information Technology, Royal Institute of Technology, Electrum 229, SE-164 40 Kista, Sweden Received October 6, 2004 We study the magneto-optic (MO) Faraday rotation in a two-dimensional square-lattice photonic crystal with a central MO defect layer in the optical wavelength range. We show that when a TM plane wave is incident upon a photonic crystal, an enhancement of Faraday rotation takes place in a region where a resonance peak appears in the photonic bandgap. In this region the mode conversion is high. © 2005 Optical Society of America OCIS codes: 230.2240, 230.3810.
Recently the application of magneto-optic (MO) materials in dielectric periodic structures to achieve large Faraday rotation (FR) has attracted much theoretical and experimental attention. MO materials, especially bismuth iron garnet (Bi3Fe5O12, BIG), have high optical transmittance and high FR in the visible and near-infrared regions. Since the pioneering work by Inoue et al.,1 it has been realized that MO FR can be enhanced at selected wavelengths in photonic crystals (PCs) that exhibit suitable photonic bandgaps. During the past few years research efforts have been devoted to the investigation of MO effects in PCs composed of MO materials. To the best of our knowledge all these experimental and theoretical studies dealt with multistack structures (onedimensional PCs) with MO defect(s), to have simultaneously enhanced MO rotation and large enough transmission or reflection.2–4 In this Letter we examine theoretically the transmission and the corresponding FR of a twodimensional (2D) MO PC, with a normally incident transverse-magnetic (TM) plane wave in the optical wavelength range. The idea is to insert a MO material (as a layer defect) into a dielectric PC, instead of placing it into a multistack material, for the purpose of enhancing the FR. The background material of the PC is an isotropic dielectric medium with air-filled cylindrical holes in a square lattice. It is assumed that the central layer defect consists of a thin BIG film. An external dc magnetic field, which is applied in the direction of the wave propagation, makes the MO medium reach saturation magnetization. We consider a 2D PC with a square-lattice configuration of circular air rods parallel to the x axis located in region II in Fig. 1. A TM plane wave, with the electric field parallel to the symmetry axis (x axis), is normally incident upon the PC surface from the left. The radii of the circular holes are r, the square-lattice constant is a, and the background is an isotropic medium with a dielectric constant ⑀b. Moreover, we introduce a MO defect of total length 3a into the middle of the structure that extends indefinitely in the y direction. At optical wavelengths the relative permeability differs only slightly from unity, 0146-9592/05/101213-3/$15.00
whereas the relative permittivity tensor ˜⑀ is given by
冤
⑀1
i⑀2 0
˜⑀ = − i⑀2 ⑀1 0
0
冥
共1兲
0 ,
⑀3
in which the saturation magnetization of the magnetic medium is directed along the z axis. One can solve the wave equations obtained from the Maxwell equations by use of a modal method based on plane-wave expansions.5 In the case of a TM electromagnetic plane wave that is incident from region I, the x components of the electric and magnetic fields in regions I and III are given by 0 0 y + kIz z兲兴 EIx共y,z兲 = E0x exp关i共kIy
+ HIx共y,z兲 =
兺n Rn exp关i共kny y + krznz兲兴,
兺n Sn exp关i共kny y + krznz兲兴,
EIII x 共y,z兲 =
兺n Tn exp兵i关kny y + ktzn共z − L兲兴其,
HIII x 共y,z兲 =
兺n Qn exp兵i关kny y + ktzn共z − L兲兴其,
共2兲
where n n 0 = kty = kny = kIy + kry
n krz =
n ktz
=
再 再
2n a
− 关共kI0兲2 − 共kny 兲2兴1/2
,
n = 0, ± 1, ± 2, ± 3, ... ,
kI0 艌 兩kny 兩
− i关共kny 兲2 − 共kI0兲2兴1/2 kI0 ⬍ 兩kny 兩 0 2 关共kIII 兲 − 共kny 兲2兴1/2
冎 冎
0 kIII 艌 兩kny 兩
0 2 1/2 0 兲 兴 kIII ⬍ 兩kny 兩 i关共kny 兲2 − 共kIII
共3兲 ,
共4兲
.
共5兲
In Eqs. (2)–(5), kI0 is the 2D incident wave vector kI0 0 = 冑⑀I / c, and kIII = 冑⑀III / c. One can now obtain the © 2005 Optical Society of America
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OPTICS LETTERS / Vol. 30, No. 10 / May 15, 2005
Fig. 1. Schematic diagram of an isotropic 2D squarelattice PC with a central defect of width 3a. The crystal and the defect, which consists of MO material, extend indefinitely in the y direction. A plane wave is incident from the left onto the structure. The lattice constant is a, and the radius of each rod is r. The distance between the PC’s surface and the first layer of circular rods is d = a / 2.
unknown plane-wave coefficients Rn , Sn , Tn, and Qn by solving the coupled-wave equations in an approximation in which the MO effects are treated as a perturbation in an isotropic medium. The method is described in detail in Ref. 6. BIG films show strong dependence of the permittivity tensor on the wavelength in the optical regime.7 For the numerical analyses below we made use of the expression for the dispersion relation of the off-diagonal elements of the permittivity tensor given explicitly by Dionne and Allen.8 In an isotropic medium the energy flows for TE and TM waves at interfaces of the PC are given by PTE = Z兩Hx兩2 / 2 and PTM = 兩Ex兩2 / 2Z, where Z is the wave impedance within the medium. The transmittance is then defined as the ratio of the output Poynting vector to the input Poynting vector. The circular rods in Fig. 1 are air, the background is an isotropic medium with dielectric constant ⑀b = 4.75, and the filling ratio is r / a = 0.43. The distance between the PC surface and the first layer of circular rods is d = a / 2. First we consider an isotropic PC without a defect (without a MO effect), consisting of 17 layers of circular rods. We make use of 3000 plane waves (Fourier components) in the fields propagating within the PC.5 Then we take a plain isotropic defect of width 3a with a dielectric constant of ⑀ = ⑀b = 4.75 in the middle of the PC (now 14 layers of rods). Figure 2 shows the zeroth-order transmittance of the TM incident plane wave at normal incidence for the PC without and with the defect. A bandgap for an incident TM mode occurs in the interval 0.28⬍ a / 2c ⬍ 0.37 [Fig. 2(a)]. Two sharp Fabry–Perot resonance peaks at a / 2c = 0.29 and 0.34 appear within the bandgap in Fig. 2(b). At other frequencies there are clear interference patterns and several opaque spectral ranges. The bandgap becomes wider in the presence of the defect. Next we introduce a defect of width 3a consisting of a MO medium in the middle of the PC structure. We take the MO material BIG as the defect medium. The numerical calculations are performed for typical val-
ues of the dispersion relation parameters extracted from a paper by Kahl and Grishin.7 We apply the parameters that correspond to a BIG film of 2560-nm thickness. The input and output regions (I and III, respectively) are air. A plane wave in TM polarization is normally incident upon the PC from region I. In Fig. 3 we show the zeroth order of the TM-mode transmittance. Since the TE and TM modes are coupled into the MO defect medium, a TE wave also exits from the PC. Figure 3 shows the TE-wave output as well. The total transmittance in zeroth order 2 at z = L is then found from the superposition of 兩E兵0其 x 兩 兵0其 2 and 兩Ey 兩 . The FR is finally obtained from the following relation9:
冉 冊 = , 1−冏 冏 2R
tan 2F
E兵y0其
E兵x0其
E兵y0其 2
共6兲
E兵x0其
where R denotes the real part. As seen from Fig. 3, in the frequency range of 0.285⬍ a / 2c ⬍ 0.306 the mode conversion within the PC-defect structure is strong, and therefore there will be a larger rotation of the polarization of the
Fig. 2. TM-mode transmittance computed for a 2D squarelattice PC with air holes (a) with an isotropic material of dielectric constant ⑀b = 4.75 as background and (b) with a centered isotropic defect of width 3a and dielectric constant ⑀ = ⑀ b.
May 15, 2005 / Vol. 30, No. 10 / OPTICS LETTERS
Fig. 3. Transmittance of TM (solid curve) and TE (dotted curve) modes with an incident TM-polarized light for a 2D PC with a MO BIG layer defect.
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According to Fig. 4, the FR is larger than 34° throughout the frequency range 0.295⬍ a / 2c ⬍ 0.306. The FR for a BIG film with an equivalent thickness is ⬃10°.7 The mode coupling influences the transmittance of TM and TE modes in relation to the isotropic case. The FR obtained in this structure is comparable with that seen in one-dimensional MO PCs.2,4 Figures 3 and 4 show that in the range 0.290⬍ a / 2c ⬍ 0.298 of high FR, the total transmission does not change appreciably. With the chosen value of the lattice constant 共a = 0.3 m兲 the bandwidth is ⬃27 nm. Such broadening of the transmission linewidth is also observed for a 2D PC with a liquid-crystal central defect, although it is not as wide as with a MO defect.10 In conclusion, we have investigated the transmission properties of a TM wave that is incident upon a 2D photonic crystal with a magneto-optic Bi3Fe5O12 layer defect. We showed that there is an enhancement of Faraday rotation in the region between bandgaps where the transmittance peak appears. On average, the FR is larger than the intrinsic FR of the BIG material by at least a factor of 3. The authors thank A. Grishin for useful discussions. A. A. Jalali’s e-mail address is
[email protected]. References
Fig. 4. FR of a TM-polarized incident light on passage through a PC with a MO BIG layer defect. FR is calculated for the zeroth-order output waves.
emitted light within this band. In this way enhancement of FR is obtained in the spectral region where the total transmittance is relatively high. The FR is larger than the intrinsic FR of a BIG film with the same dimensions.
1. M. Inoue, T. Fujii, K. Arai, and M. Abe, J. Appl. Phys. 83, 6768 (1998). 2. M. Steel, M. Levy, and R. Osgood, IEEE Photonics Technol. Lett. 12, 1171 (2000). 3. S. Kahl and A. Grishin, Appl. Phys. Lett. 84, 438 (2004). 4. M. Levy, H. C. Yang, M. J. Steel, and J. Fujita, J. Lightwave Technol. 19, 1964 (2001). 5. K. Sakoda, Phys. Rev. B 52, 8992 (1995). 6. A. A. Jalali and A. T. Friberg, “Faraday rotation in two-dimensional magneto-optic photonic crystal,” Opt. Commun. (to be published). 7. S. Kahl and A. Grishin, J. Magn. Magn. Mater. 271, 200 (2004). 8. G. F. Dionne and G. A. Allen, J. Appl. Phys. 73, 6127 (1993). 9. S. Visnovsky, K. Postava, and T. Yamaguchi, Czech. J. Phys. 51, 917 (2001). 10. H. Takeda and K. Yoshino, Phys. Rev. E 68, 046602 (2003).
