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Optics Express

  • Vol. 17, Iss. 7 — Mar. 30, 2009
  • pp: 5265–5272
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Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices

Alexander B. Khanikaev and M. J. Steel  »View Author Affiliations


Optics Express, Vol. 17, Issue 7, pp. 5265-5272 (2009)
http://dx.doi.org/10.1364/OE.17.005265


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Abstract

We develop an exact theory for light propagation in transversely-magnetized low-symmetry magnetic photonic crystals. We investigate the nature of nonreciprocal dispersion and unidirectionality in these systems and show that it is associated with boundary effects rather than propagation. We calculate the nonreciprocal response of finite structures and propose an asymmetric magneto-optical cavity as a practical building block for one-way optical components.

© 2009 Optical Society of America

1. Introduction

Magneto-optical (MO) effects such as Faraday rotation and nonreciprocal phase-shifts are widely exploited, principally in optical isolators. However, the intrinsic weakness of MO effects hinders miniaturization of such devices. It is known that micro-scale structuring of MO materials, particularly with photonic crystals (PC), is an effective approach to enhancing the response, promising compact, integrated circulators and isolators [1

1. 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). [CrossRef]

]. It has also been recently shown that magnetic nonreciprocity in plasmon [2

2. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-Free One-Way Edge Modes in a Gyro-magnetic Photonic Crystal,” Phys. Rev. Lett. 100, 01390501–01390504 (2008). [CrossRef]

] and photonic crystal [3

3. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal,” Phys. Rev. Lett. 100, 02390201–02390204 (2008). [CrossRef]

,4

4. Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. 90, 121133:1–3 (2007). [CrossRef]

] systems may result in true unidirectionality (or one-way propagation) and provide robustness against disorder. This opens up new avenues for magnetic materials in photonics, but the conditions for maximizing nonreciprocality are not well-established. In this work, we reveal the mechanisms of the unidirectionality in low symmetry MO photonic crystals (MPC).

2. Mathematical treatment

We treat the ternary photonic crystals using a transfer matrix formalism. To formulate the approach, we begin with the problem of the partial reflection of plane waves at the interface between two MO media magnetized transversely (the Cotton-Mouton or Voigt geometry) (Fig. 1). The coefficients relating amplitudes of the incident and scattered waves can be determined using the standard Maxwell boundary conditions. For MO media, these conditions yield relations different from the standard Fresnel equations [7

7. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997). [CrossRef]

]: with transverse magnetization, the electric field includes a longitudinal component along the wavevector k, and for non-normal incidence, this component contributes to the component of the field tangential to the interface. Straightforward calculations yield the expression:

ρa=ρ0a+ikxρ1a,
(1)

ρ0r=(d1ε1k2z)2(d2ε1k1z)2+kx2(d1Δ2d2Δ1)2(d1ε2k2z+d2ε1k1z)2+kx2(d1Δ2d2Δ1)2,
(2a)
ρ0t=2d2ε1k1z(d1ε2k2z+d2ε1k1z)(d1ε2k2z+d2ε1k1z)2+kx2(d1Δ2d2Δ1)2,
(2b)
ρ1t=ρ1r=(d1Δ1d2Δ2)2d2ε1k1z(d1ε1k2z+d2ε1k1z)2+kx2(d1Δ2d2Δ1)2,
(2c)
Fig. 1. Geometry of the problem: (a) ternary MPC and (b) microcavity-type nonreciprocal Fabry-Perot resonator. Magnetization M is along . Magnetic layers are in green.

where dm = ε 2 m - Δ2 m and kmx(z) is the x(z) component of the wavevector in the medium m. Thus, disregarding effects second order in Δm, it can be seen from Eqs. (1) and (2) that the main effect of MO activity is an additional term with an odd dependence on the lateral component of the wavevector kx (which is the same in every medium) and on Δm. As the coefficients ρ a 0 are purely real (we suppose the materials to be nonabsorbing), this dependence manifests mainly as a direction-dependent phase shift ϕa = tan-1(Im(ρa)/Re(ρa)), which is odd not only with respect to the angle of incidence but also with the magnetization direction (through the sign of Δm) [7

7. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997). [CrossRef]

]. As we show below, this phase shift lies at the root of the nonreciprocity in the dispersion relation of the photonic crystals reported earlier [4

4. Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. 90, 121133:1–3 (2007). [CrossRef]

].

To continue, we work in the transfer matrix formalism [8

8. P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).

]. Using the same boundary conditions that yielded (2), we can derive the interface matrix ij that connects the amplitudes of forward (H + y) and backward-going (H - y) waves on either side of the interface between layers i and j as follows:

[Hy+Hy]j=M̂ij[Hy+Hy]i=d2εjkzj[Fj*+FiFj*Fi*FjFiFj+Fi*][Hy+Hy+]i,
(3)

where Fm = (εm kzm + iΔm kxm)/dm.

2.1. Characterization of nonreciprocality

To consider propagation in the ternary PC, we impose a Bloch condition TˆH̄y = exp(iKMa)y, where KM is the Bloch vector and y = (H + y, H - y)T. This condition is satisfied when det [ - exp(iKMa)Iˆ] = 0, which is just the dispersion relation of the Bloch waves of the MPC. In the general case, this expression is rather difficult to express analytically. However, the terms linear in the lateral component kx responsible for nonreciprocity may be extracted explicitly. Limiting our consideration to the case of a single MO layer in the elementary cell (Δ1 = Δ3 = 0), after tedious but straightforward calculations we obtain the dispersion relation

cos(KMa)cos(K0a)+ε3k3zε1k1zkxk2xΔ2ε2(k1z2ε12k3z2ε32)sin(k1za1)sin(k2za2)sin(k3za3),
(4)

where KM and K 0 are Bloch wave vectors for the magnetic and corresponding non-magnetic structures and second order terms in Δ2 have been omitted. The first term in Eq. (4) is the reciprocal contribution to the dispersion containing standard nonmagnetic terms due to periodicity [8

8. P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).

]. The second explicitly shows the asymmetric contribution to the dispersion relation due to its linear dependence on the lateral wavevector component kx and Δ2. Note that reversal of the magnetization direction has the same effect as reversal of the sign of kx. (Note that a similar approximate form of the dispersion relation as Eq. (4) has been discussed before in the context of birefringence in two-component structures with longitudinal magnetization geometry [9

9. A. A. Jalali and M. Levy, “Local normal-mode coupling and energy band splitting in elliptically birefringent one-dimensional magnetophotonic crystals,” J. Opt. Soc. Am. B 25, 119–125, (2008). [CrossRef]

].)

The nonreciprocity of the dispersion in Eq. (4) is more apparent if we note the small value of the parameter Δ2, and consider the no nreciprocal contribution δK 0 to the dispersion as a small correction to its reciprocal part K 0. Substituting KM = K 0 + δ into (4) and expanding the cosine function on the left hand side of Eq. (4) gives

δε3k3zε1k1zkxk2zΔ2ε2(k1z2ε12k3z2ε32)sin(k1za1)sin(k2za2)sin(k3za3)asin(K0a).
(5)

It is evident that Eq. (5) is the term of first order in kx and Δ2 responsible for the nonreciprocity because simultaneous reversal of the wavevector k and K 0 results in a different sign of δ and therefore alters the dispersion relation for forward and backward propagation. As expected, the nonreciprocity vanishes in the symmetric configuration ε 1 = ε 3 confirming the necessity of breaking the reflection symmetry mentioned earlier. From Eq. (5), it is also obvious that the nonreciprocity reaches maximal values near the band edges where the sine in the denominator tends to zero. Figure 2 shows an exact calculation of the photonic band structure (Fig. 2(a)) using the transfer matrix technique, and the difference between eigenvectors KM for opposite propagation directions (Fig. 2(b)), and confirms our expectations. The nonreciprocity reaches its maximum at the band edges. It has a complex resonant dependence on the transverse wavevector kx due to the presence of interference in the layers, which is reflected in the sine functions in the numerator of Eqs. (4) and (5). Thus the nonreciprocity is extremely sensitive to the structure parameters and optimization will be required for every particular application. For clarity, Fig. 3 shows slices through the energy surfaces in Fig. 2 for the case of θi = 30° incidence. Parameters of the structure used in all our calculation were chosen to maximize the nonreciprocity with the short-wavelength band edge lying near optical communication wavelengths for θi = 30°. Figure 3(b) clearly shows that at this particular wavelength, the difference in the Bloch wavevectors for opposite propagation directions reaches its maximum. Note that results obtained with use of the approximate expression Eq. (5) (shown by the thick red line) are in excellent agreement with the exact result (shown by the thin blue line). In fact this approximation fails only in a very narrow (≈2 nm) wavelength range in close proximity to the band edges where it diverges (see insets to Fig. 3(b)).

Fig. 2. Photonic band structure (a) and its nonreciprocity (b) in 1D MPC with three-layer elementary cell formed from layers of SiO2 (a 1=220 nm, ε 1=2.1), Ce:BIG (a 2=130 nm, ε 2=6.25, Δ2=0.06) and Ta2O5 (a 3=110 nm, ε 3=4.7), respectively.

As was already mentioned, the results given above show considerable nonreciprocity for the structure with optimized parameters. However, the nonreciprocity could certainly be improved further. At first glance, optimization of the structure seems straight-forward because one simply has to make the correction due to the nonreciprocal term in Eq. (4) maximal. For a given angle of incidence (i.e. fixed kx) this is achieved when modulus of all the sine functions in Eqs. (4) and (5) is simultaneously equal to unity. This condition is satisfied when the thickness of each layer satisfies the condition am=(12+n)π/kzm(n=1,2,3...). However, any variation in the thickness of layers of course affect the band structure. Therefore, as the nonreciprocity is maximal at band edges, simultaneously with the conditions imposed on the thickness of layers one should guarantee that a band edge appears at the frequency of particular interest. An optimal structure must therefore satisfy sin(K(a 1,a 2,a 3)(a 1 +a 2 +a 3)) = 0, as well as the conditions on the thickness of the individual layers. The complete optimization task is therefore quite involved. To concentrate on the physical picture, we will not consider the optimization procedure any further, limiting our consideration to the particular structure introduced above (see Fig. 2).

Fig. 3. Photonic band structure of nonmagnetic (a) and nonreciprocity of magnetic (b) ternary 1D PCs, respectively. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°. Thin blue lines correspond to the exact result and the thick red line is obtained from the approximate expression Eq. (5). Insets in (b) show the nonreciprocity in the proximity of the band edges.
Fig. 4. Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) ternary 1D PCs, respectively. Figure (a) shows the whole bandgap region, while (b) shows the transmittance in the proximity of the short-wavelength band edge. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°.

3. Designing one-way structures

Fig. 5. Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) microcavities, respectively. Figure (a) shows the whole bandgap region, while (b) shows the proximity of the resonance only. The structure consists of a single Ce:BIG layer (aM=155 nm, εM=6.25, ΔM=0.06) sandwiched between two Bragg mirrors formed from a sequence often SiO2 (a 1=267 nm, ε 1=2.1) and Ta2O5 (a 2=179 nm, ε 2=4.7) layers. The asymmetry is achieved through bounding the Ce:BIG layer with different layers of the Bragg mirrors, i.e. SiO2 and Ta2O5 for left and right boundaries, respectively. The angle of incidence is 30°.

However, with the origin of the nonreciprocity in transversely-magnetized crystals clarified, we can now propose an alternative system which would be simpler for fabrication and more compact. Instead of a MPC, we propose to obtain strong nonreciprocity using an asymmetric magnetic Fabry-Perot structure: a single MO layer bounded by two non-identical non-magnetic Bragg mirrors (see schematic in Fig. 1(b)). Note that similar structures with longitudinal magnetization were proposed before for the increase of Faraday rotation [1

1. 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). [CrossRef]

,10–12

10. A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. May-dykovskiy, and O. A. Aktsipetrov, “Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects,” Phys. Rev. B 78, 193102:1–4 (2008). [CrossRef]

], but we mention again that the nature of the transverse magnetization effect we consider here is quite different. For example, in the symmetric configuration proposed before [11

11. M. Inoue, K. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers,” J. Appl. Phys. 83, 6768–6770 (1998); [CrossRef]

,12

12. M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999). [CrossRef]

], i.e. with a MO layer bound symmetrically by identical mirrors, the phase shifts at the two boundaries of the MO layer cancel each other and the effect of the nonreciprocity disappears. However, if the structure is asymmetric, the phase shifts after reflection at two opposite boundaries are different, and even if the shifts are of different sign, the net shift acquired by the wave after multiple reflections in the cavity does not vanish. Thus, a Fabry-Perot resonance of sufficient quality factor can give rise to a very large value of the total phase shift accumulated in the microcavity. This additional shift obviously alters the conventional Fabry-Perot condition in a similar way that the Bragg condition was modified in MPCs. As the phase shift is nonreciprocal, and in fact has the same absolute value but different sign for forward and backward propagation, this results in nonreciprocal resonant transmittance T(k⃗) ≠ T(-k⃗), which manifests as a frequency gap between transmission peaks for waves propagating in opposite directions (Fig. 5). It can be easily shown that the frequency gap δωkxΔM, and that the differential transmittance ∣T(k⃗) - T(-k⃗)∣ ≈ 1 if the quality factor of the resonance is sufficient to satisfy the condition δλ < δω, where δλ is bandwidth of the resonance. More precisely, the resonance condition is exactly the same as for the case of a reciprocal microcavity

ϕM=ϕ0+δϕ1(Δ1kx)+δϕ2(Δ2,kx)=2πn,
(6)

where δϕm = tan-1(Im(ρm)/Re(ρm)) is the nonreciprocal phase shift due to reflection at the interface between magnetic microcavity and the first (m = 1) or second (m = 2) nonmagnetic bounding layers (see Eqs. 2), and ϕ 0 = ϕD + ϕ 1 + ϕ 2 is the reciprocal phase which includes the dynamical phase ϕD = 2kMzdM and two boundary phases ϕm associated with reflection from the Bragg mirrors. Thus the nonreciprocal response of the structure can be optimized through: (i) an increase in the number of mirror layers which will result in narrower resonances reducing overlap of the forward and backward resonances (at the expense of the transmission bandwidth of course); or (ii) increase in the nonreciprocal phases by increasing the angle of incidence or choosing appropriate characteristics of the materials (refractive indexes and MO parameter) which will result in increase of the nonreciprocal phase shifts (according to Eqs. 2) and stronger separation of the forward and backward resonances, again reducing their overlap.

4. Conclusion

We propose the basic components described in this work as building blocks for a new class of nonreciprocal devices. Nonreciprocal MPCs of both ternary and microcavity types may be combined with waveguides to generate a new class of isolators and circulators that are compact, but simpler to fabricate than traditional 2D MO PC approaches [13

13. Z. Wang and S. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Opt. Lett. 30, 1989–1991 (2005). [CrossRef] [PubMed]

,14

14. Z. Wang and S. Fan, “Magneto-optical defects in two-dimensional photonic crystals,” Appl. Phys. B: Lasers Opt. 81, 369–375 (2005). [CrossRef]

]. Since a single magnetization is applied throughout the structure, a change in the isolation functionality could be obtained by a change in the direction of the applied magnetization. This would be extremely awkward in previously proposed systems with layers of different magnetizations. In addition, a significant advantage of MO devices based on structures of this kind is that they eliminate the polarizers that are required in the Faraday (longitudinal) configurations. At the same time, the Fabry-Perot structures proposed here may be superior to the conventional Faraday configuration, because of the complete transmission at the peak of the resonance. Nonreciprocal devices based on Faraday rotation show an intrinsic trade-off between the shift in the polarization angle and the transmission. At the peak of the Faraday rotation, the transmission of a Faraday device [11

11. M. Inoue, K. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers,” J. Appl. Phys. 83, 6768–6770 (1998); [CrossRef]

,12

12. M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999). [CrossRef]

] is never equal to 100 percent.

Our results therefore demonstrate that low-symmetry magnetic photonic crystals are excellent candidates for realization of various compact nonreciprocal photonic devices. Paradoxically perhaps, we have shown that by reducing symmetry, we may actually simplify fabrication challenges, and bring dynamically-tunable MO photonic crystal devices closer to reality.

Acknowledgments

CUDOS is an Australian Research Council Centre of Excellence.

References and links

1.

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). [CrossRef]

2.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-Free One-Way Edge Modes in a Gyro-magnetic Photonic Crystal,” Phys. Rev. Lett. 100, 01390501–01390504 (2008). [CrossRef]

3.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal,” Phys. Rev. Lett. 100, 02390201–02390204 (2008). [CrossRef]

4.

Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. 90, 121133:1–3 (2007). [CrossRef]

5.

V. Dmitriev, “Symmetry properties of 2D magnetic photonic crystals with square lattice,” Eur. Phys. J. Appl. Phys. 32, 159–165 (2005). [CrossRef]

6.

A. Figotin and I. Vitebskiy, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E 63, 066609:1–17 (2001). [CrossRef]

7.

A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997). [CrossRef]

8.

P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).

9.

A. A. Jalali and M. Levy, “Local normal-mode coupling and energy band splitting in elliptically birefringent one-dimensional magnetophotonic crystals,” J. Opt. Soc. Am. B 25, 119–125, (2008). [CrossRef]

10.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. May-dykovskiy, and O. A. Aktsipetrov, “Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects,” Phys. Rev. B 78, 193102:1–4 (2008). [CrossRef]

11.

M. Inoue, K. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers,” J. Appl. Phys. 83, 6768–6770 (1998); [CrossRef]

12.

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999). [CrossRef]

13.

Z. Wang and S. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Opt. Lett. 30, 1989–1991 (2005). [CrossRef] [PubMed]

14.

Z. Wang and S. Fan, “Magneto-optical defects in two-dimensional photonic crystals,” Appl. Phys. B: Lasers Opt. 81, 369–375 (2005). [CrossRef]

OCIS Codes
(230.3810) Optical devices : Magneto-optic systems
(230.3990) Optical devices : Micro-optical devices
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: January 21, 2009
Revised Manuscript: March 8, 2009
Manuscript Accepted: March 9, 2009
Published: March 18, 2009

Citation
Alexander B. Khanikaev and M. J. Steel, "Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices," Opt. Express 17, 5265-5272 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5265


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References

  1. MInoue, 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). [CrossRef]
  2. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008). [CrossRef]
  3. Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008). [CrossRef]
  4. Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007). [CrossRef]
  5. V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005). [CrossRef]
  6. A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001). [CrossRef]
  7. A. K. Zvezdin, V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997). [CrossRef]
  8. P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).
  9. A. A. Jalali and M. Levy, "Local normal-mode coupling and energy band splitting in elliptically birefringent one-dimensional magnetophotonic crystals," J. Opt. Soc. Am. B 25, 119-125, (2008). [CrossRef]
  10. A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008). [CrossRef]
  11. M. Inoue, K. Arai, T. Fujii and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998); [CrossRef]
  12. M. Inoue, K. Arai, T. Fujii and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999). [CrossRef]
  13. Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005). [CrossRef] [PubMed]
  14. Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005). [CrossRef]

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