## Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices

Optics Express, Vol. 17, Issue 7, pp. 5265-5272 (2009)

http://dx.doi.org/10.1364/OE.17.005265

Acrobat PDF (1011 KB)

### 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

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]

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]

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

**90**, 121133:1–3 (2007). [CrossRef]

## 2. Mathematical treatment

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

**k**, and for non-normal incidence, this component contributes to the component of the field tangential to the interface. Straightforward calculations yield the expression:

*d*=

_{m}*ε*

^{2}

_{m}- Δ

^{2}

_{m}and

*k*(

_{mx}*z*) is the

*x*(

*z*) component of the wavevector in the medium

*m*. Thus, disregarding effects second order in Δ

_{m}, 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

*k*(which is the same in every medium) and on Δ

_{x}_{m}. 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

*ϕ*= tan

_{a}^{-1}(Im(

*ρ*)/Re(

^{a}*ρ*)), which is odd not only with respect to the angle of incidence but also with the magnetization direction (through the sign of Δ

^{a}_{m}) [7

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

**90**, 121133:1–3 (2007). [CrossRef]

*Mˆ*that connects the amplitudes of forward (

_{ij}*H*

^{+}

_{y}) and backward-going (

*H*

^{-}

_{y}) waves on either side of the interface between layers

*i*and

*j*as follows:

*F*= (

_{m}*ε*

_{m}*k*+

_{zm}*i*Δ

_{m}

*k*)/

_{xm}*d*.

_{m}### 2.1. Characterization of nonreciprocality

*TˆH̄*= exp(

_{y}*iK*)

_{M}a*H̄*, where

_{y}*K*is the Bloch vector and

_{M}*H̄*= (

_{y}*H*

^{+}

_{y},

*H*

^{-}

_{y})

^{T}. This condition is satisfied when det [

*Tˆ*- exp(

*iK*)

_{M}a*I*ˆ] = 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

*k*responsible for nonreciprocity may be extracted explicitly. Limiting our consideration to the case of a single MO layer in the elementary cell (Δ

_{x}_{1}= Δ

_{3}= 0), after tedious but straightforward calculations we obtain the dispersion relation

*K*and

_{M}*K*

_{0}are Bloch wave vectors for the magnetic and corresponding non-magnetic structures and second order terms in Δ

_{2}have been omitted. The first term in Eq. (4) is the reciprocal contribution to the dispersion containing standard nonmagnetic terms due to periodicity [8]. The second explicitly shows the asymmetric contribution to the dispersion relation due to its linear dependence on the lateral wavevector component

*k*and Δ

_{x}_{2}. Note that reversal of the magnetization direction has the same effect as reversal of the sign of

*k*. (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

_{x}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]

_{2}, and consider the no nreciprocal contribution

*δ*≪

*K*

_{0}to the dispersion as a small correction to its reciprocal part

*K*

_{0}. Substituting

*K*=

_{M}*K*

_{0}+

*δ*into (4) and expanding the cosine function on the left hand side of Eq. (4) gives

*k*and Δ

_{x}_{2}responsible for the nonreciprocity because simultaneous reversal of the wavevector

**and**

*k***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

*K*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

_{M}*k*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

_{x}*θ*= 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

_{i}*θ*= 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)).

_{i}*k*) 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

_{x}*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).

## 3. Designing one-way structures

**90**, 121133:1–3 (2007). [CrossRef]

*λ*≈ 1545 nm, there is a narrow frequency range where the transmittance in the forward direction and backward direction is significantly different—the structure is transparent in the forward direction and almost opaque in the backwards direction.

**90**, 121133:1–3 (2007). [CrossRef]

*T*(

*k*⃗) -

*T*(-

*k*⃗) ∣ ≈ 1. Constructing such large stacks would be very difficult. Structures fabricated to date have been limited to around ten layers, since in order to preserve strong magnetic effects, an annealing step is required after each garnet layer is deposited, and this ultimately induces serious cracking [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]

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]

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. 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]

*T*(

*k*⃗) ≠

*T*(-

*k*⃗), 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

*δω*∝

*k*Δ

_{x}_{M}, and that the differential transmittance ∣

*T*(

*k*⃗) -

*T*(-

*k*⃗)∣ ≈ 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

*δϕ*= tan

_{m}^{-1}(Im(

*ρ*)/Re(

^{m}*ρ*)) is the nonreciprocal phase shift due to reflection at the interface between magnetic microcavity and the first (

^{m}*m*= 1) or second (

*m*= 2) nonmagnetic bounding layers (see Eqs. 2), and

*ϕ*

_{0}=

*ϕ*+

_{D}*ϕ*

_{1}+

*ϕ*

_{2}is the reciprocal phase which includes the dynamical phase

*ϕ*= 2

_{D}*k*and two boundary phases

_{Mz}d_{M}*ϕ*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.

_{m}## 4. Conclusion

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]

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]

## Acknowledgments

## 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. |

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. |

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. |

4. | Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. |

5. | V. Dmitriev, “Symmetry properties of 2D magnetic photonic crystals with square lattice,” Eur. Phys. J. Appl. Phys. |

6. | A. Figotin and I. Vitebskiy, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E |

7. | A. K. Zvezdin and V. A. Kotov, |

8. | P. Yeh, |

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 |

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 |

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. |

12. | M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. |

13. | Z. Wang and S. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Opt. Lett. |

14. | Z. Wang and S. Fan, “Magneto-optical defects in two-dimensional photonic crystals,” Appl. Phys. B: Lasers Opt. |

**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

- 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]
- 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]
- 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]
- 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]
- V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005). [CrossRef]
- A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001). [CrossRef]
- A. K. Zvezdin, V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997). [CrossRef]
- P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).
- 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]
- 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]
- 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]
- M. Inoue, K. Arai, T. Fujii and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999). [CrossRef]
- Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005). [CrossRef] [PubMed]
- 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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