## Taming the flow of light via active magneto-optical impurities |

Optics Express, Vol. 20, Issue 24, pp. 26200-26207 (2012)

http://dx.doi.org/10.1364/OE.20.026200

Acrobat PDF (1910 KB)

### Abstract

We demonstrate that the interplay of a magneto-optical layer sandwiched between two judiciously balanced gain and loss layers which are both birefringent with misaligned in-plane anisotropy, induces unidirectional electromagnetic modes. Embedding one such optically active non-reciprocal unit between a pair of birefringent Bragg reflectors, results in an exceptionally strong asymmetry in light transmission. Remarkably, such asymmetry persists regardless of the incident light polarization. This photonic architecture may be used as the building block for chip-scale non-reciprocal devices such as optical isolators and circulators.

© 2012 OSA

## 1. Introduction

4. B. E. A. Saleh and M. C. Teich, *Fundamentals of photonics* (Wiley, New York, 1991). [CrossRef]

5. There are several ways to address the problem with absorption. One approach is to replace a uniform magnetic material with a slow-wave magneto-photonic structure [6]. Under certain conditions, such a structure can enhance asymmetric transmitance effects associated with magnetism, while significantly reducing absorption. The problem with the above approach is that it does not apply to infrared and optical frequencies it can only work at MW frequencies. Another approach is to incorporate gain and loss together with non-linearity [19]. In this case, however, optical isolation occurs only for specific power ranges.

7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in *𝒫𝒯* symmetric optical lattices,” Phys. Rev. Lett. **100**, 103904 (2008). [CrossRef] [PubMed]

*𝒫𝒯*) symmetric media for which the complex index of refraction obeys the condition

*n*(

*r⃗*) =

*n*

^{*}(−

*r⃗*). Synthetic materials with

*𝒫𝒯*symmetries are shown to exhibit several intriguing features some of which have been already demonstrated in a series of recent experimental papers [9

9. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. **6**, 192 (2010). [CrossRef]

13. H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with *𝒫𝒯* symmetry,” Phys. Rev. A **85**, 062122 (2012). [CrossRef]

7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in *𝒫𝒯* symmetric optical lattices,” Phys. Rev. Lett. **100**, 103904 (2008). [CrossRef] [PubMed]

10. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with *𝒫𝒯* symmetries,” Phys. Rev. A **84**, 040101(R) (2011). [CrossRef]

11. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of *𝒫𝒯* -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. **103**, 093902 (2009). [CrossRef] [PubMed]

12. Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of *𝒫𝒯* -symmetric scattering,” Phys. Rev. A **85**, 050101(R) (2012). [CrossRef]

14. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by *𝒫𝒯*-symmetric periodic structures,” Phys. Rev. Lett **106**, 213901 (2011). [CrossRef] [PubMed]

15. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of *𝒫𝒯*-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. **38**, L171 (2005). [CrossRef]

16. S. Longhi, “Bloch oscillations in complex crystals with *𝒫𝒯* symmetry,” Phys. Rev. Lett. **103**, 123601 (2009). [CrossRef] [PubMed]

17. H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with *𝒫𝒯* symmetry,” Phys. Rev. A **85**, 013818 (2012). [CrossRef]

18. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “*𝒫𝒯* -symmetric Talbot effects,” Phys. Rev. Lett **109**, 033902 (2012). [CrossRef] [PubMed]

19. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear *𝒫𝒯* -symmetric optical structures,” Phys. Rev. A **82**, 043803 (2010). [CrossRef]

20. S. Longhi, “*𝒫𝒯* -symmetric laser absorber,” Phys. Rev. A **82**, 031801 (2010). [CrossRef]

22. Y. D. Chong, L. Ge, and A. D. Stone, “*𝒫𝒯*-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. **106**, 093902 (2011). [CrossRef] [PubMed]

23. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in *𝒫𝒯*-symmetric optical couplers,” Phys. Rev. A **82**, 043818 (2010). [CrossRef]

*𝒫𝒯*-symmetric structures and consists of three components: a central magnetic layer sandwiched between two active (one with gain and the other with loss) anisotropic layers distributed in a way that the whole structure exhibits an antilinear symmetry. The magnetic layer provides a nonreciprocal circular birefringence (magnetic Faraday effect). The role of the magnetic layer is to break the Lorentz reciprocity, which would otherwise impose the symmetry in forward and backward transmission of the layered structure. The magnetic circular birefringence does break the Lorents reciprocity, but it is not sufficient to provide asymmetry in forward and backward transmission. Another requirement is a broken space inversion symmetry. This is achieved with the use of misaligned birefringent layers as suggested in [24

24. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E **63**, 066609 (2001). [CrossRef]

*generalized*unitary relations for the energy flux conservation. The transport asymmetry can be drastically enhanced by embedding our micro-cavity inside an anisotropic Bragg grating (see Fig. 1(b)). We show the formation of broad frequency domains at the pseudo-gaps of the grating which support high-Q micro-cavity modes with enhanced transport asymmetry. The proposed architecture can be used for the creation of highly efficient on-chip non-reciprocal devices such as optical isolators and circulators.

## 2. Modeling and symmetries

*ε*̂(

*z*) is assumed to be where the variables

*z*

^{±}take values in the intervals −

*L*≤

*z*− ≤ 0 and 0 ≤

*z*

^{+}≤

*L*. Above,

*ε*=

_{xx}*ε*(

*z*)+

*δ*(

*z*)cos(2

*ϕ*(

*z*))+

*iγ*(

*z*),

*ε*=

_{xy}*δ*(

*z*)sin(2

*ϕ*(

*z*))+

*iα*(

*z*) and

*ε*=

_{yy}*ε*(

*z*)−

*δ*(

*z*)cos(2

*ϕ*(

*z*))+

*iγ*(

*z*). The function

*δ*describes the magnitude of in-plane anisotropy,

*γ*is the gain/loss parameter and the angle

*ϕ*defines the orientation of the principle axes in the

*xy*–plane. The gyrotropic parameter

*α*is responsible for the Faraday rotation. Outside the scattering region

*z*∉ [−

*L*,

*L*], we assume that the permittivity takes a constant value

*ε*

_{0}i.e

*ε*̂ =

*ε*

_{0}×

**1̂**where

**1̂**is the 2 × 2 identity matrix.

*μ*̂(

*z*) takes the form where

*μ*=

_{xx}*μ*(

*z*) =

*μ*(−

*z*),

*μ*=

_{xy}*iβ*(

*z*) =

*iβ*(−

*z*) and

*β*is another gyrotropic parameter which essentially depends on the static components of the magnetic field

*H⃗*

_{0}, as well as the frequency

*ω*.

*H⃗*

_{0}→ −

*H⃗*

_{0}and

*M⃗*

_{0}→ −

*M⃗*

_{0}implies the following transformation [24

24. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E **63**, 066609 (2001). [CrossRef]

*𝒫*̃

*𝒯*antilinear symmetry: The

*time reversal*operator

*𝒯*is an anti-linear operator which performs transpose complex conjugation while the linear operator

*𝒫*̃ =

*𝒫*Θ consist of the

*parity operator 𝒫*which represents a spatial inversion

*r⃗*→ −

*r⃗*(note though that in our case spacial inversion is the same as reflection), and the

*exchange operator*Θ which changes

*ϕ*

_{+}≡

*ϕ*(

*z*

^{+}) ↔

*ϕ*

_{−}≡

*ϕ*(

*z*

^{−}). We note that in the case of

*ϕ*

_{+}=

*ϕ*

_{−}, i.e. the two layers are not misaligned, the structure of Fig. 1(a) is

*𝒫𝒯*-symmetric. For this reason, below we will refer to our structure (with misalignment), as a generalized

*𝒫𝒯*-symmetric geometry.

## 3. Scattering formalism

*k*=

_{x}*k*= 0, the solutions of Eq. (6) for

_{y}*E⃗*(

*z*) in the left (l) and right (r) side of the scattering region, are written in terms of the forward and backward traveling waves: where The corresponding magnetic field

*H⃗*(

*z*) is

*z*̂ is the unit vector in the

*z*-direction.

**M**≡

**M**(

*α*,

*β*,

*ϕ*

_{−},

*ϕ*

_{+}) directly furnishes the relation between the electric field on the left and right sides of the scattering region (below we assume

*c*= 1 units), i.e.,

*𝒫*̃

*𝒯*on the solutions of Eq. (7) results in Since our system is invariant under the

*𝒫*̃

*𝒯*-operation then Eq. (10) is also a solution of the Maxwell equations. Applying once more the transfer matrix

**M**(

*α*,

*β*,

*ϕ*

_{−},

*ϕ*

_{+}) for the transformed solutions Eq.(10), we get: It follows from the Eq.(11) and the conjugated form of the Eq.(9) with

*α*→ −

*α*,

*β*→ −

*β*,

*ϕ*

_{−}↔

*ϕ*

_{+}that

## 4. Asymmetric transport

*𝒫*̃

*𝒯*-symmetric magneto-optical cavity of Fig. 1(a). The misalignment angle Δ

*ϕ*=

*ϕ*

_{+}−

*ϕ*

_{−}between the pair of active layers is different from 0 and

*π*/2 to ensure the asymmetry of forward and backward wave propagation [24

24. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E **63**, 066609 (2001). [CrossRef]

*R*>

^{l,r}*while in Fig. 2(b) we report the left-right transmittance difference |<*

_{p}*T*>

^{l}*− <*

_{p}*T*>

^{r}*| (where*

_{p}*R*≡ |

*r*|

^{2}and

*T*= |

*t*|

^{2}). In all these simulations the average 〈·〉

*is taken over all possible polarizations of the incoming wave. We find that the reflectances and transmittances for left/right incident waves are different from one another. Although an asymmetric left-right reflection is a characteristic property of systems with anti-linear symmetry [14*

_{p}14. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by *𝒫𝒯*-symmetric periodic structures,” Phys. Rev. Lett **106**, 213901 (2011). [CrossRef] [PubMed]

*Q*which is defined as

_{T}*𝒫*̃

*𝒯*-symmetric magneto-optical micro-cavity can be further amplified by embedding it between two identical anisotropic Bragg mirrors, see Fig. 1(b). The anisotropy at the Bragg gratings creates pseudo-gaps at the transmission spectrum as shown in Fig. 3(a). We have found that at these frequency windows the non-reciprocity is enhanced. In Fig. 3(b) we report the

*Q*-factor for a structure shown in Fig. 1(b) with a grating consisting of only 45 layers. The frequency domains of polarization independent asymmetric transport are marked with (green) shadowed areas and coincide with the pseudo-gaps of the grating.

_{T}## 5. Conclusion

26. 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 (2006). [CrossRef]

28. A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B **80**, 235106 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | L. Pavesi and D. J. Lockwood, |

2. | D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications |

3. | T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012). |

4. | B. E. A. Saleh and M. C. Teich, |

5. | There are several ways to address the problem with absorption. One approach is to replace a uniform magnetic material with a slow-wave magneto-photonic structure [6]. Under certain conditions, such a structure can enhance asymmetric transmitance effects associated with magnetism, while significantly reducing absorption. The problem with the above approach is that it does not apply to infrared and optical frequencies it can only work at MW frequencies. Another approach is to incorporate gain and loss together with non-linearity [19]. In this case, however, optical isolation occurs only for specific power ranges. |

6. | A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B |

7. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in |

8. | Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in |

9. | C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. |

10. | J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with |

11. | A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of |

12. | Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of |

13. | H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with |

14. | Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by |

15. | A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of |

16. | S. Longhi, “Bloch oscillations in complex crystals with |

17. | H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with |

18. | H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “ |

19. | H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear |

20. | S. Longhi, “ |

21. | Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. |

22. | Y. D. Chong, L. Ge, and A. D. Stone, “ |

23. | A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in |

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

25. | P. A. Mello and N. Kumar, |

26. | 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. |

27. | M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett. |

28. | A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B |

**OCIS Codes**

(000.6800) General : Theoretical physics

(130.0130) Integrated optics : Integrated optics

(230.2240) Optical devices : Faraday effect

(230.3240) Optical devices : Isolators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 31, 2012

Revised Manuscript: September 13, 2012

Manuscript Accepted: September 16, 2012

Published: November 5, 2012

**Citation**

Hamidreza Ramezani, Zin Lin, Samuel Kalish, Tsampikos Kottos, Vassilios Kovanis, and Ilya Vitebskiy, "Taming the flow of light via active magneto-optical impurities," Opt. Express **20**, 26200-26207 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26200

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### References

- L. Pavesi and D. J. Lockwood, Silicon photonics (Springer, Germany, 2004).
- D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012). [CrossRef]
- T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).
- B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New York, 1991). [CrossRef]
- There are several ways to address the problem with absorption. One approach is to replace a uniform magnetic material with a slow-wave magneto-photonic structure [6]. Under certain conditions, such a structure can enhance asymmetric transmitance effects associated with magnetism, while significantly reducing absorption. The problem with the above approach is that it does not apply to infrared and optical frequencies it can only work at MW frequencies. Another approach is to incorporate gain and loss together with non-linearity [19]. In this case, however, optical isolation occurs only for specific power ranges.
- A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008). [CrossRef]
- K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008). [CrossRef] [PubMed]
- Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).
- C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys.6, 192 (2010). [CrossRef]
- J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011). [CrossRef]
- A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009). [CrossRef] [PubMed]
- Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012). [CrossRef]
- H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012). [CrossRef]
- Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett106, 213901 (2011). [CrossRef] [PubMed]
- A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005). [CrossRef]
- S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett.103, 123601 (2009). [CrossRef] [PubMed]
- H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012). [CrossRef]
- H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012). [CrossRef] [PubMed]
- H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010). [CrossRef]
- S. Longhi, “𝒫𝒯 -symmetric laser absorber,” Phys. Rev. A82, 031801 (2010). [CrossRef]
- Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010). [CrossRef] [PubMed]
- Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011). [CrossRef] [PubMed]
- A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010). [CrossRef]
- A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001). [CrossRef]
- P. A. Mello and N. Kumar, Quantum transport in mesoscopic systems: complexity and statistical fluctuations : a maximum-entropy viewpoint, Volume 4 of Mesoscopic Physics and Nanotechnology (Oxford University Press, India, 2004).
- 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 (2006). [CrossRef]
- M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006). [CrossRef]
- A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009). [CrossRef]

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