## 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

The present rapid development of global communications and computer science demands constantly increasing capacities of data transmission and data processing. In this endeavor, photonic integrated circuits (PICs) are becoming very promising for many applications, including next generation optical networks, optical interconnects, tunable photonic oscillators, wavelength division multiplexed systems, coherent transceivers and optical buffers [1–3]. This is because PICs provide advantages, e.g., higher performance, reduced device footprint, lower component-to-component coupling losses and lower power consumption, to name a few. However, the progress on PICs is still quite limited. One challenging task that researchers are currently facing is the realization of novel classes of non-reciprocal integrated photonic devices that allow one-directional flow of information-e.g. optical isolators and circulators [4]. Currently, such unidirectional elements rely almost exclusively on magneto-optical effects, like the magnetic Faraday rotation caused by non-reciprocal circular birefringence.

At optical frequencies, all non-reciprocal effects (NRE), such as magnetic Faraday rotation, are very weak. This weakness of NRE results in prohibitively large size of most non-reciprocal devices. A natural way to enhance a weak NRE, and thus reduce the size of a structure, is to incorporate the magneto-optical material into an optical resonator, which can be a photonic structure with feature sizes comparable to the light wavelength. At the same time, such resonators cause undesirable effects, like enhanced absorption, linear birefringence, and nonlinear effects. The most deleterious of all is absorption which under resonance conditions can dramatically affect the functionality of the optical devices. For example, an enhanced absorption causes deviation of the transmitted light polarization from linear to elliptic (non-reciprocal circular dichroism), which significantly compromises the performance of optical isolator. In addition, the enhanced absorption can results in a significant power loss. Finally, even moderate absorption can lower the quality factor of the optical resonator by several orders of magnitude and, thereby, significantly compromise its performance as a Faraday rotation enhancer [5].

In this Letter we show that a strongly asymmetric transport at infrared and optical frequencies can be achieved in active magneto-photonic structures in which the spatial destribution of gain and loss displays a special, anti-linear symmetry (see Fig. 1). In classical optics, it involves the combination of delicately balanced gain and loss regions together with the modulation of the index of refraction [7]. A sub-class of such optical antilinear structures are the so–called parity-time (*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 [9–13]. These include among others, power oscillations [7–10], absorption enhanced transmission [11], unidirectional transparency and invisibility [12, 14], non-reciprocal Bloch oscillations [15, 16], a new type of conical diffraction [17] and reconfigurable Talbot effects [18]. In the nonlinear domain, such asymmetric transmittance effects can be used to realize a new generation of optical on-chip isolators and circulators [19]. Other results include the realization of coherent perfect laser-absorber [20–22] and nonlinear switching structures [23].

The micro-cavity that we investigate (see Fig. 1(a)) is a generalization of the standard *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]. Even then, the asymmetry in forward and backward transmission averaged over the input light polarization remains symmetric. In this report, we show that the interplay of the active birefringent elements with the magneto-optical layer can lead to strong asymmetry in forward and backward transmission. Although in such micro-cavities the local unitarity is violated, the balanced gain/loss design that we incorporated in our structure imposes *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

We consider the structure shown in Fig. 1(a). The non-Hermitian permittivity tensor, *ε*̂(*z*) is assumed to be

*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.

The corresponding permeability tensor, *μ*̂(*z*) takes the form

*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*.

Changing *H⃗*_{0} → −*H⃗* _{0} and *M⃗*_{0} → −*M⃗*_{0} implies the following transformation [24]

*𝒫*̃

*𝒯*antilinear symmetry:

*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

The electric and magnetic field are designated by the time-harmonic Maxwell equations:

*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:

*H⃗*(

*z*) is $\overrightarrow{H}(z)=\frac{1}{c}\widehat{z}\times \overrightarrow{E}(z)$, where

*z*̂ is the unit vector in the

*z*-direction.

The 4 × 4 transfer matrix **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.,

Performing the operation *𝒫*̃*𝒯* on the solutions of Eq. (7) results in

*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:

*α*→ −

*α*,

*β*→ −

*β*,

*ϕ*

_{−}↔

*ϕ*

_{+}that

The transmission and reflection coefficients can be expressed in terms of the transfer matrix elements as

From Eqs. (13) one can deduce the form of the scattering matrix
$\mathbf{S}\equiv \mathbf{S}(\alpha ,\beta ,{\varphi}_{-},{\varphi}_{+})=\left[\begin{array}{ll}{\mathbf{r}}^{l}\hfill & {\mathbf{t}}^{r}\hfill \\ {\mathbf{t}}^{l}\hfill & {\mathbf{r}}^{r}\hfill \end{array}\right]$, in terms of the **M**-matrix elements. After some straightforward algebra we can show that the scattering matrix should satisfy the following generalized unitary relation:

*𝒫*̃

*𝒯*-symmetric

**S**matrices in the presence of magneto-optical effects, and it is a weaker constraint than unitarity. In the latter case, each eigenvalue of the

**S**-matrix is

*unimodular*. In contrast, for generalized

*𝒫*̃

*𝒯*-symmetric structures this is not necessarily the case. Specifically, one can distinguish between two scenarios: If the

**S**and

*𝒫*̃

*𝒯*operator share the same eigenvectors |

*e*〉 with the corresponding

_{n}**S**-eigenvalues

*e*(

_{n}*α*,

*β*,

*ϕ*

_{−},

*ϕ*

_{+}) then ${e}_{n}^{*}\left(-\alpha ,-\beta ,{\varphi}_{+},{\varphi}_{-}\right){e}_{n}\left(\alpha ,\beta ,{\varphi}_{-},{\varphi}_{+}\right)=1$, and the corresponding eigenstates |

*e*〉 exhibit no net amplification or dissipation. In the opposite case |

_{n}*e*〉 is not itself

_{n}*𝒫*̃

*𝒯*- symmetric, but a

*pair*satisfies

*𝒫*̃

*𝒯*by transforming into each other |

*e*〉 ≡

_{m}*𝒫*̃

*𝒯*|

*e*〉 ≠ |

_{n}*e*〉 with one exhibiting amplification, and the other dissipation. The corresponding

_{n}**S**-matrix eigenvalues satisfy the relation ${e}_{m}^{*}\left(-\alpha ,-\beta ,{\varphi}_{+},{\varphi}_{-}\right){e}_{n}\left(\alpha ,\beta ,{\varphi}_{-},{\varphi}_{+}\right)=1$.

## 4. Asymmetric transport

Next we focus on the transport properties of the simple *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]. In the absence of gain and loss, a polarization averaged asymmetric transmission would be prohibited due to the sum rule resulting from the flux conservation and unitarity of the scattering matrix [25]. This prohibition is lifted with the introduction of loss and gain layers in Fig. 1(a).

To confirm the above expectations we present in Fig. 2(a) the result of our numerical simulation for the average (left/right) reflectance <*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], the asymmetry in the transmittances is a new element which essentially requires the presence of both a magneto-optical material and loss and/or gain materials. To further quantify the asymmetric behavior of our structure we report in Fig. 2(c) the asymmetry quality factor*

_{p}*Q*which is defined as

_{T}The asymmetry of the *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 _{T}* -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.

The enhancement of asymmetry can be understood intuitively once we realize that our micro-cavity behaves as a high-Q optical resonator filled with magneto-optical material once it is embedded in a Bragg grating. In this case, each individual photon resides in the magnetic material much longer compared to the same piece of magnetic material placed outside the resonator. Since the sign of Faraday rotation is independent of the direction of light propagation, one can assume that the total amount of Faraday rotation is proportional to the photon residence time in the magnetic material.

## 5. Conclusion

Although the enhancement of non-reciprocal effects, such as Faraday rotation, can be achieved via resonance conditions (see, for example [26–28] and references therein), the same resonance conditions also enhance the absorption, which can ruin the performance of almost any non-reciprocal device. Here we have shown that the use of a judiciously balanced gain and loss unit shown in Fig. 1, or its equivalent, can simultaneously solve two fundamental problems. Firstly, it allows a significant enhancement of the desired non-reciprocal effects without creating the energy loss issue. Secondly, the simultaneous presence of loss/gain components and a magnetic component result in strong polarization-independent transmission asymmetry, which is the defining property of an optical isolator. Our photonic architecture may be used as the building block for chip-scale non-reciprocal devices such as optical isolators and circulators.

## Acknowledgments

This research was supported by an AFOSR No. FA 9550-10-1-0433 grant and LRIR 09RY04COR grant, and by an NSF ECCS-1128571 grant.

## References and links

**1. **L. Pavesi and D. J. Lockwood, *Silicon photonics* (Springer, Germany, 2004).

**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 **1**, 1 (2012). [CrossRef]

**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, *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.

**6. **A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B **77**, 104421 (2008). [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]

**8. **Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in *𝒫𝒯* periodic potentials,” ibid. **100**, 030402 (2008).

**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]

**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]

**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]

**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]

**21. **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]

**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]

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

**25. **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).

**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]

**27. **M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett. **89**, 121113 (2006). [CrossRef]

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