## Abstract

We report a kind of broadband electromagnetic boundary mode at an interface of anti-parallel magnetized media, which can only propagate in one direction perpendicular to the magnetization and parallel to the interface. The unidirectionality of this mode originates from the permeability or permittivity tensor introduced by magnetization. We theoretically and numerically analyze the existence of the unidirectional mode, and point out that this mode can exist in both gyromagnetic and gyroelectric medium. We also propose a one-way waveguide based on this unidirectional mode, which may realize a new kind of electromagnetic isolation differing from those existing ones.

© 2010 OSA

## 1. Introduction

Nonreciprocal components, which break time-reversal symmetry, are proven to be important in electromagnetic circuits. Many theories and mechanisms have been approved to realize the nonreciprocal active components, such as circulators and isolators. The traditional way to achieve time-reversal breaking is to utilize the Faraday Effect, introduced by applying an external magnetic field. Under an external field, magneto-optical materials possess permeability or permittivity tensor with non-zero off-diagonal elements, which are imaginary. Thus the media is anisotropic, and breaks the time-reversal and space-inversion symmetries as well. Utilizing this effect, many nonreciprocal components have been developed [1–4], while some working in microwave frequencies have measured up to a commercial level. With the development of integrated optics [5,6], more and more attention is paid to the integratable nonreciprocal components [7–18], which may be suitable for on-chip optical circuits, and whose working frequencies are near infrared. Usually, on-chip isolators utilize magneto-optical (MO) effects other than Faraday rotation, such as a nonreciprocal propagation constant and nonreciprocal loss phenomenon. Recently, waveguide isolators are attempted [19–23], and isolators based on dual-waveguide, Mach-Zehnder interferometers, have also been reported [24–27]. With the help of MO photonic crystals (PhCs), analogous to edge mode in quantum hall effect, one-way waveguides have been proposed [28,29], which may provide a hopeful integratable solution. Another one-way waveguide is suggested utilizing nonreciprocal surface plasma polariton and the constraining of light by PhC [30]. Optical signals can propagate unidirectionally forward in both of these two one-way waveguides, suppressing backward scattering, if only the working frequency is in a particular range. In these one-way waveguides, MO materials play a key role in breaking time-reversal symmetry. With an external magnetic field applied, MO materials can introduce special MO effects, such as a nonreciprocal mode at the surface [31,32]. Using this nonreciprocal effect, more compact isolators suitable for integrated circuits may be achieved. Meanwhile, a magnetic domain wall, formed at an interface of anti-parallel magnetized media, has been used to obtain a nonreciprocal mode phase shift [33,34]. Combining PhCs with magnetic domain wall, an nonreciprocal waveguide can be achieved [35], which breaks time-reversal and space-inversion symmetries, respectively. However, all of these works mentioned above need complex structures or dynamic modulations, which may bring new challenges in integrated circuits. Besides, to the best of our knowledge, the existing isolations are all sensitive to frequency, which is a drawback for today’s broadband information processing. Thus it’s urgent to find a new isolating mechanism with a simple structure and broadband capacity.

Here we present and demonstrate a novel mechanism with a simple straight waveguide, in which broadband unidirectional mode exists along a magnetic domain wall formed in MO materials. Utilizing this effect, broadband isolation can be achieved with a simple structure. This nonreciprocal MO effect originates from the off-diagonal elements in permittivity or permeability tensor induced by magnetization. Different from the unidirectional mode mentioned in Ref [30], unidirectionality of this mode only depends on the signs of off-diagonal elements in the tensor. Thus, this mode can support broadband unidirectional frequencies, consequently.

## 2. Theoretical analysis

We start by analyzing the additional effects introduced by an external static magnetic field, as in this paper along the *z* axis in Cartesian coordinates, applied on homogenous MO materials. The external field changes the media’s response to an electromagnetic wave, by changing the permittivity or permeability into a tensor form, and making the media anisotropic. In frequencies near optical range, the change usually takes place in the permittivity, while in permeability near frequencies of microwave range, as below

Where ${\epsilon}_{0}$ and ${\mu}_{0}$ are the permittivity and permeability of vacuum, respectively. Here the elements of these two tensors, i.e. ${\epsilon}_{1,2}$ and ${\mu}_{1,2}$, usually fluctuate with working frequency, but in a relatively large frequency range, the signs of them keep unchanged. We will show in this paper, the signs of these tensor’s elements will determine the propagating direction of boundary mode at domain walls, across which the signs of off-diagonal elements, i.e. ${\epsilon}_{2}$ or ${\mu}_{2}$ change rapidly while signs of diagonal elements, i.e. ${\epsilon}_{1}$ or ${\mu}_{1}$, keep unchanged. Technologically, this can be realized by applying anti-parallel external magnetic field along a fixed plane [35,36].

For detail, we focus on the case of permittivity, which means the working frequency is near optical range, and the media is gyroelectronic material. The tensor form of $\overleftrightarrow{\epsilon}$ in Eq. (1) shows gyrotropic, which will only give chirality on the TM mode (with non-zero *H _{z}*,

*E*and

_{x}*E*), and leave the TE mode (with non-zero

_{y}*E*,

_{z}*H*and

_{x}*H*) nonchiral. Then for the TE mode, the media is still an isotropic one, and there is no boundary mode at domain wall. For TM mode localized at the boundary, we want their field to exponentially decay on both sides of the boundary. Either side of the domain wall is composed of MO material, with reverse sign of ${\epsilon}_{2}$. Since ${\epsilon}_{2}$ is not homogeneous in the system, we consider ${\epsilon}_{2}$ as function of position

_{y}**r**. Assuming the domain wall is at

*zx*plane in Cartesian coordinate system, as shown in Fig. (1a) , then for

*y>0*and

*y<0*we have ${\epsilon}_{2}\left(r\right)={\epsilon}_{2}$ and ${\epsilon}_{2}\left(r\right)=-{\epsilon}_{2}$, respectively. For simplicity, we neglect the material loss. We therefore take

*y*>0, and

*y*<0. Here ${\tilde{\epsilon}}_{1}={\epsilon}_{1}/({\epsilon}_{1}^{2}-{\epsilon}_{2}^{2})$ and ${\tilde{\epsilon}}_{2}={\epsilon}_{2}/({\epsilon}_{1}^{2}-{\epsilon}_{2}^{2})$,

*α*and

*β*are positive decay parameters,

*k*is the wave number along the domain wall. Then the sign of

*k*can determine propagating direction of the mode. A positive

*k*means that the mode propagates along

*+ x*axis, while a negative

*k*means the mode propagates along the

*-x*axis. In order to complete the problem, we must match the solutions in each region by the use of boundary conditions at

*y =*0 plane, that the tangential components of

**E**and the normal components of

**B**are continuous. These two conditions reduce to the results that $A=B$ and $\alpha =\beta $. Also, we finally get to a novel result that

Here ${Q}_{E}={\tilde{\epsilon}}_{2}/{\tilde{\epsilon}}_{1}={\epsilon}_{2}/{\epsilon}_{1}$, is the voigt parameter, which fluctuates with external field and working frequency, but whose sign keeps unchanged in a relative wide range of frequency. Thus, given the positive *α* and assuming positive ${\epsilon}_{1}$ and ${\epsilon}_{2}$, we therefore have positive ${Q}_{E}$ and negative *k*, which determine the mode can only propagate along *-x* direction. On the other hand, with the reversed magnetization in *y*<0 and *y*>0 regions, respectively, ${\epsilon}_{2}\left(r\right)$ changes signs and we can also get a boundary mode which can only propagate in + *x* direction. With the relation $k={n}_{\text{eff}}\omega /c$, we conclude that the transverse decay parameter *α*, which defines the intensity of localization for boundary mode, is determined by frequency *ω*, effective refractive index ${n}_{\text{eff}}$ and voigt parameter ${Q}_{E}$, with the relation $\alpha =\left|{Q}_{E}k\right|$. Therefore, the larger ${Q}_{E}$, the larger *α*, and the stronger unidirectional phenomenon will be. Also we can conclude by taking Eq. (4) into Eq. (2) and Eq. (3) that this boundary mode is actually a TEM mode, with only non-zero *H _{z}* and

*E*.

_{y}The situation is similar with gyromagnetic media, in which permeability is a tensor as $\overleftrightarrow{\mu}$ in Eq. (1). The domain wall in gyromagnetic media can support unidirectional boundary mode which is proven to be TEM mode with non-zero *H _{y}* and

*E*. The voigt parameter ${Q}_{M}={\mu}_{2}/{\mu}_{1}$ determines the decay parameter by $\alpha =\left|{Q}_{M}k\right|$, then the localization of the boundary mode at domain wall. And also, with signs of ${\mu}_{1}$ and ${\mu}_{2}$ unchanged, we have fixed-sign

_{z}*k*with relation $k=-\alpha /{Q}_{M}.$ Thus we have a broadband unidirectional boundary mode at the interface.

## 3. Numerical calculation

#### 3.1 Unidirectional boundary mode at domain wall

As shown above, the unidirectional property of this boundary mode only depends on the signs reversing of off-diagonal elements in permittivity or permeability tensors. Therefore it is robust even concerning the loss and frequency dependency of real materials. To verify our theoretical analysis about this unidirectional boundary mode, we utilize a 2D finite-difference time-domain (FDTD) method [37] to demonstrate the propagation of the unidirectional mode. As discussed above, larger voigt parameter can show greater unidirectional boundary phenomenon, we choose gyromagnetic materials mentioned in Ref [28], which show a large voigt parameter. For detail, we choose ${\mu}_{1}=14$ and ${\mu}_{2}=12.4$, while the relative permittivity is $\epsilon =15$. Here, the signs of these parameters keep unchanged in a wide frequency range, and for simplicity, we assume values of these parameters unchanged in our demonstration. As shown in Fig. 1, the nonreciprocal boundary mode can only propagate along + *x* direction, while the backward counterpart is absent. Magnetic field **H** is shown as arrows in Fig. 1(a), which is confirmed to be a TEM mode boundary at the interface. We choose five different working frequencies normalized to a unit length *a*, which are 0.050, 0.075, 0.100, 0.125 and 0.150 ($\times 2\pi c/a$), respectively. Here we use the normalized frequency instead of real frequency, for the idiomatic use in integrated optics. It is easy to transform to real frequencies, if only *a* is provided. All these frequencies can possess a unidirectional mode, which prove that this unidirectional boundary mode can work with a broadband frequency range. The localization intensity of boundary mode's energy increases with the frequency, as shown in Fig. 1(b-e). Simultaneously in our demonstration, there are bulk modes which can propagate off the domain wall, and which are reciprocal. With proper designation and with the help of a proper absorbing boundary, this reciprocal affection can be inhibited maximally.

Our discussion above bases on the conditions that magnetization over the domain wall reverses rapidly. In some occasions magnetization reversion may experience a thin layer of transition region. With transition layer, the unidirectional mode discussed above transforms into a nonreciprocal mode, which propagates alone the layer. It can be proved that with this layer’s thickness increasing, the unidirectionality of this mode decreases. Furthermore, in our discussion, the unidirectional effect relies on the anti-symmetrical profile of the domains. If the profile of the domain deviates from this situation, e.g. *y>0* half part is partly demagnetized while *y<0* half part keeps unchanged, the unidirectionality of boundary mode also decreases. In the extreme occasion, when *y>0* half part is unmagnetized, the unidirectional mode disappears. The simulation results for deviation from anti-symmetrical profile of domain wall are shown in Fig. 2
. For a partly demagnetization, we assume the parameters as ${\mu}_{1}=6$and ${\mu}_{2}=2$ for *y>*0, and keep parameters the same as above for *y<*0. Different from the anti-symmetrical case, more energy radiates into bulk modes, while still keeping unidirectional boundary mode observable. For this case the result is shown in Fig. 2(a). Another extreme occasion, when *y>0* half part is unmagnetized, we have ${\mu}_{1}=1$and${\mu}_{2}=0$. The result is shown in Fig. 2(b). In this case we cannot find any boundary mode at the interface, and all the radiation energy transforms into bulk modes. In conclusion, the nonreciprocal effect and the unidirectionality mainly depend on the rapid reversion of the magnetization.

#### 3.2 Broadband isolator based on unidirectional boundary mode

The introduction of this unidirectional mode at domain wall will provide us a potential alternative method to achieve compact nonreciprocal components such as broadband isolators, with a very simple structure. As an example, we consider a one-way waveguide composed of a straight domain wall, with anti-parallel magnetization on each side, which is shown in Fig. 3(a)
. The material’s parameters are chosen the same as above in Fig. 1. In order to prevent perturbation of bulk modes, and to achieve high isolation ratio, we clad the isolating component with absorbing layers. We here use two reciprocal waveguides as input and output ports. In order to minimize the reflection between reciprocal waveguides and one-way waveguide, we assume the reciprocal waveguide with same relative permittivity$\epsilon =15$, and a relative permeability$\mu =18.7$, for the purpose of impedance matching with one-way waveguide. The length of domain wall is chosen as $16a$, while the input and output reciprocal waveguide’s width is *a*. As discussed above, this waveguide possess broadband unidirectional boundary mode at the interface. The forward and backward propagating transient field pattern are shown in Fig. 3(b) and Fig. 3(c) respectively, with forward and backward transmission, and isolation ratio shown in Fig. 4(a)
and Fig. 4(b), respectively. We can observe a great isolation ratio in this simple isolation based on unidirectional boundary mode, and in a wide frequency range, the isolation is still robust. In Fig. 4, the fluctuating of transmissions and isolation ratio is induced by bulk mode which is reciprocal. Also we can observe ripples in both forward and backward transmissions, which are induced by reflection interference. Due to the unidirectionality of the boundary mode, backward propagating energy along the boundary is suppressed, and the only backward transmission of energy comes from bulk mode, which is reciprocal. This makes the backward transmission very low, as shown in the inside box in Fig. 4(a). The maximum of forward transmission is about 73%, due to the loss on reflections at interface with reciprocal waveguides, and also due to the loss on bulk modes which propagate off the domain wall deviating from the output port. The peak of forward transmission occurs at frequency 0.07 ($\times 2\pi c/a$), at which frequency the mode matching allows most energy from reciprocal waveguide coupled into unidirectional boundary mode. Considering the backward transmission, which is only affected by reciprocal bulk mode, the isolation ratio possesses a peak value of 25 dB at frequency 0.093 ($\times 2\pi c/a$).

## 4. Conclusion

In summary, we report the theoretical and numerical demonstration of a unidirectional boundary mode at domain wall. This unidirectional property relies on the signs reversing of off-diagonal elements in permittivity or permeability tensors, therefore it is robust in a wide frequency range. This broadband nonreciprocal behavior may be used to design compact and integratable nonreciprocal component, such as isolators and circulators. The intensity of this boundary mode is proportional to the voigt parameter, which is limited in real MO materials. Metamaterials [38,39] may provide a more tunable way to achieve higher level unidirectional performance.

## Acknowledgments

This work was supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No. 708038), and Thanks are given to Xiaofei Zang and Cai Huang for their helpful discussion.

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