## Abstract

A single-mode-waveguide optical isolator based on propagation direction dependent cut-off frequency is proposed. The isolation bandwidth is the difference between the cut-off frequencies of the lowest forward and backward propagating modes. Perturbation theory is used for analyzing the correlation between the material distribution and the bandwidth. The mode profile determines an appropriate distribution of non-reciprocal materials.

© 2008 Optical Society of America

## 1. Introduction

An optical isolator is a component that transmits light efficiently in one direction but prevents backward propagation. Integration of optical isolators with other optical devices is desired because they can improve on-chip optical systems and reduce device size and cost. On-chip optical interconnect technology requires a high data rate with a large signal-to-noise ratio, so unwanted backward propagation should be minimized. Research on optical isolators has been extensive in the past decade, leading to a wide range of isolator designs. The most traditional design consists of a Faraday rotator and two polarizers [1]. The need of polarizers makes these devices complicated for on-chip integration. To avoid this drawback, in the past few years a new generation of isolators have been developed that operate on the magneto-optical phenomenon of a nonreciprocal phase shift, rather than polarization conversion as for Faraday rotation based isolators [2–5]. A nonreciprocal phase shift is used to achieve isolation in asymmetric Mach-Zehnder interferometers (MZI), where one branch is nonreciprocal. A major obstacle is that phase shift isolators require precise wave interference from two long waveguides. Other designs include photonic crystal, single waveguide isolators where isolation is displayed in nonreciprocal band diagrams, i.e. ω(*k*)≠ω(-*k*). To generate such nonreciprocal dispersion, time reversal *T* and spatial inversion *S* symmetries need to be broken-magnetic material is capable of breaking *T*, and an asymmetric distribution of this material can break *S* [6]. One model of this phenomenon is that isolation occurs at Dirac points: frequencies where only backward propagating waves have a zero group velocity [7]. Another model of isolation using band diagrams shows that modes of backward waves are eliminated by waveguide cutoff frequencies [8, 9]. A similar design includes coupling between 2D photonic crystal waveguide modes and plasmon modes for a metal/dielectric waveguide subject to a magnetic field [10]. More recently, 2D photonic crystal waveguides have shown isolation using metal/dielectric interfaces where unidirectional modes exist that are analogous to Chiral states of the quantum Hall effect [11].

The promising features of the designs listed above include: analysis using nonreciprocal band diagrams where isolation is produced by cutoff of backward propagating modes; and the simplicity of single waveguide isolator design, rather than a rotator or MZI. This type of band diagram analysis is useful for finding unidirectional single mode operation—all other modes are unguided. The single waveguide designs given above can be further simplified by confining modes using a dielectric nanowire waveguide. Amemiya et al. have simulated and fabricated isolators using this approach [12]. However, they do not achieve isolation by modal cutoff, but by increased absorption loss for backward guided modes [13].

## 2. Proposed optical isolator designs

We propose a novel isolator design that both: (1) operates via backward propagating mode cutoff, and (2) employs a simple dielectric waveguide design. Furthermore, we achieve single mode, unidirectional propagation in simulation. Non-reciprocal material is included to break time-reversal symmetry, but the spatial inversion symmetry needs to be broken as well. This is accomplished by distributing magnetic material non-uniformly in the waveguide’s cross-section. In our isolator designs, we choose appropriate modes and an inhomogeneous distribution of non-reciprocal materials so that only the forward propagating mode(s) is guided. Figure 1 shows dispersion curves for the first and second lowest modes (solid lines labeled by Δ*ε*=0) and their forward and backward modes (dashed lines labeled by Δ*ε*≠0 indicating the incorporation of non-reciprocal material). Single mode isolation is realized between the cut-off frequencies of the lowest forward and backward modes; the isolation bandwidth is shown in the figure. It should be noted that our designs use neither the phase shift nor a difference in propagation loss between guided forward and backward modes to achieve isolation. Within the bandwidth, all modes but the lowest forward mode are unguided and quickly lose energy from a waveguide, that is, the single-mode isolation is realized.

In order to confirm this concept, we simulate multiple waveguide designs; see Fig. 2. The two-dimensional plane wave expansion (PWE) method [14] is used to calculate the waveguide dispersion. In our design, the waveguide is a nanowire on a low-index substrate. We take the z axis as the propagation direction and the permittivity is a function of x and y and independent of z. Non-reciprocal materials have the permittivity tensor:

which has nonzero off-diagonal components. We explore different waveguide shapes that include: a rectangle, a trapezoid, and a rib waveguide. A rectangle nanowire with a width *w* and a height *h* is the first design simulated. The left half of the nanowire is reciprocal material A, and the right half non-reciprocal material B-a material distribution that we call the “right-left” configuration; see Fig. 2(a). Variations on the configuration include: changing the reciprocal material on the left to non-reciprocal material, but with magnetization anti-parallel to the material on the right, as in Fig. 2(b); and stacking the materials vertically-the “up-down” design, as in Figs. 2(c), 2(d).

Figure 3(a) gives the dispersion relation for the right-left structure with the B+ and B- anti-parallel magnetizations. The dispersion diagram indicates single-mode optical isolation. The effect of off-diagonal components on the electric field is negligible, as shown in Fig. 3(b). The modeling results are summarized in Table 1. The distribution of the non-reciprocal materials is an important factor that determines the isolation bandwidth. The isolator with anti-parallel magnetization gives a larger bandwidth than an isolator with reciprocal material in one half. However, these “right-left” designs may not be simple in terms of fabrication; this could be avoided in the “up-down” configuration. Figures 2(c), 2(d) shows the “up-down” configuration, where TM mode isolation can be achieved in a waveguide with *w*<*h*. The simulation results show inversion symmetry in the dispersion relation, i.e. no isolation if we use non-reciprocal materials with the same permittivity tensors as those in the “right-left” configuration. Instead, in this “up-down” configuration different off-diagonal elements are needed to break the inversion symmetry-changing the off-diagonal elements is effectively changing the direction of the magnetization. We also analyze the structures that account for imperfections in fabrication, such as rib (Fig. 2(e)) and trapezoid waveguides (Fig. 2(g)). The trapezoidal structure shows a slightly smaller bandwidth than the rectangular structure.

## 3. Discussions

We use non-degenerate perturbation theory to understand the correlation between the inhomogeneity of nonreciprocal materials and nonreciprocal dispersion shifts. This knowledge will help us optimize the isolation bandwidth. The modal electric field is written by **E**
_{n}=E_{n}
_{0} exp[i(ωt-βz)] |*n*› where the amplitude E_{n}
_{0} is determined by normalizing |*n*›. Assuming div**E**≈0 and β≈β_{0}, the unperturbed eigenvalue equation with eigenvalue *β*
^{2} and eigenmode |*n*›=(*E*
_{x}
*E*
_{y}*E*
* _{z}*)

^{t}is:

where

and the transverse Laplacian is ∇^{2}
* _{t}*=

*∂*

_{2}/

*∂*

*x*

_{2}+

*∂*

^{2}/

*∂*

_{y}^{2}. Here, we add a small perturbation to the operator

*ω*

^{2}

*µ*

_{0}$\widehat{\Delta}$

*by changing the permittivity tensor given by*

_{ε}where *u*
* _{ij}* is real. This tensor can represent magnetization parallel to one of the Cartesian axes (x,y,z). In our modeling results, the bandwidth is small relative to the operation frequency, so β≈β

_{0}is justified. The eigenmode field change is also negligible as seen in Fig. 3(b). A square of the propagation constant is corrected to the first order:

where

isolation bandwidth obtained from practical materials is so small that the bandwidth Δ*ω* is approximated to be proportional to the dispersion shift between forward and backward waves:

Using Eq. (6) and the electric field of an unperturbed mode, we can calculate dispersion curves of forward and backward waves. Thus, we do not need to model different distributions of non-reciprocal materials in order to evaluate the bandwidth. We can visualize the contribution of non-reciprocal material to the propagation constant shift and conduct a direct comparison between different material distributions for the same unperturbed mode. Figure 4 shows two functions Im[E_{z} *E_{y}] and Im[E_{x} *E_{z}] calculated from the lowest TE and TM mode profiles. For the TE mode (Figs. 4(a), 4(b)), the largest component Im[E_{z} *E_{y}] is odd about y=0, but is almost even about a waveguide mid-plane parallel to the substrate. Therefore, it is desired that the function uyz is odd about y=0, i.e. the “right-left” anti-parallel magnetization is appropriate, and the “right-left” configuration produces a larger isolation for the lowest TE mode than for the lowest TM mode; see Fig. 3(a). Two components Im[E_{i} *E_{j}] of the lowest TM mode are shown in Figs. 4(c), 4(d). The largest component Im[E_{x} *E_{z}] is almost odd about waveguide mid-plane, but is even about y=0. Therefore, it is desired that the function u_{zx} is odd about the waveguide mid-plane, i.e. the “up-down” anti-parallel magnetization is appropriate. Table 1 also shows the value I_{yx}+I_{xz}+I_{zy}, which is approximately proportional to the bandwidth obtained from rigorous 2D PWE modeling.

Figure 5(a) shows both the dispersion curves obtained from perturbation theory and those from 2D PWE modeling. For the different isolator configurations, the deviation of our perturbation theory results from 2D PWE modeling is approximately 20% (Fig. 5(b)) and independent of the perturbation strength. The second order correction is small (the TE-TM mode conversion is small as well) and does not effectively contribute to the isolation bandwidth, because dispersion curves of forward and backward waves shift by the same amount to the same direction. Epsilon averaging is not used in our PWE modeling. We obtain the eigenvalue Eq. (2) by assuming that the gradient of the permittivity is negligible everywhere. High index contrast waveguides give a larger error than low contrast waveguides-a result of the assumption div**E**≈0. Knowing this limit of the analysis, we can use perturbation theory to optimize optical isolators.

Lastly, we show how to improve the isolator bandwidth via Im[E_{i} *E_{j}] analysis. In Fig. 4, nonreciprocal materials only exist in a rectangular waveguide region. There are large components outside of the waveguide-these components do not contribute to the dispersion curve shift. In order to use these components, we analyze two structures and calculate Im[E_{z}*E_{y}] and Im[E_{x}*E_{z}] for each case; see Fig. 6. A low-index rectangle (n=1.46) is added to each side of the high-index (n=2.5) waveguide in the panel (a). A low-index rectangle is added on top of the high-index waveguide in the panel (b). The overlap of the components with the waveguide is increased. The off-diagonal component of the permittivity tensor, for the low index material, is ±0.06i. As a result, the values I_{yx}+I_{xz}+I_{zy} are (a) 0.02738 and (b) 0.02154 at *β*̄=0.6, and the bandwidth obtained from PWE is increased to (a) 1.81% and (b) 1.17%. It is reconfirmed that the Im[E_{i}*E_{j}] analysis is useful for designing the isolation bandwidth. Although some papers discuss perturbation theory [3,4,12,13], the above method of visualizing the contribution of non-reciprocal materials to the dispersion curve shift provides a unique and versatile tool for designing isolators based on β shift [2–5,12,13].

## 4. Conclusions

We propose a waveguide isolator design based on the cut-off frequencies. Our waveguide isolator has the advantage of simplicity in design over traditional structures. Various structures are simulated in order to optimize the isolation bandwidth. A means of optimizing the isolation bandwidth is developed based on perturbation theory. Future research will include: FDTD simulation of transmission properties, and fabricating the isolators.

## Acknowledgments

The authors thank Innovation Core SEI Inc. for financial support and Akihiro Moto for discussions.

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