## Abstract

The design of an ultra-wideband waveguide magneto-optical isolator is described. The isolator is based on a Mach-Zehnder interferometer employing nonreciprocal phase shift. The ultra-wideband design is realized by adjusting the wavelength dependence of reciprocal phase difference to compensate for that of nonreciprocal phase difference in the backward direction. We obtained the ultra-wideband design that provides isolation > 35dB from 1.25μm to >1.65μm. This is the proposal of magneto-optical isolator that operates both in 1.31μm band and 1.55μm band.

© 2007 Optical Society of America

## 1. Introduction

Magneto-optical isolators contribute to establish stable high speed optical network systems by protecting optical active devices from unwanted reflections. Commercially available isolators are only bulk types. Although high isolation ratio and polarization independent operation have been realized, they are not suitable for integration. In addition, the wavelength range where

A waveguide magneto-optical isolator employing nonreciprocal phase shift is based on a Mach-Zehnder interferometer (MZI) with nonreciprocal and reciprocal phase shifter [1–5]. Recently, we have proposed a wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in MZI [6]. The operating wavelength range of the isolator

## 2. Principle of wideband design

Figure 1 shows MZI configurations of the nonreciprocal phase shift isolator for conventional and wideband design. In this diagram, the phase differences between two interferometer arms in a nonreciprocal phase shifter (NPS) and a reciprocal phase shifter (RPS) are denoted in lower parts. In spite of different reciprocal phase biases, forward and backward traveling waves become in-phase and anti-phase, respectively, in both design. So, the lightwaves propagating in two arms interfere constructively in the forward direction and destructively in the backward direction. The nonreciprocal phase difference (*θ _{N}*) and the reciprocal phase difference (

*θ*) decrease as wavelength becomes longer. Here, the sign of the nonreciprocal phase difference is important for the wideband design. In the conventional design, the nonreciprocal phase difference is set to be +π/2 for backward propagation. The reciprocal phase difference is set to be +π/2 for satisfying the anti-phase condition in the backward propagation. Therefore, the deviations of nonreciprocal and reciprocal phase difference from respective designed values are added to each other in the backward propagation. In the wideband design, the nonreciprocal phase difference is set to be -π/2 in the backward propagation. Reciprocal phase difference of 3π/2 is installed so as to satisfy the anti-phase condition for the backward propagation. In this design, the deviations of nonreciprocal and reciprocal phase difference from their designed values are canceled and the wavelength dependence of total phase difference is reduced dramatically in the backward propagation. Hence, large backward loss is obtainable in a wide wavelength range.

_{R}This adjustment is achieved by tailoring the waveguide parameters such as waveguide width and length of the reciprocal phase shifter as shown in Fig. 2. *L _{1}* and

*L*are the lengths of sections that are added to adjust a reciprocal phase difference.

_{2}*W*and

_{1}*W*are the waveguide widths of respective sections, which determine the longitudinal propagation constants

_{2}*β*and

_{1}*β*. The reciprocal phase difference

_{2}*θ*is given by

_{R}*β*-

_{2}L_{2}*β*and set to 3π/2 at the designed wavelength. Rather wideband design was achieved in our previous work [6].

_{1}L_{1}## 3. Numerical approach for ultra-wideband design

In order to realize further improvement in wideband operation, we elaborate the adjustment of reciprocal phase shifter. The phase differences *θ _{N}* and

*θ*are not linear functions of wavelength in our previous design. That is, in the wavelength concerned,

_{R}*θ*shows a convex change against the wavelength, while

_{N}*θ*is concave as shown in Fig. 3(a). The total phase difference for backward propagation becomes concave as shown in Fig. 3(b). This curve determines the backward loss. The wavelength dependence of

_{R}*θ*should be convex to minimize the wavelength dependence of total phase difference in backward propagation.

_{R}The waveguide parameters of reciprocal phase shifter affect its wavelength dependence as follows. Wider waveguide has larger propagation constant at shorter wavelength due to higher optical confinement [Fig. 4(a)]. The difference between *β _{1}* and

*β*becomes large at shorter wavelength as shown in Fig. 4(b). When the section lengths

_{2}*L*and

_{1}*L*are equal, the phase difference between two sections exhibits the wavelength dependence as is indicated by a solid curve in Fig. 4(c). When the section length

_{2}*L*is reduced keeping

_{1}*L*as well as

_{2}*W*and

_{1}*W*are constant, the wavelength dependence of the phase difference changes as shown in Fig. 4(c). Based on such a behavior, we can change the wavelength dependence of the reciprocal phase difference, from convex to concave dependence, by controlling the section length difference

_{2}*L*-

_{1}*L*. Also, by controlling section widths

_{2}*W*and

_{1}*W*, we can adjust the gradient of wavelength dependence of reciprocal phase difference.

_{2}By controlling section widths and lengths, we can approximate the wavelength dependence of nonreciprocal phase difference by that of reciprocal phase difference, while setting *θ _{R}* =3π/2 at a designed wavelength. Then, the deviation of total phase difference from π is minimized in the backward propagation, and large backward loss is obtainable in a wider wavelength range. The forward loss is slightly deteriorated in the wideband design compared with the case of conventional one, since the wavelength dependences of nonreciprocal and reciprocal phase difference add up to each other. The center wavelength where the forward loss becomes minimum is to be chosen properly to minimize the forward loss at band edges and to provide sufficient backward loss in a desired wavelength range. In the following design that covers both 1.31μm and 1.55μm.wavelength bands, we set the center wavelength at 1.43μm.

## 4. Calculation results

A CeY_{2}Fe_{5}O_{12} (Ce:YIG) is assumed as a magneto-optic garnet guiding layer, which is grown on a (111)-oriented (Ca, Mg, Zr) doped GGG substrate. SiO_{2} is deposited as an upper cladding layer. A rib waveguide formed on the guiding layer is assumed in this paper. The slab and rib heights are 0.40 μm and 0.08 μm, respectively. The waveguide width is set at the range from 2.0 μm to 3.0 μm. Since such flat waveguide can be approximated as four layered structure by an effective index method, the wavelength dependences of a nonreciprocal and a reciprocal phase shifter are calculated directly by solving the eigenvalue equation with the refractive indices of constituent layers and the Faraday rotation coefficient of Ce:YIG. Details of the calculation are given in Ref. [6]. If the waveguide has high index contrasts in the horizontal direction, a combination of a finite element method and a perturbation theory give rise to a precise result [4].

In the conventional design, the wavelength dependence of nonreciprocal phase difference *θ _{N}* is convex and that of reciprocal phase difference

*θ*is concave as shown in Fig. 3(a). To make the curve associated with

_{R}*θ*convex, both waveguide widths and lengths of the reciprocal phase shifter are tailored keeping

_{R}*β*

_{2}*L*-

_{2}*β*

_{1}*L*=3π/2 at a wavelength of 1.43μm. In the following design, we set the width of arm1 to be

_{1}*W*=2.0μm, which is the waveguide width of whole device.

_{1}Figure 5(a) shows the wavelength dependence of *θ _{R}* calculated for some different waveguide widths, where

*L*is set to be 1000μm and

_{1}*L*is slightly adjusted around 1000μm so as to realize

_{2}*θ*=3π/2 at a wavelength of 1.43μm. As

_{R}*W*becomes large, the variation of

_{2}*θ*vs wavelength becomes more convex. When the path length

_{R}*L*is varied, the wavelength dependence of

_{1}*θ*changes as is shown in Fig. 5(b). Here,

_{R}*W*is fixed at 2.4μm, and

_{2}*L*is adjusted around

_{2}*L*so as to realize

_{1}*θ*=3π/2 at a wavelength of 1.43μm. As

_{R}*L*becomes longer, the variation of

_{1}*θ*vs wavelength becomes more convex.

_{R}By tuning the widths and lengths, *θ _{R}* can be adjusted to have similar wavelength dependence to

*θ*. Table 1 shows examples of optimized parameters for some waveguide widths. Here, we set the waveguide width

_{N}*W*=2.0μm and change

_{1}*W*from 2.4μm to 3.0μm. The lengths

_{2}*L*and

_{1}*L*are chosen to achieve similar wavelength dependence to

_{2}*θ*. Figure 6 shows calculated wavelength dependences of the phase differences for

_{N}*W*=2.0μm,

_{1}*W*=3.0μm,

_{2}*L*=930μm and

_{1}*L*=930.28μm. As is observed in this figure,

_{2}*θ*has very similar wavelength dependence to

_{R}*θ*.

_{N}The calculated performances of isolators are shown in Figs. 7(a) and 7(b), which correspond to the conventional and the proposed wideband design, respectively. The MMI 3dB couplers are assumed to work ideally in the calculated wavelength range. The phase diagram shown in Fig. 6 is used to calculate the performance of wideband design shown in Fig. 7(b). In the wideband design, the backward loss is over 35dB in the calculated wavelength range. The forward loss is 0.27dB and 0.38dB at 1.31μm and 1.55μm, respectively, and becomes minimum at 1.43μm.

## 5. Conclusion

In conclusion, an ultra-wideband waveguide magneto-optical isolator is designed. A large backward loss is obtained in a wide wavelength range that covers both 1.31μm and 1.55μm band at a sacrifice of acceptable increase in a forward loss. Such wideband operation has not ever been reported in bulk isolators. However this idea may be applicable to bulk types. 45 degrees nonreciprocal Faraday rotation in a magneto-optic material is usually used for the operation. And another 45 degrees reciprocal polarization rotation by a birefringent material can be inserted to keep the operational optical axis. If the wavelength dependence of the reciprocal polarization rotator can be controlled to cancel that of Faraday rotator, the operation bandwidth of bulk isolator is extended similarly. But its adjustment with bulk optics seems to be difficult. On the other hand, a waveguide structure has many controllable parameters as mentioned above. Therefore an ultra-wideband design of waveguide isolator is achievable.

## References and links

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