## Single-mode waveguide optical isolator based on direction-dependent cutoff frequency

Optics Express, Vol. 16, Issue 20, pp. 16202-16208 (2008)

http://dx.doi.org/10.1364/OE.16.016202

Acrobat PDF (302 KB)

### 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

1. M. Levy, R. M. Osgood, H. Hegde, F. J. Cadieu, R. Wolfe, and V. J. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. **8**, 903–905 (1996). [CrossRef]

2. Y. Shoji, I. W. Hsieh, R. M. Osgood, and T. Mizumoto, “Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift,” J. Lightwave Technol. **25**, 3108–3113 (2007). [CrossRef]

5. N. Kono and M. Koshiba, “Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides,” Opt. Express **13**, 9155–9166 (2005). [CrossRef] [PubMed]

*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

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

7. A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, “Two-dimensional magnetophotonic crystal: Exactly solvable model,” Phys.Rev. B **72**, 035123 (2005). [CrossRef]

8. Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. **90**, 121133- (2007). [CrossRef]

9. M. J. Steel, M. Levy, and R. M. Osgood Jr., “High transmission enhanced Faraday rotation in onedimensional photonic crystals with defects,” IEEE Photon. Technol. Lett. **12**, 1171–1173 (2000). [CrossRef]

10. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-Way Electromagnetic Waveguide formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal,” Phys. Rev. Lett. **100**, 023902 (2008). [CrossRef] [PubMed]

11. F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. **100**, 013904 (2008). [CrossRef] [PubMed]

12. T. Amemiya, H. Shimizu, M. Yokoyama, P. N. Hai, M. Tanaka, and Y. Nakano, “1.54-um TM-mode waveguide optical isolator based on the nonreciprocal-loss phenomenon: device design to reduce insertion loss,” Appl. Opt. **46**, 5784–5791 (2007). [CrossRef] [PubMed]

13. W. Zaets and K. Ando, “Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer,” IEEE Photon. Technol. Lett. **11**, 1012–1014 (1999). [CrossRef]

## 2. Proposed optical isolator designs

*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.

14. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*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).

*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

**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:

^{2}

*=*

_{t}*∂*

_{2}/

*∂*

*x*

_{2}+

*∂*

^{2}/

*∂*

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

*ω*

^{2}

*µ*

_{0}

*by changing the permittivity tensor given by*

_{ε}*u*

*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*

_{ij}_{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:

*ω*is approximated to be proportional to the dispersion shift between forward and backward waves:

_{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.

**E**≈0. Knowing this limit of the analysis, we can use perturbation theory to optimize optical isolators.

_{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

3. J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dotsch, “Waveguide optical isolator based on Mach-Zehnder interferometer,” Appl. Phys. Lett. **76**, 2158–2160 (2000). [CrossRef]

4. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B **22**, 240–253 (2005). [CrossRef]

12. T. Amemiya, H. Shimizu, M. Yokoyama, P. N. Hai, M. Tanaka, and Y. Nakano, “1.54-um TM-mode waveguide optical isolator based on the nonreciprocal-loss phenomenon: device design to reduce insertion loss,” Appl. Opt. **46**, 5784–5791 (2007). [CrossRef] [PubMed]

13. W. Zaets and K. Ando, “Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer,” IEEE Photon. Technol. Lett. **11**, 1012–1014 (1999). [CrossRef]

2. Y. Shoji, I. W. Hsieh, R. M. Osgood, and T. Mizumoto, “Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift,” J. Lightwave Technol. **25**, 3108–3113 (2007). [CrossRef]

5. N. Kono and M. Koshiba, “Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides,” Opt. Express **13**, 9155–9166 (2005). [CrossRef] [PubMed]

12. T. Amemiya, H. Shimizu, M. Yokoyama, P. N. Hai, M. Tanaka, and Y. Nakano, “1.54-um TM-mode waveguide optical isolator based on the nonreciprocal-loss phenomenon: device design to reduce insertion loss,” Appl. Opt. **46**, 5784–5791 (2007). [CrossRef] [PubMed]

13. W. Zaets and K. Ando, “Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer,” IEEE Photon. Technol. Lett. **11**, 1012–1014 (1999). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. Levy, R. M. Osgood, H. Hegde, F. J. Cadieu, R. Wolfe, and V. J. Fratello, “Integrated optical isolators with sputter-deposited thin-film magnets,” IEEE Photon. Technol. Lett. |

2. | Y. Shoji, I. W. Hsieh, R. M. Osgood, and T. Mizumoto, “Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift,” J. Lightwave Technol. |

3. | J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dotsch, “Waveguide optical isolator based on Mach-Zehnder interferometer,” Appl. Phys. Lett. |

4. | H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B |

5. | N. Kono and M. Koshiba, “Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides,” Opt. Express |

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

7. | A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, “Two-dimensional magnetophotonic crystal: Exactly solvable model,” Phys.Rev. B |

8. | Z. Yu, Z. Wang, and S. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. |

9. | M. J. Steel, M. Levy, and R. M. Osgood Jr., “High transmission enhanced Faraday rotation in onedimensional photonic crystals with defects,” IEEE Photon. Technol. Lett. |

10. | Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-Way Electromagnetic Waveguide formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal,” Phys. Rev. Lett. |

11. | F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. |

12. | T. Amemiya, H. Shimizu, M. Yokoyama, P. N. Hai, M. Tanaka, and Y. Nakano, “1.54-um TM-mode waveguide optical isolator based on the nonreciprocal-loss phenomenon: device design to reduce insertion loss,” Appl. Opt. |

13. | W. Zaets and K. Ando, “Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer,” IEEE Photon. Technol. Lett. |

14. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.3240) Optical devices : Isolators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 11, 2008

Revised Manuscript: September 15, 2008

Manuscript Accepted: September 22, 2008

Published: September 26, 2008

**Citation**

Lingling Tang, Samuel M. Drezdzon, and Tomoyuki Yoshie, "Single-mode waveguide optical isolator based on direction-dependent cutoff frequency," Opt. Express **16**, 16202-16208 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16202

Sort: Year | Journal | Reset

### References

- M. Levy, R. M. Osgood, Jr., H. Hegde, F. J. Cadieu, R. Wolfe, and V. J. Fratello, "Integrated optical isolators with sputter-deposited thin-film magnets," IEEE Photon. Technol. Lett. 8, 903-905 (1996). [CrossRef]
- Y. Shoji, I. W. Hsieh, R. M. Osgood, and T. Mizumoto, "Polarization-Independent Magneto-Optical Waveguide Isolator Using TM-Mode Nonreciprocal Phase Shift," J. Lightwave Technol. 25, 3108-3113 (2007). [CrossRef]
- J. Fujita, M. Levy, R. M. Osgood, Jr., L. Wilkens, and H. Dotsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000). [CrossRef]
- H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, and P. Hertel, "Applications of magneto-optical waveguides in integrated optics: review," J. Opt. Soc. Am. B 22, 240-253 (2005). [CrossRef]
- N. Kono and M. Koshiba, "Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides," Opt. Express 13, 9155-9166 (2005). [CrossRef] [PubMed]
- A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001). [CrossRef]
- A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, "Two-dimensional magnetophotonic crystal: Exactly solvable model," Phys.Rev. B 72, 035123 (2005). [CrossRef]
- Z. Yu, Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133- (2007). [CrossRef]
- M. J. Steel, M. Levy, and R. M. Osgood, Jr., "High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects," IEEE Photon. Technol. Lett. 12, 1171-1173 (2000). [CrossRef]
- Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 023902 (2008). [CrossRef] [PubMed]
- F. D. M. Haldane and S. Raghu, "Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry," Phys. Rev. Lett. 100, 013904 (2008). [CrossRef] [PubMed]
- T. Amemiya, H. Shimizu, M. Yokoyama, P. N. Hai, M. Tanaka, and Y. Nakano, "1.54-um TM-mode waveguide optical isolator based on the nonreciprocal-loss phenomenon: device design to reduce insertion loss," Appl. Opt. 46, 5784-5791 (2007). [CrossRef] [PubMed]
- W. Zaets and K. Ando, "Optical Waveguide Isolator Based on Nonreciprocal Loss/Gain of Amplifier Covered by Ferromagnetic Layer," IEEE Photon. Technol. Lett. 11, 1012-1014 (1999). [CrossRef]
- S. Johnson and J. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.