## Numerical analysis of polarization splitter based on vertically coupled microring resonator

Optics Express, Vol. 14, Issue 23, pp. 11304-11311 (2006)

http://dx.doi.org/10.1364/OE.14.011304

Acrobat PDF (158 KB)

### Abstract

The performance of a polarization splitter based on vertically coupled microring resonator is rigorously investigated by a combination of a 3D full vectorial film mode matching method with a 3D full vectorial coupled mode theory. The spectral responses of the structure for TE and TM mode are calculated, together with eigenmodes of uncoupled waveguides and scattering matrix of coupling region. The result shows that the response of microring resonator is indeed strongly polarization dependent and the resonance wavelengths are different for TE and TM mode. Such property allows for the design of wavelength-sensitive integrated polarization splitter. The influence of geometrical parameters on splitting ratio is investigated and the results indicate that the structure can have a splitting ratio greater than 20dB at 1.55µm.

© 2006 Optical Society of America

## 1. Introduction

1. T. Hayakawa, S. Asakawa, and Y. Kokubun, “ARROW-B type polarization splitter with asymmetric Ybranch fabricated by a self-alignment process,” J.lightwave Technol. **15**, 1165–1170 (1997). [CrossRef]

4. B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News **11**, 24–29 (2000). [CrossRef]

5. Dion J. W. Klunder, Chris G. H. Roeloffzen, and Alfred Driessen, “A novel polarization- independent wavelength-division-multiplexing filter based on cylindrical microresonators,” IEEE J. Select. Topics Quantum Electron. **8**, 1294–1299 (2002). [CrossRef]

6. L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Select. Topics Quantum Electron. **11**, 217–223 (2005). [CrossRef]

8. R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroky, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Comm. **256**, 46–67 (2005). [CrossRef]

9. W. P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A **11**, 963–983 (1994). [CrossRef]

## 2. Theory and numerical results

*n*

_{s}=1.98 is embedded in a substrate with index

*n*

_{g}=1.45. The bent waveguide of width w=1.0µm, thickness t=0.5µm and radius R=45.01µm with refractive index

*n*

_{b}=2.0 is placed on the top of the substrate surface, covered by a material with refractive index

*n*

_{a}=1.4017. The relative position of the straight core is defined by the horizontal gap d and the vertical position b between them, where d=R-H can be positive or negative values.

10. K. R. Hiremath, “Modling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Comm. **257**, 277–297 (2006). [CrossRef]

### 2.1. Calculation of eigenmodes of straight and bent waveguides using FMM

*d*=-0.5

*µm*and vertical position

*b*=0.5

*µm*as well as the corresponding modal profiles. Table 1 lists the propagation constants of modes in those waveguides. Since an electromagnetic field propagating through a bend region would lose energy due to radiation,

*γ*

_{b}=

*β*-

*iα*is complex value, where

*β*and

*α*is phase constant and attenuation constant.

*x*

_{min},

*x*

_{max}]and numbers of slice mode

*M*. For straight waveguide, [

*x*

_{min},

*x*

_{max}]=[-6.3

*µm*,3.7

*µm*] and

*M*=100. For bent waveguide, [

*x*

_{min},

*x*

_{max}]=[-4.0

*µm*,4.0

*µm*] and

*M*=200. Operation wavelength

*λ*=1.55

*µm*.

### 2.2. Vertical coupling between straight waveguide and bent waveguide

*,*

**E**_{b}*), (*

**H**_{b}*E*

_{s},

*) are the fields of the bent and straight waveguides.*

**H**_{s}*A*

_{b},

*A*

_{s}are unknown amplitudes. Using Lorentz reciprocity theorem, we can deduce the coupled mode equation [11

11. K. R. Hiremath, “Coupled mode theory based modeling and analysis of circular optical microresonators,” ph.D. dissertation, University of Twente, the Netherlands, 2005, http://wwwhome.math.utwente.nl/~hiremathkr/research/thesis.pdf

*C*

^{-1}and

*K*are two 2×2 matrices, and their elements are based on overlap integrals of different fields [8–10

8. R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroky, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Comm. **256**, 46–67 (2005). [CrossRef]

*A*

_{s}(

*z*

_{o}),

*A*

_{s}(

*z*

_{o})) to (

*A*

_{s}(

*z*

_{i}),

*A*

_{s}(

*z*

_{i})) (see Fig. 5) [8–9

8. R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroky, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Comm. **256**, 46–67 (2005). [CrossRef]

*(z) was obtained by applying the calculation procedure to a series of computational windows with fixed lower boundary*

**S***z*

_{i}and varying upper boundary at

*z*∈[

*z*

_{i},

*z*

_{0}].

*z*and these can be considered as power evolutions in the coupling region. Taking the third picture for an example, |

*S*

_{ss}|

^{2}and |

*S*

_{bs}|

^{2}can be interpreted as the local power in the straight and bent waveguide respectively, assuming a unit power of TM polarized light is launched into the straight waveguide. It can be easily seen that initially most of the power is confined in straight waveguide, then after a certain distance, the bend mode gets excited and the interaction of the two waveguides becomes significant, and finally modes in two waveguides become stationary at the end of the coupling region. In Fig. 7, some snapshots of the process described above were shown.

*x*=20

*nm*, Δ

*y*=20

*nm*, Δ

*z*=1.0

*µm*; [

*x*

_{b},

*x*

_{t}]=[-3.0

*µm*,3.0

*µm*], [

*y*

_{i},

*y*

_{o}]=[-6.0

*µm*,4.0

*µm*], [

*y*

_{i},

*y*

_{o}]=[-18.0

*µm*,18.0

*µm*]. The interaction of waves with vacuum wavelength

*λ*=1.55

*µm*is studied.

### 2.3. Spectrum and splitting ratio of the entire structure

*S*

_{bb}=|

*S*

_{bb}| exp(

*iφ*) and

*S*

_{ss}-

*S*

_{bs}

*S*

_{sb}/

*S*

_{bb}=

*d*exp(

*iφ*), then the dropped and through power is given by the equations below [10

10. K. R. Hiremath, “Modling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Comm. **257**, 277–297 (2006). [CrossRef]

10. K. R. Hiremath, “Modling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Comm. **257**, 277–297 (2006). [CrossRef]

11. K. R. Hiremath, “Coupled mode theory based modeling and analysis of circular optical microresonators,” ph.D. dissertation, University of Twente, the Netherlands, 2005, http://wwwhome.math.utwente.nl/~hiremathkr/research/thesis.pdf

**257**, 277–297 (2006). [CrossRef]

*S*

_{m}which is expected to be slow wavelength dependent is recalculated by extracting phase gains for propagation along the bends in coupling region, and finally the actual spectrum can be obtained by complex interpolations of relevant quantities. It should be noticed that

*S*

_{m}is regarded as the scattering matrix of a coupler with zero length where the interaction takes place instantaneously at z=0 and this modification is compensated by redefining the lengths of the bends outside the coupling region as

*L*=2

*πR*. In our cases, the resonator spectrums are calculated by interpolating the bend mode propagation constants and modified scattering matrices for three different wavelengths, that is 1.545µm, 1.550µm and 1.555µm (quadratic interpolation).

*SR*

_{1}=10log

_{10}(

*SR*

_{2}=10log

_{10}(

^{TM}

_{through}).

## 3. Conclusion

## Acknowledgments

## References and links

1. | T. Hayakawa, S. Asakawa, and Y. Kokubun, “ARROW-B type polarization splitter with asymmetric Ybranch fabricated by a self-alignment process,” J.lightwave Technol. |

2. | E. Simova and I. Golub, “Polarization splitter/combiner in high index contrast Bragg reflector waveguides,” Opt. Express |

3. | K. Saitoh, Y. Sato, and M. Koshiba, “Polarization splitter in three-core photonic crystal fibers,” Opt. Express |

4. | B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photon. News |

5. | Dion J. W. Klunder, Chris G. H. Roeloffzen, and Alfred Driessen, “A novel polarization- independent wavelength-division-multiplexing filter based on cylindrical microresonators,” IEEE J. Select. Topics Quantum Electron. |

6. | L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Select. Topics Quantum Electron. |

7. | L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. |

8. | R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroky, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Comm. |

9. | W. P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A |

10. | K. R. Hiremath, “Modling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Comm. |

11. | K. R. Hiremath, “Coupled mode theory based modeling and analysis of circular optical microresonators,” ph.D. dissertation, University of Twente, the Netherlands, 2005, http://wwwhome.math.utwente.nl/~hiremathkr/research/thesis.pdf |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.1360) Optical devices : Beam splitters

(230.5440) Optical devices : Polarization-selective devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: August 10, 2006

Revised Manuscript: October 4, 2006

Manuscript Accepted: October 13, 2006

Published: November 13, 2006

**Citation**

Xinlun Cai, Dexiu Huang, and Xinliang Zhang, "Numerical analysis of polarization splitter based on vertically coupled microring resonator," Opt. Express **14**, 11304-11311 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11304

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### References

- T. Hayakawa, S. Asakawa, and Y. Kokubun, "ARROW-B type polarization splitter with asymmetric Y-branch fabricated by a self-alignment process," J. Lightwave Technol. 15, 1165-1170 (1997). [CrossRef]
- E. Simova and I. Golub, "Polarization splitter/combiner in high index contrast Bragg reflector waveguides," Opt. Express 11, 3425-3430 (2003). [CrossRef] [PubMed]
- K. Saitoh, Y. Sato, and M. Koshiba, "Polarization splitter in three-core photonic crystal fibers," Opt. Express 12, 3940-3946 (2004). [CrossRef] [PubMed]
- B. E. Little and S. T. Chu, "Toward very large-scale integrated photonics," Opt. Photonics News 11, 24-29 (2000). [CrossRef]
- D. J. W. Klunder, C. G. H. Roeloffzen, and A. Driessen, "A novel polarization- independent wavelength-division-multiplexing filter based on cylindrical microresonators," IEEE J. Sel. Tops. Quantum Electron. 8, 1294-1299 (2002). [CrossRef]
- L. Prkna, M. Hubalek, and J. Ctyroky, "Field modeling of circular microresonators by film mode matching," IEEE J. Sel. Tops. Quantum Electron. 11, 217-223 (2005). [CrossRef]
- L. Prkna, M. Hubalek, and J. Ctyroky, "Vectorial eigenmode solver for bent waveguides based on mode matching," IEEE Photon. Technol. Lett. 15, 1249-1251 (2003).
- R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroky, "Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory," Opt. Commun. 256,46-67 (2005). [CrossRef]
- W. P. Huang, "Coupled mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963-983 (1994). [CrossRef]
- K. R. Hiremath, "Modling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory," Opt. Commun. 257, 277-297 (2006). [CrossRef]
- K. R. Hiremath, "Coupled mode theory based modeling and analysis of circular optical microresonators," Ph.D. dissertation, University of Twente, the Netherlands, 2005, http://wwwhome.math.utwente.nl/~hiremathkr/research/thesis.pdf

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