## Polarization effects in tapered dielectric waveguides

Optics Express, Vol. 11, Issue 16, pp. 1931-1941 (2003)

http://dx.doi.org/10.1364/OE.11.001931

Acrobat PDF (200 KB)

### Abstract

The 3D finite-difference time-domain (FDTD) method is used to analyze the polarization effects in two kinds of linearly tapered optical waveguides: slab waveguides with only lateral tapers and rectangular cross section waveguides with both lateral and vertical tapers. For the slab waveguides, each guided mode of both the back reflected and output powers are determined and compared. For rectangular cross section waveguides, the output power of TE and TM modes with respect to taper length are computed and compared.

© 2003 Optical Society of America

## 1. Introduction

6. R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. **24**, 775–779 (1988). [CrossRef]

7. E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE **21**, 307–314 (1985). [CrossRef]

8. I. Lu, “Intrinsic modes in wedge-shaped taper above an anisotropic substrate,” IEEE J.Quantum Electron , **27**, 2373–2377 (1991). [CrossRef]

10. A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 **10**, 828–834 (1977). [CrossRef]

12. Z.N. Lu and R. Bansal, “A finite-difference third-order simplified wave equation method: an assessment and application,” IEEE Microwave Theory Technol. **42**, 132–136 (1994). [CrossRef]

13. Z.N. Lu, R. Bansal, and Peter K. Cheo, “Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,” IEEE J. Lightwave Technol. , **12**, 1373–1377 (1994). [CrossRef]

14. G.R. Hadley, “Design of tapered waveguides for improved output coupling,” IEEE Photon. Technol. Lett. **5**, 1068–1070 (1993). [CrossRef]

10. A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 **10**, 828–834 (1977). [CrossRef]

15. R.K. Winn and J.H. Harris, “Coupling from multimode to single mode linear waveguides using horn-shaped strctures,” IEEE Microwave Theory Tech. , **23**, 3012–3015 (1975). [CrossRef]

7. E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE **21**, 307–314 (1985). [CrossRef]

## 2. Tapered slab waveguide and rectangular waveguide

*TE*

^{z}or

*TM*

^{z}or

*TE*

^{z}and

*TM*

^{z}to excite the waveguide, respectively, in order to find the relation between the polarization change and the taper length. Next, we will introduce the analysis method in detail.

## 3. Analysis method

*TE*

^{z}mode in the 3D optical waveguide can be expressed as:

*m*and

*n*represent the mode number; A is the amplitude constant, B stands for the mode distribution in the cross section. In order to find the power carried by each guided mode, the amplitude has to be found. From orthogonality, we have:

*B*

_{mn}(

*y, z*) and

*C*

_{mn}are known from Eqs. (1) and (3), respectively, so the amplitude constant on each guided mode is then given from Eq. (5):

18. C. Guiffaut and K. Mahdjoubi, “A parallel FDTD algorithm using the MPI library,” IEEE Antennas and Propagation Magazine , **43**, 94–103 (2001). [CrossRef]

## 4. Results and discussion

### 4.1. Laterally tapered slab waveguide

*xy*plane. Because it is multimode-to-multimode, the field distribution is not perfectly regular. This is more obvious in

*E*

_{z}and

*H*

_{y}field components in case that they are the dominant components of

*TM*

^{z}mode. The amplitude of the dominant components of

*TE*

^{z}mode is much bigger than those of the

*TM*

^{z}mode.

*TE*modes and

*TM*modes, some is back reflected and scattered, the other is radiated and absorbed by the PML absorbing boundary. In the output power,

*TE*modes, the power is relatively small. Compared with

*TE*mode,

*TM*mode power is small and the result shows that the power is more likely to couple into higher order

*TM*mode.

*TM*mode, the value of most of the modes is small except

### 4.2. Tapered rectangular waveguide

*TM*modes have less radiation loss and better polarization stability. Figure 10 shows the radiation loss comparison and back-reflected loss comparison. As you can see, the radiation loss of

*TE*mode is almost twice the radiation loss of

*TM*mode, for taper length longer than 6µm. They reach the same level only when the taper length is equal to 3µm. As for back-reflected loss, both of them remain at thevery low level except at 3µm, the

*TM*mode outreaches

*TE*mode by 7% with respect to the source power.

*TM*source and Fig. 11(b) corresponds to the

*TE*source. Our simulation shows that for the same taper length,

*TM*mode has less polarization change compared with

*TE*mode. In other words,

*TM*mode has better polarization stability. For example, when the taper length is 9µm, if the source is

*TM*, then 81.6% power remains in

*TM*mode, 11.8% power changes into

*TE*mode, but if the source is

*TE*, only 43% power remains in

*TE*mode, 28.5% power changes into

*TM*mode. What’s more, our simulation shows that the polarization remains well with long taper length and tends to change when the taper length becomes short. This is reasonable since in the short taper, the mode profile changes so fast that the EM field no longer satisfies the adiabatic boundary conditions. As a result the polarization will change in order to adapt to the new boundary condition. As is well know, an infinitely long taper is perfectly adiabatic in which case the polarization should remain unchanged, since it is in fact a straight waveguide. Figure 11 also shows that both the

*TM*mode output power and

*TE*mode output power decrease rapidly due to the increasing radiation loss and back-reflection loss, when the taper length becomes shorter than 9µm.

## 5. Conclusions

*TM*output mode power is less than 2%, most of them are coupled into higher order modes.

*TM*mode has better polarization stability and less radiation loss. Under the same condition, the radiation loss of

*TE*mode is almost twice that of the

*TM*mode. In addition, less power is changed into other polarization if the taper is excited with

*TM*mode than

*TE*mode.

## References and Links

1. | Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski, and Dr. Dennis Prather, “Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,” SPIE’s 48th annual meeting, (to be published). |

2. | G. Agrawal, |

3. | M. Wu, P. Fan, and C. Lee, “Completely adiabatic s-shaped bent tapers in optical waveguides,” IEEE Photon. Tech. Lett. |

4. | C. Lee, M. Wu, L. Sheu, P. Fan, and J. Hsu, “Design and analysis of completely adiabatic tapered waveguides by conformal mapping,” IEEE J. Lightwave Technol. |

5. | J. Sakai and E. Marcatili, “Lossless dielectric tapers with three-dimensional geometry,” IEEE J. Lightwave Technol. |

6. | R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. |

7. | E. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J.Quantum Electron, QE |

8. | I. Lu, “Intrinsic modes in wedge-shaped taper above an anisotropic substrate,” IEEE J.Quantum Electron , |

9. | S.El Yumin, K. Komori, S. Arai, and G. Bendelli, “Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),” IEICE Tran. Electron, E77-C |

10. | A. Milton and W. Burns, “Mode coupling in optical waveguide horns,” IEEE J.Quantum Electron, QE-13 |

11. | C. Vassallo, “Analysis of tapered mode transformers for semiconductor optical amplifiers,” Opt. Quantum Electron. |

12. | Z.N. Lu and R. Bansal, “A finite-difference third-order simplified wave equation method: an assessment and application,” IEEE Microwave Theory Technol. |

13. | Z.N. Lu, R. Bansal, and Peter K. Cheo, “Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,” IEEE J. Lightwave Technol. , |

14. | G.R. Hadley, “Design of tapered waveguides for improved output coupling,” IEEE Photon. Technol. Lett. |

15. | R.K. Winn and J.H. Harris, “Coupling from multimode to single mode linear waveguides using horn-shaped strctures,” IEEE Microwave Theory Tech. , |

16. | E.A.J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell System Tech. |

17. | D.P. Rodohan and S.R Saunders, “Parallel implementations of the finite difference time domain (FDTD) method,” Computation in Electromagnetics, Second International Conference, 367–370 (1994). |

18. | C. Guiffaut and K. Mahdjoubi, “A parallel FDTD algorithm using the MPI library,” IEEE Antennas and Propagation Magazine , |

**OCIS Codes**

(000.0000) General : General

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 24, 2003

Revised Manuscript: August 4, 2003

Published: August 11, 2003

**Citation**

Ge Jin, Shouyuan Shi, Ahmed Sharkawy, and Dennis Prather, "Polarization effects in tapered dielectric waveguides," Opt. Express **11**, 1931-1941 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1931

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### References

- Thomas Dillion, Anita Balcha, Dr.Janusz Murakowski and Dr. Dennis Prather, �??Process development and application of grayscale lithography for efficient three-dimensionally profiled fiber-to-waveguide couplers,�?? SPIE�??s 48th annual meeting, (to be published).
- G.Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1992).
- M.Wu, P.Fan and C.Lee, �??Completely adiabatic s-shaped bent tapers in optical waveguides,�?? IEEE Photon. Tech. Lett. 9, 212-214 (1997). [CrossRef]
- C.Lee, M.Wu, L.Sheu, P.Fan and J.Hsu, �??Design and analysis of completely adiabatic tapered waveguides by conformal mapping,�?? IEEE J. Lightwave Technol. 15, 403-410 (1993).
- J.Sakai and E.Marcatili, �??Lossless dielectric tapers with three-dimensional geometry,�?? IEEE J. Lightwave Technol. 9, 386-393 (1991). [CrossRef]
- R.Weder, �??Dielectric three-dimensional electromagnetic tapers with no loss,�?? IEEE J. Quantum Electron. 24, 775-779 (1988). [CrossRef]
- E.Marcatili, �??Dielectric tapers with curved axes and no loss,�?? IEEE J.Quantum Electron., QE 21, 307-314 (1985). [CrossRef]
- I.Lu, �??Intrinsic modes in wedge-shaped taper above an anisotropic substrate,�?? IEEE J.Quantum Electron., 27, 2373-2377 (1991). [CrossRef]
- S. El Yumin, K. Komori, S. Arai and G. Bendelli, �??Taper-shape dependence of tapered-waveguide traveling wave semiconductor laser amplifier (TTW-SLA),�?? IEICE Tran. Electron., E77-C 4, 624-632 (1994).
- A. Milton and W. Burns, �??Mode coupling in optical waveguide horns,�?? IEEE J. Quantum Electron., QE-13 10, 828-834 (1977). [CrossRef]
- C.Vassallo, �??Analysis of tapered mode transformers for semiconductor optical amplifiers,�?? Opt. Quantum Electron. 26, 1025-1026 (1996).
- Z.N.Lu and R.Bansal, �??A finite-difference third-order simplified wave equation method: an assessment and application,�?? IEEE Microwave Theory Technol. 42, 132-136 (1994). [CrossRef]
- Z.N.Lu, R.Bansal and Peter K.Cheo, �??Radiation losses of tapered dielectric waveguides: a finite difference analysis with ridge waveguide applications,�?? IEEE J. Lightwave Technol., 12, 1373-1377 (1994). [CrossRef]
- G. R. Hadley, �??Design of tapered waveguides for improved output coupling,�?? IEEE Photon. Technol. Lett. 5, 1068-1070 (1993). [CrossRef]
- R. K. Winn and J. H. Harris, �??Coupling from multimode to single mode linear waveguides using horn-shaped strctures,�?? IEEE Microwave Theory Tech., 23, 3012-3015 (1975). [CrossRef]
- E. A. J. Marcatilli, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell System Tech. 48, 2071 (1969).
- D. P. Rodohan and S. R. Saunders, �??Parallel implementations of the finite difference time domain (FDTD) method,�?? Computation in Electromagnetics, Second International Conference, 367-370 (1994).
- C. Guiffaut and K. Mahdjoubi, �??A parallel FDTD algorithm using the MPI library,�?? IEEE Antennas and Propagation Magazine, 43, 94-103 (2001). [CrossRef]

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