## Semi-vector iterative method for modes of high-index-contrast nanoscale waveguides |

Optics Express, Vol. 21, Issue 8, pp. 9807-9812 (2013)

http://dx.doi.org/10.1364/OE.21.009807

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### Abstract

An approximate semi-analytical iterative method is presented to find vector modes of high-index contrast single mode waveguides. Present method is developed to provide improvement over scalar analysis of Vopt method. To illustrate the accuracy and efficiency of this method, modal properties of silicon strip nanoscale waveguide are studied in detail and compared with other approximate and rigorous numerical analysis.

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## 1. Introduction

1. M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. **23**, 4222–4238 (2005) [CrossRef] .

2. Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express **12**, 1622–1631 (2004) [CrossRef] [PubMed] .

3. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express **18**, 8528–8539 (2010) [CrossRef] [PubMed] .

6. K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt. **25**, 2169–2174 (1986) [CrossRef] [PubMed] .

8. A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron. **21**, 517–520 (1989) [CrossRef] .

9. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. **44**, 692–700 (1996) [CrossRef] .

10. A. Sharma, “Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction,” J. Opt. Soc. Am. A **18**, 1383–1387 (2001) [CrossRef] .

3. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express **18**, 8528–8539 (2010) [CrossRef] [PubMed] .

9. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. **44**, 692–700 (1996) [CrossRef] .

## 2. Semi-vector Vopt (SV-Vopt) method

**E**and

**H**-field are non-zero. In the semi-vector approach, we consider two types of mode: quasi-TE (or E

^{x}) and quasi-TM (or E

^{y}) mode. For the E

^{x}mode propagating in the

*z*-direction, we consider the mode field

**E**of the form (

*E*, 0,

_{x}*E*)

_{z}*exp*(

*iωt*−

*iβz*). For E

^{x}mode, this reduces full vector wave equation to following semi-vector wave equation An integral form of the semi-vector wave equation is obtained by assuming a field, separable in two transverse directions, namely the width and the depth directions. For the E

^{x}mode, let us consider a separable field of form

*E*(

_{x}*x*,

*y*) =

*χ*(

*x*)

*ϕ*(

*y*). In due course we will show that in SV-Vopt analysis

*χ*(

*x*) and

*ϕ*(

*y*) arise naturally as TM and TE mode fields of some planar index waveguides represented by index distributions

*E*(

_{x}*x*,

*y*) =

*χ*(

*x*)

*ϕ*(

*y*) in Eq. (1), multiply it by

*ϕ*(

*y*) in Eq. (2) and rearrange it to obtain

*x*-component of the

**E**-field for the E

^{x}mode is

*E*=

_{x}*χ*(

*x*)

*ϕ*(

*y*), and the propagation constant,

*β*is obtained as: Similar set of equations and procedure can be derived for E

^{y}mode by merely exchanging terms

*χ*(

*x*) with

*ϕ*(

*y*). In most cases, the convergence is reached in 2 or 3 cycles. Whether one starts with an initial approximations for

## 3. Modal analysis of Si strip waveguide

*μ*m, Si and silica refractive indices are taken as 3.5 and 1.5, respectively. For the SV-Vopt, a planar waveguide with the width same as that of the core of the strip waveguide and Si and silica as core and cladding, respectively, is taken as initial approximation for

### 3.1. Effective mode index

*n*

_{eff}=

*β*/

*k*) with waveguide core width (W) has been shown for

9. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. **44**, 692–700 (1996) [CrossRef] .

3. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express **18**, 8528–8539 (2010) [CrossRef] [PubMed] .

*n*

_{eff}than the VEIM. The SV-Vopt method is a good approximate method for waveguides operating in the single mode regime.

### 3.2. Effective mode area

*A*

_{eff}), which defined as follows: where

**E**= (

_{t}*E*,

_{x}*E*). Since the SV-Vopt approximates

_{y}*E*=

_{x}*χ*(

*x*)

*ϕ*(

*y*) and

*E*= 0 for Ex mode,

_{y}**E**= (

_{t}*E*, 0) is used here. Variation of the effective mode area,

_{x}*A*

_{eff}with core width, W, is shown in Fig. 2. Core height is kept constant at 260nm. It can be observed that value of

*A*

_{eff}is very large near the mode cut-off and attains a minimum value at some higher core width. For the fundamental mode, again the SV-Vopt results show very good agreement with the vector FEM results at all widths. This shows that, the SV-Vopt method not only gives very good estimate of propagation constant, but it also gives analytical expression for field profile with reasonable accuracy. Since the method is basically built around the fundamental mode (as is its scalar counterpart Vopt), accuracy for higher order modes is not as good, however, results can be used for making reasonable estimates.

### 3.3. Power confinement factor

_{core}with core width in Fig. 3. Core height is kept constant at 260 nm. For higher order modes, there is significant error in estimation of confinement factor near the mode cutoff. However, for fundamental mode, the SV-Vopt results show very good agreement with the vector FEM results at all widths. It can be noted that Γ

_{core}increases with increase in core width and saturates after a certain core width. An optimum core width can be chosen to have high power confinement, small effective mode area and small waveguide dimensions.

### 3.4. Modal birefringence

^{y}mode. From the propagation constants obtained for fundamental E

^{x}and E

^{y}modes, accurate estimation of birefringence can also be calculated. Birefringence is defined as difference in

*n*

_{eff}of the

*n*

_{eff}, and birefringence are plotted in Fig. 4. It can be seen that SOI strip waveguides show very large birefringence.

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. |

2. | Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express |

3. | D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express |

4. | E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. |

5. | R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in |

6. | K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt. |

7. | P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A |

8. | A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron. |

9. | K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. |

10. | A. Sharma, “Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction,” J. Opt. Soc. Am. A |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.7370) Optical devices : Waveguides

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Optical Devices

**History**

Original Manuscript: March 13, 2013

Revised Manuscript: April 5, 2013

Manuscript Accepted: April 8, 2013

Published: April 12, 2013

**Citation**

K. Gehlot and A. Sharma, "Semi-vector iterative method for modes of high-index-contrast nanoscale waveguides," Opt. Express **21**, 9807-9812 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9807

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### References

- M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol.23, 4222–4238 (2005). [CrossRef]
- Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express12, 1622–1631 (2004). [CrossRef] [PubMed]
- D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express18, 8528–8539 (2010). [CrossRef] [PubMed]
- E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).
- R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the Symposium on Submillimeter Waves, M. H. Schlam and J. Fox, (Polytechnic Press, 1970), pp. 497–516.
- K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt.25, 2169–2174 (1986). [CrossRef] [PubMed]
- P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).
- A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron.21, 517–520 (1989). [CrossRef]
- K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech.44, 692–700 (1996). [CrossRef]
- A. Sharma, “Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction,” J. Opt. Soc. Am. A18, 1383–1387 (2001). [CrossRef]

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