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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9807–9812
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Semi-vector iterative method for modes of high-index-contrast nanoscale waveguides

K. Gehlot and A. Sharma  »View Author Affiliations


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.

© 2013 OSA

1. Introduction

Recent technological advancements in complementary metal oxide semiconductor (CMOS) compatible silicon (Si) photonics have generated a great interest in Silicon-on-insulator (SOI) photonic structures [1

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

]. Small feature size, high optical field confinement and low bending loss of SOI waveguides permit to realize ultra-dense photonic integrated circuits on a single Si chip [2

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] .

].

Several numerical and approximate methods have been developed to model and characterize such kind of waveguides. Rigorous numerical methods, such as the vector finite element method (VFEM) [3

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] .

] give accurate results, but are computationally intensive. Hence there have always been efforts to develop simple and efficient approximate methods which can give fairly accurate results with less computational efforts.

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

3. Modal analysis of Si strip waveguide

3.1. Effective mode index

In Fig. 1, variation of the effective index (neff = β/k) with waveguide core width (W) has been shown for E11x, E21x, E31x and E41x modes. The analysis of SV-Vopt method and VEIM [9

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] .

] is applied to present structure and the results are plotted for comparison along with rigorous numerical results of the vector FEM analysis from [3

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] .

]. The SV-Vopt analysis gives very accurate results for the fundamental mode, even near its cut-off. For higher order modes small error is found for near cut-off modes. From the inset plot in Fig. 1, it can be observed that the SV-Vopt method gives better estimate of neff than the VEIM. The SV-Vopt method is a good approximate method for waveguides operating in the single mode regime.

Fig. 1 Variation of effective index, neff with core width, W. The inset shows the variation of error in neff of fundamental mode with W for SV-Vopt and VEIM results. Error is defined as difference in neff obtained by SV-Vopt/VEIM and VFEM.

3.2. Effective mode area

Field profiles obtained from the SV-Vopt method can be used to evaluate various other quantities of interest. One important parameter is effective mode area (Aeff), which defined as follows:
Aeff=(|Et|2dxdy)2|Et|4dxdy
(9)
where Et = (Ex, Ey). Since the SV-Vopt approximates Ex = χ(x)ϕ(y) and Ey = 0 for Ex mode, Et = (Ex, 0) is used here. Variation of the effective mode area, Aeff with core width, W, is shown in Fig. 2. Core height is kept constant at 260nm. It can be observed that value of Aeff 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.

Fig. 2 Variation of effective mode area, Aeff with core width, W.

3.3. Power confinement factor

Another important parameter is the confinement factor, which gives an estimate of the power confined in various regions of the waveguide. For fields normalized to unit power, the power confinement factor for the core of the waveguide is defined as:
Γcore=core{E*×H}zdxdy
(10)
where the integration is evaluated over the core of the waveguide. Since it is desirable to have waveguide with high power confinement in its core, we show variation of power confinement in core, Γcore 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 Γcore 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.

Fig. 3 Variation of power confinement factor in core, Γcore with core width, W.

3.4. Modal birefringence

Finally, we show that the SV-Vopt method gives equally good results for the Ey mode. From the propagation constants obtained for fundamental Ex and Ey modes, accurate estimation of birefringence can also be calculated. Birefringence is defined as difference in neff of the E11x and E11y modes. The SV-Vopt results and the reference FEM results for effective index, neff, and birefringence are plotted in Fig. 4. It can be seen that SOI strip waveguides show very large birefringence.

Fig. 4 Variation of neff for E11x and E11y mode (left scale) and modal birefringence (right scale) with core width, W.

4. Conclusion

Acknowledgments

One of the authors (Kanchan Gehlot) would like to thank the Council of Scientific and Industrial Research (CSIR), India, for providing Senior Research Fellowship.

References and links

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] .

4.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

5.

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.

6.

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

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 134, 699–702 (1987).

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] .

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

  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. Express12, 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. Express18, 8528–8539 (2010). [CrossRef] [PubMed]
  4. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).
  5. 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.
  6. K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt.25, 2169–2174 (1986). [CrossRef] [PubMed]
  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. A134, 699–702 (1987).
  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. A18, 1383–1387 (2001). [CrossRef]

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