## Finite element characterization of chromatic dispersion in nonlinear holey fibers

Optics Express, Vol. 11, Issue 13, pp. 1481-1489 (2003)

http://dx.doi.org/10.1364/OE.11.001481

Acrobat PDF (137 KB)

### Abstract

Chromatic dispersion characteristics of nonlinear photonic crystal fibers are, for the first time to our knowledge, theoretically investigated. A self-consistent numerical approach based on the full-vector finite-element method in terms of all the components of electric fields is described for the steady-state analysis of axially-nonsymmetrical nonlinear optical fibers. Electric fields obtained with this approach can be directly utilized for evaluating nonlinear refractive index distributions. To eliminate nonphysical, spurious solutions and to accurately model curved boundaries of circular air holes, curvilinear hybrid edge/nodal elements are introduced. It is found from the numerical results that under high optical intensity, chromatic dispersion characteristics become different from those of the linear state due to optical Kerr-effect nonlinearity, especially in short wavelength region.

© 2003 Optical Society of America

## 1. Introduction

1. K. Okamoto and E.A.J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect,” J. Lightwave Technol. **7**, 1988–1994 (1989). [CrossRef]

3. H.Y. Lin and H.-C. Chang, “An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,” J. Lightwave Technol. **10**, 1188–1195 (1992). [CrossRef]

4. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

5. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

5. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

6. J.C. Knight, T.A. Birks, R.F. Cregan, P.St.J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. **34**, 1347–1348 (1998). [CrossRef]

7. N.G.R. Broderick, T.M. Monro, P.J. Bennett, and D.J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. **24**, 1395–1397 (1999). [CrossRef]

8. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St.J. Russell, “Experimental measurement of group velocity in photonic crystal fiber,” Electron. Lett. **35**, 63–64 (1999). [CrossRef]

9. M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. **24**, 1017–1019 (1999). [CrossRef]

10. W.J. Wadsworth, J.C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P.St.J. Russell, “Soliton effects in photonic crystal fibres at 850 nm,” Electron. Lett. **36**, 53–55 (2000). [CrossRef]

11. W.J. Wadsworth, A. Ortigosa-Blanch, J.C. Knight, T.A. Birks, T.-P. Martin Man, and P.St.J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B **19**, 2148–2155 (2002). [CrossRef]

## 2. Full-vector finite element method

12. A.W. Snyder, Y. Chen, L. Poladian, and D.J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. **26**, 643–644 (1990). [CrossRef]

16. M.J. Holmes, D.M. Spirit, and F.P. Payne, “New gaussian-based approximation for modeling non-linear effects in optical fibers,” J. Lightwave Technol. **12**, 193–201 (1994). [CrossRef]

1. K. Okamoto and E.A.J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect,” J. Lightwave Technol. **7**, 1988–1994 (1989). [CrossRef]

3. H.Y. Lin and H.-C. Chang, “An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,” J. Lightwave Technol. **10**, 1188–1195 (1992). [CrossRef]

17. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers,” Opt. Fiber Technol. **6**, 181- (2000). [CrossRef]

21. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**, 1530–1532 (2002). [CrossRef]

17. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers,” Opt. Fiber Technol. **6**, 181- (2000). [CrossRef]

21. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**, 1530–1532 (2002). [CrossRef]

22. K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B **5**, 2494–2501 (1988). [CrossRef]

28. X.H. Wang and G.K. Cambrell, “Vectorial simulation and power-parameter characterization of nonlinear planar optical waveguides,” J. Opt. Soc. Am. B **12**, 265–274 (1995). [CrossRef]

*et al.*[29

29. S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, “Full-vectorial finite-element solution of nonlinear bistable optical waveguides,” IEEE J. Quantum Electron. **38**, 1120–1125 (2002). [CrossRef]

22. K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B **5**, 2494–2501 (1988). [CrossRef]

27. M. Zoboli, F.Di Pasquale, and S. Selleri, “Full-vectorial and scalar solutions of nonlinear optical fibers,” Opt. Comuun. **97**, 11–15 (1993). [CrossRef]

29. S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, “Full-vectorial finite-element solution of nonlinear bistable optical waveguides,” IEEE J. Quantum Electron. **38**, 1120–1125 (2002). [CrossRef]

24. X.H. Wang and G.K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear-optical channel waveguides,” J. Opt. Soc. Am. B **10**, 1090–1095 (1993). [CrossRef]

28. X.H. Wang and G.K. Cambrell, “Vectorial simulation and power-parameter characterization of nonlinear planar optical waveguides,” J. Opt. Soc. Am. B **12**, 265–274 (1995). [CrossRef]

30. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743 (2000). [CrossRef]

22. K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B **5**, 2494–2501 (1988). [CrossRef]

28. X.H. Wang and G.K. Cambrell, “Vectorial simulation and power-parameter characterization of nonlinear planar optical waveguides,” J. Opt. Soc. Am. B **12**, 265–274 (1995). [CrossRef]

29. S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, “Full-vectorial finite-element solution of nonlinear bistable optical waveguides,” IEEE J. Quantum Electron. **38**, 1120–1125 (2002). [CrossRef]

24. X.H. Wang and G.K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear-optical channel waveguides,” J. Opt. Soc. Am. B **10**, 1090–1095 (1993). [CrossRef]

**12**, 265–274 (1995). [CrossRef]

## 3. Basic equation

*xy*(transverse) plane and assume that the structure is uniform along the propagation direction (

*z*axis).

**is the electric field vector,**

*E**k*

_{0}is the free-space wavenumber, and

*n*is the refractive index. Because of the unifomity of the waveguide, we can write the electric fields

**as**

*E**n*depends on the electric field intensity and is given by

## 4. Self-consistent approach

30. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743 (2000). [CrossRef]

^{T}denotes a transpose and

*e*}

_{t}_{e}and {

*e*}

_{z}_{e}edge and nodal electric fields for each element, respectively;

*U*} and {

*V*} shape function vectors for edge elements;

*N*} shape function vector for nodal elements.

_{e}extends over all different elements. The optical power

*P*is evaluated as

*Z*

_{0}is the free-space impedance and * denotes complex conjugate. To obtain the intensity-dependent refractive index in Eq. (3), it is necessary to compute the actual electric field {

*e*}. The relation between the actual field {

*e*} and the eigenvector {

*e*’} of Eq. (5) can be written as

^{2}and {

*e*’}, respectively. Hence, one can solve it self-consistently using the following iterative scheme:

- Specify the refractive index
*n*, the wavelength λ=2π/*k*_{0}, and optical power*P*as input data. - To assign intial values to β and {
*e*’}, solve the corresponding linear problem. - To obtain a new set of β and {
*e*’}, solve Eq. (5). - Iterate the above procedures 3) and 4) until the solution converges within the desired criterion.

## 5. Chromatic dispersion characteristics of nonlinear holey fibers

### 5.1 Nonlinear optical fibers

*V*and

*b*are, respectively, the well-known normalized frequency and normalized propagation constant. For comparison, the fiber parameters are chosen to have the same values as in [1

1. K. Okamoto and E.A.J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect,” J. Lightwave Technol. **7**, 1988–1994 (1989). [CrossRef]

3. H.Y. Lin and H.-C. Chang, “An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,” J. Lightwave Technol. **10**, 1188–1195 (1992). [CrossRef]

*a*=2.5 µm, the cladding index

*n*=1.47, and

_{cl}*n*being the linear part of core index. Also,

_{co}*n*=6.4×10

_{2}P^{-14}m

^{2}is assumed, where

*n*

_{2}[m

^{2}/W] is the nonlinear coefficient,

*P*[W] is the optical power in Eq. (9), and the refractive index of nonlinear core

*n*is given by

*P*considered here is 200 kW when the fiber is made of silica glass with

*n*

_{2}=3.2×10

^{-20}m

^{2}/W and is less than 1 W when the fiber is made of organic material with

*n*

_{2}≈10

^{-14}m

^{2}/W [1

**7**, 1988–1994 (1989). [CrossRef]

2. R.A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear-optical fibers,” Opt. Lett. **16**, 70–72 (1991). [CrossRef] [PubMed]

*g*(

*V*). The calculated results for the linear case agree well with the exact solutions. Also, under high nonlinearity, our results are in good agreement with those of FEM for circulary symmetric fibers [3

**10**, 1188–1195 (1992). [CrossRef]

### 5.2 Nonlinear holey fibers

*d*is the hole diameter, Λ is the hole pitch (center-to-center distance between the holes) of triangular lattice structure, and the background nonlinear refractive index

*n*is given by

*n*being the linear part of the background index. Because of the symmetry nature of the system, only one-quarter of the fiber cross section is divided into curvilinear edge/nodal hybrid elements shown in Fig. 1. Here, the background material is assumed to be silica and the material dispersion is taken into account by using the three-term Sellmeier equation.

_{L}*d*/Λ=0.9 for different hole pitches of Λ=1.0 µm, 1.5 µm, 2.0 µm, and 2.5 µm, respectively, where the values of

*n*

_{2}

*P*is taken as a parameter. The influence of optical Kerr-effect on chromatic dispersion is larger in the shorter wavelength region and the zero-dispersion wavelength shifts to the longer wavelength region. For the shorter wavelength region, the light is strongly confined into the core region more and more with increasing optical power and so, the effect of air holes on the waveguide dispersion becomes smaller. In the longer wavelength region, on the other hand, the light confinement is not so strong, in other words, the effective core area becomes large, and so, the chromatic dispersion tends to be insensitive to the change of nonlinearity, compared to that in the shorter wavelength region. To confirm these facts, the effective core area

*A*defined as

_{eff}*d*/Λ=0.9. It can be seen clearly that in the longer wavelength region,

*A*becomes large, resulting in reducing the effective nonlinearity. As the optical power increases, because of stronger field confinement,

_{eff}*A*becomes smaller and the effective nonlinearity is enhanced.

_{eff}*d*/Λ=0.5, 0.6, 0.7, and 0.8, respectively, where the values of

*n*

_{2}

*P*is taken as a parameter. As the value of

*d*/Λ increases, because of stronger field confinement, the effective nonlinearity is enhanced and the change of zero-dispersion wavelength becomes large. Figures 7(a) and (b) show the zero-dispersion wavelength as a function of hole pitch Λ(

*d*/Λ=0.9) and of ratio of hole diameter to hole pitch

*d*/Λ(Λ=1.5 µm), respectively, where the value of

*n*

_{2}

*P*is taken as a parameter. We can see that the zero-dispersion wavelength shifts greatly to the longer wavelength region with increasing optical power, especially in smaller values of Λ and in larger values of

*d*/Λ.

## 6. Conclusion

## References and links

1. | K. Okamoto and E.A.J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect,” J. Lightwave Technol. |

2. | R.A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear-optical fibers,” Opt. Lett. |

3. | H.Y. Lin and H.-C. Chang, “An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,” J. Lightwave Technol. |

4. | J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

5. | T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

6. | J.C. Knight, T.A. Birks, R.F. Cregan, P.St.J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. |

7. | N.G.R. Broderick, T.M. Monro, P.J. Bennett, and D.J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. |

8. | M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St.J. Russell, “Experimental measurement of group velocity in photonic crystal fiber,” Electron. Lett. |

9. | M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. |

10. | W.J. Wadsworth, J.C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P.St.J. Russell, “Soliton effects in photonic crystal fibres at 850 nm,” Electron. Lett. |

11. | W.J. Wadsworth, A. Ortigosa-Blanch, J.C. Knight, T.A. Birks, T.-P. Martin Man, and P.St.J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B |

12. | A.W. Snyder, Y. Chen, L. Poladian, and D.J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. |

13. | R.A. Sammut and C. Pask, “Variational approach to nonlinear waveguides-gaussian approximations,” Electron. Lett. |

14. | R.A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B |

15. | Y. Chen, “Nonlinear fibers with arbitrary nonlinearity,” J. Opt. Soc. Am. B |

16. | M.J. Holmes, D.M. Spirit, and F.P. Payne, “New gaussian-based approximation for modeling non-linear effects in optical fibers,” J. Lightwave Technol. |

17. | F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers,” Opt. Fiber Technol. |

18. | M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. |

19. | M. Koshiba, “Full-vector analysis of photonic crystal fibers using the finite element method,” IEICE Trans. Electron. |

20. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. |

21. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. |

22. | K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B |

23. | R.D. Ettinger, F.A. Fernandez, B.M.A. Rahman, and J.B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. |

24. | X.H. Wang and G.K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear-optical channel waveguides,” J. Opt. Soc. Am. B |

25. | X.H. Wang and G.K. Cambrell, “Simulation of strong nonlinear effects in optical waveguides,” J. Opt. Soc. Am. B |

26. | S. Selleri and M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Micorwave Theory Tech. |

27. | M. Zoboli, F.Di Pasquale, and S. Selleri, “Full-vectorial and scalar solutions of nonlinear optical fibers,” Opt. Comuun. |

28. | X.H. Wang and G.K. Cambrell, “Vectorial simulation and power-parameter characterization of nonlinear planar optical waveguides,” J. Opt. Soc. Am. B |

29. | S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, “Full-vectorial finite-element solution of nonlinear bistable optical waveguides,” IEEE J. Quantum Electron. |

30. | M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 13, 2003

Revised Manuscript: June 3, 2003

Published: June 30, 2003

**Citation**

Takeshi Fujisawa and Masanori Koshiba, "Finite element characterization of chromatic dispersion in nonlinear holey fibers," Opt. Express **11**, 1481-1489 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1481

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### References

- K. Okamoto and E.A.J. Marcatili, �??Chromatic dispersion characteristics of fibers with optical Kerr-effect,�?? J. Lightwave Technol. 7, 1988-1994 (1989). [CrossRef]
- R.A. Sammut and C. Pask, �??Group velocity and dispersion in nonlinear-optical fibers,�?? Opt. Lett. 16, 70-72 (1991). [CrossRef] [PubMed]
- H.Y. Lin and H.-C. Chang, �??An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,�?? J. Lightwave Technol. 10, 1188-1195 (1992). [CrossRef]
- J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M.Atkin, �??All-silica single-mode optical fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- T.A. Birks, J.C. Knight, and P.St.J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- J.C. Knight, T.A. Birks, R.F. Cregan, P.St.J. Russell, and J.-P. de Sandro, �??Large mode area photonic crystal fiber,�?? Electron. Lett. 34, 1347-1348 (1998). [CrossRef]
- N.G.R. Broderick, T.M. Monro, P.J. Bennett, and D.J. Richardson, �??Nonlinearity in holey optical fibers : measurement and future opportunities,�?? Opt. Lett. 24, 1395-1397 (1999). [CrossRef]
- M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St. J. Russell, �??Experimental measurement of group velocity in photonic crystal fiber,�?? Electron. Lett. 35, 63-64 (1999). [CrossRef]
- M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, �??Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,�?? Opt. Lett. 24, 1017-1019 (1999). [CrossRef]
- W.J. Wadsworth, J.C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P.St.J. Russell, �??Soliton effects in photonic crystal fibres at 850 nm,�?? Electron. Lett. 36, 53-55 (2000). [CrossRef]
- W.J. Wadsworth, A. Ortigosa-Blanch, J.C. Knight, T.A. Birks, T.-P. Martin Man, and P.St.J. Russell, �??Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,�?? J. Opt. Soc. Am. B 19, 2148-2155 (2002). [CrossRef]
- A.W. Snyder, Y. Chen, L. Poladian, and D.J. Mitchell, �??Fundamental mode of highly nonlinear fibers,�?? Electron. Lett. 26, 643-644 (1990). [CrossRef]
- R.A. Sammut and C. Pask, �??Variational approach to nonlinear waveguides-gaussian approximations,�?? Electron. Lett. 26, 1131-1132 (1990). [CrossRef]
- R.A. Sammut and C. Pask, �??Gaussian and equivalent-step-index approximations for nonlinear waveguides,�?? J. Opt. Soc. Am. B 8, 395-402 (1991). [CrossRef]
- Y. Chen, �??Nonlinear fibers with arbitrary nonlinearity,�?? J. Opt. Soc. Am. B 8, 2338-2341 (1991). [CrossRef]
- M.J. Holmes, D.M. Spirit, and F.P. Payne, �??New gaussian-based approximation for modeling non-linear effects in optical fibers,�?? J. Lightwave Technol. 12, 193-201 (1994). [CrossRef]
- F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers,�?? Opt. Fiber Technol. 6, 181- (2000). [CrossRef]
- M. Koshiba and K. Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315 (2001). [CrossRef]
- M. Koshiba, �??Full-vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Trans. Electron. E85-C, 881-888 (2002).
- K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Holey fiber analysis through the finite element method,�?? IEEE Photon. Technol. Lett. 14, 1530-1532 (2002). [CrossRef]
- K. Hayata and M. Koshiba, �??Full vectorial analysis of nonlinear-optical waveguides,�?? J. Opt. Soc. Am. B 5, 2494-2501 (1988). [CrossRef]
- R.D. Ettinger, F.A. Fernandez, B.M.A. Rahman, and J.B. Davies, �??Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,�?? IEEE Photon. Technol. Lett. 3, 147-149 (1991). [CrossRef]
- X.H. Wang and G.K. Cambrell, �??Full vectorial simulation of bistability phenomena in nonlinear-optical channel waveguides,�?? J. Opt. Soc. Am. B 10, 1090-1095 (1993). [CrossRef]
- X.H. Wang and G.K. Cambrell, �??Simulation of strong nonlinear effects in optical waveguides,�?? J. Opt. Soc. Am. B 10, 2048-2055 (1993). [CrossRef]
- S. Selleri and M. Zoboli, �??An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,�?? IEEE Trans. Micorwave Theory Tech. 43, 887-892 (1995). [CrossRef]
- M. Zoboli, F.Di Pasquale, and S. Selleri, �??Full-vectorial and scalar solutions of nonlinear optical fibers,�?? Opt. Commun. 97, 11-15 (1993). [CrossRef]
- X.H. Wang and G.K. Cambrell, �??Vectorial simulation and power-parameter characterization of nonlinear planar optical waveguides,�?? J. Opt. Soc. Am. B 12, 265-274 (1995). [CrossRef]
- S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, �??Full-vectorial finite-element solution of nonlinear bistable optical waveguides,�?? IEEE J. Quantum Electron. 38, 1120-1125 (2002). [CrossRef]
- M. Koshiba and Y. Tsuji, �??Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,�?? J. Lightwave Technol. 18, 737-743 (2000). [CrossRef]

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