## Loss and dispersion analysis of microstructured fibers by finite-difference method

Optics Express, Vol. 12, Issue 15, pp. 3341-3352 (2004)

http://dx.doi.org/10.1364/OPEX.12.003341

Acrobat PDF (721 KB)

### Abstract

The dispersion and loss in microstructured fibers are studied using a full-vectorial compact-2D finite-difference method in frequency-domain. This method solves a standard eigen-value problem from the Maxwell’s equations directly and obtains complex propagation constants of the modes using anisotropic perfectly matched layers. A dielectric constant averaging technique using Ampere’s law across the curved media interface is presented. Both the real and the imaginary parts of the complex propagation constant can be obtained with a high accuracy and fast convergence. Material loss, dispersion and spurious modes are also discussed.

© 2004 Optical Society of America

## 1. Introduction

1. J. Broeng, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. **5**, 305–330 (1999). [CrossRef]

3. A. Bjarklev, J. Broeng, and A. S. Bjarklev, *Photonic crystal fibres* (Kluwer Academic Publishers, Boston/Dordrecht/London, 2003). [CrossRef]

*t*and

*z*denote respectively the transverse and longitudinal components.

4. Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express **8**, 547–554 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547 [CrossRef] [PubMed]

8. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultra-flattened dispersion in photonic crystal fibers,” Opt. Lett. **25**, 79–792 (2000). [CrossRef]

9. A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. **13**, 1795–1780 (1995). [CrossRef]

12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russel, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. **23**, 1662–1664 (1998). [CrossRef]

13. A. Cucinotta, G. Peiosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microwave Opt. Techn. Lett. **23**, 67–69 (1999). [CrossRef]

16. M. Koshiba, Y. Tsuji, and M. Hikari, “Finite element beam propagation method with perfectly matched layer boundary conditions,” IEEE Trans. Magnetics **35**, 1482–1485 (1999). [CrossRef]

17. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite-element method,” Opt. Fiber Technol. **6**, 181–191 (2000). [CrossRef]

23. D. H. Choi and W. J. R. Hoeffer, “The finite-difference time-domain method and its application to eigen-value problems,” IEEE Trans. Microwave Theory Tech. **34**, 1464–1470 (1986). [CrossRef]

28. N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonantors with curved surfaces,” IEEE Trans. Microwave Theory Tech. **45**, 1645–1649 (1997). [CrossRef]

29. K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. **34**, 1104–1113 (1986). [CrossRef]

31. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructuredd optical fibers,” Opt. Express **10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

32. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructuredd optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322 (2002). [CrossRef]

33. D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J Quantum Electron. **28**, 459–465 (1992). [CrossRef]

34. S. Guo, F. Wu, K. Ikram, and S. Albin, “Analysis of circular fibers with an arbitrary index profile by the Galerkin method,” Opt. Lett. **29**, 32–34 (2004). [CrossRef] [PubMed]

35. S. Guo, S. Albin, and R. S. Rogowski, “Comparative analysis of Bragg fibers,” Opt. Express **12**, 198–207 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-207 [CrossRef] [PubMed]

6. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

9. A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. **13**, 1795–1780 (1995). [CrossRef]

10. W. Zhi, R. Guobing, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express **11**, 980–991 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980 [CrossRef] [PubMed]

12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russel, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. **23**, 1662–1664 (1998). [CrossRef]

17. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite-element method,” Opt. Fiber Technol. **6**, 181–191 (2000). [CrossRef]

36. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. **14**, 302–307 (1966). [CrossRef]

23. D. H. Choi and W. J. R. Hoeffer, “The finite-difference time-domain method and its application to eigen-value problems,” IEEE Trans. Microwave Theory Tech. **34**, 1464–1470 (1986). [CrossRef]

28. N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonantors with curved surfaces,” IEEE Trans. Microwave Theory Tech. **45**, 1645–1649 (1997). [CrossRef]

29. K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. **34**, 1104–1113 (1986). [CrossRef]

30. P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J Lightwave Technol. **12**, 487–494 (1994). [CrossRef]

31. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructuredd optical fibers,” Opt. Express **10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

37. S. Guo, F. Wu, S. Albin, and R. S Rogowski, “Photonic band gap analysis using finite-difference frequency-domain method,” Opt. Express **12**, 1741–1746 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741 [CrossRef] [PubMed]

39. H. Y. D. Yang, “Finite difference analysis of 2D photonic crystals,” IEEE Trans. Microwave Theory Technol. **44**, 2688–2695 (1996). [CrossRef]

32. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructuredd optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322 (2002). [CrossRef]

40. N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructuredd optical fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

19. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express **11**, 843–852 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843 [CrossRef] [PubMed]

31. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructuredd optical fibers,” Opt. Express **10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

## 2. Analysis method

41. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J Comp. Phys. **114**, 185–200 (1994). [CrossRef]

13. A. Cucinotta, G. Peiosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microwave Opt. Techn. Lett. **23**, 67–69 (1999). [CrossRef]

42. E. A. Marengo, C. M. Rappaport, and E. L. Miller, “Optimum PML ABC conductivity profile in FDFD,” IEEE Trans. Magnetics **35**, 1506–1509 (1999). [CrossRef]

47. W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave & Opt. Technol. Lett. **7**, 363–369 (1997). [CrossRef]

**10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

37. S. Guo, F. Wu, S. Albin, and R. S Rogowski, “Photonic band gap analysis using finite-difference frequency-domain method,” Opt. Express **12**, 1741–1746 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741 [CrossRef] [PubMed]

_{z}as in [31

**10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

_{t}:

_{t}:

**10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

## 3. Numerical results

32. T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructuredd optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322 (2002). [CrossRef]

*a*=6.75µm, the air hole radius

*r*=2.5µm, and the refractive index of the glass is 1.45. The material dispersion is omitted since it is trivial to include it in a compact-2D FDFD scheme. According to the multipole method, the accurate effective mode index at wavelength 1.45µm would be 1.445395345+3.15×10

^{-8}i.

_{6ν}(six-fold rotation symmetry and at least one plane of reflection symmetry), and the computation region can be reduced using the symmetry properties by applying a combination of PEC and PMC (perfect magnetic conductor) [48

48. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-II:Theory,” IEEE Trans. Microwave Theory Tech. **23**, 429–433 (1975). [CrossRef]

^{nd}-order power law profile. The computation region is chosen to be 1.5

*a*along both x and y directions and the thickness of the PML layers is 10% of the thickness of the inside area along x or y direction.

**10**, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 [CrossRef] [PubMed]

*f*is the fraction of the first material ε

_{a}.

^{-5}~10

^{-6}even if a coarse mesh (for example, 30×30) is used; the accuracy is sufficient to obtain group velocity dispersion and other parameters. However, the imaginary part converges rather slowly with a relative error in the range of 10

^{-1}~10

^{-2}with almost 30% relative error using a 30×30 mesh. Though it converges to the true value with a fine mesh, the slow convergence is still not satisfactory.

_{rx}, ε

_{ry}and ε

_{rz}are the averaged dielectric constant of the cell located at the same position as E

_{x}, E

_{y}and E

_{z}.

_{z}as an example:

_{x}in yz plane and ε

_{y}in xz plane, the averaging is easy to do since the boundary is parallel to z direction and the integration cell shrinks to a line, which is shown in Fig. 3 as the dotted cells.

_{x}and E

_{y}in the xy plane as shown in Fig. 3 are not tangential to the dielectric interface, and therefore will not be continuous across the dielectric boundary. As in Fig. 4, when the integration surface for E

_{x}moves along x in the cell on the xy plane, the interface will shift, and similar is the case for E

_{y}. Since E

_{x}and E

_{y}are the average field values of the cell in xy plane, another average has to be taken and the averaged dielectric values are [28

28. N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonantors with curved surfaces,” IEEE Trans. Microwave Theory Tech. **45**, 1645–1649 (1997). [CrossRef]

49. G. W. Milton, *The theory of composites* (Cambridge University Press, Cambridge, UK, 2002). [CrossRef]

_{r}(r) and our FDFD algorithm is also applicable for these materials.

^{-3}for a moderately fine mesh.

^{nd}-order mode of the mode class 3 and 4 in the holey fiber. These modes are well confined by the single ring of the air holes and show the symmetries as discussed above. Once H

_{x}and H

_{y}(E

_{x}and E

_{y}) are solved, the other components can be obtained directly using Eqs. (6a–b).

### 3.1 Effect of dispersive and lossy/gain materials

50. R. Guobing, W. Zhi, L. Shuqin, and J. Shuisheng, “Full-vectorial analysis of complex refractive index photonic crystal fibers,” Opt. Express **12**, 1126–1135 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1135 [CrossRef]

### 3.2 Spurious modes

**45**, 1645–1649 (1997). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Broeng, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. |

2. | P. Russell, “Photonic crystal fibers,” Science |

3. | A. Bjarklev, J. Broeng, and A. S. Bjarklev, |

4. | Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express |

5. | A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A |

6. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

7. | J. Broeng, S. E. Barkou, T. Sondergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. |

8. | A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultra-flattened dispersion in photonic crystal fibers,” Opt. Lett. |

9. | A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. |

10. | W. Zhi, R. Guobing, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express |

11. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J Lightwave Technol. |

12. | D. Mogilevtsev, T. A. Birks, and P. St. J. Russel, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. |

13. | A. Cucinotta, G. Peiosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microwave Opt. Techn. Lett. |

14. | K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. |

15. | 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. |

16. | M. Koshiba, Y. Tsuji, and M. Hikari, “Finite element beam propagation method with perfectly matched layer boundary conditions,” IEEE Trans. Magnetics |

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

18. | C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J Lightwave Technol. |

19. | K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express |

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

21. | S. Guenneau, S. Lasquellec, A. Nicolet, and F. Zolla, “Design of photonic crystal fibers using finite elements,” International J. Computation and Mathematics in Electrical & Electronics Engineering COMPEL |

22. | S. Guenneau, A. Nicolet, F. Zolla, and S. Lasquellec, “Numerical and theoretical study of photonic crystal fibers,” Progress in Electromagnetics Research |

23. | D. H. Choi and W. J. R. Hoeffer, “The finite-difference time-domain method and its application to eigen-value problems,” IEEE Trans. Microwave Theory Tech. |

24. | A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron. Lett. |

25. | A. C. Cangellaris, “Numerical stability and numerical dispersion of a compact 2D FDTD method used for the dispersion analysis of waveguides,” IEEE Microwave Guided Wave Lett. |

26. | F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino, G. Tartarini, and P. Bassi, “Rigorous analysis of 3D optical and optoelectronic devices by the Compact-2D-FDTD method,” Opt. and Quantum Electron. |

27. | S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microwave Guided Wave Lett. |

28. | N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonantors with curved surfaces,” IEEE Trans. Microwave Theory Tech. |

29. | K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. |

30. | P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J Lightwave Technol. |

31. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructuredd optical fibers,” Opt. Express |

32. | T. P. White, B. T. Kuhlmey, R. C. Mcphedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructuredd optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

33. | D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J Quantum Electron. |

34. | S. Guo, F. Wu, K. Ikram, and S. Albin, “Analysis of circular fibers with an arbitrary index profile by the Galerkin method,” Opt. Lett. |

35. | S. Guo, S. Albin, and R. S. Rogowski, “Comparative analysis of Bragg fibers,” Opt. Express |

36. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. |

37. | S. Guo, F. Wu, S. Albin, and R. S Rogowski, “Photonic band gap analysis using finite-difference frequency-domain method,” Opt. Express |

38. | C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express |

39. | H. Y. D. Yang, “Finite difference analysis of 2D photonic crystals,” IEEE Trans. Microwave Theory Technol. |

40. | N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructuredd optical fibers,” J. Lightwave Technol. |

41. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J Comp. Phys. |

42. | E. A. Marengo, C. M. Rappaport, and E. L. Miller, “Optimum PML ABC conductivity profile in FDFD,” IEEE Trans. Magnetics |

43. | F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. |

44. | F. L. Teixeira and W. C. Chew, “Unified analysis of perfectly matched layers using differential forms,” Microwave Opt. Technol. Lett. |

45. | T. Tischler and W. Heinrich, “Accuracy limitations of perfectly matched layers in 3D Finite-difference frequency domain method,” IEEE Microwave Theory Tech. |

46. | U. Pekel and R. Mittra, “An application of the perfectly matched layer (PML) concept to the finite element method frequency domain analysis of scattering problems,” IEEE Microwave Guided Wave Lett. |

47. | W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave & Opt. Technol. Lett. |

48. | P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-II:Theory,” IEEE Trans. Microwave Theory Tech. |

49. | G. W. Milton, |

50. | R. Guobing, W. Zhi, L. Shuqin, and J. Shuisheng, “Full-vectorial analysis of complex refractive index photonic crystal fibers,” Opt. Express |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 28, 2004

Revised Manuscript: July 9, 2004

Published: July 26, 2004

**Citation**

Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, and Robert Rogowski, "Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express **12**, 3341-3352 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3341

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### References

- J. Broeng, "Photonic crystal fibers: a new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999). [CrossRef]
- P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
- A. Bjarklev, J. Broeng, A. S. Bjarklev, Photonic crystal fibres (Kluwer Academic Publishers, Boston/Dordrecht/London, 2003). [CrossRef]
- Z. Zhu, T. G. Brown, "Analysis of the space filling modes of photonic crystal fibers," Opt. Express 8, 547- 554 (2001). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547</a. [CrossRef] [PubMed]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, "Vector description of higher-order modes in photonic crystal fibers," J. Opt. Soc. Am. A 17, 1333-1339 (2000). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, M. V. Andres, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
- J. Broeng, S. E. Barkou, T. Sondergaard, A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, "Nearly zero ultra-flattened dispersion in photonic crystal fibers," Opt. Lett. 25, 790-792 (2000). [CrossRef]
- A. Weisshaar, J. Li, R. L. Gallawa, I. C. Goyal, "Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions," J. Lightwave Technol. 13, 1795-1780 (1995). [CrossRef]
- W. Zhi, R. Guobing, L. Shuqin, J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980</a> [CrossRef] [PubMed]
- T. M. Monro, D. J. Richardson, N. G. R. Broderick, P. J. Bennett, "Holey optical fibers: an efficient modal model," J Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
- D. Mogilevtsev, T. A. Birks, P. St. J. Russel, "Group-velocity dispersion in photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
- A. Cucinotta, G. Peiosi, S. Selleri, L. Vincetti, M. Zoboli, "Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method," Microwave Opt. Techn. Lett. 23, 67-69 (1999). [CrossRef]
- K. Saitoh, M. Koshiba, "Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides," J. Lightwave Technol. 19, 405-413 (2001). [CrossRef]
- K. Saitoh, 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]
- M. Koshiba, Y. Tsuji, M. Hikari, "Finite element beam propagation method with perfectly matched layer boundary conditions," IEEE Trans. Magnetics 35, 1482-1485 (1999). [CrossRef]
- F. Brechet, J. Marcou, D. Pagnoux, P. Roy, "Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite-element method," Opt. Fiber Technol. 6, 181-191 (2000). [CrossRef]
- C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, K. T. V. Grattan, "Loss/gain characterization of optical waveguides," J Lightwave Technol. 13, 1760-1765 (1995). [CrossRef]
- K. Saitoh, M. Koshiba, T. Hasegawa, E. Sasaoka, "Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843</a> [CrossRef] [PubMed]
- M. Koshiba, Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightwave Technol. 18, 737-743 (2000). [CrossRef]
- S. Guenneau, S. Lasquellec, A. Nicolet, F. Zolla, "Design of photonic crystal fibers using finite elements," International J. Computation and Mathematics in Electrical & Electronics Engineering COMPEL 21, 534-539 (2002). [CrossRef]
- S. Guenneau, A. Nicolet, F. Zolla, S. Lasquellec, "Numerical and theoretical study of photonic crystal fibers," Progress in Electromagnetics Research 41, 271-305 (2003).
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