## Full-vectorial finite-difference analysis of microstructured optical fibers

Optics Express, Vol. 10, Issue 17, pp. 853-864 (2002)

http://dx.doi.org/10.1364/OE.10.000853

Acrobat PDF (991 KB)

### Abstract

In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee’s 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.

© 2002 Optical Society of America

## 1. Introduction

1. K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. **26**, s113–s134 (1994). [CrossRef]

2. C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron. **29**, 95–114 (1997). [CrossRef]

3. 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]

4. 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. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. **12**, 807–809 (2000). [CrossRef]

6. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. **25**, 790–792 (2000). [CrossRef]

7. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. **13**, 588–590 (2001). [CrossRef]

8. 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]

9. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express **10**, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341 [CrossRef] [PubMed]

10. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,” Opt. Lett. **25**, 25–27 (2000). [CrossRef]

11. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express **9**, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698 [CrossRef] [PubMed]

3. 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]

4. 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]

12. J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A **15**, 748–752 (1998). [CrossRef]

13. 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]

16. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2-D photonic crystals,” J. Lightwave Technol. **17**, 2078–2081(1999). [CrossRef]

17. T.M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

18. 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]

19. G. E. Town and J. T. Lizer, “Tapered holey fibers for spot size and numerical-aperture conversion,” Opt. Lett. **26**, 1042–1044 (2001). [CrossRef]

20. M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. **30**, 327–330 (2001). [CrossRef]

21. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. **26**, 1660–1662 (2001). [CrossRef]

22. M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. **135**, 56–63 (1988). [CrossRef]

**E**

_{t}(or magnetic field

**H**

_{t})

*k*

_{0}=2π/λ is the wave number in free space, ε

_{r}is the waveguide dielectric constant, and β is the propagation constant, is directly discretized by finite difference [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

24. W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. **10**, 295–305 (1992). [CrossRef]

*et al*. [25

25. 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]

26. H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. **11**, 1559–1563 (1993). [CrossRef]

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. **12**, 487–493 (1994). [CrossRef]

28. 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]

## 2. Formulation

*i*(

*βz*-

*ωt*)]. From Maxwell’s curl equations (∇×

**E**= -∂

**B**/∂

*t*, ∇×

**H**= ∂

**D**/∂

*t*), after scaling

**E**by the free space impedance

**I**is a square identity matrix,

**ε**

_{rx},

**ε**

_{ry}and

**ε**

_{rz}are diagonal matrices determined by Eqs. (6).

**U**

_{x},

**U**

_{y},

**V**

_{x}and

**V**

_{y}are square matrices and depend on the boundary conditions of the rectangular computation window. For example, when the zero-value boundary condition is chosen for the computation window edges, we have

**V**

_{x}=-

**V**

_{y}=-

*T*denotes transpose operation. Then it is clear from Eqs. (11) and (13) that

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. **12**, 487–493 (1994). [CrossRef]

**29**, 2639–2649 (1993). [CrossRef]

**P**and

**Q**are different. One advantage of the present approach is that, once

**E**

_{t}or

**H**

_{t}is obtained, all other field components are readily obtained through Eqs. (7) and (8).

_{r}is real, then

**P**and

**Q**are real sparse matrices. The sparse matrices can be stored efficiently in computers to reduce the amount of memory required for data storage. Solving the eigenvalue equation (10) or (12) using available numerical routines provides us the effective modal index

*n*

_{eff}=

*β*/

*k*

_{0}and modal fields of the guided modes.

15. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

*ε*

_{rz}(

*j*,

*l*) =

*f*

_{a}

*ε*

_{a}+ (1-

*f*

_{a})

*ε*

_{b}is used instead of Eq. (6c) for the cell that is filled with two different dielectric media

*ε*

_{a}and

*ε*

_{b}, where

*f*

_{a}is the fraction of the cell occupied by medium

*ε*

_{a}(see Fig. 1(b)). Similarly, the averaged

*ε*

_{rx}and

*ε*

_{ry}can be calculated in the cells centered on

*E*

_{x}and

*E*

_{y}, respectively. We also apply this index averaging (IA) technique in other FD mode solvers mentioned in the Introduction. The use of the averaged refractive index of interfacial cells, as will be seen in the next section, can significantly accelerate convergence and improve the modeling accuracy for waveguides with curved interfaces, such as optical fibers and MOFs.

## 3. Numerical Results

**29**, 2639–2649 (1993). [CrossRef]

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. **12**, 487–493 (1994). [CrossRef]

### 3.1. Step-Index Fiber

*et al*.[23

**29**, 2639–2649 (1993). [CrossRef]

*et al*.[27

**12**, 487–493 (1994). [CrossRef]

*n*

_{eff}= 1.438604. We choose such a high index difference fiber in order to illustrate the applicability of the FD mode solver in modeling waveguides with curved high-index difference interfaces. In Fig.2 we show the relative error of the calculated fundamental mode index

*n*

_{eff}from the three FD solvers (both with and without index averaging) when the number of grids along the

*x*-axis is varied. As can be clearly seen, the index averaging technique can significantly stabilize and improve the accuracy for the three FD mode solvers. In Table 1, we list the calculated results from different approaches. Uniform meshes are used in all the numerical calculations.

*et al*. [23

**29**, 2639–2649 (1993). [CrossRef]

*et al*. [27

**12**, 487–493 (1994). [CrossRef]

*et al*., the current mode solver appears to get better results even with very sparse grids.

### 3.2. Microstructured Optical Fiber (MOF)

31. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express **9**, 681–686 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681 [CrossRef] [PubMed]

32. Z. Zhu and T. G. Brown, “Multipole analysis of hole-assisted optical fibers”, Opt. Commun. **206**, 333–339 (2002). [CrossRef]

*y*-polarized fundamental mode. The field amplitudes are not normalized. The relative comparison of amplitude between two cartesian components of the magnetic field indicates that

*H*

_{x}dominates in the

*y*-polarized mode. In Fig. 3(d) we show the group velocity dispersion (GVD) curves calculated from the FD method and the multipole method. The agreement is excellent. In the calculation of the GVD curves, we did not consider material dispersion which is dominant at short wavelengths and assumed constant refractive index at all wavelengths investigated, i.e.,

*n*

_{core}=1.45 and

*n*

_{clad}=1.0. While it is trivial to take into account the material dispersion by changing the refractive index at each wavelength, small differences in GVD between different approaches are better appreciated by excluding material dispersion.

*n*

_{FSM}<

*n*

_{eff}<

*n*

_{si}, where

*n*

_{si}is the refractive index of silica, and

*n*

_{FSM}is the effective index of the fundamental space-filling mode (FSM) [33

33. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**, 1031–1037 (2000). [CrossRef]

34. 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]

*n*

_{FSM}is an important parameter in the effective index model of holey fibers, which can give estimation of the number of guided modes and the effective mode areas [9

9. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express **10**, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341 [CrossRef] [PubMed]

12. J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A **15**, 748–752 (1998). [CrossRef]

*n*

_{FSM}can be obtained through a few methods including the approximate analytical solution [33

33. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**, 1031–1037 (2000). [CrossRef]

34. 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]

18. 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]

*n*

_{FSM}in the elementary region (see Fig. 4(a), region bound by B1~B4) with proper boundary conditions. For example, boundary conditions for an

*x*-polarized FSM can be: B1, B2 electric wall, B3 and B4 magnetic wall.

5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. **12**, 807–809 (2000). [CrossRef]

10. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,” Opt. Lett. **25**, 25–27 (2000). [CrossRef]

14. 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 **17**, 1333–1340 (2000). [CrossRef]

17. T.M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

*et al*. [35

35. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

*C*

_{nv}(

*n*≥3) symmetry. The modal degeneracy of air-hole assisted fiber has been numerically confirmed by the multipole method [32

32. Z. Zhu and T. G. Brown, “Multipole analysis of hole-assisted optical fibers”, Opt. Commun. **206**, 333–339 (2002). [CrossRef]

35. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

36. M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. **13**, 1313–1315 (2001). [CrossRef]

*B*=|

*n*

_{eff,x}-

*n*

_{eff,y}|, where

*n*

_{eff,x}and

*n*

_{eff,y}are the effective index for the

*x*-polarized and

*y*-polarized fundamental modes, respectively. The observed birefringence is attributed to the discretization in the FD method, which introduces different symmetry from that of the ideal waveguide structure and lifts the expected degeneracy.

*e*=0.375) in the place of the circular holes. Steel

*et al*. [37

37. M. J. Steel and R. M. Osgood Jr., “Elliptical-hole photonic crystal fibers,” Opt. Lett. **26**, 229–231 (2001). [CrossRef]

*B*=6.5×10

^{-4}at λ=1.5 μm. The indices of the

*x*-polarized and

*y*-polarized FSM are also calculated and show much larger birefringence that the fundamental modes. This has important implications for the design of polarization-maintaining MOFs because it can affect both the single-mode cutoff condition and the differential loss (e.g. bending loss or leakage). As before, the material dispersion is not considered in the calculations.

## 4. Discussion and Conclusion

## Acknowledgement

## References and Links

1. | K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. |

2. | C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron. |

3. | 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. |

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

5. | J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. |

6. | A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. |

7. | T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. |

8. | 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. |

9. | N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express |

10. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,” Opt. Lett. |

11. | B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express |

12. | J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A |

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

14. | 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 |

15. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

16. | D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2-D photonic crystals,” J. Lightwave Technol. |

17. | T.M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. |

18. | 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. |

19. | G. E. Town and J. T. Lizer, “Tapered holey fibers for spot size and numerical-aperture conversion,” Opt. Lett. |

20. | M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. |

21. | T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. |

22. | M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. |

23. | W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. |

24. | W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. |

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

26. | H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. |

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

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

29. | K. S. Kunz and R. J. Luebbers, |

30. | S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. |

31. | T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express |

32. | Z. Zhu and T. G. Brown, “Multipole analysis of hole-assisted optical fibers”, Opt. Commun. |

33. | M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. |

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

35. | M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. |

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

37. | M. J. Steel and R. M. Osgood Jr., “Elliptical-hole photonic crystal fibers,” Opt. Lett. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2400) Fiber optics and optical communications : Fiber properties

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 26, 2002

Revised Manuscript: July 29, 2002

Published: August 26, 2002

**Citation**

Zhaoming Zhu and Thomas Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express **10**, 853-864 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-17-853

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### References

- K. S. Chiang, �??Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,�?? Opt. Quantum Electron. 26, s113-s134 (1994). [CrossRef]
- C. Vassallo, �??1993-1995 optical mode solvers,�?? Opt. Quantum Electron. 29, 95-114 (1997). [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, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000). [CrossRef]
- T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588-590 (2001) [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]
- N. A. Mortensen, "Effective area of photonic crystal fibers," Opt. Express 10, 341-348 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a> [CrossRef] [PubMed]
- J. K. Ranka, R. S. Windeler, and A. J. Stentz, �??Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,�?? Opt. Lett. 25, 25-27 (2000). [CrossRef]
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