## Perturbative numerical modeling of microstructure fibers

Optics Express, Vol. 12, Issue 19, pp. 4535-4545 (2004)

http://dx.doi.org/10.1364/OPEX.12.004535

Acrobat PDF (686 KB)

### Abstract

Modeling of microstructure fibers often involves severe computational bottlenecks, in particular when a design space with many degrees of freedom must be analyzed. Perturbative versions of numerical mode-solvers can substantially reduce the computational burden of problems involving automated optimization or irregularity analysis, where perturbations arise naturally. A basic theory is presented for perturbative multipole and boundary element methods, and the speed and accuracy of the methods are demonstrated. The specific optimization results in an elliptical-hole birefringent fiber design, with substantially higher birefringence than the intuitive unoptimized design.

© 2004 Optical Society of America

## 1. Introduction

2. S. A. Diddams and D. J. Jones, et al. “Direct link between microwave and optical frequencies with a 300THz femtosecond laser comb,” Phys. Rev. Lett. , **84**, 5102–5 (2000). [CrossRef] [PubMed]

3. S. G. Johnson and M. Ibanescu, et al. “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–79 (2001). [CrossRef] [PubMed]

4. John Fini and Ryan Bise. “Progress in fabrication and modeling of microstructured optical fiber,” Jap. J. App. Phys. **43**, 5717–5730 (2004). [CrossRef]

5. A. Ferrando and E. Silvestre, et al. “Full vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–8 (1999). [CrossRef]

6. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. “Calculations of air-guiding modes in photonic crystal fibers using the multipole method,” Opt. Express, **9**, 721–32 (2001). [CrossRef]

7. F. Brechet and J. Marcou, et al. “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Tech. **6**, 181–191 (2000). [CrossRef]

8. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. “Boundary element method for analysis of holey optical fibers,” J. Lightwave. Technol. **21**, 1787–92 (2003). [CrossRef]

9. 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). software available at http://ab-initio.mit.edu/mpb. [CrossRef] [PubMed]

10. T. A. Birks, J. C. Knight, and P. S. J. Russell. “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–3 (1997). [CrossRef] [PubMed]

12. James A. West, Charlene M. Smith, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. “Surface modes in air-core photonic band-gap fibers,” Opt. Express, **12**, 1485–96 (2004). [CrossRef]

## 2. Perturbative approach

13. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli. “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. **20**, 1433–42 (2002). [CrossRef]

14. A. Peyrilloux, T. Chartier, L. Berthelot, A. Hideur, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy. “Thoeretical and experimental study of the birefringence of a photonic crystal fiber,” J. Lightwave Technol. **21**, 536–9(2003). [CrossRef]

15. I. K. Hwang, Y. J. Lee, and Y. H. Lee. “Birefringence induced by irregular structure in photonic crystal fiber,” Opt. Express **11**, 2799–2806 (2003). [CrossRef] [PubMed]

6. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. “Calculations of air-guiding modes in photonic crystal fibers using the multipole method,” Opt. Express, **9**, 721–32 (2001). [CrossRef]

### 2.1. Reformulating the multipole and boundary-element methods

*=*

**Av***λ*, the most familiar starting point for perturbation theory in physics. The mode parameters

**v***λ*and

*n*

_{eff}=

*k*

_{z}/

*k*=

*λk*

_{z}/2

*π*are instead wrapped into a mode equation in a nonlinear way:

*is a vector of coefficients representing the field, and*

**v***is a matrix imposing Maxwell’s equations for the given fiber geometry. (Other methods, such as the plane-wave or finite-element methods,*

**M***do*have the standard eigenvalue form, although it has been pointed out that care must nevertheless be taken when perturbing these equations [16

16. Steven G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink. “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E **65**, 066611 (2002). [CrossRef]

*δ*. That is, if

*is a vector of parameters describing the geometry (hole positions, radii, etc), then*

**p***=*

**p**

**p**_{0}+

*δ*

**p**_{1}. The matrix

*is directly dependent on the geometrical parameters, and the wavelength:*

**M***in orders of*

**M***δ*can be explicitly calculated using numerical approximations to the derivatives (with small ε):

*n*and field coefficients

*are unknowns of Eq. (1), not explicit functions of the geometry. They can also be expanded in orders of*

**v***δ*,

**v**_{1},

*n*

_{1}, etc) are solved by plugging into (1),

**v**_{1},

**v**_{2}, are assumed orthogonal to

**v**_{0}. In the simplest case, a first-order effective index shift can be obtained as a very simple expression,

**M**_{1}and

*′*

**M**_{0}are derivatives of

*with respect to*

**M***δ*and

*n*, respectively, and

**u**_{0}and

**v**_{0}are the left and right null vectors of the unperturbed matrix

**M**_{0}(

*n*

_{0}). In the next section we discuss details of the derivation and more complicated cases.

## 3. Perturbative estimates and irregularity analysis

**v**_{0}satisfying the unperturbed equation for

*n*

_{eff}=

*n*

_{0},

17. M. J. Steel, T. P. White, C.Martijn de Sterke, R. C. McPhedran, and L. C. Botten. “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–91 (2001). [CrossRef]

18. J. M. Fini. “Improved symmetry analysis of many-moded microstructure optical fibers,” J. Opt. Soc. Am. B, **21**, 1431–6 (2004). [CrossRef]

*δ*give an expression

*δn*

_{1}and field coefficients

*δ*

**v**_{1}. Since

**M**_{0}(

*n*

_{0}) has exactly one right null vector, it also has a single left null vector,

**u**^{†}

_{0}

**M**_{0}(

*n*

_{0}) = 0, which we use to project out the effective index perturbation:

*δn*

_{1}is obtained by projecting out the null space of

**M**_{0}, the perturbed field coefficients

*δ*

**v**_{1}are obtained by projecting out the complimentary space. On this restricted space,

**M**_{0}is invertible, and so the solution involves a matrix

**M**_{0}

^{p}that acts as an inverse of

**M**_{0}on this subspace:

*δ*

**v**_{1},

*δ*

**u**_{1}, and

**M**_{0}

^{p}from a singular-value decomposition of

**M**_{0}(

*n*

_{0}).

_{2}includes all of the second-order matrix terms:

*δ*

**u**_{1}

^{†}eliminates

*δ*

^{2}

*δ*

**v**_{2}. The remaining scalar equation can easily be solved in the remaining unknown,

*δ*

^{2}

*n*

_{2}.

*풪*{Δ

*n*

^{2}} and Error ~ 풪{Δ

*n*

^{3}} for first-order and second-order estimates, respectively), down to an apparent “noise floor” in the 10

^{-6}range. Loss estimates are also fairly accurate, and impressive when we consider that the loss is represented by an imaginary part of the effective index, with unperturbed values around

*𝓘*{

*n*

_{eff}} ≈ 6×10

^{-7}. Naturally, the imaginary part is much more difficult to estimate than the real part, and can easily be buried in noise arising from a careless implementation (poor choice of

*ε*in the numerical derivatives, etc.). The current results are improved from a previous implementation [19]: they allow accurate first-order estimates of loss for geometrical deviations up to

*σ*

_{x}=

*σ*

_{y}= .01Λ. In the Fig. 3, we see that second-order estimates can cover even larger deviations, where confinement loss is a factor of two above or below its unperturbed value.

### 3.1. Degenerate modes and birefringence

*δ*

**v**_{1}6 [Eq. (5)] near the exact solution

**v**, but now even the unperturbed solution is not completely determined

*a priori*. It could be any linear combination of the

*K*degenerate modes, and the correct combination will depend on the perturbation. This has a simple interpretation in terms of birefringence: for a symmetric unperturbed structure, we cannot even guess the axis of birefringence, let alone its magnitude, until we know something about the perturbation. Similarly, even if the unperturbed problem has degenerate modes that can be separated by symmetry, a randomly perturbed structure typically will not have the same symmetry, and a separation into symmetry classes will not be possible.

*K*-dimensional vector of superposition coefficients

*:*

**x***is a tall, thin*

**B***N*×

*K*matrix satisfying

**M**_{0}(

*n*

_{0})

*= [0]*

**B**_{N×K}. (Typically

*N*is in the hundreds or thousands and

*K*is two; accidental degeneracies can larger, but symmetry-induced degeneracies are always of order two [21

21. Paul R. McIsaac. “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” Microwave Theory and Techniques **23**, 421–9 (1975). [CrossRef]

*K*×

*N*matrix

**A**^{†}, satisfying

**A**^{†}

**M**_{0}(

*n*

_{0}) = [0]

_{K×N}. The projection of the first-order Eq. (10) gives an expression for the effective index shift,

*=*

**Px***n*

_{1}

*). Solving this for*

**Qx***n*

_{1}is a negligible computation since

*and*

**P***are small (usually 2×2).*

**Q**## 4. Application: Optimization

22. M. J. Steel and R. M. Osgood. “Elliptical-hole photonic crystal fibers,” Opt. Lett. **26**, 229–31 (2001). [CrossRef]

23. M. J. Steel and R. M. Osgood. “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. **19**, 495–503 (2001). [CrossRef]

8. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. “Boundary element method for analysis of holey optical fibers,” J. Lightwave. Technol. **21**, 1787–92 (2003). [CrossRef]

- It highlights the flexibility of the boundary element method to handle arbitrary non-circular hole shapes (in contrast to the multipole method), and
- It uses a design problem simple enough that the optimum solution might seem intuitively obvious.

^{2};

*η*= 2 = major axis/minor axis) but allow some of the holes to rotate so that they are not necessarily aligned along a common axis. For problems such as this (and aircore fibers [24

24. Charlene M. Smith, Natesan Venkataraman, Michael T. Gallagher, Dirk Müller, James A. West, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. “Low loss hollow-core silica/air photonic bandgap fibre,” Nature, **424**657–9, (2003). [CrossRef]

*= [*

**p***ϕ*

_{1}

*ϕ*

_{2}…]

^{T}, and the

*j*th element of this gradient corresponds to hole

*j*:

*n*

_{Bi}has been defined as the difference in index between the “slow” and “fast” nearly-degenerate modes (

*n*

_{slow}>

*n*

_{fast}). Once a gradient has been calculated for step

*m*, the geometry for the next step

**p**^{m+1}can be determined:

*δ*was used,

26. C. C. Su. “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” Microwave. Theory and Technol. , **33**, 1114–9, (1985). [CrossRef]

8. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. “Boundary element method for analysis of holey optical fibers,” J. Lightwave. Technol. **21**, 1787–92 (2003). [CrossRef]

23. M. J. Steel and R. M. Osgood. “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. **19**, 495–503 (2001). [CrossRef]

*δ*= .06 limits the rotations per step to Σ

_{j}Δ

^{2}).

*x*- and

*y*-aligned initial geometries result in equivalent final optimized structures, although the

*y*-aligned is rotated 120 degrees and has not completely converged after 50 iterations. As one might expect, the optimization arrives at a structure with

*C*

_{2v}symmetry (reflection on two orthogonal axes).

29. 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**, 681–6, (2001). [CrossRef] [PubMed]

30. Alexander Argyros, Ian M. Bassett, Martijn A. van Eijkelenborg, M.C.J. Large, Joseph Zagari, N.A.P. Nicorovici, Ross C. McPhedran, and C.Martijn de Sterke. “Ring structures in microstructured polymer optical fibres,” Opt. Express **9**, 813–20, (2001). [CrossRef] [PubMed]

## 5. Conclusions

## References and links

1. | P. Kaiser and H. W. Astle. “Low-loss single-matrial fibres made from pure fused silica,” Bell Syst. Tech. J. |

2. | S. A. Diddams and D. J. Jones, et al. “Direct link between microwave and optical frequencies with a 300THz femtosecond laser comb,” Phys. Rev. Lett. , |

3. | S. G. Johnson and M. Ibanescu, et al. “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express |

4. | John Fini and Ryan Bise. “Progress in fabrication and modeling of microstructured optical fiber,” Jap. J. App. Phys. |

5. | A. Ferrando and E. Silvestre, et al. “Full vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

6. | T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. “Calculations of air-guiding modes in photonic crystal fibers using the multipole method,” Opt. Express, |

7. | F. Brechet and J. Marcou, et al. “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Tech. |

8. | N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. “Boundary element method for analysis of holey optical fibers,” J. Lightwave. Technol. |

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

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

11. | T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya. “Bend-insensitive single-mode holey fibre with SMF compatibility for optical wiring applications,” In European Conference on Optical Communications, paper We2.7.3, (2003). |

12. | James A. West, Charlene M. Smith, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. “Surface modes in air-core photonic band-gap fibers,” Opt. Express, |

13. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli. “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. |

14. | A. Peyrilloux, T. Chartier, L. Berthelot, A. Hideur, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy. “Thoeretical and experimental study of the birefringence of a photonic crystal fiber,” J. Lightwave Technol. |

15. | I. K. Hwang, Y. J. Lee, and Y. H. Lee. “Birefringence induced by irregular structure in photonic crystal fiber,” Opt. Express |

16. | Steven G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink. “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E |

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

18. | J. M. Fini. “Improved symmetry analysis of many-moded microstructure optical fibers,” J. Opt. Soc. Am. B, |

19. | J. M. Fini. “Perturbative modeling of irregularities in microstructure optical fibers,” In |

20. | M. Koshiba and K. Saitoh. “Polarization-dependent confinement losses in actual holey fibers,” Photon. Technol. Lett. |

21. | Paul R. McIsaac. “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” Microwave Theory and Techniques |

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

23. | M. J. Steel and R. M. Osgood. “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. |

24. | Charlene M. Smith, Natesan Venkataraman, Michael T. Gallagher, Dirk Müller, James A. West, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. “Low loss hollow-core silica/air photonic bandgap fibre,” Nature, |

25. | William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. |

26. | C. C. Su. “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” Microwave. Theory and Technol. , |

27. | R. Bise and R. S. Windeler, et al. “Tunable photonic band gap fiber,” In |

28. | R. Bise and D. Trevor. “Sol-gel-derived microstructured fibers: fabrication and characterization,” To appear in |

29. | 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 |

30. | Alexander Argyros, Ian M. Bassett, Martijn A. van Eijkelenborg, M.C.J. Large, Joseph Zagari, N.A.P. Nicorovici, Ross C. McPhedran, and C.Martijn de Sterke. “Ring structures in microstructured polymer optical fibres,” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 25, 2004

Revised Manuscript: September 10, 2004

Published: September 20, 2004

**Citation**

John Fini, "Perturbative numerical modeling of microstructure fibers," Opt. Express **12**, 4535-4545 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4535

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### References

- P. Kaiser and H. W. Astle. �??Low-loss single-matrial fibres made from pure fused silica,�?? Bell Syst. Tech. J. 53, 1021�??39 (1974).
- S. A. Diddams, D. J. Jones, et al. �??Direct link between microwave and optical frequencies with a 300THz femtosecond laser comb,�?? Phys. Rev. Lett., 84, 5102-5 (2000). [CrossRef] [PubMed]
- S. G. Johnson, M. Ibanescu, et al. �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748-79 (2001). [CrossRef] [PubMed]
- John Fini and Ryan Bise. �??Progress in fabrication and modeling of microstructured optical fiber,�?? Jap. J. App. Phys. 43, 5717�??5730 (2004). [CrossRef]
- A. Ferrando, E. Silvestre, et al. �??Full vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276-8 (1999). [CrossRef]
- T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. �??Calculations of air-guiding modes in photonic crystal fibers using the multipole method,�?? Opt. Express, 9, 721-32 (2001). [CrossRef]
- F. Brechet, J. Marcou, et al. �??Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,�?? Opt. Fiber Tech. 6, 181-191 (2000). [CrossRef]
- N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. �??Boundary element method for analysis of holey optical fibers,�?? J. Lightwave. Technol. 21, 1787-92 (2003). [CrossRef]
- 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). software available at <a href=" http://ab-initio.mit.edu/mpb">http://ab-initio.mit.edu/mpb.</a> [CrossRef] [PubMed]
- T. A. Birks, J. C. Knight, and P. S. J. Russell. �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-3 (1997). [CrossRef] [PubMed]
- T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya. �??Bend-insensitive single-mode holey fibre with SMF compatibility for optical wiring applications,�?? In European Conference on Optical Communications, paper We2.7.3, (2003).
- James A. West, Charlene M. Smith, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. �??Surface modes in air-core photonic band-gap fibers,�?? Opt. Express, 12, 1485-96 (2004). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli. �??Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,�?? J. Lightwave Technol. 20, 1433-42 (2002). [CrossRef]
- A. Peyrilloux, T. Chartier, L. Berthelot A. Hideur, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy. �??Thoeretical and experimental study of the birefringence of a photonic crystal fiber,�?? J. Lightwave Technol. 21, 536-9(2003). [CrossRef]
- I. K. Hwang, Y. J. Lee, and Y. H. Lee. �??Birefringence induced by irregular structure in photonic crystal fiber,�?? Opt. Express 11, 2799-2806 (2003). [CrossRef] [PubMed]
- Steven G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink. �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 066611 (2002). [CrossRef]
- M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten. �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-91 (2001). [CrossRef]
- J. M. Fini. �??Improved symmetry analysis of many-moded microstructure optical fibers,�?? J. Opt. Soc. Am. B, 21, 1431-6 (2004). [CrossRef]
- J. M. Fini. �??Perturbative modeling of irregularities in microstructure optical fibers,�?? In Conference on Lasers and Electro-Optics (CLEO), TOPS vol. 96, paper CThX6, (Optical Society of America, Washington, D.C., 2004).
- M. Koshiba and K. Saitoh. �??Polarization-dependent confinement losses in actual holey fibers,�?? Photon. Technol. Lett. 15, 691-3 (2003). [CrossRef]
- Paul R. McIsaac. �??Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,�?? Microwave Theory and Techniques 23, 421-9 (1975). [CrossRef]
- M. J. Steel and R. M. Osgood. �??Elliptical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229-31 (2001). [CrossRef]
- M. J. Steel and R. M. Osgood. �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495-503 (2001). [CrossRef]
- Charlene M. Smith, Natesan Venkataraman, Michael T. Gallagher, Dirk Müller, James A. West, Nicholas F. Borrelli, Douglas C. Allan, and KarlW. Koch. �??Low loss hollow-core silica/air photonic bandgap fiber,�?? Nature, 424 657-9, (2003). [CrossRef]
- William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes in C, the art of scientific computing. (Cambridge University Press, New York, 1992).
- C. C. Su. �??A surface integral equations method for homogeneous optical fibres and coupled image lines of arbitrary cross sections,�?? Microwave. Theory and Technol., 33, 1114-9, (1985). [CrossRef]
- R. Bise, R. S. Windeler, et al. �??Tunable photonic band gap fiber,�?? In Optical Fiber Communications Conference (OFC), TOPS vol. 70, paper ThK3, (Optical Society of America, Washington, D.C., 2002).
- R. Bise and D. Trevor. �??Sol-gel-derived microstructured fibers: fabrication and characterization,�?? To appear in Optical Fiber Communications Conference (OFC), (Optical Society of America, Washington, D.C., 2005).
- 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, 681-6, (2001). [CrossRef] [PubMed]
- Alexander Argyros, Ian M. Bassett, Martijn A. van Eijkelenborg, M.C.J. Large, Joseph Zagari, N.A.P. Nicorovici, Ross C. McPhedran, and C. Martijn de Sterke. �??Ring structures in microstructured polymer optical fibres,�?? Opt. Express 9, 813-20, (2001). [CrossRef] [PubMed]

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