## A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers

Optics Express, Vol. 14, Issue 7, pp. 2979-2993 (2006)

http://dx.doi.org/10.1364/OE.14.002979

Acrobat PDF (1547 KB)

### Abstract

We describe a finite-difference numerical method that allows us to simulate the modes of air-core photonic-bandgap fibers (PBF) of any geometry in minutes on a standard PC. The modes’ effective indices and fields are found by solving a vectorial transverse magnetic-field equation in a matrix form, which can be done quickly because this matrix is sparse and because we reduce its bandwidth by rearranging its elements. The Stanford Photonic-Bandgap Fiber code, which is based on this method, takes about 4 minutes to model 20 modes of a typical PBF on a PC. Other advantage; include easy coding, faithful modeling of the abrupt discontinuities in the index profile, high accuracy, and applicability to waveguides of arbitrarily complex profile.

© 2006 Optical Society of America

## 1. Introduction

1. X. Yong, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. **25**, 1756–815 (2000). [CrossRef]

7. B. Kuhlmey, G. Renversez, and D. Maystre, “Chromatic dispersion and losses of microstructured optical fibers,” Appl. Opt. **42**, 634–639 (2003). [CrossRef] [PubMed]

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]

10. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE **5000**, 161–174, (2003). [CrossRef]

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

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

19. MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.

*et al*using a multipole decomposition method, with which they modeled all modes in a hexagonal PBF with four rings of holes at one wavelength in only about one hour on a 733-MHz personal computer. [20

20. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. Martijn de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express **8**, 721–732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721. [CrossRef]

*et al*demonstrated a polar-coordinate decomposition method that modeled the fundamental mode (effective index and fields) in 6 minutes.[21

21. L. Poladian, N. A. Issa, and T. M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express **10**, 449–454 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449. [PubMed]

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

23. C. Yu and H. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. **36**, 145–163 (2004). [CrossRef]

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

23. C. Yu and H. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. **36**, 145–163 (2004). [CrossRef]

## 2. State of the art

**r**is a vector (

*x, y*) that represents the coordinates of a particular point in a plane perpendicular to the fiber axis,

*ε*(

**r**) =

*n*

^{2}(

**r**) is the dielectric permeability of the fiber cross-section at this point, where

*n*(

**r**) is the two-dimensional refractive index profile of the fiber,

**H**(

**r**) is the mode’s magnetic field vector,

*ω*is the optical frequency, and

*c*the speed of light in vacuum. This formulation is most useful for 3D photonic-bandgap structures, and it is solved by assuming a constant value for the propagation constant

*k*

_{z}and computing the eigenfrequencies

*ω*for this value. Six basic methods have been used to solve for modes in the context of air-core PBFs. The first one is the MIT Photonic Bandgap software (MPB). [18

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

19. MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.

*ε*(

*) in spatial harmonics. Equation (1) is then written in a matrix form and solved by finding the eigenvalues of this matrix. This is done by setting*

**r***k*

_{z}to some value

*k*

_{0}and solving for the frequencies at which a mode with this wave-number occurs. The second method is the beam-propagation method (BPM). [25

25. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum. Electron. **38**, 927–933 (2002). [CrossRef]

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

24. W. Zhi, R. Guobin, 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]

27. J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, “Robust photonic band gaps for hollow core guidance in PCF made from high index glass,” Opt. Express **11**, 2854–2861 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854. [CrossRef] [PubMed]

20. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. Martijn de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express **8**, 721–732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721. [CrossRef]

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

23. C. Yu and H. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. **36**, 145–163 (2004). [CrossRef]

## 3. The SPBF method

*n*(

*x, y*) that is translation invariant along the fiber longitudinal axis (

*z*). The transverse magnetic field

**h**

_{T}(

**r**) of the waveguide modes satisfies the eigenvalue equation: [28]

*λ*

_{0}is the wavelength),

**h**

_{T}=

*h*

_{x}

*(x, y)*

**ux**+

*h*

_{y}

*(x, y)*

**uy**is the transverse magnetic field,

*h*

_{x}and

*h*

_{y}are the components of the transverse magnetic field projected onto the unit vectors

**u**

_{x}and

**u**

_{y},

*n*

_{eff}is the mode effective index. With this notation, the total magnetic field vector

**H**defined earlier is simply

**h**

_{T}+

*h*

_{z}

**u**

_{z}, where

**u**

_{z}is the unit vector along the fiber

*z*-axis. Equation (3) is satisfied by all modes (TE, TM, and hybrid modes). The longitudinal component of the magnetic field and the components of the electric field can all be calculated from the transverse magnetic field using:

*h*

_{x}(

*x, y*) and

*h*

_{y}(

*x, y*) of the transverse magnetic field, and thus they must be solved simultaneously. It is important to note that the form of Eq. (3) is such that it can be solved it by fixing the wavelength and calculating all the fiber modes at this wavelength.

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

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

*h*

_{x}(

*x, y*) and

*h*

_{y}(

*x, y*) in a plane cross-section perpendicular to the fiber axis on a grid of

*m*

_{x}points in the

*x*direction and

*m*

_{y}points in the

*y*direction. This operation yields a new magnetic field vector

*h*(

*x, y*) of dimension 1 × 2

*m*

_{x}

*m*

_{y}, which contains (arranged using some arbitrary numbering system)

*S*=

*m*

_{x}

*m*

_{y}samples of

*h*

_{x}(

*x, y*) and

*S*samples of

*h*

_{y}(

*x, y*). This sampling is done over an area that respects the crystal periodicity, i.e., over a minimum cell that is a fundamental component of the cladding’s photonic-crystal structure. Similarly, the linear operator Π is sampled using finite-difference equations (which entails, in particular, sampling the refractive index profile

*n*(

*x, y*) of the fiber), so that it is now represented by a matrix

**M**of dimension 2

*S*× 2

*S*. Equation (5) is solved subject to standard boundary conditions. Unlike with a

*k*-domain method, which relies on the Bloch theorem and thus assumes periodic boundary conditions, with this new method one of three types of boundary conditions can be used: (1) fields vanishing outside the simulation area, (2) fields periodically repeated over the entire space, with the simulation domain as a unit super cell, or (3) surrounding the simulation domain by a perfectly matched layer (for propagation loss calculation). Both the boundary conditions and the simulation domain boundary are taken into account when sampling the operatorΠ, so that this information is also contained in the matrix

**M**itself. Equation (4) is thus replaced by a single matrix equation:

*n*

_{eff}) and eigenmodes (field distributions) of the fiber modes, is a typical matrix eigenvalue problem. However, the operator matrix

**M**is typically extremely large. For example, consider the case of a photonic-bandgap fiber with a triangular pattern of circular air holes of radius

*ρ*= 0.47Λ, and a circular core of radius

*R*= 0.8Λ, 4 rows of air holes in the cladding, and a square cladding boundary. The cell is then sixteen periods on the side, i.e., its dimension is 10Λ × 10Λ. To resolve the fiber’s thin membranes and achieve sufficient accuracy, we might want to use a grid step size (or resolution) of Λ/50. The number of sampling points is then

*m*

_{x}= 10 × 50 in

*x*and

*m*

_{y}= 10 × 50 in

*y*. Thus

*S*=

*m*

_{x}

*m*

_{y}= 250,000, and

**M**will contain (2

*S*)

^{2}= 250 billion elements. Finding the eigenvalues of matrices this large is out of the range of most computers, as even storing them is problematic. However, the replacement of the differential operators by their finite-difference equivalents makes

**M**a sparse matrix. This is illustrated in Fig. 1, which shows the general form of the matrix representing the mode equation for such an example. The horizontal and vertical axes represent the coordinate of the elements in the matrix, which each go up to a maximum value of 2

*S*= 516,000. The solid dots, which merge into solid curves because their density is too high to be individually resolved on this diagram, represent nonzero elements. The blanks between them are all zero elements. In this particular example, the total number of non-zero elements

*n*

_{z}is 2,796,826.

*D*of this kind of matrix, defined as the ratio of non-zero elements to the total number of elements is

*D*≈ 1.05 × 10

^{-5}, or equivalently only about 0.001% of the matrix elements are non-zero. The implication is that in spite of the large size of the matrix, the number of elements that must be stored during computation is comparatively low, down from ~266 billion to ~2.8 million for this typical example, which is well within the range of what can be handled with standard personal computers.

**M**is its bandwidth

*B*

_{M}, defined as the maximum distance between a non-zero coefficient and the first diagonal.

**M**is proportional to S and quite large, as illustrated in Fig. 1, it is desirable to lower it. By applying a proper permutation [30] on the elements of both

**h**and

**M**, it is possible to rearrange

**M**so that all the non-zero elements are clustered near the first diagonal and the bandwidth is minimized. This process yields a new sparse matrix

**N**and a new permutated magnetic field vector

**g**, which also satisfy Eq. (6), i.e.,

**N g**=

**N**is now proportional to

*S*

^{1/2}and greatly reduced from the original value. This is illustrated in Fig. 2, which shows the matrix

**N**obtained by applying this process to the original matrix

**M**of Fig. 1. Most of the non-zero elements are now close to the diagonal. The bandwidth has been reduced from 460,320 (Fig. 1) to only 1,922 (Fig. 2), i.e., by a factor of ~240, which speeds up the calculation of the eigenvalues of the matrix by about an order of magnitude.

**N**to find all of its eigenvalues. However, this approach would require calculating and storing a very large number of eigenvalues and eigenvectors, which would take a great deal of time and memory. Furthermore, this method is generally wasteful because it calculates all of the eigenvalues. Instead, a more expedient way to solve Eq. (6) is to calculate only the few eigenvalues of interest, as done for example in Ref [24

24. W. Zhi, R. Guobin, 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]

*m*eigenvalues of

**N**that are closest to a given effective index value

*n*

_{0}, where

*m*is a small number. For example, to model the bandgap edges and effective indices of the modes within the bandgap of a particular fiber at a particular wavelength, we might be interested in calculating the

*m*

_{0}= 10 modes that are closest to

*n*

_{0}= 1, since the core modes of an air-core fiber have effective indices close to 1. The next issue is that calculating the eigenvalues of a large matrix that are closest to a given value is still a difficult task. To overcome this problem, as in Ref. [24

24. W. Zhi, R. Guobin, 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]

*m*

_{0}eigenvalues of a matrix

**N**that are closest to some value

*n*

_{0}are the same as the largest (in amplitude) eigenvalues of the matrix:[24

**11**, 980–991 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

**I**is the identity matrix. This property is useful here because the eigenvalues of maximum amplitude of a given matrix can be calculated very quickly by using what is known as the Courant-Fischer theorem.[31] This theorem restricts its eigenvalue search to small blocks of dimension

*m*

_{0}within the matrix, which is much faster than identifying all the eigenvalues. This approach is applied by first calculating a new matrix

**N**-

**I**, then inverting it using a standard LU decomposition (the matrix is expressed as the product of a lower and an upper triangular matrices) to obtain matrix

**A**, which is fast because

**N**-

**I**is also a sparse matrix with a minimum bandwidth, and finally making use of the Courant-Fischer algorithm to calculate only the

*m*

_{0}largest eigenvalues of

**A**, which are the effective indices of interest. We have therefore transformed our problem from looking for a specific set of eigenvalues around a given value, which is a difficult problem, to looking for the largest eigenvalues of a transformed matrix, which is considerably faster. [24

**11**, 980–991 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

**M**representing the action of Maxwell’s equations on the transverse magnetic field. This requires using the sampled refractive index profile distribution of the fiber as well as the wavelength and the boundary conditions. In the next step,

**M**is rearranged to yield a new sparse matrix

**N**with a minimum bandwidth to speed up the extraction of the eigenvalues. The third step is to specify the effective index

*n*

_{0}around which we want to calculate the PBF modes. We then translate the spectrum of

**N**by calculating

*N*-

*I*. The fourth step is to invert this matrix through LU decomposition. The final step is to compute the largest eigenvalues and eigenvectors of this matrix using the Courant-Fischer theorem.

## 4. Features and performance of the SPBF code

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

19. MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.

## 5. Simulation of air-core and holey fibers

*ρ*= 0.47Λ and the radius of the circular air core is

*R*= 0.8Λ. The gray areas represent air and the black areas silica. The modeling was done using periodic boundary conditions: the simulation domain (super cell) represented in Fig. 4 was repeated periodically to tile the entire space and the modes of this periodic waveguide were calculated. The separation distance between two fiber cores in this periodic tiling was selected large enough (10Λ) to ensure negligible interaction between neighboring fiber cores and thus accurate predictions. This means that the simulation domain is a rectangle 10Λ wide (a little under 5 rings), a size commonly used in simulations [14

14. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. **40**, 551–556 (2004). [CrossRef]

*R*= 1.0Λ, is shown in Fig. 7. The bandgap extends approximately from λ = 0.57Λ to λ = 0.645Λ. This bandgap supports two fundamental core modes, and no surface modes, a result consistent with the predictions of [13

13. M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express **12**, 1864–1872, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]

14. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. **40**, 551–556 (2004). [CrossRef]

*n*

_{eff}-

*n*

_{eff}|, where the reference value

*n*

_{ref}is the effective index calculated by SPBF with 600 grid points per side. The code reaches convergence to the fourth decimal place for a step size of Λ/42 (~420 points per side), and fifth decimal place for a step size of Λ/50 (~500 points per side).

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

^{-7}for grid sizes exceeding 450 points. This result confirms both the accuracy and the convergence of the SPBF code. It should be noted that the SPBF code achieves a higher accuracy (by two orders of magnitude) for each of these two fibers than for the PBF of Fig. 4 because they have a higher mode confinement, which enabled us to reduce the simulation window size, and thus the step size, which in turn improves the code’s accuracy.

*R*= 1.15Λ. This fiber exhibits surface modes, which is consistent with previously published results. [13

13. M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express **12**, 1864–1872, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]

14. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. **40**, 551–556 (2004). [CrossRef]

10. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE **5000**, 161–174, (2003). [CrossRef]

*n*

_{0}= 1 or 0.99, until a core mode is found. This first step is necessary since we do not know the effective index of core modes prior to simulation. Once the core mode effective index has been determined at a given wavelength, this effective index, or an effective index calculated from it by linear extrapolation, can be used as the target effective index at the next wavelength. The number of modes solved at this wavelength can thus be lowered, to typically less than 20, since in general only the core modes, surface modes, and a few bulk modes at the edges of the bandgap need to be determined. This procedure reduces the computation time by tracking a particular mode of interest, and it centers the calculated modes on the core modes.

## 6. Conclusions

## Acknowledgments

## References and links

1. | X. Yong, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. |

2. | P.V. Kaiser and H. W. Astle, “Low-loss single material fibers made from pure fused silica,” Bell Syst. Tech. J. |

3. | J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber 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. | J. C. Knight, T. A. Birks, R .F. Cregan, P. St. J. Russell, and J. P. Sandro, “Photonic crystals as optical fibers-physics and applications,” Opt. Mater. |

6. | R. S. Windeler, J. L Wagener, and D. J. Giovanni, “Silica-air microstructured fibers: Properties and applications,” Optical Fiber Communications conference, San Diego (1999). |

7. | B. Kuhlmey, G. Renversez, and D. Maystre, “Chromatic dispersion and losses of microstructured optical fibers,” Appl. Opt. |

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

9. | G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. |

10. | D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE |

11. | K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express |

12. | J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express |

13. | M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express |

14. | H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. |

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

16. | D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express |

17. | K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express |

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

19. | MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/. |

20. | T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. Martijn de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express |

21. | L. Poladian, N. A. Issa, and T. M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express |

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

23. | C. Yu and H. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. |

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

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

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

27. | J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, “Robust photonic band gaps for hollow core guidance in PCF made from high index glass,” Opt. Express |

28. | A. W. Snyder and J. D. Love, |

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

30. | A. George and J. Liu, |

31. | R. A. Horn and C. R. Johnson, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2310) Fiber optics and optical communications : Fiber optics

(230.7370) Optical devices : Waveguides

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: December 19, 2005

Revised Manuscript: March 16, 2006

Manuscript Accepted: March 17, 2006

Published: April 3, 2006

**Citation**

Vinayak Dangui, Michel J. F. Digonnet, and Gordon S. Kino, "A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers," Opt. Express **14**, 2979-2993 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2979

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### References

- X. Yong, R. K. Lee, and A. Yariv, "Asymptotic analysis of Bragg fibers," Opt. Lett. 25, 1756-815 (2000). [CrossRef]
- P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).
- J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999). [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]
- J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999). [CrossRef]
- R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications conference, San Diego (1999).
- B. Kuhlmey, G. Renversez, and D. Maystre, "Chromatic dispersion and losses of microstructured optical fibers," Appl. Opt. 42, 634-639 (2003). [CrossRef] [PubMed]
- 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]
- G. Renversez, B. Kuhlmey, and R. McPhedran, "Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses," Opt. Lett. 28, 989-991 (2003). [CrossRef] [PubMed]
- D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, "Surface modes and loss in air-core photonic band-gap fibers," in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE 5000, 161-174, (2003). [CrossRef]
- K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12, 394-400 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]
- J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485. [CrossRef] [PubMed]
- M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers," Opt. Express 12, 1864-1872, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]
- H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551-556 (2004). [CrossRef]
- T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt. Lett. 26, 1660-1662 (2001). [CrossRef]
- D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express 10, 1285-1290 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314.
- K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100. [CrossRef] [PubMed]
- S. G. Johnson, and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in plane wave basis," Opt. Express 8, 173-190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]
- MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.
- T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721. [CrossRef]
- L. Poladian, N. A. Issa, and T. M. Monro, "Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry," Opt. Express 10, 449-454 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449. [PubMed]
- Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853. [PubMed]
- C. Yu and H. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004). [CrossRef]
- W. Zhi, R. Guobin, 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]
- K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum. Electron. 38, 927-933 (2002). [CrossRef]
- M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-30 (2001). [CrossRef]
- J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, "Robust photonic band gaps for hollow core guidance in PCF made from high index glass," Opt. Express 11, 2854-2861 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854. [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983), Chap. 29.
- N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997). [CrossRef]
- A. George and J. Liu, Computer Solution of Large Sparse Positive Definite Systems, (Prentice-Hall, 1981).
- R. A. Horn and C. R. Johnson, Matrix Analysis, (Cambridge University Press, 1990), Chap. 4.

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