## A vector boundary matching technique for efficient and accurate determination of photonic bandgaps in photonic bandgap fibers |

Optics Express, Vol. 19, Issue 13, pp. 12582-12593 (2011)

http://dx.doi.org/10.1364/OE.19.012582

Acrobat PDF (1211 KB)

### Abstract

A vector boundary matching technique has been proposed and demonstrated for finding photonic bandgaps in photonic bandgap fibers with circular nodes. Much improved accuracy, comparing to earlier works, comes mostly from using more accurate cell boundaries for each mode at the upper and lower edges of the band of modes. It is recognized that the unit cell boundary used for finding each mode at band edges of the 2D cladding lattice is not only dependent on whether it is a mode at upper or lower band edge, but also on the azimuthal mode number and lattice arrangements. Unit cell boundaries for these modes are determined by mode symmetries which are governed by the azimuthal mode number as well as lattice arrangement due to mostly geometrical constrains. Unit cell boundaries are determined for modes at both upper and lower edges of bands of modes dominated by m = 1 and m = 2 terms in their longitudinal field Fourier-Bessel expansion series, equivalent to LP0s and LP1s modes in the approximate LP mode representations, for hexagonal lattice to illustrate the technique. The novel technique is also implemented in vector form and incorporates a transfer matrix algorithm for the consideration of nodes with arbitrary refractive index profiles. Both are desired new capabilities for further explorations of advanced new designs of photonic bandgap fibers.

© 2011 OSA

## 1. Introduction

1. C. B. Olausson, C. I. Falk, J. K. Lyngsø, B. B. Jensen, K. T. Therkildsen, J. W. Thomsen, K. P. Hansen, A. Bjarklev, and J. Broeng, “Amplification and ASE suppression in a polarization-maintaining ytterbium-doped all-solid photonic bandgap fibre,” Opt. Express **16**(18), 13657–13662 (2008). [CrossRef] [PubMed]

2. A. Shirakawa, C. B. Olausson, H. Maruyama, K. Ueda, J. K. Lyngsø, and J. Broeng, “High power ytterbium fiber lasers at extremely long wavelengths by photonic bandgap fiber technology,” Opt. Fiber Technol. **16**(6), 449–457 (2010). [CrossRef]

2. A. Shirakawa, C. B. Olausson, H. Maruyama, K. Ueda, J. K. Lyngsø, and J. Broeng, “High power ytterbium fiber lasers at extremely long wavelengths by photonic bandgap fiber technology,” Opt. Fiber Technol. **16**(6), 449–457 (2010). [CrossRef]

3. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. **29**(20), 2369–2371 (2004). [CrossRef] [PubMed]

6. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express **13**(21), 8452–8459 (2005). [CrossRef] [PubMed]

7. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express **14**(13), 6291–6296 (2006). [CrossRef] [PubMed]

8. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B **71**(19), 195108 (2005). [CrossRef]

7. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express **14**(13), 6291–6296 (2006). [CrossRef] [PubMed]

9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers, I. formulation,” J. Opt. Soc. Am. B **19**(10), 2322–2330 (2002). [CrossRef]

10. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers, II. implementation and results,” J. Opt. Soc. Am. B **19**(10), 2331–2340 (2002). [CrossRef]

9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers, I. formulation,” J. Opt. Soc. Am. B **19**(10), 2322–2330 (2002). [CrossRef]

10. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers, II. implementation and results,” J. Opt. Soc. Am. B **19**(10), 2331–2340 (2002). [CrossRef]

11. T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express **9**(13), 721–732 (2001). [CrossRef] [PubMed]

## 2. Fields around a circular node with a complex refractive index profile

9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers, I. formulation,” J. Opt. Soc. Am. B **19**(10), 2322–2330 (2002). [CrossRef]

_{m}is the Bessel function of the first kind; H

^{1}

_{m}is the Hankel function of the first kind, r is radial coordination centered at the node center in the transverse plane; and θ is the corresponding azimuthal coordination. The first term of each field representation in Eq. (1) and Eq. (2) can be viewed as incoming wave originated from outside of the node and the second term as outgoing scattered wave from the node as described in [9]. We also havewhere k is vacuum wave vector; n

_{p}is refractive index of layer p; and β is the propagation constant of the mode. It is worth noting that we have allowed β>kn

_{p}in Eq. (3), resulting in imaginary k

_{p}r for J

_{m}and H

^{1}

_{m}in Eq. (1) and (2) in this case. The fields in Eq. (1) and (2) are still valid in this case considering the transformation K

_{m}(-iz) = (π/2)i

^{m + 1}H

^{1}

_{m}(z) and I

_{m}(-iz) = i

^{˗m}J

_{m}(z) where I

_{m}and K

_{m}are modified Bessel function of the first and second kind respectively. H in Eq. (2) is a normalized magnetic field component for easier representations. It is connected to regular magnetic field h by

_{m}= D

_{m}= 0 for p = 0. It can now be understood that all the four field coefficients for the outmost region, i.e. a

_{m}, b

_{m}, C

_{m}and D

_{m}for p = N, can be determined by the only two non-zero components a

_{m}and C

_{m}of the innermost circle for each m. Since our following analysis is going to mostly involve the field in the outmost region, we will write them down.

## 3. Vector boundary matching technique for the first band of modes

_{z}and h

_{z}fields. Unlike in the scalar mode approximations, other m terms are non-zero, but smaller in comparison. The z components are dominated by two anti-phase symmetric lobes over the node (see Fig. 2 and Fig. 3 ). This gives a transverse field over the node resembles the quasi-Gaussian mode with which we are familiar. The orientations of each z field components need to be such so that the transverse fields are in-phase between adjacent nodes for modes determining the upper edge of this band of modes. This can be easily achieved for this mode as shown in Fig. 2.

_{z}, h

_{z}and transverse electric field e

_{t}are given in Fig. 2 for a hexagonal lattice with d/Λ = 0.19 where d is node diameter and Λ is node spacing. The node normalized frequency V is 0.9 in this case, where normalized frequency V is defined as for conventional fibers. In this case, this upper edge mode has an identical distribution over each node. The hexagonal boundary of each unit cell is apparent (see the white cell boundary in Fig. 2). Each corner of this unit cell is located at the center of the equilateral triangle formed by the three adjacent nodes. The transverse field is in-phase between any adjacent nodes. The hexagonal boundary of the center unit cell is illustrated in the e

_{t}plot along with radial lines from center of coordination of this cell. In this case, there are 24 radial lines equally spaced by π/12 degree angle. At the hexagonal cell boundary, we should have ∂\e

_{t}\/∂r = 0 and ∂\h

_{t}\/∂r = 0, due to the symmetry of this mode. This allows us to write a total of 48 boundary equations in this case, two at each intersection point for each radial line with the cell boundary. Bearing in mind that the field parameters around the node can be derived from a

_{m}and C

_{m}at the center of the node for each m from Eq. (6). We can assemble the following cell boundary matching equations.

_{t}\/∂r = 0 and ∂\h

_{t}\/∂r = 0 respectively at each intersection points, making a total 48 rows. We used ∂\e

_{x}\/∂r = 0 and ∂\h

_{x}\/∂r = 0 in our implementations. This is adequate to constrain all the field coefficients, as e

_{y}and h

_{y}are similarly constrained. The m is chosen from –m

_{max}to m

_{max}, making a total of 2(2m

_{max}+ 1) variables. It is worth noting that we are looking for modes with a minimum, i.e. trench, along the cell boundary. ∂\e

_{x}\/∂r = 0 and ∂\h

_{x}\/∂r = 0 are valid even though most of the radial lines are not crossing the cell boundary at a right angle.

_{max}= 12 in our case, making a total of 50 variables in this case, giving M a dimension of 48 × 50. Typically, if a band from the equivalent scalar LP

_{rs}mode needs to be resolved, m

_{max}needs to be chosen so that it is slightly larger than r + 1 to allow accurate resolution of the desired band. The number of radial lines can be chosen accordingly to allow efficient computation as well as accurate resolution of all desired bands. It is not necessary to have equally spaced radial lines. It is also worth noting that the boundary conditions in Eq. (12) lead to coupling among various m terms. This allows the field distributions arising from summation of all m-terms in Eq. (8) and Eq. (9) to produce the desired non-circular field distributions. Search for singular value of M provides solutions for mode propagation constant β from eigenvalue and the associated field coefficients from the corresponding eigenvalue vector, which allow calculation of all fields around a node.

_{t}= 0 and h

_{t}= 0 on the triangular unit cell boundary of the transverse electric field. We again used 24 radial lines in our case with m

_{max}= 12 to set up a matrix M with a dimension of 48 × 50 to find this band edge.

## 4. Determination of higher order bandgaps

_{rs}, and lattice symmetry. The edge modes of the bands of higher order modes which are dominated by m = 1 term, equivalent to the scalar LP

_{0s}modes, are, therefore, expected to have similar symmetry as the edge modes of the first band for the same lattice which was studied in the last section for a hexagonal lattice. From previous works [12

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**(20), 9483–9490 (2006). [CrossRef] [PubMed]

_{0s}and LP

_{1s}modes in a typical photonic bandgap fibers, equivalent to modes dominated by m = 1 and m = 2 terms respectively. Modes with higher azimuthal mode number, i.e. s>1 (equivalent to field being dominated by terms with m>2), tend to be more confined to the nodes. Related bands of these modes tend to be narrow and do not contribute as much to the bandgaps.

_{z}, h

_{z}and e

_{t}of a mode on the upper edge of the second band of modes are given in Fig. 4 for a hexagonal lattice. This mode symmetry is complicated by the geometry constrains of the fields dominated by m = 2 terms. The resulting transverse fields have alternating field distribution from one node to an adjacent node. This makes precise determination of the boundary cell very difficult for this case. We resort to use a similar hexagonal unit cell as that for the upper edge of the first band of modes in this case. The hexagonal unit cell is illustrated in Fig. 4. On this unit cell boundary, we have ∂\e

_{t}\/∂r = 0 and ∂\h

_{t}\/∂r = 0. A total number of 24 equally spaced radial lines (not illustrated) similar to that in last section is used to generate 48 boundary equations in this study.

_{z}, h

_{z}and e

_{t}of a mode on the lower edge of the second band of modes are given in Fig. 5 for a hexagonal lattice of nodes. In this case, the field symmetry is straight forward, resulting in a hexagonal unit cell (see Fig. 5). Transverse fields should be zero over the boundary of this unit cell due to the anti-phase symmetries. A total number of 24 equally spaced radial lines (not illustrated) similar to that in last section is used to generate 48 boundary equations in this study. The mode symmetry of the analysis in this section equally applied to all other modes dominated by m = 2 terms in the field summations as it has been pointed out earlier in the discussion of modes dominated by m = 1 terms.

## 5. Impact of unit cell boundaries of the edge modes

## 6. Full bandgap simulations

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**(20), 9483–9490 (2006). [CrossRef] [PubMed]

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express **14**(20), 9483–9490 (2006). [CrossRef] [PubMed]

**14**(20), 9483–9490 (2006). [CrossRef] [PubMed]

**14**(20), 9483–9490 (2006). [CrossRef] [PubMed]

## 7. Bandgap of lattice with nodes of graded index

## 8. Summary

## 9. Appendix

_{z}, H

_{z}, e

_{θ}and H

_{θ}are continuous at each boundary p, resulting in four equations for adjacent layers p and p + 1 for each m. The transfer matrix can then be obtained. The following matrix elements are for a specific m; but subscript m is omitted for simplicity. where

_{θ}and H

_{θ}for p = N describe the azimuthal fields around the node and can be derived similarly to these in Eq. (14). Radial field components e

_{r}and H

_{r}for p = N can be obtained through the following equations.

## References and links

1. | C. B. Olausson, C. I. Falk, J. K. Lyngsø, B. B. Jensen, K. T. Therkildsen, J. W. Thomsen, K. P. Hansen, A. Bjarklev, and J. Broeng, “Amplification and ASE suppression in a polarization-maintaining ytterbium-doped all-solid photonic bandgap fibre,” Opt. Express |

2. | A. Shirakawa, C. B. Olausson, H. Maruyama, K. Ueda, J. K. Lyngsø, and J. Broeng, “High power ytterbium fiber lasers at extremely long wavelengths by photonic bandgap fiber technology,” Opt. Fiber Technol. |

3. | F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. |

4. | A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express |

5. | A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express |

6. | G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express |

7. | J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express |

8. | G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B |

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

10. | B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers, II. implementation and results,” J. Opt. Soc. Am. B |

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

12. | T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express |

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

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

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

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 18, 2011

Revised Manuscript: May 31, 2011

Manuscript Accepted: June 7, 2011

Published: June 14, 2011

**Citation**

Liang Dong, "A vector boundary matching technique for efficient and accurate determination of photonic bandgaps in photonic bandgap fibers," Opt. Express **19**, 12582-12593 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12582

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### References

- C. B. Olausson, C. I. Falk, J. K. Lyngsø, B. B. Jensen, K. T. Therkildsen, J. W. Thomsen, K. P. Hansen, A. Bjarklev, and J. Broeng, “Amplification and ASE suppression in a polarization-maintaining ytterbium-doped all-solid photonic bandgap fibre,” Opt. Express 16(18), 13657–13662 (2008). [CrossRef] [PubMed]
- A. Shirakawa, C. B. Olausson, H. Maruyama, K. Ueda, J. K. Lyngsø, and J. Broeng, “High power ytterbium fiber lasers at extremely long wavelengths by photonic bandgap fiber technology,” Opt. Fiber Technol. 16(6), 449–457 (2010). [CrossRef]
- F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29(20), 2369–2371 (2004). [CrossRef] [PubMed]
- A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13(1), 309–314 (2005). [CrossRef] [PubMed]
- A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005). [CrossRef] [PubMed]
- G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005). [CrossRef] [PubMed]
- J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006). [CrossRef] [PubMed]
- G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71(19), 195108 (2005). [CrossRef]
- T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers, I. formulation,” J. Opt. Soc. Am. B 19(10), 2322–2330 (2002). [CrossRef]
- B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers, II. implementation and results,” J. Opt. Soc. Am. B 19(10), 2331–2340 (2002). [CrossRef]
- T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9(13), 721–732 (2001). [CrossRef] [PubMed]
- T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006). [CrossRef] [PubMed]
- A. W. Snyder, and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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