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.

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References

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  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]
  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]
  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 13(1), 309–314 (2005).
    [CrossRef] [PubMed]
  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 13(7), 2503–2511 (2005).
    [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]
  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]
  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]
  13. A. W. Snyder, and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

2010 (1)

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]

2008 (1)

2006 (2)

2005 (4)

2004 (1)

2002 (2)

2001 (1)

Argyros, A.

Bigot, L.

Bird, D. M.

Birks, T. A.

Bjarklev, A.

Botten, L.

Botten, L. C.

Bouwmans, G.

Broeng, J.

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]

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]

Cordeiro, C. M. B.

de Sterke, C. M.

Douay, M.

Falk, C. I.

George, A. K.

Hansen, K. P.

Hedley, T. D.

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]

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]

Jensen, B. B.

Knight, J. C.

Kuhlmey, B. T.

Leon-Saval, S. G.

Lopez, F.

Luan, F.

Lyngsø, J. K.

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]

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]

Maruyama, H.

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]

Maystre, D.

McPhedran, R.

McPhedran, R. C.

Olausson, C. B.

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]

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]

Pearce, G. J.

Provino, L.

Quiquempois, Y.

Renversez, G.

Russell, P. St. J.

Shirakawa, A.

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]

Smith, G.

St J Russell, P.

Stone, J. M.

Therkildsen, K. T.

Thomsen, J. W.

Ueda, K.

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]

White, T.

White, T. P.

J. Opt. Soc. Am. B (2)

Opt. Express (7)

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. 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]

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]

Opt. Fiber Technol. (1)

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]

Opt. Lett. (1)

Phys. Rev. B (1)

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]

Other (1)

A. W. Snyder, and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (8)

Fig. 1
Fig. 1

Illustration of a photonic lattice.

Fig. 2
Fig. 2

Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 0.9 and d/Λ = 0.19. This is a mode located at upper edge of the first band.

Fig. 3
Fig. 3

Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 0.8 and d/Λ = 0.17. This is a mode located at lower edge of the first band.

Fig. 4
Fig. 4

Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 2.6 and d/Λ = 0.46. This is a mode located at upper edge of the second band.

Fig. 5
Fig. 5

Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 2.65 and d/Λ = 0.47. This is a mode located at lower edge of the second band.

Fig. 6
Fig. 6

Lower band edges determined using the triangular unit cell (black dots) and hexagonal unit cell (red circles) for the fiber in [12] (d/Λ = 0.41, node index n0 = 1.48716 and background index nb = 1.458.).

Fig. 7
Fig. 7

Bandgap diagram determined by the vector boundary matching technique the fiber in [12] (d/Λ = 0.41, node index n0 = 1.48716 and background index nb = 1.458.). The upper band edges are determined by the hexagonal unit cell (black dots) and lower edges are determined by a combination of the triangular and hexagonal unit cells (red circles).

Fig. 8
Fig. 8

Bandgap diagrams for hexagonal lattices with (a) nodes of a step index profile and (b) nodes of a parabolic index profile. Peak index difference is 0.003, d/Λ = 0.4, nb = 1.45, M = 6, and 12 equally spaced radial lines. Upper band edges are marked by black dots and lower band edges red circles.

Equations (33)

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e z p = m = + [ a m p J m ( k p r ) + b m p H m 1 ( k p r ) ] e i m θ
H z p = m = + [ C m p J m ( k p r ) + D m p H m 1 ( k p r ) ] e i m θ
k p 2 = k 2 n p 2 β 2
H = μ 0 ε 0 h
[ a m p + 1 b m p + 1 C m p + 1 D m p + 1 ] = M m p [ a m p b m p C m p D m p ]
[ a m N b m N C m N D m N ] = M m [ a m 0 b m 0 C m 0 D m 0 ]
M m = p = 0 N 1 M m p
e z N = m = + [ a m N J m ( k p r ) + b m N H m 1 ( k p r ) ] e i m θ
H z N = m = + [ C m N J m ( k p r ) + D m N H m 1 ( k p r ) ] e i m θ
e x N = e r N cos ( θ ) e θ N sin ( θ )
e y N = e r N sin ( θ ) + e θ N cos ( θ )
M [ a M 0 ... a M 0 b M 0 ... b M 0 ] = 0
e θ p = m = { m β r k p 2 [ a m p J m ( k p r ) + b m p H m 1 ( k p r ) ] i k k p [ C m p J m 1 ( k p r ) J m + 1 ( k p r ) 2 + D m p H m 1 ( k p r ) H m + 1 1 ( k p r ) 2 ] } e i m θ
H θ p = m = { m β r k p 2 [ C m p J m ( k p r ) + D m p H m 1 ( k p r ) ] + i k n p 2 k p [ a m p J m 1 ( k p r ) J m + 1 ( k p r ) 2 + b m p H m 1 ( k p r ) H m + 1 1 ( k p r ) 2 ] } e i m θ
M 11 p = 1 M 0 { J m ( k p ρ p ) [ H m 1 1 ( k p + 1 ρ p ) H m + 1 1 ( k p + 1 ρ p ) ] n p 2 k p + 1 n p + 1 2 k p H m 1 ( k p + 1 ρ p ) [ J m 1 ( k p ρ p ) J m + 1 ( k p ρ p ) ] }
M 12 p = 1 M 0 { H m 1 ( k p ρ p ) [ H m 1 1 ( k p + 1 ρ p ) H m + 1 1 ( k p + 1 ρ p ) ] n p 2 k p + 1 n p + 1 2 k p H m 1 ( k p + 1 ρ p ) [ H m 1 1 ( k p ρ p ) H m + 1 1 ( k p ρ p ) ] }
M 13 p = 1 M 0 i 2 m β k p + 1 ρ p n p + 1 2 k ( 1 k p 2 1 k p + 1 2 ) H m 1 ( k p + 1 ρ p ) J m ( k p ρ p )
M 14 p = 1 M 0 i 2 m β k p + 1 ρ p n p + 1 2 k ( 1 k p 2 1 k p + 1 2 ) H m 1 ( k p + 1 ρ p ) H m 1 ( k p ρ p )
M 21 p = 1 M 0 { J m ( k p ρ p ) [ J m 1 ( k p + 1 ρ p ) J m + 1 ( k p + 1 ρ p ) ] n p 2 k p + 1 n p + 1 2 k p J m ( k p + 1 ρ p ) [ J m 1 ( k p ρ p ) J m + 1 ( k p ρ p ) ] }
M 22 p = 1 M 0 { H m 1 ( k p ρ p ) [ J m 1 ( k p + 1 ρ p ) J m + 1 ( k p + 1 ρ p ) ] n p 2 k p + 1 n p + 1 2 k p J m ( k p + 1 ρ p ) [ H m 1 1 ( k p ρ p ) H m + 1 1 ( k p ρ p ) ] }
M 23 p = 1 M 0 i 2 m β k p + 1 ρ p n p + 1 2 k ( 1 k p 2 1 k p + 1 2 ) J m ( k p + 1 ρ p ) J m ( k p ρ p )
M 24 p = 1 M 0 i 2 m β k p + 1 ρ p n p + 1 2 k ( 1 k p 2 1 k p + 1 2 ) J m ( k p + 1 ρ p ) H m 1 ( k p ρ p )
M 31 p = 1 M 0 i 2 m β k p + 1 ρ p k ( 1 k p 2 1 k p + 1 2 ) H m 1 ( k p + 1 ρ p ) J m ( k p ρ p )
M 32 p = 1 M 0 i 2 m β k p + 1 ρ p k ( 1 k p 2 1 k p + 1 2 ) H m 1 ( k p + 1 ρ p ) H m 1 ( k p ρ p )
M 33 p = 1 M 0 { J m ( k p ρ p ) [ H m 1 1 ( k p + 1 ρ p ) H m + 1 1 ( k p + 1 ρ p ) ] k p + 1 k p H m 1 ( k p + 1 ρ p ) [ J m 1 ( k p ρ p ) J m + 1 ( k p ρ p ) ] }
M 34 p = 1 M 0 { H m 1 ( k p ρ p ) [ H m 1 1 ( k p + 1 ρ p ) H m + 1 1 ( k p + 1 ρ p ) ] k p + 1 k p H m 1 ( k p + 1 ρ p ) [ H m 1 1 ( k p ρ p ) H m + 1 1 ( k p ρ p ) ] }
M 41 p = 1 M 0 i 2 m β k p + 1 ρ p k ( 1 k p 2 1 k p + 1 2 ) J m ( k p + 1 ρ p ) J m ( k p ρ p )
M 42 p = 1 M 0 i 2 m β k p + 1 ρ p k ( 1 k p 2 1 k p + 1 2 ) J m ( k p + 1 ρ p ) H m 1 ( k p ρ p )
M 43 p = 1 M 0 { J m ( k p ρ p ) [ J m 1 ( k p + 1 ρ p ) J m + 1 ( k p + 1 ρ p ) ] k p + 1 k p J m ( k p + 1 ρ p ) [ J m 1 ( k p ρ p ) J m + 1 ( k p ρ p ) ] }
M 44 p = 1 M 0 { H m 1 ( k p ρ p ) [ J m 1 ( k p + 1 ρ p ) J m + 1 ( k p + 1 ρ p ) ] k p + 1 k p J m ( k p + 1 ρ p ) [ H m 1 1 ( k p ρ p ) H m + 1 1 ( k p ρ p ) ] }
M 0 = J m ( k p + 1 ρ p ) [ H m 1 1 ( k p + 1 ρ p ) H m + 1 1 ( k p + 1 ρ p ) ] H m 1 ( k p + 1 ρ p ) [ J m 1 ( k p + 1 ρ p ) J m 1 ( k p + 1 ρ p ) ]
e r N = m = { m k r k b 2 [ C m N J m ( k b r ) + D m N H m 1 ( k b r ) ] + i β k b [ a m N J m 1 ( k b r ) J m + 1 ( k b r ) 2 + b m N H m 1 ( k b r ) H m + 1 1 ( k b r ) 2 ] } e i m θ
H r N = m = { m k n b 2 r k b 2 [ a m N J m ( k b r ) + b m N H m 1 ( k b r ) ] + i β k b [ C m N J m 1 ( k b r ) J m + 1 ( k b r ) 2 + D m N H m 1 ( k b r ) H m + 1 1 ( k b r ) 2 ] } e i m θ

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