Abstract

Propagation of light in a square-lattice hollow-core photonic crystal fibre is analysed as a model of guidance in a class of photonic crystal fibres that exhibit broad-band guidance without photonic bandgaps. A scalar governing equation is used and analytic solutions based on transfer matrices are developed for the full set of modes. It is found that an exponentially localised fundamental mode exists for a wide range of frequencies. These analytic solutions of an idealised structure will form the basis for analysis of guidance in a realistic structure in a following paper.

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  1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
    [CrossRef] [PubMed]
  2. P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005).
    [CrossRef] [PubMed]
  3. D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
    [CrossRef] [PubMed]
  4. A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008).
    [CrossRef] [PubMed]
  5. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006).
    [CrossRef] [PubMed]
  6. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
    [CrossRef] [PubMed]
  7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
    [CrossRef] [PubMed]
  8. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
    [CrossRef] [PubMed]
  9. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007).
    [CrossRef] [PubMed]
  10. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010).
    [CrossRef] [PubMed]
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  12. S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009).
    [CrossRef] [PubMed]
  13. L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).
  14. L. Chen and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres II: Perturbation theory for a realistic fibre structure,” Submitted to Opt. Express (2011).
  15. T. Birks, D. Bird, T. Hedley, J. Pottage, and P. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express 12, 69–74 (2004).
    [CrossRef] [PubMed]
  16. S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644–1650 (2002).
    [CrossRef]
  17. A. Kumar, A. N. Kaul, and A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. 10, 86–88 (1985).
    [CrossRef] [PubMed]
  18. P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.
  19. 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, 195108 (2005).
  20. G. Pearce, J. Pottage, D. Bird, P. Roberts, J. Knight, and P. Russell, “Hollow-core pcf for guidance in the mid to far infra-red,” Opt. Express 13, 6937–6946 (2005).
    [CrossRef] [PubMed]
  21. J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express 11, 2854–2861 (2003).
    [CrossRef] [PubMed]
  22. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006).
    [CrossRef] [PubMed]
  23. 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, 6291–6296 (2006).
    [CrossRef] [PubMed]
  24. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002).
    [CrossRef]

2010 (1)

2009 (1)

2008 (2)

2007 (3)

2006 (3)

2005 (2)

2004 (1)

2003 (2)

J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express 11, 2854–2861 (2003).
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

2002 (2)

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

1985 (1)

1951 (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, 195108 (2005).

Abeeluck, A. K.

Ahmad, F. R.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Argyros, A.

Beaudou, B.

Benabid, F.

Bhagwat, A. R.

Bird, D.

Bird, D. M.

Birks, T.

Birks, T. A.

F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006).
[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, 6291–6296 (2006).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Burger, S.

Couny, F.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Eggleton, B. J.

Farr, L.

Février, S.

Gaeta, A. L.

A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008).
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Gallagher, M. T.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

George, A. K.

Ghatak, A. K.

Headley, C.

Hedley, T.

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, 195108 (2005).

Herrmann, J.

Husakou, A.

Im, S.-J.

Kaul, A. N.

Kawakami, S.

Knight, J.

Knight, J. C.

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, 6291–6296 (2006).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Koch, K. W.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Kumar, A.

Light, P. S.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006).
[CrossRef] [PubMed]

Litchinitser, N. M.

Lloyd-Lucas, F. D.

P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Love, J. D.

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

Luan, F.

Mangan, B.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Mason, M.

Mller, D.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Ouzounov, D. G.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Pearce, G.

Pearce, G. J.

Pla, J.

Pottage, J.

Poulton, C. G.

Raymer, M. G.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

Roberts, P.

Roberts, P. J.

F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Russell, P.

Russell, P. S. J.

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007).
[CrossRef] [PubMed]

P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Sabert, H.

Silcox, J.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Snyder, A. W.

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

Stone, J. M.

Thomas, M. G.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Tomlinson, A.

Venkataraman, N.

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Viale, P.

Wiederhecker, G. S.

Williams, D.

J. Lightwave Technol. (1)

Opt. Express (12)

T. Birks, D. Bird, T. Hedley, J. Pottage, and P. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express 12, 69–74 (2004).
[CrossRef] [PubMed]

F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
[CrossRef] [PubMed]

S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009).
[CrossRef] [PubMed]

P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005).
[CrossRef] [PubMed]

A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008).
[CrossRef] [PubMed]

A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
[CrossRef] [PubMed]

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007).
[CrossRef] [PubMed]

S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010).
[CrossRef] [PubMed]

G. Pearce, J. Pottage, D. Bird, P. Roberts, J. Knight, and P. Russell, “Hollow-core pcf for guidance in the mid to far infra-red,” Opt. Express 13, 6937–6946 (2005).
[CrossRef] [PubMed]

J. Pottage, D. Bird, T. Hedley, J. Knight, T. Birks, P. Russell, and P. Roberts, “Robust photonic band gaps for hollow core guidance in pcf made from high index glass,” Opt. Express 11, 2854–2861 (2003).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006).
[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, 6291–6296 (2006).
[CrossRef] [PubMed]

Opt. Lett. (3)

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, 195108 (2005).

Science (3)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Mller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of Megawatt Optical Solitons in Hollow-Core Photonic Band-Gap Fibers,” Science 301, 1702–1704 (2003).
[CrossRef] [PubMed]

Other (4)

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

L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).

L. Chen and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres II: Perturbation theory for a realistic fibre structure,” Submitted to Opt. Express (2011).

P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Application , E. Burstein and C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

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Figures (7)

Fig. 1
Fig. 1

Schematic of transverse planes of hollow-core PCFs. The cladding structure in (a) is a Kagome lattice, (b) is a square-lattice hollow-core PCF. Corresponding scanning electron micrographs can be found in Refs. [6] and [7]. Rectangular model PCFs are shown in (c) and (d). (c) is a realistic model PCF in which the dielectric constant of the intersections is equal to that of the glass strips, as indicated in the inset. (d) is an ideal model structure for the application of the scalar governing equation; the inset shows that the intersections have a higher dielectric constant than the strips.

Fig. 2
Fig. 2

(a) Sketch of a general one-dimensional arrangement of air and glass regions. The light and dark colours represent air and glass, respectively. The widths of the Nth air and (N +1) th glass regions are h a N and h g N + 1 . Here, x a N and x g N + 1 are two arbitrary points within the air and glass. All the quantities are made dimensionless through division by Λ. (b) and (c) Examples of even and odd solutions in one dimension of an 8×8 supercell of the model structure.

Fig. 3
Fig. 3

(a) PDOS and fundamental guided mode for the ideal model PCF structure of Fig. 1(d) and the scalar governing equation. The red line shows the propagation constant of the fundamental mode. The green and black lines are two frequencies for which the field of the modes are plotted in (b) and (c,d), respectively. (b) The fundamental mode with βΛ = 16.031 at k 0Λ = 16.5. (c) and (d) The fundamental and selected cladding mode with equivalent βΛ = 13.941 at k 0Λ = 14.5.

Fig. 4
Fig. 4

Field plots in the transverse plane for symmetric modes (i.e. C = 0) close to the air-line. The normalised frequency is k 0Λ = 40, and the (βΛ)2 values of these modes vary from 1572.73 to 1587.12. The fundamental mode is shown in (a).

Fig. 5
Fig. 5

(a) Schematic diagram of the two regions in which the field is concentrated for high-index modes in an 8×8 supercell. (b–e) Four modes with fields localised in the blue regions in (a). Modes (b) and (c) have even waves in both directions; (d) and (e) have odd waves along the axes. Four other modes have a similar field pattern, but in the green regions of (a).

Fig. 6
Fig. 6

Example plots of symmetric solutions with delocalised fields. The (βΛ)2 values are 535.01, 498.25 and −344.78 for (a), (b) and (c), respectively. Note that only the region close to the central defect is shown.

Fig. 7
Fig. 7

Two examples of pairs of non-symmetric modes. (a) and (b) Air-guided modes generated by the symmetric modes shown in Figs. 4(a) and 4(e). (c) and (d) Glass-guided modes formed by the combination of the high-index mode of Fig. 5(b) and the delocalised mode of Fig. 6(a). The (βΛ)2 values are 1579.93 for modes (a) and (b), and 2073.28 for modes (c) and (d).

Equations (18)

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( t 2 + n 2 k 0 2 β 2 ) h t = ( t × h t ) × t ln n 2 ,
( t 2 + n 2 k 0 2 β 2 ) h t = 0 ,
n 2 ( x , y ) = n a 2 + Δ n 2 ( x ) + Δ n 2 ( y ) ,
d 2 X ( x ) d x 2 + [ p x ( x ) Λ ] 2 X ( x ) = 0 , with [ p x ( x ) Λ ] 2 = ( k 0 Λ ) 2 [ n a 2 + Δ n 2 ( x ) ] ( β Λ ) 2 ξ
d 2 Y ( y ) d y 2 + [ p y ( y ) Λ ] 2 Y ( y ) = 0 , with [ p y ( y ) Λ ] 2 = ( k 0 Λ ) 2 Δ n 2 ( y ) + ξ ,
( p x g Λ ) 2 = K g 2 cos 2 θ and ( p y g Λ ) 2 = K g 2 sin 2 θ ,
( p x g Λ ) 2 = K g 2 ( 1 + C ) / 2 , ( p y g Λ ) 2 = K g 2 ( 1 C ) / 2 .
( p x a Λ ) 2 = K g 2 ( 1 + C ) / 2 ( k 0 Λ ) 2 ( n g 2 n a 2 )
( p y a Λ ) 2 = K g 2 ( 1 C ) / 2 ( k 0 Λ ) 2 ( n g 2 n a 2 ) .
X j N ( x ) = a j N cos [ ( p x j Λ ) ( x x j N ) ] + b j N sin [ ( p x j Λ ) ( x x j N ) ] / ( p x j Λ ) ,
( cos [ ( p x a Λ ) h a R N ] sin [ p x a Λ ] h a R N / ( p x a Λ ) ( p x a Λ ) sin [ ( p x a Λ ) h a R N ] cos [ ( p x a Λ ) h a R N ] ) ( a a N b a N ) = ( cos [ ( p x a Λ ) h g L N + 1 ] sin [ ( p x g Λ ) h g L N + 1 ] / ( p x g Λ ) ( p x g Λ ) sin [ ( p x g Λ ) h g L N + 1 ] cos [ p x g Λ h g L N + 1 ] ) ( a g N + 1 b g N + 1 ) ,
a _ g N + 1 = m _ _ ( h g N + 1 2 , p x g Λ ) 1 m _ _ ( h a N 2 , p x a Λ ) a _ a N ,
m _ _ ( h , p ) = ( cos ( h p ) sin ( h p ) / p p sin ( h p ) cos ( h p ) )
a _ j N = ( a j N , b j N ) T , a _ j N + 1 = ( a j N + 1 , b j N + 1 ) T .
( 1 + C ) K g 2 = K g 0 v 2
( 1 C ) K g 2 = K g 0 u 2 .
K g 2 = ( K g 0 v 2 + K g 0 u 2 ) / 2 .
( β Λ ) 2 = [ ( β 0 v Λ ) 2 + ( β 0 u Λ ) 2 ] / 2

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