Abstract

An approximate method for finding the band structure of simple photonic bandgap fibres is presented. Our simple model is an isolated high-index rod in a circular unit cell with two alternative boundary conditions. Band plots calculated this way are found to correspond closely to calculations using an accurate numerical method.

© 2006 Optical Society of America

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References

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  1. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, D. J. Trevor, "Tunable photonic band gap fiber," in Proceedings of the Optical Fiber Communications Conference (2002), 466-468.
  2. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, P. St.J. Russell, "All-solid photonic band gap fiber," Opt. Lett. 29, 2369-2371 (2004).
    [CrossRef] [PubMed]
  3. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, P. St.J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005).
    [CrossRef] [PubMed]
  4. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, 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, 8452-8459 (2005).
    [CrossRef] [PubMed]
  5. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, D. M. Bird, "Bend loss in all-solid bandgap fibres," Opt. Express 14, 5688-5698 (2006).
    [CrossRef] [PubMed]
  6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, D. M. Bird, "An improved photonic bandgap fiber based on an array of rings," Opt. Express 14, 6291-6296 (2006).
    [CrossRef] [PubMed]
  7. A. Wang, A. K. George, J. C. Knight, "Three-level neodymium fiber laser incorporating photonic bandgap fiber," Opt. Lett. 31, 1388-1390 (2006).
    [CrossRef] [PubMed]
  8. C. K. Nielsen, K. G. Jespersen, S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 µm using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 6063-6068 (2006).
    [CrossRef] [PubMed]
  9. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, B. J. Eggleton, "Resonance and scattering in microstructured optical fibers," Opt. Lett. 27, 1977-1979 (2002).
    [CrossRef]
  10. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
    [CrossRef] [PubMed]
  11. J. Lægsgaard, "Gap formation and guided modes in photonic band gap fibres with high-index rods," J. Opt. A: Pure Appl. Opt. 6, 798-804 (2004).
    [CrossRef]
  12. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  13. S. L. Altmann, Band Theory of Solids: An Introduction from the Point of View of Symmetry (Clarendon Press, 1994).
  14. P. W. Atkins, Molecular Quantum Mechanics (Oxford University Press, 1983).
  15. T. A. Birks, J. C. Knight, P. St.J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  16. sections I and II of G. J. Pearce, T. D. Hedley, D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell's equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
    [CrossRef]
  17. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, P. St.J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004).
    [CrossRef] [PubMed]
  18. T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991).
    [CrossRef]

2006 (4)

2005 (3)

2004 (3)

2003 (1)

2002 (1)

1997 (1)

1991 (1)

T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991).
[CrossRef]

Argyros, A.

Bigot, L.

Bird, D. M.

Birks, T. A.

Bouwmans, G.

Cordeiro, C. M. B.

de Sterke, C. M.

Douay, M.

Dunn, S. C.

Eggleton, B. J.

George, A. K.

Hedley, T. D.

Hussey, C. D.

T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991).
[CrossRef]

Jespersen, K. G.

Keiding, S. R.

Knight, J. C.

Lægsgaard, J.

J. Lægsgaard, "Gap formation and guided modes in photonic band gap fibres with high-index rods," J. Opt. A: Pure Appl. Opt. 6, 798-804 (2004).
[CrossRef]

Leon-Saval, S. G.

Li, Y. W.

T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991).
[CrossRef]

Litchinitser, N. M.

Lopez, F.

Luan, F.

McPhedran, R. C.

Nielsen, C. K.

Pearce, G. J.

Pottage, J. M.

Provino, L.

Quiquempois, Y.

Russell, P. St.J.

Stone, J. M.

Usner, B.

Wang, A.

White, T. P.

J. Opt. A: Pure Appl. Opt. (1)

J. Lægsgaard, "Gap formation and guided modes in photonic band gap fibres with high-index rods," J. Opt. A: Pure Appl. Opt. 6, 798-804 (2004).
[CrossRef]

Opt. Commun. (1)

T. A. Birks, Y. W. Li, C. D. Hussey, "Waveguides with delta function layers," Opt. Commun. 83, 203-209 (1991).
[CrossRef]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rev. B (1)

sections I and II of G. J. Pearce, T. D. Hedley, D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell's equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Other (4)

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, D. J. Trevor, "Tunable photonic band gap fiber," in Proceedings of the Optical Fiber Communications Conference (2002), 466-468.

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

S. L. Altmann, Band Theory of Solids: An Introduction from the Point of View of Symmetry (Clarendon Press, 1994).

P. W. Atkins, Molecular Quantum Mechanics (Oxford University Press, 1983).

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Figures (6)

Fig. 1.
Fig. 1.

SEM image of an all-solid silica bandgap fibre [5]. The light grey regions are Ge-doped raised-index rods. The fibre’s bandgap-guiding core is the site of a missing rod in the centre.

Fig. 2.
Fig. 2.

(a) A rod of radius a in a hexagonal unit cell of width Λ, with local co-ordinate s normal to the cell boundary. (b) The rod in a circular unit cell of radius b, with radial co-ordinate r.

Fig. 3.
Fig. 3.

Plots of band structure for the example bandgap fibre described in the text. The bandgaps are shown in red. The rod modes from which the bands arise are labelled along the top. (a) DOS calculated using the plane-wave method, with light grey corresponding to high DOS. The yellow curve is the “fundamental” core-guided mode. (b) Band edges calculated using the method of Section 2 and filled in grey between top and bottom edges. The edges of the LP04 band (considered in Fig. 4) are drawn thicker.

Fig. 4.
Fig. 4.

Normalised radial intensity plots |Ψ|2 within a circular unit cell for the states at the top (red) and bottom (blue) of the LP04 band for frequencies kΛ of (left to right) 115, 145 and 170.

Fig. 5.
Fig. 5.

Approximate band structure for the fibre cladding of Fig. 3 but with d/Λ of (left to right) 0.6, 0.4 and 0.2. The horizontal axes have been scaled so that the same range of ka is shown. The rod modes above cutoff can be identified by comparing the middle graph with Fig. 3.

Fig. 6.
Fig. 6.

(a) Band plot for the thin-ring cladding described in the text. (b) Cutoff V-values of the modes of an isolated thick ring as a function of the inner to outer radius ratio c/a. m=1 modes with l>6 are omitted for clarity.

Tables (1)

Tables Icon

Table 1. Calculated frequency widths of some m=2 bands at cutoff, in kΛ units. The plane-wave value for the LP22 band* is imprecise because (see Fig. 3) this band crosses the LP03 band at cutoff.

Equations (18)

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V Δ V = { 4 l f l l 0 2 ln f l = 0
Ψ ( r ) = { J l ( U r a ) r a A K l ( W r a ) + B I l ( W r a ) r > a , β k n l o > 0 C J l ( Q r a ) + D Y l ( Q r a ) r > a , β k n l o < 0 E ( r a ) l + F ( r a ) 1 r > a , β k n l o = 0 , l 0 G + H ln ( r a ) r > a , β k n l o = 0 , l = 0
V 2 = k 2 a 2 ( n h i 2 n l o 2 )
W 2 = a 2 ( β 2 k 2 n l o 2 ) = Q 2
U 2 = a 2 ( k 2 n h i 2 β 2 ) = V 2 W 2
Ψ ' ( b ) = 0 ( top of band )
Ψ ( b ) = 0 ( bottom of band )
g ( V , W 2 ) = 0 .
g top ( V , W 2 ) = { [ A K ' l ( α W ) + B I ' l ( α W ) ] W U l W 2 > 0 [ C J ' l ( α Q ) + D Y ' l ( α Q ) ] Q U l W 2 < 0 [ E α l F α l ] l α V l W 2 = 0 , l 0 H α W 2 = 0 , l = 0
g bottom ( V , W 2 ) = { [ A K l ( α W ) + B I l ( α W ) ] U l W 2 > 0 [ C J l ( α Q ) + D Y l ( α Q ) ] U l W 2 < 0 [ E α l + F α 1 ] V l 1 W 2 = 0 , l 0 G + H ln α W 2 = 0 , l = 0
A = W I l + 1 ( W ) J l ( U ) + U J l + 1 ( U ) I l ( W )
B = W K l + 1 ( W ) J l ( U ) U J l + 1 ( U ) K l ( W )
C = [ Q Y l + 1 ( Q ) J l ( U ) + U J l + 1 ( U ) Y l ( Q ) ] π 2
D = [ Q J l + 1 ( Q ) J l ( U ) U J l + 1 ( U ) J l ( Q ) ] π 2
E = V J l 1 ( V ) 2 l
F = V J l + 1 ( V ) 2 l
G = J 0 ( V )
H = V J 1 ( V )

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