Abstract

An analytical method for band structure calculations of photonic crystal fibers with liquid crystal infiltrations is presented. The scalar eigenvalue equation is extended to treat both isotropic and anisotropic materials by introducing a coefficient to describe the index contrast between the extraordinary and ordinary refractive index of the liquid crystal. The simple model provides a fast insight into bandgap formation in photonic crystal fibers filled with anisotropic material such as liquid crystal, which would be useful to aid the design based on such fibers.

© 2008 Optical Society of America

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References

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  1. 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]
  2. P. St. J. Russell, "Photonic crystal fibers," J. Lightwave Technol. 24, 4729-4749 (2006).
    [CrossRef]
  3. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic band gap fiber," in OSA Trends in Optics and Photonics (TOPS) 70, Optical Fiber Communication Conference Technical Digest, Postconference Edition (Optical Society of America, Washington, DC, 2002), 466-468.
  4. 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, 8452-8459 (2005).
    [CrossRef] [PubMed]
  5. B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific, 1990).
    [CrossRef]
  6. M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
    [CrossRef]
  7. G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
    [CrossRef]
  8. S. P. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express 11, 167-175 (2003).
    [CrossRef] [PubMed]
  9. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. d. Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003).
    [CrossRef] [PubMed]
  10. J. Lagsgaard, "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]
  11. 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]
  12. J. Sun and C. C. Chan, "Effect of liquid crystal alignment on bandgap formation in photonic bandgap fibers," Opt. Lett. 32, 1989-1991 (2007).
    [CrossRef] [PubMed]
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  14. C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
    [CrossRef]
  15. Lixon Information, (Chisso Corporation, Chiba, Japan, (1995).

2008 (1)

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

2007 (1)

2006 (2)

2005 (2)

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

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, 8452-8459 (2005).
[CrossRef] [PubMed]

2004 (1)

J. Lagsgaard, "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]

2003 (2)

1996 (1)

1994 (1)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Albin, S.

Atkin, D. M.

Bigot, L.

Bird, D. M.

Birks, T. A.

Bjarklev, A.

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

Bouwmans, G.

Chan, C. C.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

d. Sterke, C. M.

Douay, M.

Dunn, S. C.

Eggleton, B. J.

Engan, H. E.

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

Gong, Y.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

Guo, S. P.

Haakestad, M. W.

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

Hu, J.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

Huang, W. P.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Knight, J. C.

Lagsgaard, J.

J. Lagsgaard, "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]

Larsen, T. T.

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

Litchinitser, N. M.

Lopez, F.

McPhedran, R. C.

Nielsen, M. D.

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

Pearce, G. J.

Provino, L.

Quiquempois, Y.

Ren, G.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

Russell, P. St. J.

Shum, P.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

Sun, J.

Usner, B.

Wang, G.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

White, T. P.

Xu, C. L.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Yu, X.

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

IEEE Photon. Lett. (1)

M. W. Haakestad, T. T. Larsen, M. D. Nielsen, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Lett. 17, 819- 821 (2005).
[CrossRef]

J. Lightwave Technol. (2)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

P. St. J. Russell, "Photonic crystal fibers," J. Lightwave Technol. 24, 4729-4749 (2006).
[CrossRef]

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

J. Lagsgaard, "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)

G. Ren, P. Shum, X. Yu, J. Hu, G. Wang, and Y. Gong, "Polarization dependent guiding in liquid crystal filled photonic crystal fibers," Opt. Commun. 281, 1598-1606 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Other (4)

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic band gap fiber," in OSA Trends in Optics and Photonics (TOPS) 70, Optical Fiber Communication Conference Technical Digest, Postconference Edition (Optical Society of America, Washington, DC, 2002), 466-468.

B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific, 1990).
[CrossRef]

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

Lixon Information, (Chisso Corporation, Chiba, Japan, (1995).

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Figures (4)

Fig.1.
Fig.1.

(a). LCPCF with triangular lattices. The shaded region in cladding is filled NLC, the background is silica. (b) Hexagonal unit cell of the NLC rods in silica. (c) Approximated circular unit cell of the NLC rods in silica

Fig. 2.
Fig. 2.

(a). Bandgap map of the LCPCF obtained from PWE method. (b). Bandgap map of the LCPCF obtained from the analytical method. The total bandgaps are the overlap zone between two polarization bandgaps.

Fig. 3.
Fig. 3.

Width of the first bandgap for both polarizations, obtained by the analytical scalar method and PWE method.

Fig. 4.
Fig. 4.

(a). The gap width plot as a function of ne 2 /no 2 . (b). The bandgap variations at two ne 2 /no 2 points. The insets show the variation of the position and the width of the gap. (c). The gap width plot as a function of ne 2 /nsi 2 . (d). The bandgap variations in terms of the normalized wavelength. The insets show the variation of the position and the width of the gap.

Equations (14)

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2 ψ x y 2 + x [ 1 n zz 2 x ( n xx 2 ψ x ) ] + ( n xx 2 k 2 β 2 ) ψ x = 2 ψ y x y x [ 1 n zz 2 y ( n yy 2 ψ y ) ]
2 ψ y x 2 + y [ 1 n zz 2 y ( n yy 2 ψ y ) ] + ( n yy 2 k 2 β 2 ) ψ y = 2 ψ x x y y [ 1 n zz 2 x ( n xx 2 ψ x ) ]
2 ψ r 2 + f r ψ r + ( k 0 2 n i 2 β 2 f l 2 r 2 ) ψ = 0
R 2 2 Z R 2 + R Z R + ( R 2 m 2 ) Z = 0
ψ ( r ) = { J m ( U r a ) ( U r a ) θ r a [ A K l ( W r a ) + B I l ( W r a ) ] r > a , β k n lo > 0 V 2 = k 0 2 a 2 ( n hi 2 n lo 2 ) [ C J l ( Qr a ) + D Y l ( Qr a ) ] r > a , β k n lo < 0 where W 2 = a 2 ( β 2 k 0 2 n lo 2 ) = Q 2 [ E ( r a ) l + F ( r a ) l ] r > a , β k n lo = 0 , l 0 U 2 = a 2 ( k 0 2 n hi 2 β 2 ) = V 2 W 2 [ G + H ln ( r a ) ] r > a , β k n lo = 0 , l = 0
A = [ W J m ( U ) I l + 1 ( W ) + U I l ( W ) J m + 1 ( U ) J m ( U ) I l ( W ) ( m θ l ) ] U θ
B = [ W J m ( U ) K l + 1 ( W ) U K l ( W ) J m + 1 ( U ) + J m ( U ) K l ( W ) ( m θ l ) ] U θ
C = [ UY l ( Q ) J m + 1 ( U ) Q J m ( U ) Y l + 1 ( Q ) Y l ( Q ) J m ( U ) ( m θ l ) ] π 2 U θ
D = [ QJ m ( U ) J l + 1 ( Q ) U J l ( Q ) J m + 1 ( U ) + J l ( Q ) J m ( U ) ( m θ l ) ] π 2 U θ
E = ( J m 1 ( V ) V + J m ( V ) ( l m θ ) ) 2 l V θ
F = ( J m + 1 ( V ) V + J m ( V ) ( l m + θ ) ) 2 l V θ
G = J θ ( V ) V θ
H = J θ + 1 ( V ) V θ 1
f top ( V , W 2 ) = { A K l ( W α ) + B I l ( W α ) C J l ( Q α ) + D Y l ( Q α ) E α l F α l H α f bottom ( V , W 2 ) = { A K l ( W α ) + B I l ( W α ) W 2 > 0 C J l ( Q α ) + D Y l ( Q α ) W 2 < 0 E α l + F α l W 2 = 0 , l 0 G + H ln α W 2 = 0 , l = 0

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