Abstract

We consider chromatic dispersion of capillary tubes and photonic crystal fibers infiltrated with liquid crystals. A perturbative scheme for inclusion of material dispersion of both liquid crystal and the surrounding waveguide material is derived. The method is used to calculate the chromatic dispersion at different temperatures.

© 2006 Optical Society of America

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  1. C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
    [CrossRef]
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    [CrossRef]
  3. K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006

2005

2004

Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85, 5182-5184 (2004).
[CrossRef]

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

2003

2002

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers," J. Opt. Soc. Am. B 19, 753-764 (2002).
[CrossRef]

2001

2000

1991

J. D. Dai, and C. K. Jen, "Analysis of cladded uniaxial single-crystal fibers," J. Opt. Soc. Am. A 8, 2021-2025 (1991).
[CrossRef]

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 1511-1522 (1991).
[CrossRef]

1989

1986

N. V. Tabiryan, A. V. Sukhov, and B. Y. Zel'dovich, "Orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Alkeskjold, T.

Alkeskjold, T. T.

T. T. Alkeskjold, "Optical devices based on liquid crystal photonic bandgap fibers," Ph.D. thesis (Department of Communication, Optics & Materials, Technical University of Denmark, 2005).

Anawati, A.

Andrés, P.

Baehr-Jones, T.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Bassi, P.

Bjarklev, A.

Broeng, J.

Brugioni, S.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Chau, A. H. L.

Coen, S.

Dai, J. D.

Demokan, M. S.

Dolinski, M.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Du, F.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Eggleton, B.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Faetti, S.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Ferrando, A.

Green, M.

Hansen, T. P.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
[CrossRef]

Harvey, J. D.

Hermann, D.

Hermann, D. S.

Ho, H. L.

Hochberg, M.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Hoo, Y. L.

Huang, Y.

Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85, 5182-5184 (2004).
[CrossRef]

Jen, C. K.

Jin, W.

Joannopoulos, J. D.

Johnson, S. G.

Kerbage, C.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Knight, J. C.

Kriezis, E. E.

Lægsgaard, J.

Larsen, T. T.

Lee, M. A.

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 1511-1522 (1991).
[CrossRef]

Leonhardt, R.

Li, J.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Libori, S. E. B.

Lin, H.

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 1511-1522 (1991).
[CrossRef]

Loncar, M.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Lu, Y.-Q.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Mach, P.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Madden, S. J.

Maune, B.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Meucci, R.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Miret, J. J.

Nielsen, K.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
[CrossRef]

Nielsen, M.

Noordegraaf, D.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Palffy-Muhoray, P.

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 1511-1522 (1991).
[CrossRef]

Psaltis, D.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Qiu, Y.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Ranka, J. K.

Riishede, J.

Rogers, J.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Russell, P. St. J.

Scherer, A.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Scolari, L.

Silvestre, E.

Sørensen, T.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
[CrossRef]

Stentz, A. J.

Sukhov, A. V.

N. V. Tabiryan, A. V. Sukhov, and B. Y. Zel'dovich, "Orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Tabiryan, N. V.

N. V. Tabiryan, A. V. Sukhov, and B. Y. Zel'dovich, "Orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Tsiboukis, T. D.

Wadsworth, W. J.

Windeler, R.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Windeler, R. S.

Witzens, J.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Wu, S. T.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Wu, S.-T.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Xiao, L.

Xu, Y.

Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85, 5182-5184 (2004).
[CrossRef]

Yariv, A.

Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85, 5182-5184 (2004).
[CrossRef]

Zel'dovich, B. Y.

N. V. Tabiryan, A. V. Sukhov, and B. Y. Zel'dovich, "Orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Zhao, C.

Zografopoulos, D. C.

Appl. Opt.

Appl. Phys. Lett.

Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85, 5182-5184 (2004).
[CrossRef]

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

J. Appl. Phys.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, "Infrared refractive indices of liquid crystals," J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

J. Opt. A Pure Appl. Opt.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, "Selective filling of photonic crystal fibers," J. Opt. A Pure Appl. Opt. 7, L13-L20 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

N. V. Tabiryan, A. V. Sukhov, and B. Y. Zel'dovich, "Orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 1511-1522 (1991).
[CrossRef]

Opt. Commun.

C. Kerbage, R. Windeler, B. Eggleton, P. Mach, M. Dolinski, and J. Rogers, "Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber," Opt. Commun. 204, 179-184 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Other

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

T. T. Alkeskjold, "Optical devices based on liquid crystal photonic bandgap fibers," Ph.D. thesis (Department of Communication, Optics & Materials, Technical University of Denmark, 2005).

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Figures (8)

Fig. 1
Fig. 1

(a) Parallel and (b) axial alignment of LC molecules in a cylindrical geometry. Figures reproduced with permission (Ref. [11]).

Fig. 2
Fig. 2

Dielectric constants of silica and E7. The curves for silica and E7 are based on the Sellmeier expression given in Eq. (18) and the Cauchy polynomial given in Eq. (19), respectively. The parameters given in Table 1 are used.

Fig. 3
Fig. 3

Chromatic dispersion of the fundamental HE 11 mode for planar alignment of LC molecules and different radii of the capillary tube. The curves show the chromatic dispersion found using the exact analytical result for the dispersion relation, and markers show the result found numerically together with our perturbative method. The inset shows the absolute difference between the two results, i.e., Δ D = D exact D perturbative .

Fig. 4
Fig. 4

Chromatic dispersion for axial alignment of LC molecules and different radii of the capillary tube. The self-consistent frequencies are found using the perturbative method. Examples of H 2 for the TM 01 and HE 11 modes are shown in the insets.

Fig. 5
Fig. 5

Representative contour plots of the real part of the x component of H for the TM 01 mode (left) and the HE 11 mode (right) for the axial alignment in a capillary tube. The white circle shows the boundary of the LC infiltrated capillary.

Fig. 6
Fig. 6

Chromatic dispersion for the PCF structure shown in the inset with different radii of the central hole. The ratios R Λ = 0.6 and r Λ = 0.3 are fixed. The LC molecules are planarly aligned. A zoom around the ZDWs is shown in Fig. 8

Fig. 7
Fig. 7

Chromatic dispersion for the PCF structure shown in the inset in Fig. 6 with different radii of the central hole. The LC molecules are aligned axially. Examples of H 2 for the TM and HE modes are shown in the insets.

Fig. 8
Fig. 8

Chromatic dispersion at 25 ° C (solid curves) and at 50 ° C (dashed curves) for the HE 11 modes. (a), (b) Capillary tube ( R = 1.0 μ m ) with the LC parallel and axially aligned, respectively. (c), (d) Selectively filled PCF structure as shown in inset in Fig. 6 with the LC parallel and axially aligned, respectively. In (c) the solid circles (●) and the open circles (엯) are for R = 0.75 μ m and R = 1.0 μ m , respectively. The dispersion profile in (d) is for R = 1.0 μ m .

Tables (1)

Tables Icon

Table 1 Parameters for Cauchy and Sellmeier Polynomials Given in Eqs. (18, 19) a

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ϵ ̿ = [ ϵ r r 0 ϵ r z 0 ϵ ϕ ϕ 0 ϵ z r 0 ϵ z z ] ,
Φ h = ω 2 c 2 h ,
Φ = β × ϵ ̿ 1 β × ,
D = ω 2 2 π c v g 2 d v g d ω ,
h , Φ h h , h = ω 2 c 2 .
d Φ d β = Φ β + v g j Φ ϵ j ϵ j ω ,
Φ β = β × ϵ ̿ 1 ( 0 0 i ) × + ( 0 0 i ) × ϵ ̿ 1 β × ,
Φ ϵ j = β × ϵ ̿ 1 ϵ ̿ ϵ j ϵ ̿ 1 β × .
v g = v g 0 { 1 + ω 2 [ e , d ϵ S d ω e S h , h + e , ( ϵ ̿ ϵ d ϵ d ω + ϵ ̿ ϵ d ϵ d ω ) e LC h , h ] } 1 ,
v g 0 = c 2 2 ω h , Φ β h h , h = c Re ( e * × h ) z h , h ,
ω sc ω 0 + j ω ϵ j Δ ϵ j = ω 0 ( 1 j E j Δ ϵ j ) ,
E S = 1 2 e , e S h , h ,
E = 1 2 e , ϵ ̿ ϵ e LC h , h ,
E = 1 2 e , ϵ ̿ ϵ e LC h , h .
ϵ i ( ω sc ) ϵ i ( ω 0 ) + d ϵ i d ω ω 0 ( ω sc ω 0 ) = ϵ i ( ω 0 ) d ϵ i d ω ω 0 j E j Δ ϵ j ω 0 = ϵ i , 0 + Δ ϵ i .
v g 0 ϵ j = 2 ω ϵ j β = β ω E j = ( v g 0 E j + ω E j β ) .
v g sc = v g 0 j ( v g 0 E j + ω 0 E j β ) Δ ϵ j 1 + ω sc j E j d ϵ i d ω ω sc .
ϵ Si O 2 = 1 + j = 1 3 a j λ 2 λ 2 b j ,
ϵ , = ( A 1 , + A 2 , λ 2 + A 3 , λ 4 ) 2 ,

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