Abstract

Chromatic dispersion characteristics of nonlinear photonic crystal fibers are, for the first time to our knowledge, theoretically investigated. A self-consistent numerical approach based on the full-vector finite-element method in terms of all the components of electric fields is described for the steady-state analysis of axially-nonsymmetrical nonlinear optical fibers. Electric fields obtained with this approach can be directly utilized for evaluating nonlinear refractive index distributions. To eliminate nonphysical, spurious solutions and to accurately model curved boundaries of circular air holes, curvilinear hybrid edge/nodal elements are introduced. It is found from the numerical results that under high optical intensity, chromatic dispersion characteristics become different from those of the linear state due to optical Kerr-effect nonlinearity, especially in short wavelength region.

© 2003 Optical Society of America

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References

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Electron. Lett. (5)

J.C. Knight, T.A. Birks, R.F. Cregan, P.St.J. Russell, and J.-P. de Sandro, �??Large mode area photonic crystal fiber,�?? Electron. Lett. 34, 1347-1348 (1998).
[CrossRef]

M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St. J. Russell, �??Experimental measurement of group velocity in photonic crystal fiber,�?? Electron. Lett. 35, 63-64 (1999).
[CrossRef]

A.W. Snyder, Y. Chen, L. Poladian, and D.J. Mitchell, �??Fundamental mode of highly nonlinear fibers,�?? Electron. Lett. 26, 643-644 (1990).
[CrossRef]

R.A. Sammut and C. Pask, �??Variational approach to nonlinear waveguides-gaussian approximations,�?? Electron. Lett. 26, 1131-1132 (1990).
[CrossRef]

W.J. Wadsworth, J.C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P.St.J. Russell, �??Soliton effects in photonic crystal fibres at 850 nm,�?? Electron. Lett. 36, 53-55 (2000).
[CrossRef]

IEEE J. Quantum Electron. (2)

S.S.A. Obayya, B.M.A. Rahman, K.T.V. Grattan, and H.A. El-Mikati, �??Full-vectorial finite-element solution of nonlinear bistable optical waveguides,�?? IEEE J. Quantum Electron. 38, 1120-1125 (2002).
[CrossRef]

K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Holey fiber analysis through the finite element method,�?? IEEE Photon. Technol. Lett. 14, 1530-1532 (2002).
[CrossRef]

R.D. Ettinger, F.A. Fernandez, B.M.A. Rahman, and J.B. Davies, �??Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,�?? IEEE Photon. Technol. Lett. 3, 147-149 (1991).
[CrossRef]

M. Koshiba and K. Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315 (2001).
[CrossRef]

IEEE Trans. Micorwave Theory Tech. (1)

S. Selleri and M. Zoboli, �??An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,�?? IEEE Trans. Micorwave Theory Tech. 43, 887-892 (1995).
[CrossRef]

IEICE Trans. Electron. (1)

M. Koshiba, �??Full-vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Trans. Electron. E85-C, 881-888 (2002).

J. Lightwave Technol. (4)

K. Okamoto and E.A.J. Marcatili, �??Chromatic dispersion characteristics of fibers with optical Kerr-effect,�?? J. Lightwave Technol. 7, 1988-1994 (1989).
[CrossRef]

H.Y. Lin and H.-C. Chang, �??An efficient method for determining the chromatic dispersion characteristics of nonlinear single-mode optical fibers,�?? J. Lightwave Technol. 10, 1188-1195 (1992).
[CrossRef]

M.J. Holmes, D.M. Spirit, and F.P. Payne, �??New gaussian-based approximation for modeling non-linear effects in optical fibers,�?? J. Lightwave Technol. 12, 193-201 (1994).
[CrossRef]

M. Koshiba and Y. Tsuji, �??Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,�?? J. Lightwave Technol. 18, 737-743 (2000).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (1)

M. Zoboli, F.Di Pasquale, and S. Selleri, �??Full-vectorial and scalar solutions of nonlinear optical fibers,�?? Opt. Commun. 97, 11-15 (1993).
[CrossRef]

Opt. Fiber Technol. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers,�?? Opt. Fiber Technol. 6, 181- (2000).
[CrossRef]

Opt. Lett. (5)

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Figures (7)

Fig. 1.
Fig. 1.

Curvilinear high-order hybrid edge/nodal element.

Fig. 2.
Fig. 2.

Group velocity dispersion of circular-core optical fiber with core nonlinearlity.

Fig. 3.
Fig. 3.

Holey fiber.

Fig. 4.
Fig. 4.

Chromatic dispersion characteristics of nonlinear holey fibers with d/Λ=0.9 for different hole pitches of (a) Λ=1.0 µm, (b) Λ=1.5 µm, (c) Λ=2.0 µm, (d) Λ=2.5 µm.

Fig. 5.
Fig. 5.

Effective core areas of nonlinear holey fibers with Λ=1.5 µm and d/Λ=0.9.

Fig. 6.
Fig. 6.

Chromatic dispersion characteristics of nonlinear holey fibers with Λ=1.5 µm for different hole pitches of (a) d/Λ=0.5, (b) d/Λ=0.6, (c) d/Λ=0.7, (d) d/Λ=0.8.

Fig. 7.
Fig. 7.

Zero-dispersion wavelength of nonlinear holey fibers as a function of (a) hole pitch Λ and (b) of ratio of diameter to hole pitch d/Λ.

Equations (23)

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

× ( × E ) k 0 2 n 2 E = 0
E ( x , y , z ) = e ( x , y ) exp ( j βz )
n = n ( x , y ; e 2 ) .
E = ( i x e x + i y e y + i z e z ) exp ( j βz )
= ( i x { U } T + i y { V } T ) { e t } e exp ( j βz ) + i z j β { N } T { e z } e exp ( j βz )
[ K ( e ) ] { e } β 2 [ M ( e ) ] { e } = { 0 }
{ e } = [ { e t } { e z } ]
[ K ( e ) ] = [ [ K tt ( e ) ] [ 0 ] [ 0 ] [ 0 ] ]
[ K tt ( e ) ] = e e [ { U } y { U } T y + { U } y { V } T x + { V } x { V } T y
{ V } x { V } T x + k 0 2 n 2 { U } { U } T + k 0 2 n 2 { V } { V } T ]
[ M ( e ) ] = [ [ M tt ] [ M tz ] [ M zt ] [ M zz ( e ) ] ]
[ M tt ] = e e [ { U } { U } T + { V } { V } T ] dx dy
[ M tz ] = [ M zt ]
= e e [ { U } { N } T x + { V } { N } T y ] dx dy
[ M zz ( e ) ] = e e [ { N } x { N } T x + { N } y { N } T y k 0 2 n 2 { N } { N } T ] dx dy
P = 1 2 ( E × H * ) · i z dx dy
= β 2 k 0 Z 0 ( { e t } T [ M tt ] { e t } + { e t } T [ M tz ] { e z } )
{ e } = { e } P P
P = β 2 k 0 Z 0 ( { e t } T [ M tt ] { e ' t } + { e t } T [ M tz ] { e z } )
g ( V ) = V d 2 ( V b ) d V 2
n 2 = n co 2 + n co 2 n 2 Z 0 e 2 .
n 2 = n L 2 + n L 2 n 2 Z 0 e 2
A eff = ( E 2 dx dy ) 2 E 4 dx dy

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