Abstract

A finite-element-based vectorial optical mode solver is used to analyze microstructured optical waveguides. By employing 1st-order Bayliss-Gunzburger-Turkel-like transparent boundary conditions, both the real and imaginary part of the modal indices can be calculated in a relatively small computational domain. Results for waveguides with either circular or non-circular microstructured holes, solid- or air-core will be presented, including the silica-air Bragg fiber recently demonstrated by Vienne et al. (Post-deadline Paper PDP25, OFC 2004). The results of solid-core structures are in good agreement with the results of other methods while the results of air-core structure agree to the experimental results.

© 2004 Optical Society of America

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  1. J.C. Knight, T.A. Birks, P.S.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. Russell, �??Photonic Crystal Fibers,�?? Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  3. Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, �??Guiding optical light in air using an all-dielectric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999).
    [CrossRef]
  4. A. Ferrando, E. Silvestre, J.J. Miret, and P. Andres, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000).
    [CrossRef]
  5. J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.S.J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? Photonics Technol. Lett. 12, 807-809 (2000).
    [CrossRef]
  6. T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, �??Confinement losses in microstructured optical fibers,�?? Opt. Lett. 26, 1660-1662 (2001).
    [CrossRef]
  7. T.A. Birks, J.C. Knight, and P.S.J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  8. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748-779 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>.
    [CrossRef] [PubMed]
  9. I.M. Basset and A. Argyros, �??Elimination of polarization degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342</a>.
    [CrossRef]
  10. A. Argyros, N. Issa, I. Basset, and M. van Eijkelenborg, �??Microstructured optical fiber for single-polarization air guidance,�?? Opt. Lett. 29, 20-22 (2004).
    [CrossRef] [PubMed]
  11. A. Ferrando, E. Solvestre, J.J. Miret, P. Andres, M.V. Andres, �??Full-vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276-278 (1999).
    [CrossRef]
  12. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980-991 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980</a>.
    [CrossRef] [PubMed]
  13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, �??Multipole method for microstructured optical fibers. I. Formulation,�?? J. Opt. Soc. Am B 19, 2322-2330 (2002).
    [CrossRef]
  14. N.A. Issa and L. Poladian, �??Vector wave expansion method for leaky modes of microstructured optical fibers,�?? J. Lightwave Technol. 21, 1005-1012 (2003).
    [CrossRef]
  15. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, �??Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,�?? Opt. Express 10, 54-59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>.
    [CrossRef] [PubMed]
  16. K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,�?? J. Quantum Electron. 38, 927-933 (2002).
    [CrossRef]
  17. C.P. Yu and H.C. Chang, �??Applications of the finite difference mode solution method to photonic crystal structures,�?? Opt. Quantum Electron. 36, 145-163 (2004).
    [CrossRef]
  18. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers by finite element method,�?? Optical Fiber Technol. 6, 181-191 (2000).
    [CrossRef]
  19. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Holey fiber analysis through the finite element method,�?? Photonics Tech. Lett. 14, 1530-1532 (2002).
    [CrossRef]
  20. M. Koshiba, �??Full-vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Trans. Electron. E85-C, 881-888 (2002).
  21. D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, S. Selleri, �??Leakage properties of photonic crystal fibers,�?? Opt. Express 10, 1314-1319 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314</a>.
    [CrossRef] [PubMed]
  22. K. Saitoh and M. Koshiba, �??Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,�?? Opt. Express 11, 3100-3109 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100</a>
    [CrossRef] [PubMed]
  23. H.P. Uranus, H.J.W.M. Hoekstra, and E. van Groesen, �??Galerkin finite element scheme with Bayliss-Gunzburger-Turkel-like boundary conditions for vectorial optical mode solver,�?? J. Nonlinear Optical Phys. Materials (to be published).
  24. A. Bayliss, M. Gunzburger, and E. Turkel, �??Boundary conditions for the numerical solution of elliptic equations in exterior regions,�?? SIAM J. Appl. Math. 42, 430-451 (1982).
    [CrossRef]
  25. H.E. Hernandez-Figueroa, F.A. Fernandez, Y. Lu, and J.B. Davies, �??Vectorial finite element modeling of 2D leaky waveguides,�?? Trans. Magnetics 31, 1710-1713 (1995).
    [CrossRef]
  26. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fibers,�?? Opt. Express 11, 1310-1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>.
    [CrossRef] [PubMed]
  27. M. Koshiba and K. Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? Photonics Technol. Lett. 13, 1313-1315 (2001).
    [CrossRef]
  28. I.H. Malitson, �??Interspecimen comparison of the refractive index of fused silica,�?? J. Opt. Soc. Am. 55, 1205-1209 (1965).
    [CrossRef]
  29. L. Poladian, N.A. Issa, and T.M. Monro, �??Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,�?? Opt. Express 10, 449-454 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449</a>.
    [CrossRef] [PubMed]
  30. T.M. Monro D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, �??Modeling large air fraction holey optical fibers,�?? J. Lightwave Technol. 18, 50-56 (2000).
    [CrossRef]
  31. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, �??First demonstration of air-silica Bragg fiber,�?? Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22-27 Feb. 2004.

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. (3)

J. Nonlinear Optical Phys. Mat. (1)

H.P. Uranus, H.J.W.M. Hoekstra, and E. van Groesen, �??Galerkin finite element scheme with Bayliss-Gunzburger-Turkel-like boundary conditions for vectorial optical mode solver,�?? J. Nonlinear Optical Phys. Materials (to be published).

J. Opt. Soc. Am B (1)

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Renversez, C.M. de Sterke, and L.C. Botten, �??Multipole method for microstructured optical fibers. I. Formulation,�?? J. Opt. Soc. Am B 19, 2322-2330 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quantum Electron. (1)

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

OFC (1)

G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen, J.B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N.A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, �??First demonstration of air-silica Bragg fiber,�?? Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22-27 Feb. 2004.

Opt. Express (8)

L. Poladian, N.A. Issa, and T.M. Monro, �??Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,�?? Opt. Express 10, 449-454 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449</a>.
[CrossRef] [PubMed]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fibers,�?? Opt. Express 11, 1310-1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>.
[CrossRef] [PubMed]

F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, �??Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,�?? Opt. Express 10, 54-59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>.
[CrossRef] [PubMed]

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, S. Selleri, �??Leakage properties of photonic crystal fibers,�?? Opt. Express 10, 1314-1319 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314</a>.
[CrossRef] [PubMed]

K. Saitoh and M. Koshiba, �??Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,�?? Opt. Express 11, 3100-3109 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100</a>
[CrossRef] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980-991 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980</a>.
[CrossRef] [PubMed]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748-779 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>.
[CrossRef] [PubMed]

I.M. Basset and A. Argyros, �??Elimination of polarization degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342</a>.
[CrossRef]

Opt. Lett. (6)

Opt. Quantum Electron. (1)

C.P. Yu and H.C. Chang, �??Applications of the finite difference mode solution method to photonic crystal structures,�?? Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Optical Fiber Technol. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers by finite element method,�?? Optical Fiber Technol. 6, 181-191 (2000).
[CrossRef]

Photonics Tech. Lett. (1)

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

Photonics Technol. Lett. (2)

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.S.J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? Photonics Technol. Lett. 12, 807-809 (2000).
[CrossRef]

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

Science (1)

P. Russell, �??Photonic Crystal Fibers,�?? Science 299, 358-362 (2003).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

A. Bayliss, M. Gunzburger, and E. Turkel, �??Boundary conditions for the numerical solution of elliptic equations in exterior regions,�?? SIAM J. Appl. Math. 42, 430-451 (1982).
[CrossRef]

Trans. Magnetics (1)

H.E. Hernandez-Figueroa, F.A. Fernandez, Y. Lu, and J.B. Davies, �??Vectorial finite element modeling of 2D leaky waveguides,�?? Trans. Magnetics 31, 1710-1713 (1995).
[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

The structure with 6 circular holes, its computational domain and triangulation.

Fig. 2.
Fig. 2.

(a) The real part of the mode effective indices and (b) the dispersion parameter of the structure with 6 circular holes.

Fig. 3.
Fig. 3.

The confinement loss of the structure with circular holes. (a) Confinement loss of the first-ten modes of the 1-ring (6-hole) structure. (b) The effect of adding more rings of holes in the cladding.

Fig. 4.
Fig. 4.

Structure with 3 annular-shaped holes.

Fig. 5.
Fig. 5.

The transverse magnetic fields of the first-six modes of the structure with 3 annular-shaped holes. (a) HE 11 a -, (b) HE 11 b -, (c)) TE 01 -, (d) TM 01 -, (e) HE 21 a -, and (f) HE 21 b -like modes.

Fig. 6.
Fig. 6.

(a) The real part of the effective indices and (b) the dispersion parameter of the modes of structure with 3 annular-shaped holes.

Fig. 7.
Fig. 7.

(a) The imaginary part of the effective indices and (b) the confinement loss of the modes of structure with 3 annular-shaped holes.

Fig. 8.
Fig. 8.

The model of the air-core structure with 3 rings of annular-shaped holes in the cladding.

Fig. 9.
Fig. 9.

The transverse magnetic fields of the modes of the air-core structure: (a) HE 11 a -, (b) HE 11 b -, (c)) TE 01 -, (d) TM 01 -, (e) HE 21 a -, and (f) HE 21 b -like modes.

Fig. 10.
Fig. 10.

The real part of the field profiles of (a) & (b) the HE 11 a -like and (c) & (d) the TE 01 -like modes of the air-core structure.

Fig. 11.
Fig. 11.

The longitudinal component of the time averaged Poynting vector of (a) the HE 11 a - and (b) the TE 01 -like modes of the air-core structure. The color-coded scale is in arbitrary unit.

Tables (3)

Tables Icon

Table 1. Computational results of the structure with 6 circular holes and their comparison with other methods.

Tables Icon

Table 2. Calculated mode effective indices of the structure with 3 annular-shaped holes.

Tables Icon

Table 3. The computational results of the first-six modes of the air-core structure.

Equations (15)

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

[ y [ 1 n zz 2 ( x H y y H x ) ] x [ 1 n zz 2 ( x H y y H x ) ] ] [ 1 n yy 2 x ( x H x + y H y ) 1 n xx 2 y ( x H x + y H y ) ] + k 0 2 n eff 2 [ 1 n yy 2 H x 1 n xx 2 H y ] = k 0 2 [ H x H y ]
BoundaryElement e { Γ e 1 n zz 2 w y ( x H y y H x ) d y Γ e 1 n zz 2 w x ( x H y y H x ) d x
Γ e 1 n yy 2 w x ( x H x + y H y ) d y + Γ e 1 n xx 2 w y ( x H x + y H y ) d x }
+ InterfaceElement e { Γ int , e 1 n yy 2 w x ( x H x + y H y ) d y + Γ int , e 1 n xx 2 w y ( x H x + y H y ) d x }
+ TriangularElement e Ω e { 1 n zz 2 ( x w y y w x ) ( x H y y H x ) + [ x ( 1 n yy 2 w x ) + y ( 1 n xx 2 w y ) ] ( x H x + y H y )
+ k 0 2 n eff 2 ( 1 n yy 2 w x H x + 1 n xx 2 w y H y ) k 0 2 ( w x H x + w y H y ) } d x d y = 0
H ( r , θ ) Γ = [ H x H y ] Γ = p = 0 1 r p + 1 2 [ H x , p ( θ ) exp ( j k r , x r ) H y , p ( θ ) exp ( j k r , y r ) ]
B 1 ( [ H x H y ] ) Γ = { ( r + 1 2 r ) [ H x H y ] + j [ k r , x H x k r , y H y ] } Γ = O ( r 5 2 )
k r , x Γ = k 0 n xx 2 n eff 2 Γ
k r , y Γ = k 0 n yy 2 n eff 2 Γ
x H x Γ = r ̂ x ̂ ( j k r , x + 1 2 r ) H x Γ + O ( r 5 2 )
y H x Γ = r ̂ y ̂ ( j k r , x + 1 2 r ) H x Γ + O ( r 5 2 )
x H y Γ = r ̂ x ̂ ( j k r , y + 1 2 r ) H y Γ + O ( r 5 2 )
y H y Γ = r ̂ y ̂ ( j k r , y + 1 2 r ) H y Γ + O ( r 5 2 )
Dispersion = λ c 2 λ 2 [ Re ( n eff ) ]

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