Abstract

A numerical approach based on the scalar finite element method is applied to analyse the modal properties of photonic crystal fibers having a solid core and a cladding region with either circular or non-circular microstructured holes. A correction which accounts for the polarization effects due to the large refractive index difference between silica materials and air holes is included in the analysis. Numerical results show that the proposed technique is an efficient and accurate alternative to vector ones.

© 2006 Optical Society of America

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  1. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, "All silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1999).
    [CrossRef]
  2. T. A. Birks, J. C. Knight, P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  3. W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, "Soliton effects in photonic crystal fibers at 850 nm," Electron. Lett. 36, 53-55 (2000).
    [CrossRef]
  4. K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, "Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding," Opt. Express 9, 714-720 (2001).
    [CrossRef] [PubMed]
  5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, "Anomalous dispersion in photonic crystal fiber," IEEE Photon. Technol. Lett. 12, 807-809 (2000).
    [CrossRef]
  6. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
    [PubMed]
  7. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003).
    [CrossRef] [PubMed]
  8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, M. V. Andrés, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999).
    [CrossRef]
  9. 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]
  10. 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).
    [PubMed]
  11. 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]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  13. S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, "Differential multipole method for microstructured optical fibers," J. Opt. Soc. Am. B 21, 1919-1928 (2004).
    [CrossRef]
  14. Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers, "Opt. Express 10, 853-864 (2002).
    [PubMed]
  15. 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]
  16. M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-330 (2001).
    [CrossRef]
  17. K. Saitoh and M. Koshiba, "Numerical modeling of photonic crystal fibers," J. Lightwave Technol. 23, 3580-3580 (2005).
    [CrossRef]
  18. T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers: An efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999).
    [CrossRef]
  19. C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000).
    [CrossRef] [PubMed]
  20. J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
    [CrossRef]
  21. V. H. Aristizabal, F. J. Vélez, and P. Torres, "Modelling of photonic crystal fibers with the scalar finite element method," in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Laser and their Applications, A. Marcano and J. L. Paz, eds., Proc. SPIE 5622, 849-854 (2004).
    [CrossRef]
  22. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004).
    [CrossRef] [PubMed]
  23. N. A. Mortensen, "Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers," Opt. Lett. 30, 1455-1457 (2005).
    [CrossRef] [PubMed]
  24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).
  25. H. P. Uranus and H. J. W. M. Hoekstr, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004).
    [CrossRef] [PubMed]
  26. 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]
  27. K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003).
    [CrossRef]

2005

2004

2003

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003).
[CrossRef] [PubMed]

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003).
[CrossRef]

2002

2001

2000

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

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

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]

C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000).
[CrossRef] [PubMed]

1999

1997

Aktin, D. M.

Andrés, M. V.

Andrés, P.

Arriaga, J.

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

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

Bassi, P.

Bellanca, G.

Bennett, P. J.

Bird, D. M.

Birks, T. A.

Botten, L. C.

Broderick, N. G. R.

Brown, T. G.

Campbell, S.

Chang, H. C.

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]

Cucinotta, A.

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]

Eggleton, B. J.

Ferrando, A.

Fogli, F.

Furusawa, K.

Hasegawa, T.

Hedley, T. D.

Hoekstr, H. J. W. M.

Kerbage, C. E.

Knight, J. C.

W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[PubMed]

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

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

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, "All silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1999).
[CrossRef]

T. A. Birks, J. C. Knight, P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

Koshiba, M.

K. Saitoh and M. Koshiba, "Numerical modeling of photonic crystal fibers," J. Lightwave Technol. 23, 3580-3580 (2005).
[CrossRef]

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003).
[CrossRef] [PubMed]

K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003).
[CrossRef]

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]

Kuhlmey, B. T.

Lægsgaard, J. N.

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Malinowski, A. N.

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Miret, J. J.

Monro, T. M.

Monro, T. N.

Mortensen, N. A.

N. A. Mortensen, "Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers," Opt. Lett. 30, 1455-1457 (2005).
[CrossRef] [PubMed]

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Nilsson, J.

Ortigosa-Blanch, A.

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

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

Pottage, J. M.

Price, J. H. V.

Qiu, M.

M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-330 (2001).
[CrossRef]

Reeves, W. H.

Renversez, G.

Richardson, D. J.

Riishede, J.

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Roberts, P. J.

Russell, P. St. J.

Saccomandi, L.

Sahu, J. K.

Saitoh, K.

K. Saitoh and M. Koshiba, "Numerical modeling of photonic crystal fibers," J. Lightwave Technol. 23, 3580-3580 (2005).
[CrossRef]

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003).
[CrossRef] [PubMed]

K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003).
[CrossRef]

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]

Sasaoka, E.

Selleri, S.

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]

Silvestre, E.

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

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, M. V. Andrés, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999).
[CrossRef]

Trillo, S.

Uranus, H. P.

Vincetti, L.

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]

Wadsworth, W. J.

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

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

Westbrook, P. S.

White, T. P.

Windeler, R. S.

Yu, C. P.

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]

Zhu, Z.

Zoboli, M.

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]

Electron. Lett.

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

IEEE Photon. Technol. Lett.

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

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]

K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003).
[CrossRef]

J. Lightwave Technol.

J. Opt. A: Pure Appl. Opt.

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

J. Opt. Soc. Am. B

J. Quantum Electron.

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]

Microwave Opt. Technol. Lett.

M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-330 (2001).
[CrossRef]

Opt. Express

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).
[PubMed]

Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers, "Opt. Express 10, 853-864 (2002).
[PubMed]

W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[PubMed]

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003).
[CrossRef] [PubMed]

K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, "Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding," Opt. Express 9, 714-720 (2001).
[CrossRef] [PubMed]

C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000).
[CrossRef] [PubMed]

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004).
[CrossRef] [PubMed]

H. P. Uranus and H. J. W. M. Hoekstr, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Quantum Electron.

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]

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

V. H. Aristizabal, F. J. Vélez, and P. Torres, "Modelling of photonic crystal fibers with the scalar finite element method," in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Laser and their Applications, A. Marcano and J. L. Paz, eds., Proc. SPIE 5622, 849-854 (2004).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

(a). PCF with triangular lattice of air holes. (b). Effective index of the fundamental mode for this structure where Λ=2.3 µm and d/Λ is taken as a parameter. The dotted and dash dotted curves are the predictions of scalar and corrected scalar solution, respectively. Comparison with vector results reported in Ref. [11] (solid lines).

Fig. 2.
Fig. 2.

(a). Cross section of the 3 annular-shaped holes PCF. (b). Effective index of the HE11- like mode for this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [25].

Fig. 3.
Fig. 3.

(a). Cross section of the cobweb PCF. (b). Effective index of the fundamental mode for this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [26].

Fig. 4.
Fig. 4.

(a). Cross section of the SPSM-PCF. (b). Modal dispersion for the slow-axis and fastaxis modes of this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [27].

Equations (6)

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

E ( x , y , z ) = { E t + E z z ̂ } exp ( i β z ) ,
{ t 2 + k 0 2 n 2 β 2 } E t = t { E t · t ln n 2 } ,
{ t 2 + k 0 2 n 2 β ˜ 2 } E ˜ t = 0 .
[ A ] { e } = β ˜ 2 [ B ] { e } ,
δ β 2 = β 2 β ˜ 2 = A ( t · E ˜ t ) E t · t ln n 2 d A A E ˜ t · E t d A ,
β i 2 β ˜ i 2 = A 1 n 2 x n i 2 x i E i x i E i d A A E i 2 d A ,

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