Abstract

A fully-vectorial mode solver based on the finite element method is employed in a combination with the downhill simplex method for the dispersion optimization of photonic crystal fibers made from highly nonlinear glasses. The nonlinear fibers are designed for telecom applications such as parametric amplification, wavelength conversion, ultra-fast switching and regeneration of optical signals. The optimization is carried out in terms of the zero dispersion wavelength, dispersion magnitude and slope, nonlinear coefficient and confinement loss in the wavelength range around 1.55µm. We restrict our work to the index-guiding fiber structures with a small number of hexagonally arrayed air holes.

©2008 Optical Society of America

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References

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  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
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    [Crossref]
  4. V. V. R. Kumar, A. K. George, J. C. Knight, and P. S. J. Russell, “Tellurite photonic crystal fiber,” Opt. Express 11, 2641–2645 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2641.
    [Crossref] [PubMed]
  5. H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. M. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5082.
    [Crossref] [PubMed]
  6. K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  14. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
    [Crossref]
  15. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-11-1310.
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  20. E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).
  21. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (2003).
    [Crossref] [PubMed]
  22. F. Poletti, K. Furusawa, Z. Yusoff, N. G. R. Broderick, and D. J. Richardson, “Nonlinear tapered holey fibers with high stimulated Brillouin scattering threshold and controlled dispersion,” J. Opt. Soc. Am. B 24, 2185–2194 (2007).
    [Crossref]
  23. Schott Optical Glass Catalogue 2007, http://www.schott.com.
  24. I. I. Oprea, H. Hesse, and K. Betzler, “Optical properties of bismuth borate glasses,” Opt. Mater. 26, 235–237 (2004).
    [Crossref]
  25. G. Ghosh, “Sellmeier-coefficients and chromatic dispersions for some tellurite glasses,” J. Am. Ceram. Soc. 78, 2828–2830 (1995).
    [Crossref]
  26. A. V. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003).
    [Crossref]
  27. http://www.mathworks.com.
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2007 (3)

2005 (1)

2004 (3)

2003 (7)

V. V. R. Kumar, A. K. George, J. C. Knight, and P. S. J. Russell, “Tellurite photonic crystal fiber,” Opt. Express 11, 2641–2645 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2641.
[Crossref] [PubMed]

P. Petropoulos, H. Ebendorff-Heidepriem, V. Finazzi, R. C. Moore, K. Frampton, D. J. Richardson, and T. M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref] [PubMed]

V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlineariry and confinement loss trade-offs,” J. Opt. Soc. Am. B 20, 1427–1436 (2003).
[Crossref]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-11-1310.
[Crossref] [PubMed]

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003), http://www.opticsexpress. org/abstract.cfm?URI=OPEX-11-8-843.
[Crossref] [PubMed]

G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (2003).
[Crossref] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003).
[Crossref]

2002 (4)

E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

V. V. R. Kumar, A. K. George, W. H. Reeves, J. C. Knight, P. S. J. Russell, F. G. Omenetto, and A. J. Taylor, “Extruded soft glass photonic crystal fiber for ultrabroad supercontinuum generation,” Opt. Express 10, 1520–1525 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-25-1520.
[PubMed]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

2000 (1)

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibers,” Electron. Lett. 36, 1998–2000 (2000).
[Crossref]

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

1997 (1)

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[Crossref]

1995 (1)

G. Ghosh, “Sellmeier-coefficients and chromatic dispersions for some tellurite glasses,” J. Am. Ceram. Soc. 78, 2828–2830 (1995).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995).

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Asimakis, S.

Asobe, M.

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[Crossref]

Betzler, K.

I. I. Oprea, H. Hesse, and K. Betzler, “Optical properties of bismuth borate glasses,” Opt. Mater. 26, 235–237 (2004).
[Crossref]

Bigot, L.

Broderick, N. G. R.

Chow, K. K.

Douay, M.

Ebendorff- Heidepriem, H.

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Ebendorff-Heidepriem, H.

Feng, X.

S. Asimakis, P. Petropoulos, F. Poletti, J. Y. Y. Leong, R. C. Moore, K. E. Frampton, X. Feng, W. H. Loh, and D. J. Richardson, “Towards efficient and broadband four-wave-mixing using short-length dispersion tailored lead silicate holey fibers,” Opt. Express 15, 596–601 (2007) http://www.opticsexpress.org/abstract. cfm?URI=OPEX-15-2-596.
[Crossref] [PubMed]

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Finazzi, V.

Frampton, K.

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. M. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5082.
[Crossref] [PubMed]

P. Petropoulos, H. Ebendorff-Heidepriem, V. Finazzi, R. C. Moore, K. Frampton, D. J. Richardson, and T. M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref] [PubMed]

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Frampton, K. E.

Furusawa, K.

George, A. K.

Ghosh, G.

G. Ghosh, “Sellmeier-coefficients and chromatic dispersions for some tellurite glasses,” J. Am. Ceram. Soc. 78, 2828–2830 (1995).
[Crossref]

Guobin, R.

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Hasegawa, T.

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Herrmann, J.

A. V. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003).
[Crossref]

Hesse, H.

I. I. Oprea, H. Hesse, and K. Betzler, “Optical properties of bismuth borate glasses,” Opt. Mater. 26, 235–237 (2004).
[Crossref]

Hewak, D. W.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibers,” Electron. Lett. 36, 1998–2000 (2000).
[Crossref]

Hewak, D.W.

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

Hsueh, Y. L.

E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

Hu, E. S.

E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003).
[Crossref]

Kazovsky, L. G.

E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

Kerrinckx, E.

Kiang, K. M.

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

Kikuchi, K.

Kim, J.-K.

J.-K. Kim, “Investigation of high-nonlinearity glass fibers for potential applications in ultrafast nonlinear fiber devices,” Ph.D. thesis (Virginia Polytechnic Institute and State University, Blacksburg, VA., 2005), http://en.scientificcommons.org/1530271.

Knight, J. C.

Koizumi, F.

Koshiba, M.

Kuhlmey, B.

Kumar, V. V. R.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Leong, J.

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Leong, J. Y. Y.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Loh, W.

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Loh, W. H.

Marhic, M. E.

E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

McPhedran, R.

Monro, T.

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Monro, T. M.

F. Poletti, V. Finazzi, T. M. Monro, N. G. R. Broderick, V. Tse, and D. J. Richardson, “Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers,” Opt. Express 13, 3728–3736 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3728.
[Crossref] [PubMed]

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. M. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5082.
[Crossref] [PubMed]

P. Petropoulos, H. Ebendorff-Heidepriem, V. Finazzi, R. C. Moore, K. Frampton, D. J. Richardson, and T. M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref] [PubMed]

V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlineariry and confinement loss trade-offs,” J. Opt. Soc. Am. B 20, 1427–1436 (2003).
[Crossref]

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibers,” Electron. Lett. 36, 1998–2000 (2000).
[Crossref]

Moore, R.

K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002).
[Crossref]

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Moore, R. C.

Nagashima, T.

Ohara, S.

Omenetto, F. G.

Oprea, I. I.

I. I. Oprea, H. Hesse, and K. Betzler, “Optical properties of bismuth borate glasses,” Opt. Mater. 26, 235–237 (2004).
[Crossref]

Petropoulos, P.

Poletti, F.

Quiquempois, Y.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Reeves, W. H.

Renversez, G.

Richardson, D.

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. M. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5082.
[Crossref] [PubMed]

J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

Richardson, D. J.

F. Poletti, K. Furusawa, Z. Yusoff, N. G. R. Broderick, and D. J. Richardson, “Nonlinear tapered holey fibers with high stimulated Brillouin scattering threshold and controlled dispersion,” J. Opt. Soc. Am. B 24, 2185–2194 (2007).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Computational window with applied boundary conditions and finite element mesh.
Fig. 2.
Fig. 2. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with lead silicate glass named in Tables 2 and 3 as Fiber F1.
Fig. 3.
Fig. 3. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with bismuth glass named in Tables 2 and 3 as Fiber F2.
Fig. 4.
Fig. 4. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with tellurite glass named in Tables 2 and 3 as Fiber F3.
Fig. 5.
Fig. 5. Chromatic dispersion curve (black) as a function of wavelength for the single-mode optimized PCF with chalcogenide glass named in Tables 2 and 3 as Fiber F4.
Fig. 6.
Fig. 6. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for optimized multi-mode PCF with lead silicate glass named in Tables 2 and 3 as Fiber F5.
Fig. 7.
Fig. 7. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with bismuth glass named in Tables 2 and 3 as Fiber F6.
Fig. 8.
Fig. 8. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with tellurite glass named in Tables 2 and 3 as Fiber F7.
Fig. 9.
Fig. 9. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with chalcogenide glass named in Tables 2 and 3 as Fiber F8.
Fig. 10.
Fig. 10. Nelder-Mead simplices after a reflection and an expansion step. The original simplex is shown with a dashed line.
Fig. 11.
Fig. 11. Nelder-Mead simplices after an outside contraction, an inside contraction, and a shrink. The original simplex is shown with a dashed line.

Tables (3)

Tables Icon

Table 1. Selected nonlinear glasses.

Tables Icon

Table 2. Geometrical parameters and confinement loss CL (calculated at 1.55 µm) of the optimized fibers.

Tables Icon

Table 3. Dispersion D, and dispersion slope S, effective area Aeff and nonlinear coefficient γ (calculated at 1.55 µm) of the optimized fibers.

Equations (8)

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γ = 2 π n 2 λ A eff ,
D = λ c d 2 Re [ n eff ] d λ 2 ,
CL = 8.686 Im [ 2 π λ · n eff ]
F = λ i = 1.5 μ m 1.6 μ m D ( λ i ) ,
n 2 = 1 + B 1 λ 2 λ 2 C 1 + B 2 λ 2 λ 2 C 2 + B 3 λ 2 λ 2 C 3 ,
n 2 = A 0 + A 1 x + A 2 x 2 + B 1 x λ 2 C 0 C 1 x D 0 λ 2 ,
n 2 = A + B λ 2 λ 2 C + D λ 2 λ 2 E ,
n 2 = 1 + B 1 λ 2 λ 2 C 1 2 + B 2 λ 2 λ 2 C 2 2 + B 2 λ 2 λ 2 C 3 2 + B 4 λ 2 λ 2 C 4 2 + B 5 λ 2 λ 2 C 5 2 ,

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