Abstract

In order to simply design a photonic crystal fiber (PCF), we provide numerically based empirical relations for V parameter and W parameter of PCFs only dependent on the two structural parameters — the air hole diameter and the hole pitch. We demonstrate the accuracy of these expressions by comparing the proposed empirical relations with the results of full-vector finite element method. Through the empirical relations we can easily evaluate the fundamental properties of PCFs without the need for numerical computations.

© 2005 Optical Society of America

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References

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  1. P.St.J.  Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  2. S.G.  Johnson, J.D.  Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
    [CrossRef] [PubMed]
  3. T.P.  White, B.T.  Kuhlmey, R.C.  McPhedran, D.  Maystre, G.  Renversez, C.M.  de Sterke, L.C.  Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [CrossRef]
  4. M.  Koshiba, “Full-vector analysis of photonic crystal fibers using the finite element method,” IEICE Trans. Electron. E85-C, 881–888 (2002).
  5. K.  Saitoh, M.  Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933 (2002).
    [CrossRef]
  6. M.  Koshiba, K.  Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
    [CrossRef] [PubMed]
  7. T.A.  Birks, J.C.  Knight, P.St.J.  Russell, “Endlessy single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
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  9. M.D.  Nielsen, N.A.  Mortensen, “Photonic crystal fiber design based on the V-parameter,” Opt. Express 11, 2762–2768 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762.
    [CrossRef] [PubMed]
  10. N.A.  Mortensen, M.D.  Nielsen, J.R.  Folkenberg, A.  Petersson, H.R.  Simonsen, “Improved large-mode-area endlessly single-mode photonic crystal fibers,” Opt. Lett. 28, 393–395 (2003).
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  11. D.  Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
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2004 (1)

2003 (4)

2002 (3)

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

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

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

1997 (1)

1971 (1)

Birks, T.A.

Botten, L.C.

de Sterke, C.M.

Folkenberg, J.R.

Gloge, D.

Hansen, K.P.

Joannopoulos, J.D.

S.G.  Johnson, J.D.  Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
[CrossRef] [PubMed]

Johnson, S.G.

S.G.  Johnson, J.D.  Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
[CrossRef] [PubMed]

Knight, J.C.

Koshiba, M.

M.  Koshiba, K.  Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
[CrossRef] [PubMed]

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

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

Kuhlmey, B.T.

Maystre, D.

McPhedran, R.C.

Mortensen, N.A.

Nielsen, M.D.

Petersson, A.

Renversez, G.

Russell, P.St.J.

Saitoh, K.

M.  Koshiba, K.  Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
[CrossRef] [PubMed]

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

Simonsen, H.R.

White, T.P.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

K.  Saitoh, M.  Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933 (2002).
[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. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (4)

Science (1)

P.St.J.  Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Other (1)

S.G.  Johnson, J.D.  Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
[CrossRef] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

Index-guiding photonic crystal fiber.

Fig. 2.
Fig. 2.

Effective V parameter as a function of λ/Λ.

Fig. 3.
Fig. 3.

Effective cladding index as a function of λ/Λ.

Fig. 4.
Fig. 4.

Effective W parameter as a function of λ/Λ.

Fig. 5.
Fig. 5.

Effective index of the fundamental mode neff as a function of λ/Λ.

Fig. 6.
Fig. 6.

Chromatic dispersion as a function of wavelength for (a) Λ=2.0 µm, (b) Λ=2.5 µm, and (c) Λ=3.0 µm. Solid curves, results of empirical relations; dashed curves, results of vector FEM.

Tables (2)

Tables Icon

Table 1. Fitting coefficients in Eq. (6).

Tables Icon

Table 2. Fitting coefficients in Eq. (8).

Equations (9)

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V = 2 π λ a eff n co 2 n FSM 2 = U 2 + W 2
U = 2 π λ a eff n co 2 n eff 2
W = 2 π λ a eff n eff 2 n FSM 2
V eff = 2 π λ Λ n eff 2 n FSM 2
V ( λ Λ , d Λ ) = A 1 + A 2 1 + A 3 exp ( A 4 λ Λ )
A i = a i 0 + a i 1 ( d Λ ) b i 1 + a i 2 ( d Λ ) b i 2 + a i 3 ( d Λ ) b i 3
W ( λ Λ , d Λ ) = B 1 + B 2 1 + B 3 exp ( B 4 λ Λ )
B i = c i 0 + c i 1 ( d Λ ) d i 1 + c i 2 ( d Λ ) d i 2 + c i 3 ( d Λ ) d i 3
D = λ c d 2 n eff d λ 2 + D m

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