Abstract

Approximate empirical relations for nonlinear photonic crystal fibers (PCFs) are newly proposed. Replacing a PCF with a conventional step-index fiber, closed form expressions for the effective refractive index and the effective core area of nonlinear PCFs are derived. To define the equivalent cladding index, the effective index of the so-called fundamental space-filling mode, which is calculated using empirical relations for the effective normalized frequency, is introduced, and thus, nonlinear guided waves propagating in PCFs can be easily characterized without the need for numerical computations. The validity of the method proposed here is ensured by comparing the calculated results with those obtained by a full-vector finite-element method.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2005 (2)

2004 (2)

2003 (2)

2000 (1)

1991 (2)

1964 (1)

R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Andrees, P.

Binosi, D.

Chen, Y.

Chiao, R.Y.

R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

de Cordoba, P.F.

Ferrando, A.

Fibich, G.

Fujisawa, T.

Gaeta, A.L.

Garmire, E.

R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Koshiba, M.

Monsoriu, J.A.

Pask, C.

Saitoh, K.

Sammut, R.A.

Townes, C.H.

R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Zacares, M.

J. Opt. Soc. Am. B (2)

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

(a) A geometry of PCF and (b) its equivalent classical optical fiber model.

Fig. 2.
Fig. 2.

Effective refractive index with nonsaturable nonlinearity as a function of P/Pc for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

Fig. 3.
Fig. 3.

Field distributions of nonlinear PCFs with d/Λ=0.4 and P/Pc =0.9, for λ/Λ=(a) 0.1 and (b) 0.4.

Fig. 4.
Fig. 4.

Errors of approximate empirical relations as a function of the normalized power for λ/Λ=(a) 0.1, (b) 0.4, and (c) 1.5.

Fig. 5.
Fig. 5.

Normalized effective core area with nonsaturable nonlinearity as a function of P/Pc for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

Fig. 6.
Fig. 6.

Effective refractive index with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.

Fig. 7.
Fig. 7.

Effective refractive index with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4

Fig. 8.
Fig. 8.

Normalized effective core area with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.

Fig. 9.
Fig. 9.

Normalized effective core area with saturable nonlinearity as a function of P/Pc for d/Λ(a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4.

Equations (22)

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V ( λ Λ , d Λ ) = 2 π λ a eff n co 2 n cl 2
= A 1 ( d Λ ) + A 2 ( d Λ ) 1 + A 3 ( d Λ ) exp { A 4 ( d Λ ) λ Λ }
n 2 = n L 2 + n L 2 n 2 Z 0 E 2
ϕ = A exp ( r 2 w 2 )
n eff 2 = n co 2 1 k 0 2 ( 1 a eff 2 + 2 w 2 ) + n co 2 n 2 w 2 A 2 8 Z 0 ( 1 a eff 2 + 4 w 2 ) .
w = a eff ln V NL
V NL = V 1 k 0 2 n co 2 n 2 w 2 A 2 8 Z 0 .
P = 0 2 π 0 n co ϕ 2 2 Z 0 r d r d θ = π n co w 2 A 2 4 Z 0
P c = λ 2 2 π n co n 2 ,
n eff 2 = n co 2 ( λ 2 π a eff ) 2 [ 1 + 2 ln V NL P P c ( 1 + 4 ln V NL ) ] .
A eff = π w 2 = π a eff 2 ln V NL .
V NL = V 1 P P c .
n 2 = ( n sat 2 n cl 2 ) { 1 exp [ n cl 2 n 2 Z 0 ( n sat 2 n cl 2 ) E 2 ] }
n eff 2 = n co 2 ( λ 2 π a eff ) 2 [ 1 R 0 2 + V 2 exp ( 1 R 0 2 ) ]
+ ( λ 2 π a eff ) 2 V sat 2 { 1 1 Q [ 1 exp ( Q ) ] }
R 0 = w 2 a eff ,
V sat = k 0 a eff n sat 2 n cl 2 ,
Q = 4 P V sat 2 R 0 2 P c ,
1 V 2 [ 1 P P c m = 0 ( Q ) m ( 1 + 0.5 m ) 2 m ! ] = exp ( 1 R 0 2 )
A eff = 2 π R 0 2 a eff 2 = 2 π R 0 2 3 Λ 2 .
Δ n eff = n eff n eff , FEM n eff , FEM
n eff 2 = n co 2 1 2 ( 3 λ 2 π Λ ) 2 .

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