Abstract

We investigated the modal properties of complex refractive-index core photonic crystal fibers (PCFs) with the supercell model. The validity of the approach is shown when we compare our results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCFs, which means that the power flows out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating the gain or loss coefficients of these fibers. The numerical results obtained by the approximation formula agree well with the full-vectorial results.

© 2004 Optical Society of America

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References

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Appl. Opt.

Electron. Lett.

K. G. Hougaard, J. Broeng, and A. Bjarklev, �??Low pump power photonic crystal fibre amplifiers,�?? Electron. Lett., 39, 599-600 (2003)
[CrossRef]

Electronics Lett.

W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J Arriaga, "Yb3+-doped photonic crystal fibre laser," Electronics Lett. 36, 1452 �?? 1454 (2000)
[CrossRef]

IEEE J. Quantum Electron.

R. Singh Sunanda, and E. Khular Sharma, �??Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,�?? IEEE J. Quantum Electron. 37, 635-640 (2001).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818-823 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818</a>
[CrossRef] [PubMed]

W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, �??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]

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]

W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542</a>
[CrossRef] [PubMed]

B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where�??s the edge?" Opt. Express 10, 1285-1290 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a>
[CrossRef] [PubMed]

Opt. Lett.

Science

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, �??An all-dielectric coaxial waveguide,�?? Science 289, 415-419(2000).
[CrossRef] [PubMed]

Other

A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

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Figures (8)

Fig. 1.
Fig. 1.

Cross section of PCF with complex refractive-index core.

Fig. 2.
Fig. 2.

Real part and imaginary part of modal field HE11.

Fig. 3.
Fig. 3.

Real part and imaginary part of modal field TM01.

Fig. 4.
Fig. 4.

Phase distribution of mode HE11x.

Fig. 5.
Fig. 5.

Phase distribution along x=0 axis with different ni for d/Λ=0.3, λ=1550 nm.

Fig. 6.
Fig. 6.

Real part (ner ) and imaginary part (nei ) of the mode index of fundamental mode versus relative air hole size d/Λ at 980, 1310, and 1550 nm.

Fig. 7.
Fig. 7.

(a) Mode indices and difference of fibers A and B, (b) power confinement factors and difference of fibers A and B. For mode index ne and power confinement factor Γ, the solid curves indicate fiber A and the curves with circles indicate fiber B.

Fig. 8.
Fig. 8.

(a) Imaginary part of the mode indices versus relative air hole size d/Λ at different wavelengths. (b) Imaginary part of the mode indices versus wavelength with different relative air hole size d/Λ. The solid curves show the numerical results from full-vectorial method, and dotted curves with circles indicate the results from Eq. (11).

Tables (1)

Tables Icon

Table 1. Comparison of complex mode index ne between Ref. [5] and our method for complex index step fibers

Equations (12)

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e x ( x , y ) mn = a , b = 0 F 1 ε ab x ψ 2 a + m ( x ) ψ 2 b + n ( y ) ,
e y ( x , y ) mn ¯ = a , b = 0 F 1 ε ab y ψ 2 a + m ¯ ( x ) ψ 2 b + n ¯ ( y ) ,
ε ( r ) = ε ( x , y ) = a , b = 0 P 1 P ab cos 2 π ax D x cos 2 π by D y ,
ln ε ( r ) = ln ε ( x , y ) = a , b = 0 P 1 P ab ln cos 2 π ax D x cos 2 π by D y ,
L mn [ ε x ε y ] [ [ M 1 + k 2 M 2 + M 3 x ] mn [ M 4 x ] mn [ M 4 y ] m n ¯ [ M 1 + k 2 M 2 + M 3 y ] m n ¯ ] [ ε x ε y ] = β 2 [ ε x ε y ] ,
L mn * [ ε x * ε y * ] = β 2 * [ ε x * ε y * ] .
γ = 2 β i = 2 k ( ε 0 μ 0 ) 1 2 A n r n i e 2 dA Re { A e × h * · z ̂ dA } ,
A n r 2 e × h * · z ̂ dA = β r k ( ε 0 μ 0 ) 1 2 A n r 2 e 2 dA ,
A e × h * · z ̂ dA n er ( ε 0 μ 0 ) 1 2 A e 2 dA ,
n ei = β i / k 1 n er A n r n i e 2 dA A e 2 dA ,
n ei 1 n er k n kr n ki Γ k ,
n ei Γ n corer n er n corei ,

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