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Full-vectorial analysis of complex refractive-index photonic crystal fibers

Ren Guobin, Wang Zhi, Lou Shuqin, Liu Yan, and Jian Shuisheng

Open Access Open Access

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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Figures (8)
Fig. 1.
Fig. 1. Cross section of PCF with complex refractive-index core.

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Fig. 2.
Fig. 2. Real part and imaginary part of modal field HE11.

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Fig. 3.
Fig. 3. Real part and imaginary part of modal field TM01.

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Fig. 4.
Fig. 4. Phase distribution of mode HE11x.

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Fig. 5.
Fig. 5. Phase distribution along x=0 axis with different ni for d/Λ=0.3, λ=1550 nm.

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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.

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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.

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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).

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Tables (1)

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

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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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