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Mode classification and degeneracy in photonic crystal fibers

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

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Abstract

A novel full vector model based on the supercell lattice method is presented to analyze photonic crystal fiber (PCF). From symmetry analysis waveguide modes, we classify PCF modes into nondegenerate or degenerate pairs according to the minimum waveguide sectors and their appropriate boundary conditions. We describe how the modes of the PCF can be labeled by step-index fiber analogs, with the exception of modes that have the same symmetry as the PCF. When the doublet of the degenerate pairs both have the same symmetry as the PCF, they will be split into two nondegenerate modes.

© 2003 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. Reconstruction of the dielectric structure of a triangular lattice TIR-PCF with the parameters Λ=2.3µm, d/Λ=0.6, P 1=50, P 2=500. (a) Three-dimensional (3-D) refractive-index reconstruction. (b) Cross section along the y=0 axis of index reconstruction.

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Fig. 2.
Fig. 2. Minimum sectors for waveguides with C6ν symmetry. The modes of waveguides are classified into eight classes (p=1,….8). Solid lines indicate short-circuit boundary conditions, dashed lines indicate open-circuit boundary conditions.

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Fig. 3.
Fig. 3. Mode index of the first 24 modes in TIR-PCF with structural parameters Λ=2.3µm, d/Λ=0.8 at λ=633nm.

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Fig. 4.
Fig. 4. Total modal intensity distribution of HE11 mode (a) and 2-D electric vector distributions of polarization doublet modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.8, at wavelength λ=633nm.

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Fig. 5.
Fig. 5. Electric vector distributions for modes 3, 4, 5, 6 corresponding to (a) TE01, (b), (c) HE21, and (d) TM01 mode of step-index fiber.

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Fig. 6.
Fig. 6. Intensity distribution of the combination of modes 4 and 5 (HE21).

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Fig. 7.
Fig. 7. Electric vector distributions of mode 7–14.

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Fig. 8
Fig. 8 Intensity distributions of nondegenerate mode HE311, degenerate mode EH11, HE12 and EH21.

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Fig. 9.
Fig. 9. Modal intensity distribution of EH311 mode (a) and 2-D electric vector distributions of EH311 and EH312 modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.9, at wavelength λ=0.633µm.

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Fig. 10.
Fig. 10. Minimum sectors for waveguides with C4ν symmetry. The waveguides modes are divided into eight classes (p=1,….6). Solid lines indicate short-circuit boundary conditions; dashed lines indicate open-circuit boundary conditions.

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Fig. 11.
Fig. 11. Electric vector distributions for modes 3–6 of the square lattice PCF.

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Fig. 12.
Fig. 12. intensity distributions of nondegenerate modes HE211 and TE01 of a square lattice PCF.

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Tables (2)

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Table 1. Mode-index (n eff ), mode class (p), degeneracy, computation error (Δn) and label for first 14 modes

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Table 2. Mode index (n eff ), mode class (p), degeneracy, computation error (Δn), and label for first eight modes

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Equations (16)

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E → j ( x , y , z ) = [ e → j t ( x , y ) + e → j z ( x , y ) ] e j ( β j z − ω t ) ,
( ∇ t 2 − β j 2 + k 2 n 2 ) e x = − ∂ ∂ x ( e x ∂ In n 2 ∂ x + e y ∂ In n 2 ∂ y ) ,
( ∇ t 2 − β j 2 + k 2 n 2 ) e y = − ∂ ∂ y ( e x ∂ In n 2 ∂ x + e y ∂ In n 2 ∂ y ) ,
e x ( x , y ) = ∑ a , b = 0 F − 1 ε a b x ψ a ( x ) ψ b ( y ) ,
e y ( x , y ) = ∑ a , b = 0 F − 1 ε a b y ψ a ( x ) ψ b ( y ) ,
ψ i ( s ) = 2 − i 2 π − 1 4 ( i ) ! ω exp ⁡ ( − s 2 2 ω 2 ) H i ( s ω ) ,
n 2 ( x , y ) = ∑ a , b = 0 P 1 − 1 P 1 a b cos ⁡ 2 π a x l 1 x cos ⁡ 2 π b y l 1 y + ∑ a , b = 0 P 2 − 1 P 2 a b cos ⁡ 2 π a x l 2 x cos ⁡ 2 π b y l 2 y ,
In n 2 ( x , y ) = ∑ a , b = 0 P 1 − 1 P 1 a b In cos ⁡ 2 π a x l 1 x cos ⁡ 2 π b y l 1 y + ∑ a , b = 0 P 2 − 1 P 2 ab ln cos ⁡ 2 π a x l 2 x cos ⁡ 2 π b y l 2 y ,
L [ e x e y ] = [ I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) x I abcd ( 4 ) x I abcd ( 4 ) y I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) y ] [ e x e y ] = β j 2 [ e x e y ] ,
I abcd ( 1 ) = ∬ − ∞ + ∞ ψ a ( x ) ψ b ( y ) ∇ t 2 [ ψ c ( x ) ψ d ( y ) ] dx dy ,
I abcd ( 2 ) = ∬ − ∞ + ∞ n 2 ψ a ( x ) ψ b ( y ) ψ c ( x ) ψ d ( y ) dx dy ,
I abcd ( 3 ) x = ∬ − ∞ + ∞ ψ a ( x ) ψ b ( y ) ∂ ∂ x [ ψ c ( x ) ψ d ( y ) ∂ In n 2 ∂ x ] dx dy ,
I abcd ( 3 ) y = ∬ − ∞ + ∞ ψ a ( x ) ψ b ( y ) ∂ ∂ y [ ψ c ( x ) ψ d ( y ) ∂ In n 2 ∂ y ] dx dy ,
I abcd ( 4 ) x = ∬ − ∞ + ∞ ψ a ( x ) ψ b ( y ) ∂ ∂ x [ ψ c ( x ) ψ d ( y ) ∂ In n 2 ∂ y ] dx dy ,
I abcd ( 4 ) y = ∬ − ∞ + ∞ ψ a ( x ) ψ b ( y ) ∂ ∂ y [ ψ c ( x ) ψ d ( y ) ∂ In n 2 ∂ x ] dx dy .
e ⇀ t = i k 2 n 2 − β 2 { β ∇ t e z − ( μ 0 ε 0 ) 1 2 k z ̂ × ∇ t h z } .
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