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Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers

Chin-Ping Yu and Hung-Chun Chang

Open Access Open Access

Abstract

Eigenvalue equations for solving full-vector modes of optical waveguides are formulated using Yee-mesh-based finite difference algorithms and incorporated with perfectly matched layer absorbing boundary conditions. The established method is thus able to calculate the complex propagation constants and the confinement losses of leaky waveguide modes. Proper matching of dielectric interface conditions through the Taylor series expansion of the fields is adopted in the formulation to achieve high numerical accuracy. The method is applied to the study of the holey fiber with triangular lattice, the two-core holey fiber, and the air-guiding photonic crystal fiber.

©2004 Optical Society of America

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Figures (14)
Fig. 1.
Fig. 1. The cross-section of an arbitrary waveguide problem with the PMLs placed at the edges of the computing domain.

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Fig. 2.
Fig. 2. Yee’s 2-D mesh for the FDFD method.

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Fig. 3.
Fig. 3. The situation of a dielectric interface lying between two sampled points.

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Fig. 4.
Fig. 4. Relative errors in the modal index of the fundamental TE-like mode for the square channel waveguide using the present method with three different schemes in dealing with the dielectric interface.

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Fig. 5.
Fig. 5. Relative errors in the modal index of the fundamental mode and the corresponding computation time for the strongly guiding optical fiber using the present method with the proper BC matching scheme and the stair-case approximation.

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Fig. 6.
Fig. 6. The computing window with PMLs for the holey fiber with three-ring air holes.

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Fig. 7.
Fig. 7. (a) The effective indices and (b) the losses of the x-polarized fundamental guided modes in the three-ring holey fiber with a=2.3 µm.

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Fig. 8.
Fig. 8. Modal birefringence of the one-ring HF using the present method with the proper BC matching scheme and the stair-case approximation.

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Fig. 9.
Fig. 9. The computing window with PMLs for the two-core holey fiber.

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Fig. 10.
Fig. 10. (a) The effective indices and (b) the losses of the even modes in the two-core holey fiber with r/a=0.25, 0.30, and 0.35.

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Fig. 11.
Fig. 11. (a) The effective indices and (b) the losses of the odd modes in the two-core holey fiber with r/a=0.25, 0.30, and 0.35.

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Fig. 12.
Fig. 12. The computing window with PMLs for a hollow PCF with six rings of air holes in the cladding.

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Fig. 13.
Fig. 13. (a) The effective indices of the surface modes and fundamental core modes of the hollow PCF of Fig. 11 with air filling fraction f=0.7. (b) Losses of the core modes and the surface modes.

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Fig. 14.
Fig. 14. The field distributions of the surface modes and fundamental core modes at points A, B, C, and D indicated in Fig. 13(a).

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Tables (2)

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Table 1. The values of sx and sy for the PML region I, II, and III.

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Table 2. The calculated modal index as a function of the number of grid points along the x-axis and the reference result is n eff,ref=1.445395345-3.15×10-8 j [8].

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Equations (29)

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∇ ′ × E ¯ = − j ω μ 0 H ¯ ∇ ′ × H ¯ = j ω ε 0 ε r E ¯
∇ ′ = x 1 s x ∂ ∂ x + y ̂ 1 s y ∂ ∂ y + z ̂ ∂ ∂ z
− j ω μ 0 ( H x ) i , j + 1 ⁄ 2 = [ 1 s y ∂ E z ∂ y + j β E y ] i , j + 1 ⁄ 2
− j ω μ 0 ( H y ) i + 1 ⁄ 2 , j = [ − j β E x − 1 s x ∂ E z ∂ x ] i + 1 ⁄ 2 , j
− j ω μ 0 ( H z ) i + 1 / 2 , j + 1 / 2 = [ 1 s x ∂ E y ∂ x − 1 s y ∂ E x ∂ y ] i + 1 / 2 , j + 1 / 2
j ω ε 0 ( ε x E x ) i + 1 / 2 , j = [ 1 s y ∂ H z ∂ y + j β H y ] i + 1 / 2 , j
j ω ε 0 ( ε y E y ) i , j + 1 / 2 = [ − j β H x − 1 s x ∂ H z ∂ x ] i , j + 1 / 2
j ω ε 0 ( ε z E z ) i , j = [ 1 s x ∂ H y ∂ x − 1 s y ∂ H x ∂ y ] i , j
s = 1 − j σ e ω ε 0 n 2 = 1 − j σ m ω μ 0
σ e ε 0 n 2 = σ m μ 0
σ e ( ρ ) = σ max ( ρ d ) m
R = exp [ − 2 σ max ε 0 cn ∫ 0 d ( ρ ⁄ d ) m d ρ ]
σ max = m + 1 2 ε 0 cn d ln 1 R
s = 1 − j 3 λ 4 π nd ( ρ d ) 2 ln 1 R .
− j ω μ 0 [ H x H y H z ] = [ 0 j β I A y − j β I 0 − A x − B y B x 0 ] [ E x E y E z ]
j ω ε 0 [ ε x 0 0 0 ε y 0 0 0 ε z ] [ E x E y E z ] = [ 0 j β I C y − j β I 0 − C x − D y D x 0 ] [ H x H y H z ]
( A x E z ) i , j = E z , ( i + 1 , j ) − E z , ( i , j ) s x , ( i + 1 ⁄ 2 , j ) Δ x ( A y E z ) i , j = E z , ( i , j + 1 ) − E z , ( i , j ) s y , ( i , j + 1 ⁄ 2 ) Δ y ( B x E y ) i , j = E y , ( i + 1 , j + 1 ⁄ 2 ) − E y , ( i , j + 1 ⁄ 2 ) s x , ( i + 1 ⁄ 2 , j + 1 ⁄ 2 ) Δ x ( B y E x ) i , j = E x , ( i + 1 ⁄ 2 , j + 1 ) − E x , ( i + 1 ⁄ 2 , j ) s y , ( i + 1 ⁄ 2 , j + 1 ⁄ 2 ) Δ y ( C x H z ) i , j = H z , ( i + 1 ⁄ 2 , j + 1 ⁄ 2 ) − H z , ( i − 1 ⁄ 2 , j + 1 ⁄ 2 ) s x , ( i , j + 1 ⁄ 2 ) Δ x ( C y H z ) i , j = H z , ( i + 1 ⁄ 2 , j + 1 ⁄ 2 ) − H z , ( i + 1 ⁄ 2 , j − 1 ⁄ 2 ) s y , ( i + 1 ⁄ 2 , j ) Δ y ( D x H y ) i , j = H y , ( i + 1 ⁄ 2 , j ) − H y , ( i − 1 ⁄ 2 , j ) S x , ( i , j ) Δ x ( D y H x ) i , j = H x , ( i , j + 1 ⁄ 2 ) − H x , ( i , j − 1 ⁄ 2 ) s y , ( i , j ) Δ y
Q [ H x H y ] = [ Q xx Q xy Q yx Q yy ] [ H x H y ] = β 2 [ H x H y ]
Q xx = − k 0 − 2 A x D y C x ε z − 1 B y + ( ε y + k 0 − 2 A x D x ) ( k 0 2 I + C y ε z − 1 B y )
Q yy = − k 0 − 2 A y D x C y ε z − 1 B x + ( ε x + k 0 − 2 A y D y ) ( k 0 2 I + C x ε z − 1 B x )
Q xy = k 0 − 2 A x D y ( k 0 2 I + C x ε z − 1 B x ) − ( ε y + k 0 − 2 A x D x ) C y ε z − 1 B x
Q yx = k 0 − 2 A y D x ( k 0 2 I + C y ε z − 1 B y ) − ( ε x + k 0 − 2 A y D y ) C x ε z − 1 B y .
∂ E y ∂ x ∣ i + 1 ⁄ 2 , j + 1 ⁄ 2 = 2 ( 2 r x + 1 ) Δ x { E y , L − E y , ( i , j + 1 ⁄ 2 ) }
E y , L = ε 2 ε 1 n y 2 + ε 2 n x 2 E y , R + ( ε 2 − ε 1 ) n x n y ε 1 n y 2 + ε 2 n x 2 E x , L
E y , R = ( 3 2 − r x ) E y , ( i + 1 , j + 1 ⁄ 2 ) − ( 1 2 − r x ) E y , ( i + 2 , j + 1 ⁄ 2 )
E x , L = 1 + r x 2 ( E x , ( i + 1 ⁄ 2 , j + 1 ) + E x , ( i + 1 ⁄ 2 , j ) ) − r x 2 ( E x , ( i − 1 ⁄ 2 , j + 1 ) + E x , ( i − 1 ⁄ 2 , j ) ) .
∂ H z ∂ x ∣ i + 1 , j + 1 ⁄ 2 = 2 ( r x − 1 ) Δ x { H z , ( i + 3 ⁄ 2 , j + 1 ⁄ 2 ) − H z , R } .
H z , R = H z , L = ( 3 2 − r x ) H z , ( i + 1 ⁄ 2 , j + 1 ⁄ 2 ) − ( 1 2 − r x ) H z , ( i − 1 ⁄ 2 , j + 1 ⁄ 2 ) .
ε z ( i , j ) = ε 1 · f + ε 2 · ( 1 − f )
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