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Full-vectorial finite-difference analysis of microstructured optical fibers

Zhaoming Zhu and Thomas G. Brown

Open Access Open Access

Abstract

In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee’s 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.

©2002 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. (a) Yee’s 2-D mesh; (b) Mesh cells across a curved interface.

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Fig. 2.
Fig. 2. Relative error in the calculated fundamental mode index neff of a step-index circular optical fiber. The fiber has a core diameter 6 μm, a core refractive index of 1.45 at wavelength 1.5 μm, and air cladding with unity refractive index. The calculation window is chosen to be the first quadrant of the fiber cross section with a computation window size of 6μm by 6 μm. The left boundary is magnetic wall, the bottom is electric wall; all others are zero-value boundaries.

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Fig. 3.
Fig. 3. (a) Schematic of an air-hole assisted optical fiber; (b) Relative errors of calculated fundamental mode index from different mode solvers when the number of grids along the x-axis is varied (fiber parameters are the same as in Table 2); (c) Magnetic field plots of the fundamental mode (y-polarization); (d) Calculated GVD curves (material dispersion not considered).

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Fig. 4.
Fig. 4. (a) Calculation window of the holey fiber; (b) Comparison between different FD mode solvers (Λ=2.3 μm, r a=0.5 μm, silica refractive index of 1.45 is assumed at λ=1.5 μm, calculation window of 3Λ by 3Λ is the first quadrant bounded by C1~C4. C1: magnetic wall, C2~C4: electric wall); (c) Field plots of the y-polarized fundamental mode; (d) Calculated effective modal index β/k 0, nFSM and GVD (Silica refractive index of 1.45 is assumed at all wavelengths).

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Fig. 5.
Fig. 5. Convergence of modal birefringence of the fundamental modes in (a) air-hole assisted optical fiber and in (b) holey optical fiber. The parameters for the two fibers are listed in Table 2 and Fig. 4(b), respectively.

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Fig. 6.
Fig. 6. (a) Calculated effective index and GVD of the x- and y-polarized fundamental modes for a holey fiber with elliptical air holes. The air holes have their major axis along the y-direction. The semimajor axis is 0.8 μm; the semiminor axis is 0.5 μm. Other calculation parameters are same as in Fig. 4. The effective indices of x-polarized and y-polarized FSM are also shown. (b) Contour plots of major magnetic field components for x-polarized (left) and y-polarized (right) fundamental modes at wavelength of 1.5 μm.

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Tables (3)

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Table 1. Calculated fundamental mode indices of a step-index fiber from different FD mode solvers. The fiber parameters are the same as in Fig. 2. The analytical solution for the fundamental mode index is 1.438604.

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Table 2. Calculated fundamental mode indices for air-hole-assisted optical fiber shown in Fig. 3(a). Core index 1.45, silica cladding index 1.42, r 0=2μm, r a=2μm, Λ=5μm, wavelength=1.5 μm, first quadrant window size 8μm by 8μm, C1: magnetic wall, C2~C4: electric wall. The multipole analysis [32] gives neff =1.4353607.

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Table 3. Calculated fundamental mode indices for the holey fiber shown in Fig. 4(a) from different methods. The fiber parameters are same as in Fig. 4(b). The number of grids along both x-axis and y-axis is 120.

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Equations (32)

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( ∇ t 2 + k 0 2 ε r ) E t + ∇ t ( ε r − 1 ∇ t ε r · E t ) = β 2 E t
( ∇ t 2 + k 0 2 ε r ) H t + ε r − 1 ∇ t ε r × ( ∇ t × H t ) = β 2 H t
i k 0 H x = ∂ E z ∂ y − iβ E y ,
i k 0 H y = iβ E x − ∂ E z ∂ x ,
i k 0 H z = ∂ E y ∂ x − ∂ E x ∂ y ,
− i k 0 ε r E x = ∂ H z ∂ y − iβ H y ,
− i k 0 ε r E y = iβ H x − ∂ H z ∂ x ,
− i k 0 ε r E z = ∂ H y ∂ x − ∂ H x ∂ y .
i k 0 H x ( j , l ) = [ E z ( j , l + 1 ) − E z ( j , l ) ] Δ y − iβ E y ( j , l ) ,
i k 0 H y ( j , l ) = iβ E x ( j , l ) − [ E z ( j + 1 , l ) − E z ( j , l ) ] Δ x ,
i k 0 H z ( j , l ) = [ E y ( j + 1 , l ) − E y ( j , l ) ] Δ x − [ E x ( j , l + 1 ) − E x ( j , l ) ] Δ y ,
− i k 0 ε rx ( j , l ) E x ( j , l ) = [ H z ( j , l ) − H z ( j , l − 1 ) ] Δ y − iβ H y ( j , l ) ,
− i k 0 ε ry ( j , l ) E y ( j , l ) = iβ H x ( j , l ) − [ H z ( j , l ) − H z ( j − 1 , l ) ] Δ x ,
− i k 0 ε rz ( j , l ) E z ( j , l ) = [ H y ( j , l ) − H y ( j − 1 , l ) ] Δ x − [ H x ( j , l ) − H x ( j , l − 1 ) ] Δ y ,
ε rx ( j , l ) = [ ε r ( j , l ) + ε r ( j , l − 1 ) ] 2 ,
ε ry ( j , l ) = [ ε r ( j , l ) + ε r ( j − 1 , l ) ] 2 ,
ε rz ( j , l ) = [ ε r ( j , l ) + ε r ( j − 1 , l − 1 ) + ε r ( j , l − 1 ) + ε r ( j − 1 , l ) ] 4 .
i k 0 [ H x H y H z ] = [ 0 − iβ I U y iβ I 0 − U x − U y U x 0 ] [ E x E y E z ] ,
− i k 0 [ ε rx 0 0 0 ε ry 0 0 0 ε rz ] [ E x E y E z ] = [ 0 − iβ I V y iβ I 0 − V x − V y V x 0 ] [ H x H y H z ] ,
U x = 1 Δ x [ − 1 1 − 1 1 ⋱ ⋱ ⋱ ⋱ − 1 1 − 1 ] , U y = 1 Δ y [ − 1 1 − 1 ⋱ ⋱ 1 ⋱ − 1 1 ] ,
V x = 1 Δ x [ 1 − 1 1 − 1 ⋱ ⋱ ⋱ − 1 1 − 1 1 ] , V y = 1 Δ y [ 1 1 ⋱ − 1 ⋱ ⋱ 1 − 1 1 ] .
P [ E x E y ] = [ P xx P xy P yx P yy ] [ E x E y ] = β 2 [ E x E y ] ,
P xx = − k 0 − 2 U x ε rz − 1 V y V x U y + ( k 0 2 I + U x ε rz − 1 V x ) ( ε rx + k 0 − 2 V y U y ) ,
P yy = − k 0 − 2 U y ε rz − 1 V x V y U x + ( k 0 2 I + U y ε rz − 1 V y ) ( ε ry + k 0 − 2 V x U x ) ,
P xy = U x ε rz − 1 V y ( ε ry + k 0 − 2 V x U x ) − k 0 − 2 ( k 0 2 I + U x ε rz − 1 V x ) V y U x ,
P yx = U y ε rz − 1 V x ( ε rx + k 0 − 2 V y U y ) − k 0 − 2 ( k 0 2 I + U y ε rz − 1 V y ) V x U 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 V x U y U x ε rz − 1 V y + ( ε ry + k 0 − 2 V x U x ) ( k 0 2 I + U y ε rz − 1 V y ) ,
Q yy = − k 0 − 2 V y U x U y ε rz − 1 V x + ( ε rx + k 0 − 2 V y U y ) ( k 0 2 I + U x ε rz − 1 V x ) ,
Q xy = − ( ε ry + k 0 − 2 V x U x ) U y ε rz − 1 V x + k 0 − 2 V x U y ( k 0 2 I + U x ε rz − 1 V x ) ,
Q yx = − ( ε rx + k 0 − 2 V y U y ) U x ε rz − 1 V y + k 0 − 2 V y U x ( k 0 2 I + U y ε rz − 1 V y ) .
Q xx = P yy T , Q yy = P xx T , Q xy = − P xy T , Q yx = − P yx T .
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