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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References

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Asia-Pacific Engin Res Forum on Micro & Emag Thry (1)

H. C. Chang and C. P. Yu, ???Yee-mesh-based finite difference optical waveguide eigenmode solver with perfectly matched layer absorbing boundaries conditions,??? in Proceedings of 5th Asia-Pacific Engineering Research Forum on Microwaves and Electromagnetic Theory, pp. 158???164, Kyushu University, Fukuoka, Japan, July 29???30, 2004.
[PubMed]

IEEE J. Quantum Electron. (2)

K. Saitoh and M. Koshiba, ???Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,??? IEEE J. Quantum Electron. 38, 927???933 (2002).
[CrossRef]

H. K. Kim, J. Shin, S. Fan, M. J. F. Digonnet, and G. S. Kino, ???Designing air-core photonic-bandgap fibers free of surface modes,??? IEEE J. Quantum Electron. 40, 551???556 (2004).
[CrossRef]

IEEE Microwave Guided Wave Lett. (2)

R. Mittra and U. Pekel, ???A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,??? IEEE Microwave Guided Wave Lett. 5, 84???86 (1995).
[CrossRef]

C. M. Rappaport, ???Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,??? IEEE Microwave Guided Wave Lett. 5, 90???92 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Koshiba and K. Saitoh, ???Numerical verification of degeneracy in hexagonal photonic crystal fibers,??? IEEE Photon. Technol. Lett. 13, 1313???1315 (2001).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

K. S. Yee, ???Numerical solution of initial boundary value problems involving Maxwell???s equations in isotropic media,??? IEEE Trans. Antennas Propagat. AP-14, 302???307 (1966).

Inst. Elec. Eng. Proc.-J (1)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, ???Full-vectorial mode calculations by finite difference method,??? Inst. Elec. Eng. Proc.-J 141, 281???286 (1994).
[CrossRef]

Integr. Photon. Res. Dig. (1)

H. C. Chang and C. P. Yu, ???Yee-mesh-based finite difference eigenmode analysis algorithms for optical waveguides and photonic crystals,??? in OSA Integr. Photon. Res. Dig., 2004, paper IFE4.

J. Computat. Phys. (1)

J. P. Berenger, ???A perfectly matched layer for the absorption of electromagnetic waves,??? J. Computat. Phys. 114, 185???200 (1994).
[CrossRef]

J. Lightwave Technol. (6)

J. Opt. Soc. Am. B (1)

Microwave Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, ???A 3D perfectly matched medium from modified Maxwell???s equations with stretched coordinates,??? Microwave Opt. Technol. Lett. 7, 599???604 (1994).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Progress in Electromagnetics Research (1)

Ditkowski, J. S. Hesthaven, and C. H. Teng, ???Modeling dielectric interfaces in the FDTD-method: A comparative study,??? in 2000 Progress in Electromagnetics Research (PIERS 2000) Proceedings, Cambridge, Massachusetts, July 5???14, 2000.

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, ???Singlemode photonic band gap guidance of light in air,??? Science 285, 1537???1539 (1999).
[CrossRef] [PubMed]

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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.

Fig. 2.
Fig. 2.

Yee’s 2-D mesh for the FDFD method.

Fig. 3.
Fig. 3.

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

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.

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.

Fig. 6.
Fig. 6.

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

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.

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.

Fig. 9.
Fig. 9.

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

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.

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.

Fig. 12.
Fig. 12.

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

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.

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).

Tables (2)

Tables Icon

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

Tables Icon

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].

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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