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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References

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    [CrossRef]
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    [CrossRef]
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  20. M. Qiu, �??Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,�?? Microwave Opt. Technol. Lett. 30, 327-330 (2001).
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    [CrossRef]
  25. K. Bierwirth, N. Schulz, and F. Arndt, �??Finite-difference analysis of rectangular dielectric waveguide structures,�?? IEEE Trans. Microwave Theory Tech. 34, 1104-1113 (1986).
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  26. H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, �??Vectorial integrated finite-difference analysis of dielectric waveguides,�?? J. Lightwave Technol. 11, 1559-1563 (1993).
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    [CrossRef]
  28. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
    [CrossRef]
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  30. S. Dey and R. Mittra, �??A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,�?? IEEE Trans. Microwave Theory Tech. 47, 1717-1739 (1999).
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  32. Z. Zhu and T. G. Brown, �??Multipole analysis of hole-assisted optical fibers,�?? Opt. Commun. 206, 333-339 (2002).
    [CrossRef]
  33. M. Midrio, M. P. Singh, and C. G. Someda, �??The space filling mode of holey fibers: an analytical vectorial solution,�?? J. Lightwave Technol. 18, 1031-1037 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  35. M. J. Steel, T. P. White, C. M. de Sterke, R. C.McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-490 (2001).
    [CrossRef]
  36. M. Koshiba and K.Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315 (2001).
    [CrossRef]
  37. M. J. Steel and R. M. Osgood, Jr., �??Elliptical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229-231 (2001).
    [CrossRef]

IEEE J. Quantum Electron. (1)

W. P. Huang and C. L. Xu, �??Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,�?? IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? IEEE Photon. Technol. Lett. 12, 807-809 (2000).
[CrossRef]

T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588-590 (2001)
[CrossRef]

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

IEEE Proc. J. Optoelectron. (1)

M. S. Stern, �??Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,�?? IEE Proc. J. Optoelectron. 135, 56-63 (1988).
[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. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

K. Bierwirth, N. Schulz, and F. Arndt, �??Finite-difference analysis of rectangular dielectric waveguide structures,�?? IEEE Trans. Microwave Theory Tech. 34, 1104-1113 (1986).
[CrossRef]

S. Dey and R. Mittra, �??A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,�?? IEEE Trans. Microwave Theory Tech. 47, 1717-1739 (1999).

J. Lightwave Technol. (6)

M. Midrio, M. P. Singh, and C. G. Someda, �??The space filling mode of holey fibers: an analytical vectorial solution,�?? J. Lightwave Technol. 18, 1031-1037 (2000).
[CrossRef]

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, �??Vectorial integrated finite-difference analysis of dielectric waveguides,�?? J. Lightwave Technol. 11, 1559-1563 (1993).
[CrossRef]

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, �??Analysis of vectorial mode fields in optical waveguides by a new finite difference method,�?? J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, �??The finite-difference vector beam propagation method. Analysis and Assessment,�?? J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, �??Localized function method for modeling defect modes in 2-D photonic crystals,�?? J. Lightwave Technol. 17, 2078-2081(1999).
[CrossRef]

T.M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, �??Modeling large air fraction holey optical fibers,�?? J. Lightwave Technol. 18, 50-56 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Microwave Opt. Technol. Lett. (1)

M. Qiu, �??Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,�?? Microwave Opt. Technol. Lett. 30, 327-330 (2001).
[CrossRef]

Opt. Commun. (1)

Z. Zhu and T. G. Brown, �??Multipole analysis of hole-assisted optical fibers,�?? Opt. Commun. 206, 333-339 (2002).
[CrossRef]

Opt. Express (5)

Opt. Fiber Technol. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method,�?? Opt. Fiber Technol. 6, 181-191 (2000).
[CrossRef]

Opt. Lett. (10)

G. E. Town and J. T. Lizer, �??Tapered holey fibers for spot size and numerical-aperture conversion,�?? Opt. Lett. 26, 1042-1044 (2001).
[CrossRef]

J. K. Ranka, R. S. Windeler, and A. J. Stentz, �??Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,�?? Opt. Lett. 25, 25-27 (2000).
[CrossRef]

A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, �??Full-vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276-278 (1999).
[CrossRef]

N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, �??Nonlinearity in holey optical fibers: measurement and future opportunities,�?? Opt. Lett. 24, 1395-1397 (1999).
[CrossRef]

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000).
[CrossRef]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, �??All-silica single mode optical fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996).
[CrossRef] [PubMed]

T. A. Birks, J. C. Knight, and P. St. J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

M. J. Steel, T. P. White, C. M. de Sterke, R. C.McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-490 (2001).
[CrossRef]

T. P.White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, �??Confinement losses in microstructured optical fibers,�?? Opt. Lett. 26, 1660-1662 (2001).
[CrossRef]

M. J. Steel and R. M. Osgood, Jr., �??Elliptical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229-231 (2001).
[CrossRef]

Opt. Quantum Electron. (2)

K. S. Chiang, �??Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,�?? Opt. Quantum Electron. 26, s113-s134 (1994).
[CrossRef]

C. Vassallo, �??1993-1995 optical mode solvers,�?? Opt. Quantum Electron. 29, 95-114 (1997).
[CrossRef]

Other (1)

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).

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Figures (6)

Fig. 1.
Fig. 1.

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

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.

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).

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).

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.

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.

Tables (3)

Tables Icon

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.

Tables Icon

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.

Tables Icon

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.

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 E y ,
i k 0 H y = 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 H y ,
i k 0 ε r E y = 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 E y ( j , l ) ,
i k 0 H y ( j , l ) = 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 H y ( j , l ) ,
i k 0 ε ry ( j , l ) E y ( j , l ) = 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 I U y 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 I V y 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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