Abstract

The dispersion and loss in microstructured fibers are studied using a full-vectorial compact-2D finite-difference method in frequency-domain. This method solves a standard eigen-value problem from the Maxwell’s equations directly and obtains complex propagation constants of the modes using anisotropic perfectly matched layers. A dielectric constant averaging technique using Ampere’s law across the curved media interface is presented. Both the real and the imaginary parts of the complex propagation constant can be obtained with a high accuracy and fast convergence. Material loss, dispersion and spurious modes are also discussed.

© 2004 Optical Society of America

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    [CrossRef]

Electron. Lett. (1)

A. Asi, L. Shafai, "Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD," Electron. Lett. 28, 1451-1452 (1992).
[CrossRef]

IEEE J Quantum Electron. (2)

K. Saitoh, 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]

D. Marcuse, "Solution of the vector wave equation for general dielectric waveguides by the Galerkin method," IEEE J Quantum Electron. 28, 459-465 (1992).
[CrossRef]

IEEE Microwave Guided Wave Lett. (4)

F. L. Teixeira, W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microwave Guided Wave Lett. 7, 371-373 (1997).
[CrossRef]

U. Pekel, R. Mittra, "An application of the perfectly matched layer (PML) concept to the finite element method frequency domain analysis of scattering problems," IEEE Microwave Guided Wave Lett. 5, 258-260 (1995).
[CrossRef]

A. C. Cangellaris, "Numerical stability and numerical dispersion of a compact 2D FDTD method used for the dispersion analysis of waveguides," IEEE Microwave Guided Wave Lett. 3, 3-5 (1993).
[CrossRef]

S. Xiao, R. Vahldieck, H. Jin, "Full-wave analysis of guided wave structures using a novel 2-D FDTD," IEEE Microwave Guided Wave Lett. 2, 165-167 (1992).
[CrossRef]

IEEE Microwave Theory Tech. (1)

T. Tischler, W. Heinrich, "Accuracy limitations of perfectly matched layers in 3D Finite-difference frequency domain method," IEEE Microwave Theory Tech. 50, 1885-1888 (2002).

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. Magnetics (2)

E. A. Marengo, C. M. Rappaport, E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magnetics 35, 1506-1509 (1999).
[CrossRef]

M. Koshiba, Y. Tsuji, M. Hikari, "Finite element beam propagation method with perfectly matched layer boundary conditions," IEEE Trans. Magnetics 35, 1482-1485 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

N. Kaneda, B. Houshmand, T. Itoh, "FDTD analysis of dielectric resonantors with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

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

D. H. Choi, W. J. R. Hoeffer, "The finite-difference time-domain method and its application to eigen-value problems," IEEE Trans. Microwave Theory Tech. 34, 1464-1470 (1986).
[CrossRef]

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-II:Theory," IEEE Trans. Microwave Theory Tech. 23, 429-433 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

H. Y. D. Yang, "Finite difference analysis of 2D photonic crystals," IEEE Trans. Microwave Theory Technol. 44, 2688-2695 (1996).
[CrossRef]

Int. J. Computation Math. Electrical Ele (1)

S. Guenneau, S. Lasquellec, A. Nicolet, F. Zolla, "Design of photonic crystal fibers using finite elements," International J. Computation and Mathematics in Electrical & Electronics Engineering COMPEL 21, 534-539 (2002).
[CrossRef]

J Comp. Phys. (1)

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

J Lightwave Technol. (3)

T. M. Monro, D. J. Richardson, N. G. R. Broderick, P. J. Bennett, "Holey optical fibers: an efficient modal model," J Lightwave Technol. 17, 1093-1102 (1999).
[CrossRef]

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, K. T. V. Grattan, "Loss/gain characterization of optical waveguides," J Lightwave Technol. 13, 1760-1765 (1995).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

J. Lightwave Technol. (4)

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

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

Microwave & Opt. Technol. Lett. (1)

W. C. Chew, J. M. Jin, E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave & Opt. Technol. Lett. 7, 363-369 (1997).
[CrossRef]

Microwave Opt. Techn. Lett. (1)

A. Cucinotta, G. Peiosi, S. Selleri, L. Vincetti, M. Zoboli, "Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method," Microwave Opt. Techn. Lett. 23, 67-69 (1999).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

F. L. Teixeira, W. C. Chew, "Unified analysis of perfectly matched layers using differential forms," Microwave Opt. Technol. Lett. 20, 124-126 (1999).
[CrossRef]

Opt. and Quantum Electron. (1)

F. Zepparelli, P. Mezzanotte, F. Alimenti, L. Roselli, R. Sorrentino , G.Tartarini and P . Bassi, "Rigorous analysis of 3D optical and optoelectronic devices by the Compact-2D-FDTD method," Opt. and Quantum Electron. 31, 827-841 (1999).
[CrossRef]

Opt. Express (8)

Z. Zhu, T. G. Brown, "Full-vectorial finite-difference analysis of microstructuredd optical fibers," Opt. Express 10, 853-864 (2002). <a href ="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853</a>
[CrossRef] [PubMed]

K. Saitoh, M. Koshiba, T. Hasegawa, E. Sasaoka, "Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843</a>
[CrossRef] [PubMed]

W. Zhi, R. Guobing, L. Shuqin, J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980</a>
[CrossRef] [PubMed]

S. Guo, S. Albin, R. S. Rogowski, "Comparative analysis of Bragg fibers," Opt. Express 12, 198-207 (2004). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-198">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-207</a>
[CrossRef] [PubMed]

R. Guobing, W. Zhi, L. Shuqin, and J. Shuisheng, "Full-vectorial analysis of complex refractive index photonic crystal fibers," Opt. Express 12, 1126-1135 (2004). <a href= http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1135">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1135</a>
[CrossRef]

C. P. Yu, H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of twodimensional photonic crystals," Opt. Express 12, 1397-1408 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1397">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1397</a>
[CrossRef] [PubMed]

S. Guo, F. Wu, S. Albin, R. S Rogowski, "Photonic band gap analysis using finite-difference frequency-domain method," Opt. Express 12, 1741-1746 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1741</a>
[CrossRef] [PubMed]

Z. Zhu, T. G. Brown, "Analysis of the space filling modes of photonic crystal fibers," Opt. Express 8, 547- 554 (2001). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547</a.
[CrossRef] [PubMed]

Opt. Fiber Technol. (2)

J. Broeng, "Photonic crystal fibers: a new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

F. Brechet, J. Marcou, D. Pagnoux, 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. (5)

Progress in Electromagnetics Research (1)

S. Guenneau, A. Nicolet, F. Zolla, S. Lasquellec, "Numerical and theoretical study of photonic crystal fibers," Progress in Electromagnetics Research 41, 271-305 (2003).

Science (1)

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other (2)

A. Bjarklev, J. Broeng, A. S. Bjarklev, Photonic crystal fibres (Kluwer Academic Publishers, Boston/Dordrecht/London, 2003).
[CrossRef]

G. W. Milton, The theory of composites (Cambridge University Press, Cambridge, UK, 2002).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

The PCF under study. A quarter of the PCF is used in calculation, which can obtain the third and fourth mode classes with a 90-degree rotation symmetry.

Fig. 2.
Fig. 2.

The relative error of the calculated complex mode index of the fundamental mode. The y-axis is the relative error of the real and imaginary part of the mode index of the fundamental mode. Note the different scales of the two y-axes.

Fig. 3.
Fig. 3.

The six field components and the discretization of the transverse index profile in the x-y plane. The E and H components are in red and blue colors respectively. The orange line denotes the curved interface across the cells, and the dotted cells show the integration plane for Ex and Ey respectively.

Fig. 4.
Fig. 4.

Calculation of Ex, Ey and Ez using Ampere’s law. The orange line denotes the dielectric boundary in the integration plane. From left to right are the integration cells for Ex, Ey and Ez respectively.

Fig. 5.
Fig. 5.

The accuracy and convergence of the complex effective mode index using a more reasonable averaging technique. Note that the scale of the right y-axis is at least an order of magnitude smaller than the corresponding one in Fig. 2.

Fig. 6.
Fig. 6.

The mode field patterns of the fundamental mode (top three) and 2nd-order mode (bottom three) in the degenerate mode class 3 and 4.

Fig. 7.
Fig. 7.

Some spurious cladding modes created by the artificial waveguide between the PML + zero boundary and the air holes. These modes are weak and highly lossy.

Tables (2)

Tables Icon

Table 1. Calculated mode index of the fundamental mode. The accurate value is 1.445395345+3.15×10-8i (by multipole method in [32])

Tables Icon

Table 2. The complex mode index with a lossy core material

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

ξ ( x , y , z , t ) = { ξ t ( x , y ) + ξ z ( x , y ) } exp [ j ( ω t β z ) ]
( t 2 + k 0 2 n 2 β 2 ) E t = t ( E t · t ln n 2 )
( t 2 + k 0 2 n 2 β 2 ) H t = ( t × H t ) × t ln n 2
( t 2 + k 0 2 n 2 β 2 ) E z = j β E t · t ln n 2
( t 2 + k 0 2 n 2 β 2 ) H z = ( t H z + j β H t ) · t ln n 2
j k 0 s ε r E = × H
j k 0 s μ r H = × E
s = [ s y s x s x s y s x s y ]
s x = 1 σ x j w ε 0 , s y = 1 σ y j w ε 0
j k 0 [ s y s x ε rx s x s y ε ry s x s y ε rz ] [ E x E y E z ] = [ 0 j β I V y j β I 0 V x V y V x 0 ] [ H x H y H z ]
j k 0 [ s y s x μ rx s x s y μ ry s x s y μ rz ] [ H x H y H z ] = [ 0 j β I U y j β I 0 U x U y U x 0 ] [ E x E y E z ]
ε rx = s y s x ε rx , ε ry = s x s y ε ry , ε rz = s x s y ε rz
μ rx = s y s x μ rx , μ ry = s x s y μ ry , μ rz = s x s y μ rz
[ Q xx Q xy Q yx Q yy ] [ H x H y ] = β 2 [ H x H y ]
[ P xx P xy P yx P yy ] [ E x E y ] = β 2 [ E x E y ]
P xx = μ ry V y μ rz 1 U y + U x ε rz 1 V x ε rx + k 0 2 μ ry ε rx + k 0 2 U x ε rz 1 { V x V y V y V x } μ rz 1 U y
P xy = μ ry V y μ rz 1 U x + U x ε rz 1 V y ε ry + k 0 2 U x ε rz 1 { V y V x V x V y } μ rz 1 U x
P yx = μ rx V x μ rz 1 U y + U y ε rz 1 V x ε rx + k 0 2 U y ε rz 1 { V x V y V y V x } μ rz 1 U y
P yy = μ rx V x μ rz 1 U x + k 0 2 μ rx ε ry + U y ε rz 1 V y ε ry + k 0 2 U y ε rz 1 { V y V x V x V y } μ rz 1 U x
Q xx = ε ry U y ε rz 1 V y + V x μ rz 1 U x μ rx + k 0 2 ε ry μ rx + k 0 2 V x μ rz 1 { U x U y U y U x } ε rz 1 V y
Q xy = ε ry U y ε rz 1 V x + V x μ rz 1 U y μ ry + k 0 2 V x μ rz 1 { U y U x U x U y } ε rz 1 V x
Q yx = ε rx U x ε rz 1 V y + V y μ rz 1 U x μ rx + k 0 2 V y μ rz 1 { U x U y U y U x } ε rz 1 V y
Q yy = ε rx U x ε rz 1 V x + k 0 2 ε rx μ ry + V y μ rz 1 U y μ ry + k 0 2 V y μ rz 1 { U y U x U x U y } ε rz 1 V x
ε = ε a f + ε b ( 1 f )
t A 1 ε x E x dydz = L 1 H · dl
t A 2 ε y E y dydz = L 2 H · dl
t A 1 ε z E z dydy = L 3 H · dl
t ( ε ¯ z E ¯ z Δ x Δ y ) = L 3 H · dl and ε ¯ z = f ε a + ( 1 f ) ε b
ε ¯ x = [ 1 Δ x x x + Δ x 1 ε a f ( x ) + ε b ( 1 f ( x ) ) dx ] 1
ε ¯ y = [ 1 Δy y y + Δ y 1 ε a f ( y ) + ε b ( 1 f ( y ) ) dy ] 1

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