Abstract

A meshless method for the solution of full vectorial optical mode fields has been applied to micro-structured optical waveguides. The Finite Cloud Method is used to approximate the solution using a point distribution and material definitions. Presented are two methods of defining material interfaces, one which implements a step index and a second which uses a graded index. Coupled field equations are used to solve for both transverse components of the magnetic field as well as the guided wavelength and effective index of refraction. Comparing results for a ridge waveguide, solid core, micro-structured and air core structures with commercial FEM solvers highlight the methods versatility, accuracy and efficiency.

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References

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2010 (1)

D. Burke, S. Moslemi-Tabrizi, and T. Smy, “Simulation of inhomogeneous models using the finite cloud method,” Materialwiss. Werkstofftech.41, 336–340 (2010).
[CrossRef]

2006 (1)

2004 (2)

2001 (1)

N. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation,” Int. J. Numer. Methods Eng.50, 2373–2410 (2001).
[CrossRef]

2000 (1)

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

1997 (1)

K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid eigenvalue solver for mode calculation of planar optical waveguides,” IEEE Photon. Technol. Lett.9, 967–969 (1997).
[CrossRef]

1994 (1)

P. Lusse, P. Stuwe, J. Schule, and 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]

1975 (1)

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-i: Summary of results,” IEEE Trans. Microwave Theory Tech.23(5), 421–429 (1975).
[CrossRef]

Aluru, N.

N. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation,” Int. J. Numer. Methods Eng.50, 2373–2410 (2001).
[CrossRef]

Burke, D.

D. Burke, S. Moslemi-Tabrizi, and T. Smy, “Simulation of inhomogeneous models using the finite cloud method,” Materialwiss. Werkstofftech.41, 336–340 (2010).
[CrossRef]

Chen, J.-S.

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

Fini, J.M.

Hoekstra, H.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Li, G.

N. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation,” Int. J. Numer. Methods Eng.50, 2373–2410 (2001).
[CrossRef]

Liu, W. K.

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

Lusse, P.

K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid eigenvalue solver for mode calculation of planar optical waveguides,” IEEE Photon. Technol. Lett.9, 967–969 (1997).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and 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]

McIsaac, P.

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-i: Summary of results,” IEEE Trans. Microwave Theory Tech.23(5), 421–429 (1975).
[CrossRef]

Moslemi-Tabrizi, S.

D. Burke, S. Moslemi-Tabrizi, and T. Smy, “Simulation of inhomogeneous models using the finite cloud method,” Materialwiss. Werkstofftech.41, 336–340 (2010).
[CrossRef]

Ramm, K.

K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid eigenvalue solver for mode calculation of planar optical waveguides,” IEEE Photon. Technol. Lett.9, 967–969 (1997).
[CrossRef]

Russell, P. S.

Schule, J.

P. Lusse, P. Stuwe, J. Schule, and 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]

Smy, T.

D. Burke, S. Moslemi-Tabrizi, and T. Smy, “Simulation of inhomogeneous models using the finite cloud method,” Materialwiss. Werkstofftech.41, 336–340 (2010).
[CrossRef]

Stuwe, P.

P. Lusse, P. Stuwe, J. Schule, and 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]

Unger, H.-G.

K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid eigenvalue solver for mode calculation of planar optical waveguides,” IEEE Photon. Technol. Lett.9, 967–969 (1997).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and 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]

Uranus, H.

Wang, H.-P.

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

Yoon, S.

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (1)

J.-S. Chen, S. Yoon, H.-P. Wang, and W. K. Liu, “An improved reproducing kernel particle method for nearly incompressible finite elasticity,” Comput. Methods Appl. Mech. Eng.181, 117–145 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid eigenvalue solver for mode calculation of planar optical waveguides,” IEEE Photon. Technol. Lett.9, 967–969 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-i: Summary of results,” IEEE Trans. Microwave Theory Tech.23(5), 421–429 (1975).
[CrossRef]

Int. J. Numer. Methods Eng. (1)

N. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation,” Int. J. Numer. Methods Eng.50, 2373–2410 (2001).
[CrossRef]

J. Lightwave Technol. (2)

P. Lusse, P. Stuwe, J. Schule, and 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]

P. S. Russell, “Photonic-crystal fibers,” J. Lightwave Technol.24, 4729–4749 (2006).
[CrossRef]

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

Materialwiss. Werkstofftech. (1)

D. Burke, S. Moslemi-Tabrizi, and T. Smy, “Simulation of inhomogeneous models using the finite cloud method,” Materialwiss. Werkstofftech.41, 336–340 (2010).
[CrossRef]

Opt. Express (1)

Other (5)

MathWorks (2011), http://www.mathworks.com/products/matlab/ .

GNU Octave (2011), http://www.gnu.org/s/octave/ .

COMSOL Multiphysics, Version 4.1 Comsol Inc. (2011), http://www.comsol.com .

Rsoft FemSim, Version 3.3 Rsoft Inc. (2011), http://www.rsoftdesign.com/products.php?sub=Component+Design$\&$itm=FemSIM .

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (8)

Fig. 1
Fig. 1

Parameters and dimensions for (a) a ridge waveguide (b) a solid core step-index fiber.

Fig. 2
Fig. 2

The first six modes of the ridge waveguide.

Fig. 3
Fig. 3

The first six modes of the solid core waveguide.

Fig. 4
Fig. 4

Parameters and dimensions for (a) a Bragg diffraction air core fiber and (b) a photonic crystal fiber with 6 circular air holes.

Fig. 5
Fig. 5

The first six modes of the Bragg diffraction air core structure.

Fig. 6
Fig. 6

The first six modes of the circular air hole photonic crystal fiber.

Fig. 7
Fig. 7

Dispersion parameter comparison between COMSOL MultiPhysics and the FCM for modes 1 and 3 of the photonic crystal fiber.

Fig. 8
Fig. 8

Mode amplitudes for two structures. (a) Field magnitude of a 2nd order mode for ridge (b) Hx plotted for a 2nd order mode for the air-hole fiber.

Tables (6)

Tables Icon

Table 1 Comparison of effective index of refraction for the first six modes of the ridge waveguide. The s-FCM and g-FCM being the step-index FCM and the graded-index FCM, compared with results from Rsoft FemSIM and COMSOL MultiPhysics.

Tables Icon

Table 2 Comparison of effective index of refraction for the first six modes of the solid core waveguide. The s-FCM and g-FCM being the step-index FCM and the graded-index FCM, compared with results from Rsoft FemSIM and COMSOL MultiPhysics.

Tables Icon

Table 3 Comparison of effective index of refraction for the first six modes of the step fiber waveguide. The s-FCM and g-FCM being the step-index FCM and the graded-index FCM, compared with half and quarter symmetry using PEC and PMC boundaries. *Simulation time using a dual core iMac at 2.4GHz.

Tables Icon

Table 4 Comparison of effective index of refraction for the first six modes of the Bragg diffraction air core structure. The s-FCM and g-FCM being the step-index FCM and the graded-index FCM, compared with results from Rsoft FemSIM and COMSOL Multi-Physics.

Tables Icon

Table 5 Comparison of effective index of refraction for the first six modes of the air hole waveguide. The s-FCM and g-FCM being the step-index FCM and the graded-index FCM, compared with results from Rsoft FemSIM and COMSOL MultiPhysics.

Tables Icon

Table 6 Comparison of effective index of refraction for the first six modes of the air hole waveguide. The s-FCM and g-FCM being the air-hole FCM and the graded-index FCM, compared with half and quarter symmetry using PEC and PMC boundaries.

Equations (21)

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

u ( x ) a = I = 1 N P ζ ( x , x I ) φ ( x K x I ) u I
P T ( s ) = [ 1 , s , s 2 ] , m = 3
ζ ( x , x I ) = P T ( x I ) C ( x )
p i ( x ) = I = 1 N P ζ ( x , x I ) φ ( x K x I ) p i ( x I ) , i = 1 , , m .
M C ( x ) = P ( x ) .
N I ( x ) = P T ( x ) M 1 P ( x I ) φ ( x K x I )
σ ( 2 u x 2 + 2 u y 2 ) = ρ ( x , y ) ,
σ ( N xx U + N yy U ) = R
2 H x x 2 + 2 H x y 2 + ( ε r k 0 2 β 2 ) H x = 0 2 H y x 2 + 2 H y y 2 + ( ε r k 0 2 β 2 ) H y = 0
H x x | a + H y y | a = H x x | b + H y y | b
1 ε a H x y | a 1 ε a H y x | a = 1 ε b H x y | b 1 ε b H y x | b
2 H x x 2 + 2 H x y 2 + ε r k 0 2 H x = β 2 H x 2 H y x 2 + 2 H y y 2 + ε r k 0 2 H y = β 2 H y
[ N x x + N y y + I ε r k 0 2 0 0 N x x + N y y + I ε r k 0 2 ] [ H x H y ] = β 2 [ H x H y ]
M a x ( a 2 H x + k 0 2 ε a H x ) + M b x ( b 2 H x + k 0 2 ε b H x ) + ( H x x | a + H y y | a ) ( H x x | b + H y y | b ) = β 2 H x
M a y ( a 2 H y + k 0 2 ε a H y ) + M b y ( b 2 H y + k 0 2 ε b H y ) + ( 1 ε a H x y | a 1 ε a H y x | a ) ( 1 ε b H x y | b 1 ε b H y x | b ) = β 2 H y .
M n x = [ ( ε n ε a + ε b ) cos 2 ϕ + ( 1 2 sin 2 ϕ ) ] M n y = [ ( ε n ε a + ε b ) sin 2 ϕ + ( 1 2 cos 2 ϕ ) ]
2 H x x 2 + 2 H x y 2 1 ε ε y H x y + 1 ε ε y H y x + ε r k 0 2 H x = β 2 H x 2 H y x 2 + 2 H y y 2 1 ε ε x H y x + 1 ε ε x H x y + ε r k 0 2 H y = β 2 H y
H = 0
H | | = 0
Dispersion = λ c 2 λ 2 [ ( n eff ) ]
f ( x ) = f ( a ) + f ( a ) 1 ! ( x a ) + f ( a ) 2 ! ( x a ) 2

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