Abstract

A highly accurate radiation boundary condition for finite difference analysis of open waveguides is introduced. The boundary condition is applicable to the structures embedded in a homogeneous medium and fitted to the cross section of the structure. The numerical tests carried out for a few types of waveguides including microstructured fibers showed that the proposed approach improves the accuracy by about an order of magnitude in comparison with the PML technique and eliminates all its disadvantages.

© 2007 Optical Society of America

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References

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  1. Z. Zhu, T.G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853.
    [PubMed]
  2. S. Guo, Feng Wu, S. Albin, H. Tai, R. S. Rogowski,"Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express 12, 3341-3352 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341.
    [CrossRef] [PubMed]
  3. P. Kowalczyk, M. Wiktor, M. Mrozowski, "Efficient finite difference analysis of microstructured optical fibers," Opt. Express 13, 10349-10359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10349.
    [CrossRef] [PubMed]
  4. A. Taflove, S.C. Hagness, "Computational electrodynamics: the finite-difference time-domain method," Artech House, Boston (2005), 3rd edn.
  5. J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  6. G.R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron.,  28, 363-370 (1992).
    [CrossRef]
  7. H.P. Uranus, H.J.W.M. Hoekstra, "Modeling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795.
    [CrossRef] [PubMed]
  8. H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
    [CrossRef]
  9. H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer," J. Lightwave Technol.,  20, 1141 - 1148 (2002).
    [CrossRef]
  10. E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).
  11. C.D. Meyer, "Matrix analysis and applied linear algebra", SIAM, Philadelphia (2000).
  12. N. Kaneda, B. Houshmand, T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
    [CrossRef]
  13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Ranversez, C.M. de Sterke, L.C. Botten, M.J. Steel, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  14. N.A. Issa, L. Poladian, "Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers," J. Lightwave Technol. 21, 1005-1012 (2003).
    [CrossRef]

2005 (2)

E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).

P. Kowalczyk, M. Wiktor, M. Mrozowski, "Efficient finite difference analysis of microstructured optical fibers," Opt. Express 13, 10349-10359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10349.
[CrossRef] [PubMed]

2004 (2)

2003 (1)

2002 (3)

2001 (1)

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
[CrossRef]

1997 (1)

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

1994 (1)

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

1992 (1)

G.R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron.,  28, 363-370 (1992).
[CrossRef]

Berenger, J.P.

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

Botten, L.C.

Brown, T.G.

de Sterke, C.M.

De Zutter, D.

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer," J. Lightwave Technol.,  20, 1141 - 1148 (2002).
[CrossRef]

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
[CrossRef]

Feng Wu, S.

Guo, S.

Hadley, G.R.

G.R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron.,  28, 363-370 (1992).
[CrossRef]

Hanson, G.W.

E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).

Hoekstra, H.J.W.M.

Houshmand, B.

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

Issa, N.A.

Itoh, T.

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

Kaneda, N.

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

Kartchevski, E.M.

E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).

Kowalczyk, P.

Kuhlmey, B.T.

Maystre, D.

McPhedran, R.C.

Mrozowski, M.

Nosich, A.I.

E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).

Poladian, L.

Ranversez, R.

Rogier, H.

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer," J. Lightwave Technol.,  20, 1141 - 1148 (2002).
[CrossRef]

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
[CrossRef]

Steel, M.J.

Uranus, H.P.

White, T.P.

Wiktor, M.

Zhu, Z.

IEEE J. Quantum Electron. (1)

G.R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron.,  28, 363-370 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
[CrossRef]

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

J. Appl. Math. (1)

E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).

J. Comput. Phys. (1)

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

J. Lightwave Technol. (2)

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

Opt. Express (4)

Other (2)

A. Taflove, S.C. Hagness, "Computational electrodynamics: the finite-difference time-domain method," Artech House, Boston (2005), 3rd edn.

C.D. Meyer, "Matrix analysis and applied linear algebra", SIAM, Philadelphia (2000).

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Figures (8)

Fig. 1.
Fig. 1.

A cross section of the slab dielectric waveguide covered with Yee’s mesh.

Fig. 2.
Fig. 2.

Tangential electric field of Yee’s mesh. A fragment of the computational domain in a neighborhood of the boundary.

Fig. 3.
Fig. 3.

The field distribution for neff = 0.99526-0.00134j.

Fig. 4.
Fig. 4.

The field distribution for neff = 0.98086-0.00533j.

Fig. 5.
Fig. 5.

The field distribution for neff = 0.95621-0.01196j.

Fig. 6.
Fig. 6.

The computational domain for PML technique - left hand side and for analytical boundary conditions (ABC) - right hand side.

Fig. 7.
Fig. 7.

Two different types of a microstructured optical fibers.

Fig. 8.
Fig. 8.

Berenger and leaky modes for structure presented in fig. 7a. Leaky modes are surrounded by a circle.

Tables (8)

Tables Icon

Table 1. The results obtained for a structure with parameters ε r1 = ε r3 = 1.21, ε r2 = 1, b = 1nm, λ0 = 0.2nm. The percentage error of the real and imaginary part is given in brackets.

Tables Icon

Table 2. The results obtained for a structure with parameters ε r1 = ε r3 = 9, ε r2 = 1, b= 1nm for λ 0 = 1.5nm. The percentage error of the real and imaginary part is given in brackets.

Tables Icon

Table 3. The results obtained for an analysis of the optical fiber. The percentage error of the real and imaginary part is given in brackets.

Tables Icon

Table 4. The real part of the effective indices of the structure with 6 circular holes obtained by different methods. The percentage error (in brackets) is calculated in reference to Multipole Method (MM) [13] (10 multipole moments).

Tables Icon

Table 5. The imaginary part of the effective indices of the structure with 6 circular holes obtained by different methods. The percentage error (in brackets) is calculated in reference to Multipole Method.

Tables Icon

Table 6. The real part of the effective indices of the structure with 3 angular-shaped holes obtained by different methods. The percentage error (in brackets) is calculated in reference to Vector FDM-ABC Scheme (70 Azimuthal, 54 radial terms) [14].

Tables Icon

Table 7. The imaginary part of the effective indices of the structure with 3 angular-shaped holes obtained by different methods. The percentage error (in brackets) is calculated in reference to FDM-ABC method.

Tables Icon

Table 8. The convergence of the fundamental mode for structure presented in fig. 7a.

Equations (37)

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{ E y x = j ω μ 0 H z H z x = ( γ 2 j ω μ 0 + j ω ε ) E y .
{ e m + 1 e m Δ x = j ω μ 0 h m h m h m 1 Δ x = ( γ 2 j ω μ 0 + j ω ε m ) e m
[ R yz ( h ) R zy ( e ) ω μ 0 ε ] e = γ 2 e
[ R zy ( e ) ] m , n = Δ x 1 { 1 , m = n 1 , m = n + 1 , 0 , otherwise
[ R yz ( h ) ] m , n = Δ x 1 { 1 , m = n 1 , m = n 1 , 0 , otherwise
[ ε ] m , n = { ε m , m = n 0 , otherwise ,
E y ( x ) = A e κ 1 x , x < b 2
E y ( x ) = A + e κ 3 x , x > b 2
e 1 = a e 2 e M = a + e M 1 ,
Ae = γ 2 e ,
{ A 1,1 e 1 + A 1,2 e 2 + + A 1 , M 1 e M 1 + A 1 , M e M = γ 2 e 1 A 2,1 e 1 + A 2,2 e 2 + + A 2 , M 1 e M 1 + A 2 , M e M = γ 2 e 2 A 3,1 e 1 + A 3,2 e 2 + + A 3 , M 1 e M 1 + A 3 , M e M = γ 2 e 3 A M 2,1 e 1 + A M 2,2 e 2 + + A M 2 , M 1 e M 1 + A M 2 , M e M = γ 2 e M 2 A M 1,1 e 1 + A M 1,2 e 2 + + A M 1 , M 1 e M 1 + A M 1 , M e M = γ 2 e M 1 A M , 1 e 1 + A M , 2 e 2 + + A M , M 1 e M 1 + A M , M e M = γ 2 e M .
{ ( A 2,1 a + A 2,2 ) e 2 + + ( A 2 , M 1 + A 2 , M a + ) e M 1 = γ 2 e 2 ( A 3,1 a + A 3,2 ) e 2 + + ( A 3 , M 1 + A 3 , M a + ) e M 1 = γ 2 e 3 ( A M 2,1 a + A M 2,2 ) e 2 + + ( A M 2 , M 1 + A M 2 , M a + ) e M 1 = γ 2 e M 2 ( A M 1,1 a + A M 1,2 ) e 2 + + ( A M 1 , M 1 + A M 1 , M a + ) e M 1 = γ 2 e M 1 .
A ˜ ( γ ) e ˜ = γ 2 e ˜ .
A ˜ ( γ i ) e ˜ = γ i + 1 2 e ˜ i = 0,1 ,
A E t = γ 2 E t .
E ρ ρ φ z = γ κ 2 E z ρ φ z ρ j ω μ ρ κ 2 H z ρ φ z φ ,
E φ ρ φ z = γ ρ κ 2 E z ρ φ z φ + j ω μ κ 2 H z ρ φ z ρ ,
H ρ ρ φ z = j ω ε ρ κ 2 E z ρ φ z φ γ κ 2 H z ρ φ z ρ ,
H φ ρ φ z = j ω ε κ 2 E z ρ φ z ρ γ ρ κ 2 H z ρ φ z φ .
E z ρ φ z = m = 0 [ A m H m ( 2 ) ( κ ρ ) sin ( m φ ) + B m H m ( 2 ) ( κ ρ ) cos ( m φ ) ] e γ z ,
H z ρ φ z = m = 0 j η [ C m H m ( 2 ) ( κ ρ ) sin ( m φ ) + D m H m ( 2 ) ( κ ρ ) cos ( m φ ) ] e γ z ,
E x x y z = E ρ ρ φ z cos ( φ ) E φ ρ φ z sin ( φ )
E y x y z = E ρ ρ φ z sin ( φ ) + E φ ρ φ z cos ( φ ) ,
E x x y 0
= m = 0 Q { A m γ κ [ m ρ κ H m ( 2 ) ( κ ρ ) cos ( m φ ) sin ( φ ) H m ( 2 ) ( κ ρ ) sin ( m φ ) cos ( φ ) ]
B m γ κ [ m ρ κ H m ( 2 ) ( κ ρ ) sin ( m φ ) sin ( φ ) + H m ( 2 ) ( κ ρ ) cos ( m φ ) cos ( φ ) ]
+ C m ω μ κ η [ m ρ κ H m ( 2 ) ( κ ρ ) cos ( m φ ) cos ( φ ) + H m ( 2 ) ( κ ρ ) sin ( m φ ) sin ( φ ) ]
D m ω μ κ η [ m ρ κ H m ( 2 ) ( κ ρ ) sin ( m φ ) cos ( φ ) + H m ( 2 ) ( κ ρ ) cos ( m φ ) sin ( φ ) ] }
E y x y 0
= m = 0 Q { A m γ κ [ m ρ κ H m ( 2 ) ( κ ρ ) cos ( m φ ) cos ( φ ) H m ( 2 ) ( κ ρ ) sin ( m φ ) sin ( φ ) ]
B m γ κ [ m ρ κ H m ( 2 ) ( κ ρ ) sin ( m φ ) cos ( φ ) + H m ( 2 ) ( κ ρ ) cos ( m φ ) sin ( φ ) ]
+ C m ω μ κ η [ m ρ κ H m ( 2 ) ( κ ρ ) cos ( m φ ) sin ( φ ) H m ( 2 ) ( κ ρ ) sin ( m φ ) cos ( φ ) ]
D m ω μ κ η [ m ρ κ H m ( 2 ) ( κ ρ ) sin ( m φ ) sin ( φ ) + H m ( 2 ) ( κ ρ ) cos ( m φ ) cos ( φ ) ] } .
E t C = M C C
E t B = M B C
E t B = M B M C ( inv ) E t C
n eff = γ j k 0

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