Abstract

We introduce and develop a new explicit vector beam propagation method, based on the iterated Crank–Nicolson scheme, which is an established numerical method in the area of computational relativity. The proposed approach results in a fast and robust method, characterized by simplicity, efficiency, and versatility. It is free of limitations inherent in implicit beam propagation methods, which are associated with poor convergence or uneconomical use of memory in the solution of large sparse linear systems, and thus it can tackle problems of considerable size and complexity. The advantages offered by this approach are demonstrated by analyzing a multimode interference coupler and a twin-core photonic crystal fiber. A possible wide-angle generalization is also provided.

© 2009 Optical Society of America

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References

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  1. C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam-propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).
    [CrossRef]
  2. W. P. Huang and C. L. Xu, “Simulation of 3-dimensional optical wave-guides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639-2649 (1993).
    [CrossRef]
  3. Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am. A 23, 2014-2019 (2006).
    [CrossRef]
  4. D. Schulz, C. Glingener, M. Bludszuweit, and E. Voges, “Mixed finite element beam propagation method,” J. Lightwave Technol. 16, 1336-1342 (1998).
    [CrossRef]
  5. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405-413 (2001).
    [CrossRef]
  6. L. Vincetti, A. Cucinotta, S. Selleri, and M. Zoboli, “Three-dimensional finite-element beam propagation method: assessments and developments,” J. Opt. Soc. Am. A 17, 1124-1131 (2000).
    [CrossRef]
  7. Y. Chung and N. Dagli, “Analysis of z-invariant and z-variant semiconductor rib wave-guides by explicit finite-difference beam propagation method with nonuniform mesh configuration,” IEEE J. Quantum Electron. 27, 2296-2305 (1991).
    [CrossRef]
  8. Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
    [CrossRef]
  9. Y. C. Chung and N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photonics Technol. Lett. 6, 540-542 (1994).
    [CrossRef]
  10. F. Xiang and G. L. Yip, “An explicit and stable finite-difference 2-d vector beam-propagation method,” IEEE Photonics Technol. Lett. 6, 1248-1250 (1994).
    [CrossRef]
  11. H. M. Masoudi and J. M. Arnold, “Spurious modes in the DuFort-Frankel finite-difference beam propagation method,” IEEE Photonics Technol. Lett. 9, 1382-1384 (1997).
    [CrossRef]
  12. P. Sewell, T. M. Benson, and A. Vukovic, “A stable DuFort-Frankel beam-propagation method for lossy structures and those with perfectly matched layers,” J. Lightwave Technol. 23, 374-381 (2005).
    [CrossRef]
  13. S. A. Teukolsky, “Stability of the iterated Crank-Nicholson method in numerical relativity,” Phys. Rev. D 61, 087501 (2000).
    [CrossRef]
  14. G. Leiler and L. Rezzolla, “Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity,” Phys. Rev. D 73, 044001 (2006).
    [CrossRef]
  15. Y. Shi and D. Dai, “Design of a compact multimode interference coupler based on deeply-etched SiO2 ridge waveguides,” Opt. Commun. 271, 404-407 (2007).
    [CrossRef]
  16. K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11, 3188-3195 (2003).
    [CrossRef] [PubMed]

2007 (1)

Y. Shi and D. Dai, “Design of a compact multimode interference coupler based on deeply-etched SiO2 ridge waveguides,” Opt. Commun. 271, 404-407 (2007).
[CrossRef]

2006 (2)

G. Leiler and L. Rezzolla, “Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity,” Phys. Rev. D 73, 044001 (2006).
[CrossRef]

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am. A 23, 2014-2019 (2006).
[CrossRef]

2005 (1)

2003 (1)

2001 (1)

2000 (2)

1998 (1)

1997 (1)

H. M. Masoudi and J. M. Arnold, “Spurious modes in the DuFort-Frankel finite-difference beam propagation method,” IEEE Photonics Technol. Lett. 9, 1382-1384 (1997).
[CrossRef]

1994 (2)

Y. C. Chung and N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photonics Technol. Lett. 6, 540-542 (1994).
[CrossRef]

F. Xiang and G. L. Yip, “An explicit and stable finite-difference 2-d vector beam-propagation method,” IEEE Photonics Technol. Lett. 6, 1248-1250 (1994).
[CrossRef]

1993 (2)

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam-propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

W. P. Huang and C. L. Xu, “Simulation of 3-dimensional optical wave-guides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

1991 (2)

Y. Chung and N. Dagli, “Analysis of z-invariant and z-variant semiconductor rib wave-guides by explicit finite-difference beam propagation method with nonuniform mesh configuration,” IEEE J. Quantum Electron. 27, 2296-2305 (1991).
[CrossRef]

Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
[CrossRef]

Arnold, J. M.

H. M. Masoudi and J. M. Arnold, “Spurious modes in the DuFort-Frankel finite-difference beam propagation method,” IEEE Photonics Technol. Lett. 9, 1382-1384 (1997).
[CrossRef]

Benson, T. M.

Bludszuweit, M.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam-propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

Chung, Y.

Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
[CrossRef]

Y. Chung and N. Dagli, “Analysis of z-invariant and z-variant semiconductor rib wave-guides by explicit finite-difference beam propagation method with nonuniform mesh configuration,” IEEE J. Quantum Electron. 27, 2296-2305 (1991).
[CrossRef]

Chung, Y. C.

Y. C. Chung and N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photonics Technol. Lett. 6, 540-542 (1994).
[CrossRef]

Cucinotta, A.

Dagli, N.

Y. C. Chung and N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photonics Technol. Lett. 6, 540-542 (1994).
[CrossRef]

Y. Chung and N. Dagli, “Analysis of z-invariant and z-variant semiconductor rib wave-guides by explicit finite-difference beam propagation method with nonuniform mesh configuration,” IEEE J. Quantum Electron. 27, 2296-2305 (1991).
[CrossRef]

Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
[CrossRef]

Dai, D.

Y. Shi and D. Dai, “Design of a compact multimode interference coupler based on deeply-etched SiO2 ridge waveguides,” Opt. Commun. 271, 404-407 (2007).
[CrossRef]

Farrell, G.

Glingener, C.

Huang, W. P.

W. P. Huang and C. L. Xu, “Simulation of 3-dimensional optical wave-guides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam-propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

Koshiba, M.

Leiler, G.

G. Leiler and L. Rezzolla, “Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity,” Phys. Rev. D 73, 044001 (2006).
[CrossRef]

Masoudi, H. M.

H. M. Masoudi and J. M. Arnold, “Spurious modes in the DuFort-Frankel finite-difference beam propagation method,” IEEE Photonics Technol. Lett. 9, 1382-1384 (1997).
[CrossRef]

Rezzolla, L.

G. Leiler and L. Rezzolla, “Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity,” Phys. Rev. D 73, 044001 (2006).
[CrossRef]

Saitoh, K.

Sato, Y.

Schulz, D.

Selleri, S.

Semenova, Y.

Sewell, P.

Shi, Y.

Y. Shi and D. Dai, “Design of a compact multimode interference coupler based on deeply-etched SiO2 ridge waveguides,” Opt. Commun. 271, 404-407 (2007).
[CrossRef]

Teukolsky, S. A.

S. A. Teukolsky, “Stability of the iterated Crank-Nicholson method in numerical relativity,” Phys. Rev. D 61, 087501 (2000).
[CrossRef]

Thylen, L.

Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
[CrossRef]

Vincetti, L.

Voges, E.

Vukovic, A.

Wang, Q.

Xiang, F.

F. Xiang and G. L. Yip, “An explicit and stable finite-difference 2-d vector beam-propagation method,” IEEE Photonics Technol. Lett. 6, 1248-1250 (1994).
[CrossRef]

Xu, C. L.

W. P. Huang and C. L. Xu, “Simulation of 3-dimensional optical wave-guides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam-propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).
[CrossRef]

Yip, G. L.

F. Xiang and G. L. Yip, “An explicit and stable finite-difference 2-d vector beam-propagation method,” IEEE Photonics Technol. Lett. 6, 1248-1250 (1994).
[CrossRef]

Zoboli, M.

Electron. Lett. (1)

Y. Chung, N. Dagli, and L. Thylen, “Explicit finite-difference vectorial beam propagation method,” Electron. Lett. 27, 2119-2121 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. P. Huang and C. L. Xu, “Simulation of 3-dimensional optical wave-guides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

Y. Chung and N. Dagli, “Analysis of z-invariant and z-variant semiconductor rib wave-guides by explicit finite-difference beam propagation method with nonuniform mesh configuration,” IEEE J. Quantum Electron. 27, 2296-2305 (1991).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

Y. C. Chung and N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photonics Technol. Lett. 6, 540-542 (1994).
[CrossRef]

F. Xiang and G. L. Yip, “An explicit and stable finite-difference 2-d vector beam-propagation method,” IEEE Photonics Technol. Lett. 6, 1248-1250 (1994).
[CrossRef]

H. M. Masoudi and J. M. Arnold, “Spurious modes in the DuFort-Frankel finite-difference beam propagation method,” IEEE Photonics Technol. Lett. 9, 1382-1384 (1997).
[CrossRef]

J. Lightwave Technol. (4)

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

Opt. Commun. (1)

Y. Shi and D. Dai, “Design of a compact multimode interference coupler based on deeply-etched SiO2 ridge waveguides,” Opt. Commun. 271, 404-407 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. D (2)

S. A. Teukolsky, “Stability of the iterated Crank-Nicholson method in numerical relativity,” Phys. Rev. D 61, 087501 (2000).
[CrossRef]

G. Leiler and L. Rezzolla, “Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity,” Phys. Rev. D 73, 044001 (2006).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic layout of the 1 × 4 MMI coupler. (b) Cross-section of the deeply etched Si O 2 ridge input waveguide. (c) Fundamental x-polarized mode at λ 0 = 1550 nm . The minor field component has a maximum value equal to 3.7 × 10 3 when the maximum value of the major component is 1.

Fig. 2
Fig. 2

(a) Distribution of the dominant electric field component ( E x ) over the whole extent of the MMI coupler, on the a b plane marked in Fig. 1b. (b) Distribution of the dominant electric field component ( E x ) on the exit plane of the 1 × 4 MMI coupler. Calculations were performed with the ICN-FD-VBPM.

Fig. 3
Fig. 3

Variation of the dominant electric field component ( E x ) along the a b line on the exit plane of the 1 × 4 MMI coupler. Calculations were performed with both the ICN-FD-VBPM and the conventional CN-FD-VBPM.

Fig. 4
Fig. 4

(a) Structural layout of a twin-core PCF coupler: Λ is the lattice constant, d is the cladding hole diameter. (b) Fundamental y-polarized mode of the single core waveguide at λ 0 = 1550 nm . The minor field component has a maximum value equal to 0.065 when the maximum value of the major component is 1.

Fig. 5
Fig. 5

The y-polarized mode field distribution at (a) z = 0 μ m , (b) z = L c 2 = 147.25 μ m , (c) z = L c = 294.5 μ m .

Fig. 6
Fig. 6

Guided power in core A versus propagation distance, calculated by the ICN-FD-VBPM, the conventional CN-FD-VBPM, and the CN-FE-VBPM.

Fig. 7
Fig. 7

(a) Top view of the tilted ridge waveguide: the cross section is that of Fig. 1b. (b) Dominant electric field component recorded on the c d plane for the paraxial, second-order wide-angle, and third-order wide-angle schemes.

Tables (2)

Tables Icon

Table 1 Simulation Times for the MMI Coupler

Tables Icon

Table 2 Simulation Times and Coupling Length for the Twin-Core PCF Coupler

Equations (36)

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

× × E k 0 2 n 2 E = 0 ,
2 E + k 0 2 n 2 E ( E ) = 0 .
2 E x + x [ 1 n 2 x ( n 2 E x ) ] + x [ 1 n 2 y ( n 2 E y ) ] 2 E x x 2 2 E y x y + k 0 2 n 2 E x = 0 ,
2 E y + y [ 1 n 2 y ( n 2 E y ) ] + y [ 1 n 2 x ( n 2 E x ) ] 2 E y y 2 2 E x y x + k 0 2 n 2 E y = 0 ,
2 E z + z [ 1 n 2 x ( n 2 E x ) ] + z [ 1 n 2 y ( n 2 E y ) ] 2 E x z x 2 E y z y + k 0 2 n 2 E z = 0 .
E = E ̃ exp ( j k ref z ) ,
z { [ E ̃ x E ̃ y ] } = [ A x x A x y A y x A y y ] [ E ̃ x E ̃ y ] ,
E ̃ z z = A z z E ̃ z + B ( E ̃ x , E ̃ y ) ,
A x x E ̃ x = 1 j 2 k ref { x [ 1 n 2 x ( n 2 E ̃ x ) ] + 2 E ̃ x y 2 + k 0 2 ( n 2 n ref 2 ) E ̃ x } ,
A x y E ̃ y = 1 j 2 k ref { x [ 1 n 2 y ( n 2 E ̃ y ) ] 2 E ̃ y x y } ,
A y x E ̃ x = 1 j 2 k ref { y [ 1 n 2 x ( n 2 E ̃ x ) ] 2 E ̃ x y x } ,
A y y E ̃ y = 1 j 2 k ref { y [ 1 n 2 y ( n 2 E ̃ y ) ] + 2 E ̃ y x 2 + k 0 2 ( n 2 n ref 2 ) E ̃ y } ,
A z z E ̃ z = 1 j 2 k ref [ 2 E ̃ z x 2 + 2 E ̃ z y 2 + k 0 2 ( n 2 n ref 2 ) E ̃ z ] ,
B ( E ̃ x , E ̃ y ) = 1 j 2 k ref ( z j k ref ) [ 1 n 2 x ( n 2 E ̃ x ) + 1 n 2 y ( n 2 E ̃ y ) E ̃ x x E ̃ y y ] .
z E ̃ t = A t E ̃ t ,
z E ̃ z = A z E ̃ z + B E ̃ t ,
E ̃ t l + 1 ( 1 ) E ̃ t l Δ z = A t E ̃ t l .
E ̃ t l + 1 2 ( 1 ) = a 1 E ̃ t l + 1 ( 1 ) + ( 1 a 1 ) E ̃ t l ,
E ̃ t l + 1 ( 2 ) E ̃ t l Δ z = A t E ̃ t l + 1 2 ( 1 ) ,
E ̃ t l + 1 2 ( 2 ) = a 2 E ̃ t l + 1 ( 2 ) + ( 1 a 2 ) E ̃ t l .
E ̃ t l + 1 E ̃ t l Δ z = A t E ̃ t l + 1 2 ( 2 ) .
E ̃ t l + 1 = [ I + Δ z A t + a 2 ( Δ z A t ) 2 + a 1 a 2 ( Δ z A t ) 3 ] E ̃ t l .
E ̃ z l + 1 ( 1 ) E ̃ z l Δ z = A z E ̃ z l + B E ̃ t l + 1 2 ,
E ̃ z l + 1 2 ( 1 ) = a 1 E ̃ z l + 1 ( 1 ) + ( 1 a 1 ) E ̃ z l ,
E ̃ z l + 1 ( 2 ) E ̃ z l Δ z = A z E ̃ z l + 1 2 ( 1 ) + B E ̃ t l + 1 2 ,
E ̃ z l + 1 2 ( 2 ) = a 2 E ̃ z l + 1 ( 2 ) + ( 1 a 2 ) E ̃ z l ,
E ̃ z l + 1 E ̃ z l Δ z = A z E ̃ z l + 1 2 ( 2 ) + B E ̃ t l + 1 2 .
ξ = ρ ( I + Δ z A t + a 2 Δ z 2 A t 2 + a 1 a 2 Δ z 3 A t 3 ) ,
Δ z 2 ρ ( j A t ) ,
Δ z min { [ 1 k ref ( 1 Δ x 2 + 1 Δ y 2 ) k 2 k ref 2 4 k ref ] 1 } .
ξ = ρ ( k = 0 n + 1 ( i = 2 k a n i + 2 ) Δ z k A t k ) .
1 j 2 k ref 2 E ̃ t z 2 + E ̃ t z = A t E ̃ t ,
E ̃ t z = j k ref ( I I + P ) E ̃ t ,
I + P = I + 1 2 P 1 8 P 2 + 1 16 P 3 + ,
z E ̃ t = A t WA E ̃ t ,
A t WA = j k ref ( 1 2 P 1 8 P 2 + 1 16 P 3 + ) .

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