Abstract

We extend the method of lines [ T. Itoh, ed., Numerical Techniques for Microwave and Millimeter Wave Passive Structures ( Wiley, New York, 1989), pp. 381– 446] for application as a beam-propagation algorithm with absorbing boundary conditions. We thus avoid reflections from the computational window edges. The new algorithm has in principle none of the restrictions of standard beam-propagation methods (propagation under paraxial conditions in low-contrast media and relatively small propagation steps) and is applied to planar waveguides with high-refractive-index steps (e.g., straight or S-bend slab waveguides) and to rib waveguides.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. M. D. Feit and J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. of America A 7, 73–79 (1990).
    [CrossRef]
  3. P. Lagasse and R. Baets, “The beam propagation method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed., No. 86 of Series E of North Atlantic Treaty Organization Advanced Science Institute Series (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
    [CrossRef]
  4. T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
    [CrossRef]
  5. D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [CrossRef]
  6. R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.
  7. U. Rogge and R. Pregla, “The method of lines for the analysis of strip-loaded optical waveguides,” J. Opt. Soc. Am. B.459–463 (1991).
    [CrossRef]
  8. J. G. Blaschak and G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,” J. Computat. Phys. 77, 109–139 (1988).
    [CrossRef]
  9. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–77 (1975).
    [CrossRef]
  10. P. Lagasse and et al., “COST-216 comparative study of S-bend and directional coupler analysis methods,” in Proceedings of 16th European Conference on Optical Communication (Nijhoff, Amsterdam, 1990), Vol. 1, pp. 175–178.

1991 (1)

U. Rogge and R. Pregla, “The method of lines for the analysis of strip-loaded optical waveguides,” J. Opt. Soc. Am. B.459–463 (1991).
[CrossRef]

1990 (1)

M. D. Feit and J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. of America A 7, 73–79 (1990).
[CrossRef]

1989 (2)

T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

1988 (1)

J. G. Blaschak and G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,” J. Computat. Phys. 77, 109–139 (1988).
[CrossRef]

1978 (1)

1975 (1)

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–77 (1975).
[CrossRef]

Baets, R.

P. Lagasse and R. Baets, “The beam propagation method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed., No. 86 of Series E of North Atlantic Treaty Organization Advanced Science Institute Series (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

Blaschak, J. G.

J. G. Blaschak and G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,” J. Computat. Phys. 77, 109–139 (1988).
[CrossRef]

Davies, J. B.

T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Feit, M. D.

M. D. Feit and J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. of America A 7, 73–79 (1990).
[CrossRef]

M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. of America A 7, 73–79 (1990).
[CrossRef]

M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Harris, J.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–77 (1975).
[CrossRef]

Heiblum, M.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–77 (1975).
[CrossRef]

Hermansson, B.

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Koch, T. B.

T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Kriegsmann, G. A.

J. G. Blaschak and G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,” J. Computat. Phys. 77, 109–139 (1988).
[CrossRef]

Lagasse, P.

P. Lagasse and et al., “COST-216 comparative study of S-bend and directional coupler analysis methods,” in Proceedings of 16th European Conference on Optical Communication (Nijhoff, Amsterdam, 1990), Vol. 1, pp. 175–178.

P. Lagasse and R. Baets, “The beam propagation method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed., No. 86 of Series E of North Atlantic Treaty Organization Advanced Science Institute Series (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

Pascher, W.

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Pregla, R.

U. Rogge and R. Pregla, “The method of lines for the analysis of strip-loaded optical waveguides,” J. Opt. Soc. Am. B.459–463 (1991).
[CrossRef]

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Rogge, U.

U. Rogge and R. Pregla, “The method of lines for the analysis of strip-loaded optical waveguides,” J. Opt. Soc. Am. B.459–463 (1991).
[CrossRef]

Wickramasinghe, D.

T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Yevick, D.

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

T. B. Koch, J. B. Davies, and D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75–77 (1975).
[CrossRef]

J. Computat. Phys. (1)

J. G. Blaschak and G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,” J. Computat. Phys. 77, 109–139 (1988).
[CrossRef]

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

U. Rogge and R. Pregla, “The method of lines for the analysis of strip-loaded optical waveguides,” J. Opt. Soc. Am. B.459–463 (1991).
[CrossRef]

J. Opt. Soc. of America A (1)

M. D. Feit and J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. of America A 7, 73–79 (1990).
[CrossRef]

Other (3)

P. Lagasse and R. Baets, “The beam propagation method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed., No. 86 of Series E of North Atlantic Treaty Organization Advanced Science Institute Series (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

P. Lagasse and et al., “COST-216 comparative study of S-bend and directional coupler analysis methods,” in Proceedings of 16th European Conference on Optical Communication (Nijhoff, Amsterdam, 1990), Vol. 1, pp. 175–178.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

One-dimensional discretization for the MoL–BPM.

Fig. 2
Fig. 2

Partition of a ridge-waveguide structure in basic medium nb and remaining difference index nd for a 3D MoL–BPM.

Fig. 3
Fig. 3

Discretization for a 3D MoL–BPM.

Fig. 4
Fig. 4

Propagation of a Gaussian beam with sloping incidence in a homogeneous medium. a, Standard BPM; b, BPM based on the MoL. λ = 1.55 μm, w = 7 μm, N = 64, α = 45°, Δz = 0.6 μm, zmax = 15 μm, n1 = n2 = n3 = 1.0.

Fig. 5
Fig. 5

Beam propagation in slab waveguides, calculated with the MoL. a, Monomode slab waveguide, excited with a Gaussian beam. λ = 1.55 μm, w = 7 μm, N = 27, Δz = 20 μm, zmax = 980 μm, n1 = 1.0, n2 = 3.2584, n3 = 3.1694, wf = 1.0 μm. b, Multimode slab waveguide, excited with two eigenmodes. λ = 1.55 μm, w = 7 μm, N = 32, Δz = 2 μm, zmax = 60 μm, n1 = 1.0, n2 = 3.2584, n3 = 3.1694, wf = 1.8 μm.

Fig. 6
Fig. 6

Loss calculations for rib-waveguide S bends.

Fig. 7
Fig. 7

Intensity distribution of a TE0 mode in a rib waveguide, calculated with the MoL–BPM. Waveguide parameters are as in Fig. 8, λ = 1.286 μm, Δz = 0.3 μm, z = 1.95 mm. Excitation is with a Gaussian beam at z = 0.

Fig. 8
Fig. 8

Lateral TE field distribution of a rib waveguide, calculated with the MoL–BPM. Rib-waveguide parameters and dimensions are ws = 2.0 μm, wt = 0.3 μm, wf = 1.0 μm. Simulation parameters are λ = 1.286 μm, Δz = 0.3 μm, z = 1.95 mm. Excitation is a Gaussian beam at z = 0.

Equations (41)

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

2 z ¯ 2 ψ h + 2 x ¯ 2 ψ h + ɛ r ( x ) ψ h = 0 ,
2 z ¯ 2 ψ e + ɛ r ( x ) x ¯ [ 1 ɛ r ( x ) x ¯ ψ e ] + ɛ r ( x ) ψ e = 0.
ψ e ψ e , ψ h ψ h , ɛ r ɛ h             ( on dashed lines ) , ɛ r ɛ e             ( on solid lines ) , h ¯ ɛ r ( x ) x ¯ [ h ¯ ɛ r ( x ) x ¯ ψ e ] - ɛ e D t ɛ h - 1 D a P e ɛ ψ e , h ¯ 2 2 x ¯ ψ h D t D a P h ψ h ,
d 2 d z ¯ 2 ψ h - ( h ¯ - 2 P h - ɛ h ) ψ h = 0 ,
d 2 d z ¯ 2 ψ e - ( h ¯ - 2 P e ɛ - ɛ e ) ψ e = 0 ,
L ψ e , h = L + L - ψ e , h = 0 ,
L ± = x ¯ ± j ɛ r 1 / 2 ( 1 + 1 ɛ r 2 z ¯ 2 ) 1 / 2
x ¯ ɛ r ( x , z ) = 0
L - ψ = 0
L + ψ = 0
L ± = x ¯ ± j ɛ r p 1 / 2 ( 1 + 1 2 ɛ r p d 2 d z ¯ 2 ) ,             p = r , l ,
L D ± D x ± j ɛ r p 1 / 2 ( 1 + 1 2 ɛ r p d 2 d z ¯ 2 ) I ,
ψ 0 = - a 1 ψ 1 + b 1 ψ 2 , ψ N + 1 = b r ψ N - 1 - a r ψ N ,
a p = 2 + n d 2 1 + j n d , b p = 1 - j n d 1 + j n d , p = r , l , n d = h ɛ ¯ r p 1 / 2 .
ψ 0 [ ψ 1 ψ 2 ψ N - 1 , ψ N ] ψ N + 1 h ψ x [ 1 - 1 1 - 1 1 - 1 ] - 1 1 D [ a l - b l - 1 1 - 1 1 b r - a r ] D a ,
d 2 d z ¯ 2 ψ e , h - Q e , h ψ e , h = 0 ,
Q e = h ¯ - 2 P e ɛ - ɛ e , Q h = h ¯ - 2 P h - ɛ h .
ψ ¯ e , h = T e , h - 1 ψ e , h ,             with T e , h - 1 Q e , h T = λ ¯ e , h 2 ,
d 2 d z ¯ 2 ψ ¯ e , h - λ ¯ e , h 2 ψ ¯ e , h = 0 ,
ψ ¯ e , h ( z + Δ z ) = exp ( - λ ¯ e , h Δ z ¯ ) ψ ¯ e , h ( z ) .
ψ ¯ e , h ( z + Δ z ) = T e , h ψ ¯ e , h ( z + Δ z ) .
2 ψ + k 0 2 n 2 ( x , y , z ) ψ = 0.
ψ ( x , y , z 0 + Δ z ) = ϕ ( x , y , z 0 + Δ z ) exp Γ .
2 ϕ + k 0 2 n b 2 ( x , y ) ϕ = 0
Γ = - j k 0 z z + Δ z n d ( x , y , z ) d z .
n 2 ( x , y , z ) = n b 2 ( x , y ) + n d 2 ( x , y , z ) .
n b 2 = n x 2 ( x ) + n y 2 ( y ) .
n x 2 ( x ) = 0 , n m 2 = ½ ( n 0 2 + n t 2 )             or n m 2 = n t 2 .
ϕ ϕ , n b 2 ϕ n y 2 ϕ + ϕ n x 2 .
d 2 d z ¯ 2 ϕ - h ¯ y - 2 P y ϕ - h ¯ x - 2 P x + n y 2 ϕ + ϕ n x 2 = 0.
d 2 d z ¯ 2 ϕ - Q y ϕ - ϕ Q x = 0 ,
Q p = h ¯ p - 2 P p - n p 2 ,             h ¯ p = k 0 h p ,             p = x , y .
T p - 1 Q p T p = λ ¯ p 2 ,             p = x , y .
ϕ ¯ = T y - 1 ϕ T x .
d 2 d z ¯ 2 ϕ ¯ - λ ¯ y 2 ϕ ¯ - ϕ ¯ λ x 2 = 0.
d 2 d z ¯ 2 ϕ ¯ ˜ - λ ^ 2 ϕ ¯ ˜ = 0 ,
λ ^ 2 = λ ^ x 2 + λ ^ y 2
λ ^ x 2 = λ ¯ x 2 I y ,             λ ^ y 2 = I x λ ¯ y 2 ,
ϕ ¯ ˜ ( z + Δ z ) = exp ( - λ ^ Δ z ¯ ) ϕ ¯ ˜ ( z ) .
ϕ ^ = ( T x - 1 T y ) ϕ ¯ ˜ .
ψ ( z + Δ z ) = ϕ ( z + Δ z ) · exp Γ ,

Metrics