Abstract

The solution of the scalar wave equation in the parabolic approximation is considered through the finite-difference and the Fourier-transform (i.e., beam propagation method) techniques. Examples are taken from the field of integrated optics and include propagation in straight, tapered, Y-branched, and coupled waveguides. A comparison of numerical results obtained by the two methods is presented, and a comparison with other analytical or numerical methods is also given. In the numerous cases studied it is shown that the finite-difference method yields a large, order-of-magnitude range improvement in accuracy or computational speed when compared with the Fourier-transform method.

© 1991 Optical Society of America

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References

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  1. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  2. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  3. C. Yeh, W. P. Brown, R. Szejn, “Multimode inhomogeneous fiber couplers,” Appl. Opt. 18, 489–495 (1979).
    [CrossRef] [PubMed]
  4. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,”J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  5. P. E. Lagasse, R. Baets, “The beam propagating method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
    [CrossRef]
  6. J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976).
  7. R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
    [CrossRef]
  8. M. Kuznetsov, “Radiation loss in dielectric waveguide Y-branch structures,” IEEE J. Lightwave Technol. LT-3, 674–677 (1985).
    [CrossRef]
  9. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 361.
  10. D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
    [CrossRef]
  11. D. Yevick, B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
    [CrossRef]
  12. Y. Chung, N. Dagli, “Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett. 26, 711–713 (1990).
    [CrossRef]

1990 (3)

R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
[CrossRef]

D. Yevick, B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
[CrossRef]

Y. Chung, N. Dagli, “Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett. 26, 711–713 (1990).
[CrossRef]

1989 (1)

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

1985 (1)

M. Kuznetsov, “Radiation loss in dielectric waveguide Y-branch structures,” IEEE J. Lightwave Technol. LT-3, 674–677 (1985).
[CrossRef]

1981 (1)

1979 (1)

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Baets, R.

P. E. Lagasse, R. Baets, “The beam propagating method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

Brown, W. P.

Chung, Y.

Y. Chung, N. Dagli, “Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett. 26, 711–713 (1990).
[CrossRef]

Claerbout, J. F.

J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976).

Dagli, N.

Y. Chung, N. Dagli, “Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett. 26, 711–713 (1990).
[CrossRef]

Feit, M. D.

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

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

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

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
[CrossRef]

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, “Radiation loss in dielectric waveguide Y-branch structures,” IEEE J. Lightwave Technol. LT-3, 674–677 (1985).
[CrossRef]

Lagasse, P. E.

J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,”J. Opt. Soc. Am. 71, 803–810 (1981).
[CrossRef]

P. E. Lagasse, R. Baets, “The beam propagating method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Osgood, R. M.

R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
[CrossRef]

Podlesnik, D. V.

R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
[CrossRef]

Szejn, R.

van der Donk, J.

Van Roey, J.

Yeh, C.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 361.

Yevick, D.

D. Yevick, B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
[CrossRef]

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Electron. Lett. (2)

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

Y. Chung, N. Dagli, “Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett. 26, 711–713 (1990).
[CrossRef]

IEEE J. Lightwave Technol. (1)

M. Kuznetsov, “Radiation loss in dielectric waveguide Y-branch structures,” IEEE J. Lightwave Technol. LT-3, 674–677 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Yevick, B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

R. Scarmozzino, D. V. Podlesnik, R. M. Osgood, “Losses of tapered dielectric slab waveguides with axial variations in index of refraction,” IEEE Trans. Microwave Theory Tech. 38, 141–147 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (3)

P. E. Lagasse, R. Baets, “The beam propagating method in integrated optics,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Dordrecht, The Netherlands, 1984), pp. 375–393.
[CrossRef]

J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 361.

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Figures (9)

Fig. 1
Fig. 1

Computational error in straight waveguide propagation as a function of the normalized index difference (NfNl)/Nl obtained from the FD method and the BPM. Both the peak (symbols) and steady-state average (solid lines) on-axis errors are shown. The waveguide parameters are k0 = 1, Nl = 1, Nf as indicated in the figure, d chosen for single mode as described in the text, and L = 500. The numerical parameters are Δx = 0.1d and h = 5. The boundary conditions are periodic with αn = 0.1, xα = 6.9/γ, and wb = 128Δx, and k ¯ = k0Nl was used.

Fig. 2
Fig. 2

Propagation in a tapered waveguide by (a) the FD method (5.3% loss), (b) the BPM (37% loss). The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.1, d0 = 1, dL = 0.5, and L = 100. The numerical parameters are Δx = 0.1 and h = 5. The boundary conditions are periodic with αn = 0.1, xα = 37, and wb = 51.2, and k ¯ = β was used.

Fig. 3
Fig. 3

Radiation loss in tapered waveguides as a function of taper length L obtained from the FD method and the BPM. Shown for comparison is the result obtained from a LNMA as in Ref. 7. Note the good agreement of the FD method with the LNMA. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.1, d0 = 1, dL = 0.5, and L = 100. The numerical parameters are Δx = (0.1, 0.05) for L = (1 − 300,400), and h = (0.2, 2, 5) for L = (1, 10, 30 − 400). The boundary conditions are periodic with αn = 0.1, xα = 37, and wb = 51.2, and k ¯ = β was used.

Fig. 4
Fig. 4

Propagation in a Y-branched waveguide by (a) the FD method (4.3% loss), (b) the BPM (15% loss). The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 5. The branch half-angle is ϕ = 2.5°, and the propagation distance is L = 2000. The numerical parameters are Δx = 0.5 and h = 20. The boundary conditions are periodic with αn = 0.1, xα = 88, and wb = 256, and k ¯ = was used.

Fig. 5
Fig. 5

Radiation loss in Y-branched waveguides as a function of half-angle ϕ obtained from the FD method. Shown for comparison is the result obtained in Ref. 8 by the VCM method as well as the approximate result obtained from |t1|2 as described in the text. Note the good agreement of the FD method with the VCM method. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 5. The Y-branch length L varied with angle as described in the text. The numerical parameters are Δx = 0.5 and h = 5. The boundary conditions are periodic with αn = 0.1, xα = 88, and wb = 256, and k ¯ = β was used.

Fig. 6
Fig. 6

Radiation loss in Y-branched waveguides as a function of half-angle ϕ obtained from the FD method and the BPM with large step size. Shown for comparison is the result obtained in Fig. 5 by the FD method with a small step size. Note that at this larger step size the FD method suffers a small error, while the BPM suffers a much larger error. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 5. The Y-branch length L varied with angle as described in the text. The numerical parameters are Δx = 0.5 and h as indicated in the figure. The boundary conditions are periodic with αn = 0.1, xα = 88, and wb = 256, and k ¯ = β was used.

Fig. 7
Fig. 7

Radiation loss in Y-branched waveguides as a function of Y-branch length L and for various branching half-angles ϕ obtained from the FD method and the BPM. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 5. The numerical parameters are Δx = 0.5 and h = 20. The boundary conditions are periodic with αn = 0.1, xα = 88, and wb = 256, and k ¯ = β was used.

Fig. 8
Fig. 8

Propagation in coupled slab waveguides by (a) the FD method, (b) the BPM. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 10. The waveguide separation is s = 20, and the propagation distance is L = 10,000. The numerical parameters are Δx = 1 and h = 100. The boundary conditions are periodic with αn = 0.1, xα = 63, and wb = 128, and k ¯ = β was used.

Fig. 9
Fig. 9

Coupling length Lc as a function of waveguide separation s by the FD method and the BPM. Shown for comparison is the analytic result obtained from coupled-mode theory. Note the good agreement of the FD method with the coupled-mode analysis. The waveguide parameters are k0 = 1, Nl = 1, Nf = 1.01, and d = 10. The numerical parameters are Δx = 1 and h = (10, 20, 50, 100, 200) for s = (0 − 1, 2 − 5, 10 − 15, 20, 30). The boundary conditions are periodic with αn = 0.1, xα = 63, and wb = 128, and k ¯ = β was used.

Equations (12)

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2 ϕ x 2 + 2 ϕ z 2 + ω 2 c 2 n 2 ( x , z ) ϕ = 0.
2 i k ¯ u z + u x x + ( k 2 - k ¯ 2 ) u = 0.
u ( x , z 0 + Δ z ) = ψ ( x , z 0 + Δ z ) exp ( i Γ ) .
Γ = k 2 - k ¯ 2 2 k ¯ Δ z
ψ ( x i , z ) = 1 N n = - N / 2 + 1 N / 2 Ψ n ( z ) exp ( - i κ n x i ) , κ n = 2 π n N Δ x , Ψ n ( z 0 + Δ z ) = Ψ n ( z 0 ) exp ( - i κ n 2 2 k ¯ Δ z ) ,
u z = A ( x , z ) u x x + B ( x , z ) u ,
u z u s r + 1 - u s r Δ z , A u x x 1 2 { u s - 1 r - 2 u s r + u s + 1 r Δ x 2 + u s - 1 r + 1 - 2 u s r + 1 + u s + 1 r + 1 Δ x 2 } A s r + 1 / 2 , B u 1 2 { u s r + u s r + 1 } B s r + 1 / 2 .
a s u s - 1 r + 1 + b s u s r + 1 + c s u s + 1 r + 1 = d s ,             s = 1 , N - 1 , a s = c s = - ρ A s r + 1 / 2 , b s = 2 ( 1 + ρ A s r + 1 / 2 ) - h B s r + 1 / 2 , d s = [ 2 ( 1 - ρ A s r + 1 / 2 ) + h B s r + 1 / 2 ] u s r + ρ A s r + 1 / 2 ( u s - 1 r + u s + 1 r ) .
u 0 = u N ,             a 0 u N - 1 + b 0 u 0 + c 0 u 1 = d 0 .
( b 1 - a 1 c 0 b 0 ) u 1 + c 1 u 2 - a 1 a 0 b 0 u N - 1 = d 1 - a 1 d 0 b 0 ,             - c N - 1 c 0 b 0 u 1 + a N - 1 u N - 2 + ( b N - 1 - c N - 1 a 0 b 0 ) u N - 1 = d N - 1 - c N - 1 d 0 b 0 .
n = N l ,             x < D ( z ) + x α , n = N l ( 1 + i α n ) ,             D ( z ) + x α x w b .
L c = π 2 κ ,             κ = p 2 q 2 exp ( - q s ) β ( 1 + q d ) ( p 2 + q 2 ) .

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