Abstract

A novel beam-propagation method, based on diagonalization of the Hermitian operator that generates the solution to the Helmholtz equation in media with real refractive indices, is described. This method solves some of the problems of the conventional split-step beam-propagation method, namely, the difficulty of handling large index steps and large spatial frequencies, at the expense of higher computational requirements in structures with a refractive-index distribution that varies in the direction of propagation.

© 1992 Optical Society of America

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. 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]
  3. L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
    [CrossRef]
  4. B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
    [CrossRef]
  5. L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.
  6. P. Danielsen, “Two-dimensional propagating beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
    [CrossRef]
  7. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [CrossRef] [PubMed]
  8. K. Koai, P. L. Liu, “Modelling of Ti:LiNbO3waveguide devices: part I—directional couplers,” IEEE J. Lightwave Technol. 7, 533–539 (1989).
    [CrossRef]
  9. L. Thylén, P. Svensson, W. K. Burns, “Beam propagation method analysis of the digital switch,” in Integrated and Guided-Wave Optics, Vol. 4 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper WBB3, pp. 229–232.
  10. D. Yevick, B. Hermansson, “The accuracy of band-structure calculations based on the split-step fast Fourier transform method,”J. Phys. C 18, 4303–4314 (1985).
    [CrossRef]
  11. I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
    [CrossRef]
  12. M. D. Feit, J. A. Fleck, “Computation of mode eigenfunctions in graded index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980).
    [CrossRef] [PubMed]
  13. D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated optics structures,” Electron. Lett. 25, 1611–1613 (1989).
    [CrossRef]
  14. L. Thylén, “Integrated optics in LiNbO3: recent developments in devices for telecommunications,” IEEE J. Lightwave Technol. 6, 847–861 (1988).
    [CrossRef]
  15. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  16. Sun sparc station is a trademark of Sun Microsystems Incorporated.
  17. L. Thylén, E. M. Wright, G. I. Stegeman, C. T. Seaton, J. V. Moloney, “Beam propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
    [CrossRef] [PubMed]

1990 (1)

I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
[CrossRef]

1989 (2)

K. Koai, P. L. Liu, “Modelling of Ti:LiNbO3waveguide devices: part I—directional couplers,” IEEE J. Lightwave Technol. 7, 533–539 (1989).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated optics structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

1988 (1)

L. Thylén, “Integrated optics in LiNbO3: recent developments in devices for telecommunications,” IEEE J. Lightwave Technol. 6, 847–861 (1988).
[CrossRef]

1986 (1)

1985 (1)

D. Yevick, B. Hermansson, “The accuracy of band-structure calculations based on the split-step fast Fourier transform method,”J. Phys. C 18, 4303–4314 (1985).
[CrossRef]

1984 (2)

B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

P. Danielsen, “Two-dimensional propagating beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[CrossRef]

1983 (1)

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

1978 (1)

Arvidsson, G.

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

Baets, R.

Burns, W. K.

L. Thylén, P. Svensson, W. K. Burns, “Beam propagation method analysis of the digital switch,” in Integrated and Guided-Wave Optics, Vol. 4 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper WBB3, pp. 229–232.

Danielsen, P.

P. Danielsen, “Two-dimensional propagating beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[CrossRef]

Djupsjöbacka, A.

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

Feit, M. D.

Fleck, J. A.

Glasner, M.

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated optics structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

Gustafson, T. K.

I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “The accuracy of band-structure calculations based on the split-step fast Fourier transform method,”J. Phys. C 18, 4303–4314 (1985).
[CrossRef]

B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Kim, I.

I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
[CrossRef]

Koai, K.

K. Koai, P. L. Liu, “Modelling of Ti:LiNbO3waveguide devices: part I—directional couplers,” IEEE J. Lightwave Technol. 7, 533–539 (1989).
[CrossRef]

Lagasse, P. E.

Lagerström, B.

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

Liu, P. L.

K. Koai, P. L. Liu, “Modelling of Ti:LiNbO3waveguide devices: part I—directional couplers,” IEEE J. Lightwave Technol. 7, 533–539 (1989).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Moloney, J. V.

Seaton, C. T.

Stegeman, G. I.

Svensson, P.

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

L. Thylén, P. Svensson, W. K. Burns, “Beam propagation method analysis of the digital switch,” in Integrated and Guided-Wave Optics, Vol. 4 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper WBB3, pp. 229–232.

Thylén, L.

I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
[CrossRef]

L. Thylén, “Integrated optics in LiNbO3: recent developments in devices for telecommunications,” IEEE J. Lightwave Technol. 6, 847–861 (1988).
[CrossRef]

L. Thylén, E. M. Wright, G. I. Stegeman, C. T. Seaton, J. V. Moloney, “Beam propagation method analysis of a nonlinear directional coupler,” Opt. Lett. 11, 739–741 (1986).
[CrossRef] [PubMed]

B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

L. Thylén, P. Svensson, W. K. Burns, “Beam propagation method analysis of the digital switch,” in Integrated and Guided-Wave Optics, Vol. 4 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper WBB3, pp. 229–232.

van der Donk, J.

Van Roey, J.

Wright, E. M.

Yevick, D.

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated optics structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

D. Yevick, B. Hermansson, “The accuracy of band-structure calculations based on the split-step fast Fourier transform method,”J. Phys. C 18, 4303–4314 (1985).
[CrossRef]

B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

I. Kim, T. K. Gustafson, L. Thylén, “Analysis of quantum confined structures using the beam propagation method,” Appl. Phys. Lett. 57, 285–287 (1990).
[CrossRef]

Electron. Lett. (1)

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated optics structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

IEEE J. Lightwave Technol. (2)

L. Thylén, “Integrated optics in LiNbO3: recent developments in devices for telecommunications,” IEEE J. Lightwave Technol. 6, 847–861 (1988).
[CrossRef]

K. Koai, P. L. Liu, “Modelling of Ti:LiNbO3waveguide devices: part I—directional couplers,” IEEE J. Lightwave Technol. 7, 533–539 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Danielsen, “Two-dimensional propagating beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. C (1)

D. Yevick, B. Hermansson, “The accuracy of band-structure calculations based on the split-step fast Fourier transform method,”J. Phys. C 18, 4303–4314 (1985).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

B. Hermansson, L. Thylén, D. Yevick, “A propagating beam method analysis of nonlinear effects in optical waveguides,” Opt. Quantum Electron. 16, 525–534 (1984).
[CrossRef]

Other (4)

L. Thylén, B. Lagerström, P. Svensson, A. Djupsjöbacka, G. Arvidsson, “Computer analysis and design of Ti:LiNbO3integrated optics devices and comparison with experiments,” in Proceedings of the 9th European Conference on Optical Communications (Elsevier, Amsterdam, 1983), pp. 425–428.

L. Thylén, P. Svensson, W. K. Burns, “Beam propagation method analysis of the digital switch,” in Integrated and Guided-Wave Optics, Vol. 4 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper WBB3, pp. 229–232.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Sun sparc station is a trademark of Sun Microsystems Incorporated.

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Figures (5)

Fig. 1
Fig. 1

Optical field evolution on a logarithmic scale through a symmetric step-index single-mode waveguide of width 0.1 μm excited by its fundamental mode at wavelength λ = 1.55 μm. The propagation step and total propagation length are 200 μm and 10 mm, respectively. The refractive index of the film is 3.1246, and the substrate index is 1.0.

Fig. 2
Fig. 2

Optical field evolution through a symmetric step-index single-mode waveguide excited by a Gaussian noneigenmode at wavelength λ = 1.55 μm. Waveguide parameters are unchanged compared with Fig. 1. The propagation step and total propagation length are 0.1 and 6.0 μm, respectively. The Gaussian beam spot size at z = 0 is 0.1 μm.

Fig. 3
Fig. 3

Optical field evolution through a multimode cosh−2 profile waveguide excited by the first-order bound mode at wavelength λ = 1.55 μm. The nominal guide thickness h is 0.5 μm. The substrate index ns and the peak index deviations Δn are 3.6 and 0.3124, respectively. The propagation step is 200 μm, and the total propagation length is 10 mm.

Fig. 4
Fig. 4

(a) Optical field evolution of a Gaussian beam traveling at a 45° angle to the optical axis in a homogeneous medium of refractive index n = 1.5. The Gaussian beam spot size at z = 0 is 10 μ m. Wavelength λ is 0.633 μm. The propagation step and the total propagation length are 0.3 and 30 μm, respectively. (b) Contour plot of the optical field levels 0.7, 0.6, 0.5, and 0.4.

Fig. 5
Fig. 5

(a) Power exchange in a directional coupler of identical step-index-profile single-mode waveguides of width 2 μm. Refractive indices of the film and the substrate are 3.1796 and 3.17, respectively. Separation of the waveguides, from edge to edge, is 5 μm. The propagation step is 200 μm, and the total propagation length is 10 mm. The wavelength is 1.55 μm. (b) Power exchange in a narrow directional coupler of identical step-index-profile single-mode waveguides of width 0.1 μm. Refractive indices of the film and the substrate are 4.3373 and 3.17, respectively. Separation of the waveguides, from edge to edge, is 0.6 μm. The propagation step is 1 μm, and the total propagation length is 0.1 mm. The wavelength is 1.55 μm.

Equations (10)

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2 E z 2 + 2 E + ( n 2 k 0 2 - n 0 2 k 0 2 ) E = 0
2 2 x 2 ,
E ( x , z ) = E 0 ( x , z ) exp ( - i n 0 k 0 z ) ,
2 E ^ 0 z 2 - 2 i n 0 k 0 E ^ 0 z - κ 2 E ^ 0 + k 0 2 · n ^ 2 * E ^ 0 = 0 ,
( 2 z 2 - 2 i n 0 k 0 z ) E ^ 0 = M E ^ 0 ,
E ^ 0 = E ^ 0 ( κ , z = 0 ) = [ E ^ 0 ( κ 1 ) , E ^ 0 ( κ 2 ) , , E ^ 0 ( κ N ) ] , M = Diag ( κ 1 2 κ 1 2 κ N 2 ) - k 0 2 D , D i j = f ^ ( κ i - κ j ) / N ,
E ^ ( κ , z ) = exp ( - i z B ) E ^ ( κ , z = 0 ) ,
U BU = B d
E ^ ( κ , z ) = U exp ( - i z B d ) U E ^ ( κ , z = 0 ) .
{ 1 x < x ie cos 2 ( π 2 x - x ie x oe - x ie ) x ie x x oe 0 x oe < x < x b ,

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