Abstract

We derive a general method for solving second-order wave equations on the basis of a fourth-order Taylor expansion, with respect to time, of the field and its time derivative. The method requires explicit evaluation of space derivatives up to fourth order of the field and its time derivative, and the time step is subject to a stability criterion, similar to the one routinely applied in conjunction with the time-domain–finite-difference method. The Taylor-series method is generalizable to vector wave equations, to wave equations involving dissipation, and to the Helmholtz wave equation. The method is equally applicable to situations involving large or small variations in the refractive index. Sample numerical solutions in one and two space dimensions are described.

© 1998 Optical Society of America

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  1. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  2. A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
    [CrossRef]
  3. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  4. A. Taflove, K. R. Umashanker, “The finite-difference time-domain method for numerical modeling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering. (Elsevier, New York, 1990).
  5. S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
    [CrossRef]
  6. W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
    [CrossRef]
  7. D. M. Young, R. T. Gregory, A Survey of Numerical Methods (Addison-Wesley, Reading, Mass., 1971), Vol. 1.
  8. M. D. Feit, J. A. Fleck, “Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662–664 (1989).
    [CrossRef] [PubMed]
  9. A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
    [CrossRef]
  10. R. P. Ratowsky, J. A. Fleck, “Treatment of angular derivatives in the Schrödinger equation by the finite Fourier series method,” J. Comput. Phys. 93, 376–387 (1991).
    [CrossRef]
  11. The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  12. For a discussion of the sampling theorem and its implications on the accuracy of band-limited functions and their derivatives see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  13. M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
    [CrossRef]
  14. M. R. Hermann, J. A. Fleck, “Split operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates,” Phys. Rev. A 38, 6000–6012 (1988).
    [CrossRef] [PubMed]
  15. J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, New York, 1976).
  16. D. Y. Lee, Y. Saad, M. H. Schultz, “An efficient method for solving the three-dimensional wide angle wave equation,” in Computational Acoustics, Vol 1—Wave Propagation, D. Lee, R. I. Sternberg, M. H. Schultz, eds. (North-Holland, Amsterdam, 1988).
  17. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate solution of the Helmholtz equation by Lanczos orthoganalization for media with loss or gain,” Opt. Lett. 17, 10–12 (1992).
    [CrossRef] [PubMed]
  18. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate description of ultra wide-angle beam propation in homogeneous media by Lanczos orthoganalization,” Opt. Lett. 19, 1284–1286 (1994).
    [CrossRef] [PubMed]
  19. J. A. Fleck “Solution of the scalar Helmholtz wave equation by Lanczos reduction,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices: Part II: Waves and Interactions, W. P. Huang, ed. (EMW Publishing, Cambridge, Mass., 1995).
  20. A. Taflove, M. E. Baldwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
    [CrossRef]
  21. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 22–23.
  22. See, for example, M. D. Feit, J. A. Fleck, “Filament formation and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–644 (1988) and references therein.
    [CrossRef]

1994 (1)

1992 (1)

1991 (4)

S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, “Treatment of angular derivatives in the Schrödinger equation by the finite Fourier series method,” J. Comput. Phys. 93, 376–387 (1991).
[CrossRef]

1989 (1)

1988 (3)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

M. R. Hermann, J. A. Fleck, “Split operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates,” Phys. Rev. A 38, 6000–6012 (1988).
[CrossRef] [PubMed]

See, for example, M. D. Feit, J. A. Fleck, “Filament formation and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–644 (1988) and references therein.
[CrossRef]

1982 (1)

M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

1976 (1)

The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

1975 (1)

A. Taflove, M. E. Baldwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Baldwin, M. E.

A. Taflove, M. E. Baldwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 22–23.

Brigham, E. O.

For a discussion of the sampling theorem and its implications on the accuracy of band-limited functions and their derivatives see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Chaudari, S. K.

S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
[CrossRef]

Chaudhuri, S. K.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Chu, S. T.

S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

Claerbout, J. F.

J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, New York, 1976).

Feit, M. D.

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate description of ultra wide-angle beam propation in homogeneous media by Lanczos orthoganalization,” Opt. Lett. 19, 1284–1286 (1994).
[CrossRef] [PubMed]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate solution of the Helmholtz equation by Lanczos orthoganalization for media with loss or gain,” Opt. Lett. 17, 10–12 (1992).
[CrossRef] [PubMed]

M. D. Feit, J. A. Fleck, “Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662–664 (1989).
[CrossRef] [PubMed]

See, for example, M. D. Feit, J. A. Fleck, “Filament formation and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–644 (1988) and references therein.
[CrossRef]

M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Fleck, J. A.

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate description of ultra wide-angle beam propation in homogeneous media by Lanczos orthoganalization,” Opt. Lett. 19, 1284–1286 (1994).
[CrossRef] [PubMed]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate solution of the Helmholtz equation by Lanczos orthoganalization for media with loss or gain,” Opt. Lett. 17, 10–12 (1992).
[CrossRef] [PubMed]

R. P. Ratowsky, J. A. Fleck, “Treatment of angular derivatives in the Schrödinger equation by the finite Fourier series method,” J. Comput. Phys. 93, 376–387 (1991).
[CrossRef]

M. D. Feit, J. A. Fleck, “Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662–664 (1989).
[CrossRef] [PubMed]

M. R. Hermann, J. A. Fleck, “Split operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates,” Phys. Rev. A 38, 6000–6012 (1988).
[CrossRef] [PubMed]

See, for example, M. D. Feit, J. A. Fleck, “Filament formation and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–644 (1988) and references therein.
[CrossRef]

M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. A. Fleck “Solution of the scalar Helmholtz wave equation by Lanczos reduction,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices: Part II: Waves and Interactions, W. P. Huang, ed. (EMW Publishing, Cambridge, Mass., 1995).

Goss, A.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

Gregory, R. T.

D. M. Young, R. T. Gregory, A Survey of Numerical Methods (Addison-Wesley, Reading, Mass., 1971), Vol. 1.

Hermann, M. R.

M. R. Hermann, J. A. Fleck, “Split operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates,” Phys. Rev. A 38, 6000–6012 (1988).
[CrossRef] [PubMed]

Huang, W. P.

S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

Lee, D. Y.

D. Y. Lee, Y. Saad, M. H. Schultz, “An efficient method for solving the three-dimensional wide angle wave equation,” in Computational Acoustics, Vol 1—Wave Propagation, D. Lee, R. I. Sternberg, M. H. Schultz, eds. (North-Holland, Amsterdam, 1988).

Majd, M.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Morris, J. R.

The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Petermann, K.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Ratowsky, R. P.

Saad, Y.

D. Y. Lee, Y. Saad, M. H. Schultz, “An efficient method for solving the three-dimensional wide angle wave equation,” in Computational Acoustics, Vol 1—Wave Propagation, D. Lee, R. I. Sternberg, M. H. Schultz, eds. (North-Holland, Amsterdam, 1988).

Schultz, M. H.

D. Y. Lee, Y. Saad, M. H. Schultz, “An efficient method for solving the three-dimensional wide angle wave equation,” in Computational Acoustics, Vol 1—Wave Propagation, D. Lee, R. I. Sternberg, M. H. Schultz, eds. (North-Holland, Amsterdam, 1988).

Splett, A.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Steiger, A.

M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

Taflove, A.

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

A. Taflove, M. E. Baldwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

A. Taflove, K. R. Umashanker, “The finite-difference time-domain method for numerical modeling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering. (Elsevier, New York, 1990).

Umashanker, K. R.

A. Taflove, K. R. Umashanker, “The finite-difference time-domain method for numerical modeling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering. (Elsevier, New York, 1990).

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 22–23.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Young, D. M.

D. M. Young, R. T. Gregory, A Survey of Numerical Methods (Addison-Wesley, Reading, Mass., 1971), Vol. 1.

Appl. Phys. (1)

The split operator beam-propagation method takes advantage of the ease and accuracy of computing spatial derivatives by the FFT method. See, for example, 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);J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353–363 (1978); andM. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Comput. Phys. Pub. (1)

S. T. Chu, W. P. Huang, S. K. Chaudari, “Simulation and analysis of waveguide based optical integrated circuits,” Comput. Phys. Pub. 68, 451–484 (1991).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, “A scalar finite-difference time-domain approach for guided-wave optics,” IEEE Photonics Technol. Lett. 3, 524–526 (1991).
[CrossRef]

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,” IEEE Photonics Technol. Lett. 3, 466–468 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

IEEE Trans. Microwave Theory Tech. (1)

A. Taflove, M. E. Baldwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

J. Comput. Phys. (2)

R. P. Ratowsky, J. A. Fleck, “Treatment of angular derivatives in the Schrödinger equation by the finite Fourier series method,” J. Comput. Phys. 93, 376–387 (1991).
[CrossRef]

M. D. Feit, J. A. Fleck, A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. A (1)

M. R. Hermann, J. A. Fleck, “Split operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinates,” Phys. Rev. A 38, 6000–6012 (1988).
[CrossRef] [PubMed]

Wave Motion (1)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

Other (8)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

A. Taflove, K. R. Umashanker, “The finite-difference time-domain method for numerical modeling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering. (Elsevier, New York, 1990).

D. M. Young, R. T. Gregory, A Survey of Numerical Methods (Addison-Wesley, Reading, Mass., 1971), Vol. 1.

J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, New York, 1976).

D. Y. Lee, Y. Saad, M. H. Schultz, “An efficient method for solving the three-dimensional wide angle wave equation,” in Computational Acoustics, Vol 1—Wave Propagation, D. Lee, R. I. Sternberg, M. H. Schultz, eds. (North-Holland, Amsterdam, 1988).

J. A. Fleck “Solution of the scalar Helmholtz wave equation by Lanczos reduction,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices: Part II: Waves and Interactions, W. P. Huang, ed. (EMW Publishing, Cambridge, Mass., 1995).

For a discussion of the sampling theorem and its implications on the accuracy of band-limited functions and their derivatives see, for example, E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 22–23.

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Figures (22)

Fig. 1
Fig. 1

Plot of ratio of the local group velocity vg(k) to the phase velocity vp as a function of k(normalized)=k/κmax, where κmax=π/Δx. The wider the variation of vg(k) from the phase velocity vp the more the constituent Fourier components will dephase with respect to one another and the more the wave packet will spread over the integration cycle time.

Fig. 2
Fig. 2

Step refractive-index profile, modeled with an l=22 super-Lorentz distribution.

Fig. 3
Fig. 3

Plot of reciprocal permittivity, 1/n(x)2, and a Gaussian-field amplitude distribution versus position, with refractive-index profile from Fig. 2.

Fig. 4
Fig. 4

Plot of reciprocal permittivity or 1/n(x)2 and (2/x2)×(1/n2)(2E/x2), labeled fourth derivative, for the finest zoning case with N=128 grid points across figure. Field and refractive index are those used in Fig. 3. Analytical and numerical computations coincide to within the accuracy of the plot.

Fig. 5
Fig. 5

Plot of analytic reciprocal permittivity or 1/n(x)2 and (2/x2)(1/n2)(2E/x2), with N=64. Numerical values are plotted as points.

Fig. 6
Fig. 6

Plot of analytic reciprocal permittivity or 1/n(x)2 and (2/x2)(1/n2)(2E/x2), with N=32. Numerical values are plotted as points.

Fig. 7
Fig. 7

Initial Gaussian field distribution and refractive-index distribution, which makes an abrupt change from n=1 to n=3.

Fig. 8
Fig. 8

Gray-scale contour image of pulse propagating in a medium where the refractive index undergoes an abrupt change of refractive index.

Fig. 9
Fig. 9

As the field passes through the refractive-index transition region, it breaks up into transmitted and reflected contributions. The reflected field has undergone phase reversal.

Fig. 10
Fig. 10

Scattered and transmitted fields at ct=80. The difference in width of scattered and transmitted fields is due to wavelength scaling by the refractive index in the two regions.

Fig. 11
Fig. 11

Electric-field energy density u for ct=0 as a function of position.

Fig. 12
Fig. 12

Electric-field energy density u for ct=80 as a function of position.

Fig. 13
Fig. 13

Gray-scale contour image of the electric-field energy density.

Fig. 14
Fig. 14

Total electric-field energy on the grid as a function of time.

Fig. 15
Fig. 15

Gray-scale contour image of the initial field, indicating its placement in the y junction device.

Fig. 16
Fig. 16

Initial on-axis field as a function of axial distance.

Fig. 17
Fig. 17

Gray-scale contour image of the intensity after propagating 72 time steps in y junction device.

Fig. 18
Fig. 18

Initial field and circular waveguide configurations. Contour lines define waveguide boundaries.

Fig. 19
Fig. 19

Propagation in a circular waveguide. Field configuration after 50 time steps.

Fig. 20
Fig. 20

Propagation in a circular waveguide. Intensity configuration after 50 time steps.

Fig. 21
Fig. 21

Solution of the Helmholtz equation for the electric field, ψ(x, z), obtained with the help of the fourth-order Taylor-series algorithm for a waveguide with Δn=1.

Fig. 22
Fig. 22

Solution of the Helmholtz equation for a waveguide with Δn=0.02.

Equations (131)

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

E(r, t)=n=0NnEtntn+O(tN+1).
2Ez2+2E+ω2c2n2(r)=0,
i ψz=Hψ,
H=12k2+k2[n2(r)-1],
ψ(r)=exp(-ikz)E(r)
ψ(z, r)=exp{±ikz[1-(1+2H/k)1/2]}ψ(0, r).
2Et2=c2n22E,
E(r, t)=E(0)+Et0t+122Et20t2+163Et30t3+1244Et40t4+O(t5),
E(r, t)=E(r, t)t.
2Et2=c2n22E.
E(r, t)=E(0)+Et0t+122Et20t2+163Et30t3+1244Et40t4+O(t5),
E(r, t)=E(0)+E(0)t+122Et20t2+162Et20t3+1244Et40t4+O(t5).
E(r, t)=E(0)+2Et20t+122Et20t2+164Et40t3+1244Et40t4+O(t5).
2Et2=c2n22E,
4Et4=c2n22 2Et2=c2n22 c2n22E,
2Et2=c2n22E,
4Et4=c2n22 c2n22E.
E(r, t)=E(0)+E(0)t+12c2n22E(0)t2+16c2n22E(0)t3+124c2n2×2 c2n22E(0)t4+O(t5),
E(r, t)=E(0)+c2n22E(0)t+12c2n22E(0)t2+16c2n22 c2n22E(0)t3+124c2n22c2n2×2E(0)t4+O(t5).
p2=c2n22,
E(t)=cosh(pt)E(0)+1psinh(pt)E(0).
E(t)=p sinh(pt)E(0)+cosh(pt)E(0).
E(t)=E(0)+E(0)t+12p2E(0)t2+16p2E(0)t3+124p4E(0)t4+O(t5),
E(t)=E(0)+p2E(0)t+12p2E(0)t2+16p4E(0)t3+124p4E(0)t4+O(t5),
E(k, t)=E(k, 0)coscntκx2+κy2+κz2+ncκx2+κy2+κz2E(k, 0)×sincntκx2+κy2+κz2.
E(k, t)=E(k, 0)12expicntκx2+κy2+κz2+12exp-icntκx2+κy2+κz2.
E(x, t)=12fx-cnt+fx+cnt,
E(k, 0)=±i cnκx2+κy2+κz2E(k, 0).
E(0)=-cnf(x),
E(x, t)=fx-cnt.
×H=1cDt,
×E=-1cHt.
1c2Dt2=-××E,
2Et2=c2n2[2E-(·E)],
·(E)=0
2Et2=c2n22E+1n2E·(n2).
t(·E)=0.
E(t)=cosh(pt)E(0)+1psinh(pt)E(0).
E(t)=p sinh(pt)E(0)+cosh(pt)E(0),
p2=-c2n2××.
E(t)=E(0)+E(0)t+12p2E(0)t2+16p2E(0)t3+124p4E(0)t4+O(t5),
E(t)=E(0)+p2E(0)t+12p2E(0)t2+16p4E(0)t3+124p4E(0)t4+O(t5).
V2=p2E(0)=c2n2{2E(0)-[·E(0)]},
V4=p4E(0)=p2[p2E(0)]=p2V2=c2n2[2V2-(·V2)],
n2c22Et2+σc2Et=2E,
2Et2+σn2Et-c2n22E=0,
E(t)=exp(-bt)cosh(Pt)+bPsinh(Pt)E(0)+1Psinh(Pt)E(0),
b=σ2n2,
P2=σ2n22+c2n22E.
E(t)=exp(-bt)E(0)+[E(0)+bE(0)]t+12P2E(0)t2+16[P2E(0)+bP2E(0)]t3+124P4E(0)t4+1120[P4E(0)+bP4E(0)]t5.
E(t)=-bE(t)+exp(-bt)E(0)+bE(0)+P2E(0)t+12[P2E(0)+bP2E(0)]t2+16P4E(0)t3+124[P4E(0)+bP4E(0)]t4,
E(r, t)=exp(iωt)ψ(r)
2ψ+ω2c2n2ψ=0,
2ψz2+Q2ψ=0,
Q2=2+ω2c2n2,
2=2x2+2y2.
ψ(z)=cos(Qz)ψ(0)+1Qsin(Qz)ψ(0),
ψ(z)=ψz=-sin(Qz)ψ(0)+cos(Qz)ψ(0),
ψ(z)=ψ(x, y, z),
ψ(0)=ψ(x, y, z=0),
ψ(0)=ψzz=0.
ψ(z)=ψ(0)+ψ(0)z-12Q2ψ(0)z2-16Q2ψ(0)z3+124Q4ψ(0)z4,
ψ(z)=ψ(0)-Q2ψ(0)z-12Q2ψ(0)z2+16Q4ψ(0)z3+124Q4ψ(0)z4.
f(x)=n=-N/2+1N/2fn expi 2πnxL,
fn=1L-L/2L/2f(x)exp-i 2πnxLdx.
fn=1Nn=-N/2+1N/2f(nΔx)exp-i 2πnnN,
fˆn=n=-0N-1f(nΔx)exp-i 2πnN
f(x)=-n=-N/2+1N/22πnL2fn expi 2πnxL.
f(nΔx)=-n=-N/2+1N/22πnNΔx2fn expi 2πnnN.
2f(mΔx, nΔy)=-m=-N/2+1N/2n=-N/2+1N/2×2πmNΔx2+2πnNΔy2×fmn expi 2π(mm+nn)N,
fmn=1N2m=-N/2+1N/2n=-N/2+1N/2f(mΔx, nΔy)×exp-i 2π(mm+nn)N.
cos(u)=1-12u2+124u4-1720u6+O(u8),
1usin(u)=1-16u2+1120u4-15400u6+O(u8),
u=cntκx2+κy2+κz2.
cΔtnπ2Δx2+π2Δy2+π2Δz21/2π2
cΔtn121Δx2+1Δy2+1Δz2-1/2:3D.
cΔtn121Δx2+1Δy2-1/2:2D,
cΔtnΔx2:1D.
cΔtnΔx4.
ψ(x, y, z)=ψ0(κx, κy, z)exp[i(κxx+κyy)].
d2ψ0dz2=-(k02n2-k2)ψ0,
k02=ω2c2,
k2=κx2+κy2.
ψ0(k, z)=ψ0(k, 0)cos[(k02n2-k2)1/2z]+ψ0(k, 0)(k02n2-k2)1/2sin[(k02n2-k2)1/2z],
k=(κx, κy).
ψ0(k, z)=ψ0(k, 0)cosh[(k2-k02n2)1/2z]+ψ0(k, 0)(k2-k02n2)1/2×sinh[(k2-k02n2)1/2z]ψ(0).
k0nΔzπ/2,
Δzλ/4n,
1Δx2+1Δy2-1/2λ2n:2TD,
Δxλ2n:1TD.
E(k, t)=E(k, 0)coscntκx2+κy2+κz2,
E(k, t)=E(k, 0)cos(ωt),
ω=cnk,
k=κx2+κy2+κz2.
vp=ωk=cn.
cos ωΔt=1-12c2n2(Δt)2k2+124c4n4(Δt)4k4
ω=1Δtarccos1-12c2n2(Δt)2k2+124c4n4(Δt)4k4,
cΔt/n=Δx/2,
ω=2cnΔxarccos1-12kΔx22+124kΔx24.
vg(k)=ωk=cnR(k),
R(k)=kΔx2-16kΔx231-1-12kΔx22+124kΔx2421/2.
L(x, l)=σl(x-x0)l+σl,=1[(x-x0)/σ]l+1,
h(x-x0, σ)=[u(x-x0+σ)-u(x-x0-σ)],
u(x)=0,x01,x>0.
n(x)=1+Δn[(x-x0)/σ]l+1,
2x21n22Ex2,
E=A exp-x22σ12,
n(y)=1+Δn[(y-y0)/σ]l+1,
n(x, y)=1+Δn{[(x-x0)/σx]2+[(y-y0)/σy]2}l+1,
n(x, y)=1+Δn[(x-x0)/σx]l+[(y-y0)/σy]l+1,
n(x, y)=1+Δn{[-x sin θ+y cos θ-(-x0 sin θ+y0 cos θ)]/σ}l+1,
n(x, y)=1+Δn{(y-y0)/[σ0+(x-x0)tan θ]}l+1,
u=18π(E2+μH2),
S=c4πE×H.
u=4πE2=μ4πH2,
S=c4πμ1/2E2=c4πμ1/2H2.
u=E2=n2E2
I=c1/2E2=(c/1/2)u.
2ki Fz=2F+k2n2n02-1F,
I=|F|2.
n(z)=1+Δn[(z-z0)/σ]l+1,
1n2E·(n2)=1n2Ez n2z
2Ext2=c2n22Exz2.
n2c22Ht2=2H+( log )×(×H),
n2c22Hyt2=2Hyz2- log zHyz.
Ex(z, 0)=f(z)=A exp[-(z-z1)2/2σ2],
u=Ex2.
Etot=1Nj=0N-1(jΔz)Ex2(jΔz),
E(x, z, 0)=A exp[-(z-z0)4/2σz4]×exp[-(x-x0)2/2σx2]cos[2π(z-z0)/λ].
n(x, z)=1+Δn1+(x-x0)2σ12+(z-z0)2σ12l-Δn1+(x-x0)2σ22+(z-z0)2σ22l,
n(x)=1+Δn[(x-x0)/σ]l+1.

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