Abstract

The combination of complex scaling with the (t, t′) representation of the time-dependent Schrödinger equation [ J. Chem. Phys. 99, 4590 ( 1993)] permits the design of graded-index multimode fiber to control the distribution of power among the modes. The differential modal losses are associated with the imaginary parts of the complex eigenvalues of a complex scaled Floquet-type operator. Although the illustrative numerical calculations are given here for the case in which the index of refraction is periodically varied along the fiber axis, the method is applicable for a more general coordinate-dependent index-of-refraction case.

© 1995 Optical Society of America

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References

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  1. See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).
  2. H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
    [CrossRef]
  3. C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
    [CrossRef]
  4. H. Tal Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time-dependent Schrödinger equation," J. Chem. Phys. 81, 3967 (1984).
    [CrossRef]
  5. C. Cerjan and R. Kosloff, "Efficient variable time-dependent scheme for intense field–atom interactions," Phys. Rev. A 47, 1852 (1993).
    [CrossRef] [PubMed]
  6. D. Kosloff and R. Kosloff, "A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics," J. Comput. Phys. 52, 35 (1983).
    [CrossRef]
  7. M. D. Feit, J. A. Fleck, Jr., and A. Steiger, "Solution of the Schrödinger equation by a spectral method," J. Comput. Phys. 47, 412 (1982).
    [CrossRef]
  8. C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
    [CrossRef]
  9. M. D. Feit and J. A. Fleck, Jr., "Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms," Opt. Lett. 14, 662 (1989).
    [CrossRef] [PubMed]
  10. S. Leasure and R. E. Wyatt, "Floquet theory of the interaction of a molecule with a laser field: techniques and an application," Opt. Eng. 19, 46 (1980).
    [CrossRef]
  11. S. Banerjee and A. Sharma, "Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields," J. Opt. Soc. Am. A 6, 1884 (1989).
    [CrossRef]
  12. J. Van Roey, Van der Douk, and P. E. Lagasse, "Beampropagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803 (1980).
    [CrossRef]
  13. U. Peskin and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: theory, computational algorithm and applications," J. Chem. Phys. 99, 4590 (1993).
    [CrossRef]
  14. U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
    [CrossRef]
  15. U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
    [CrossRef]
  16. E. Balslev and J. M. Combes, "Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions," Commun. Math. Phys. 22, 280 (1971).
    [CrossRef]
  17. B. Simon, "Quadratic form techniques and the Balslev– Combes theorem," Commun. Math. Phys. 27, 1 (1972); Ann. Math. 97, 247 (1973).
    [CrossRef]
  18. W. P. Reinhardt, "Complex coordinates in the theory of atomic and molecular structure and dynamics," Ann. Rev. Phys. Chem. 33, 223 (1982).
    [CrossRef]
  19. B. R. Junker, "Recent computational developments in the use of complex scaling in resonance phenomena," Adv. At. Mol. Phys. 18, 207 (1982).
    [CrossRef]
  20. Y. K. Ho, "The method of complex coordinate rotation and its application to atomic collision processes," Phy. Rep. C 99, 1 (1983).
    [CrossRef]
  21. N. Moiseyev, "Resonances, cross sections, and partial widths by the complex coordinate method," Isr. J. Chem. 31, 311 (1991).
  22. N. Moiseyev and H. J. Korsch, "Metastable quasienergy positions and widths for time-periodic Hamiltonians by the complex-coordinate method," Phys. Rev. A 41, 498 (1990); Isr. J. Chem. 30, 107 (1990).
    [CrossRef] [PubMed]
  23. I. Vorobeichik, U. Peskin, and N. Moiseyev, "Propagation of light beam in optical fiber by the (t, t) method," Nonlinear Opt. (to be published).

1994 (2)

U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
[CrossRef]

U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
[CrossRef]

1993 (2)

U. Peskin and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: theory, computational algorithm and applications," J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

C. Cerjan and R. Kosloff, "Efficient variable time-dependent scheme for intense field–atom interactions," Phys. Rev. A 47, 1852 (1993).
[CrossRef] [PubMed]

1992 (1)

H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
[CrossRef]

1991 (3)

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
[CrossRef]

N. Moiseyev, "Resonances, cross sections, and partial widths by the complex coordinate method," Isr. J. Chem. 31, 311 (1991).

1990 (1)

N. Moiseyev and H. J. Korsch, "Metastable quasienergy positions and widths for time-periodic Hamiltonians by the complex-coordinate method," Phys. Rev. A 41, 498 (1990); Isr. J. Chem. 30, 107 (1990).
[CrossRef] [PubMed]

1989 (2)

1984 (1)

H. Tal Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time-dependent Schrödinger equation," J. Chem. Phys. 81, 3967 (1984).
[CrossRef]

1983 (2)

D. Kosloff and R. Kosloff, "A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics," J. Comput. Phys. 52, 35 (1983).
[CrossRef]

Y. K. Ho, "The method of complex coordinate rotation and its application to atomic collision processes," Phy. Rep. C 99, 1 (1983).
[CrossRef]

1982 (3)

W. P. Reinhardt, "Complex coordinates in the theory of atomic and molecular structure and dynamics," Ann. Rev. Phys. Chem. 33, 223 (1982).
[CrossRef]

B. R. Junker, "Recent computational developments in the use of complex scaling in resonance phenomena," Adv. At. Mol. Phys. 18, 207 (1982).
[CrossRef]

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

1980 (2)

J. Van Roey, Van der Douk, and P. E. Lagasse, "Beampropagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803 (1980).
[CrossRef]

S. Leasure and R. E. Wyatt, "Floquet theory of the interaction of a molecule with a laser field: techniques and an application," Opt. Eng. 19, 46 (1980).
[CrossRef]

1972 (1)

B. Simon, "Quadratic form techniques and the Balslev– Combes theorem," Commun. Math. Phys. 27, 1 (1972); Ann. Math. 97, 247 (1973).
[CrossRef]

1971 (1)

E. Balslev and J. M. Combes, "Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions," Commun. Math. Phys. 22, 280 (1971).
[CrossRef]

Alon, O. E.

U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
[CrossRef]

Balslev, E.

E. Balslev and J. M. Combes, "Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions," Commun. Math. Phys. 22, 280 (1971).
[CrossRef]

Banerjee, S.

Bisseling, R.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Cerjan, C.

C. Cerjan and R. Kosloff, "Efficient variable time-dependent scheme for intense field–atom interactions," Phys. Rev. A 47, 1852 (1993).
[CrossRef] [PubMed]

H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
[CrossRef]

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Cherin, A. H.

See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).

Combes, J. M.

E. Balslev and J. M. Combes, "Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions," Commun. Math. Phys. 22, 280 (1971).
[CrossRef]

Douk, Van der

Ezer, H. Tal

H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
[CrossRef]

H. Tal Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time-dependent Schrödinger equation," J. Chem. Phys. 81, 3967 (1984).
[CrossRef]

Feit, M. D.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

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

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

Fleck, J. A.

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

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

Friesner, R.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Guldberg, A.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Hammerich, A. D.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Ho, Y. K.

Y. K. Ho, "The method of complex coordinate rotation and its application to atomic collision processes," Phy. Rep. C 99, 1 (1983).
[CrossRef]

Julicard, G.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Junker, B. R.

B. R. Junker, "Recent computational developments in the use of complex scaling in resonance phenomena," Adv. At. Mol. Phys. 18, 207 (1982).
[CrossRef]

Karrlein, W.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Korsch, H. J.

N. Moiseyev and H. J. Korsch, "Metastable quasienergy positions and widths for time-periodic Hamiltonians by the complex-coordinate method," Phys. Rev. A 41, 498 (1990); Isr. J. Chem. 30, 107 (1990).
[CrossRef] [PubMed]

Kosloff, D.

D. Kosloff and R. Kosloff, "A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics," J. Comput. Phys. 52, 35 (1983).
[CrossRef]

Kosloff, R.

U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
[CrossRef]

C. Cerjan and R. Kosloff, "Efficient variable time-dependent scheme for intense field–atom interactions," Phys. Rev. A 47, 1852 (1993).
[CrossRef] [PubMed]

H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
[CrossRef]

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

H. Tal Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time-dependent Schrödinger equation," J. Chem. Phys. 81, 3967 (1984).
[CrossRef]

D. Kosloff and R. Kosloff, "A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics," J. Comput. Phys. 52, 35 (1983).
[CrossRef]

Lagasse, P. E.

Leasure, S.

S. Leasure and R. E. Wyatt, "Floquet theory of the interaction of a molecule with a laser field: techniques and an application," Opt. Eng. 19, 46 (1980).
[CrossRef]

Leforestier, C.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Lipkin, N.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Meyer, H. Dieter

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Moiseyev, N.

U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
[CrossRef]

U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
[CrossRef]

U. Peskin and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: theory, computational algorithm and applications," J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

N. Moiseyev, "Resonances, cross sections, and partial widths by the complex coordinate method," Isr. J. Chem. 31, 311 (1991).

N. Moiseyev and H. J. Korsch, "Metastable quasienergy positions and widths for time-periodic Hamiltonians by the complex-coordinate method," Phys. Rev. A 41, 498 (1990); Isr. J. Chem. 30, 107 (1990).
[CrossRef] [PubMed]

I. Vorobeichik, U. Peskin, and N. Moiseyev, "Propagation of light beam in optical fiber by the (t, t) method," Nonlinear Opt. (to be published).

Peskin, U.

U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
[CrossRef]

U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
[CrossRef]

U. Peskin and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: theory, computational algorithm and applications," J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

I. Vorobeichik, U. Peskin, and N. Moiseyev, "Propagation of light beam in optical fiber by the (t, t) method," Nonlinear Opt. (to be published).

Qian, J.

C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
[CrossRef]

Reinhardt, W. P.

W. P. Reinhardt, "Complex coordinates in the theory of atomic and molecular structure and dynamics," Ann. Rev. Phys. Chem. 33, 223 (1982).
[CrossRef]

Roey, J. Van

Roncero, O.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Sharma, A.

Simon, B.

B. Simon, "Quadratic form techniques and the Balslev– Combes theorem," Commun. Math. Phys. 27, 1 (1972); Ann. Math. 97, 247 (1973).
[CrossRef]

Steiger, A.

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

Tannor, D. J.

C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
[CrossRef]

Vorobeichik, I.

I. Vorobeichik, U. Peskin, and N. Moiseyev, "Propagation of light beam in optical fiber by the (t, t) method," Nonlinear Opt. (to be published).

Williams, C. J.

C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
[CrossRef]

Wyatt, R. E.

S. Leasure and R. E. Wyatt, "Floquet theory of the interaction of a molecule with a laser field: techniques and an application," Opt. Eng. 19, 46 (1980).
[CrossRef]

Adv. At. Mol. Phys. (1)

B. R. Junker, "Recent computational developments in the use of complex scaling in resonance phenomena," Adv. At. Mol. Phys. 18, 207 (1982).
[CrossRef]

Ann. Rev. Phys. Chem. (1)

W. P. Reinhardt, "Complex coordinates in the theory of atomic and molecular structure and dynamics," Ann. Rev. Phys. Chem. 33, 223 (1982).
[CrossRef]

Commun. Math. Phys. (2)

E. Balslev and J. M. Combes, "Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions," Commun. Math. Phys. 22, 280 (1971).
[CrossRef]

B. Simon, "Quadratic form techniques and the Balslev– Combes theorem," Commun. Math. Phys. 27, 1 (1972); Ann. Math. 97, 247 (1973).
[CrossRef]

Isr. J. Chem. (1)

N. Moiseyev, "Resonances, cross sections, and partial widths by the complex coordinate method," Isr. J. Chem. 31, 311 (1991).

J. Chem. Phys. (5)

U. Peskin and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: theory, computational algorithm and applications," J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

U. Peskin, O. E. Alon, and N. Moiseyev, "The solution of the time-dependent Schrödinger equation by the (t, t) method: multiphoton ionization/dissociation probabilities in different gauges of the electromagnetic potentials," J. Chem. Phys. 100, 7310 (1994).
[CrossRef]

U. Peskin, R. Kosloff, and N. Moiseyev, "The solution of the time dependent Schrödinger equation by the (t, t) method: the use of global polynomial propagators for time dependent Hamiltonians," J. Chem. Phys. 100, 8849 (1994).
[CrossRef]

H. Tal Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time-dependent Schrödinger equation," J. Chem. Phys. 81, 3967 (1984).
[CrossRef]

C. J. Williams, J. Qian, and D. J. Tannor, "Dynamics of triatomic photodissociation in the interaction representation. I. Methodology," J. Chem. Phys. 95, 1721 (1991).
[CrossRef]

J. Comput. Phys. (4)

D. Kosloff and R. Kosloff, "A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics," J. Comput. Phys. 52, 35 (1983).
[CrossRef]

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

H. Tal Ezer, R. Kosloff, and C. Cerjan, "Low-order polynomial approximation of propagation for the time-dependent Schrö dinger equation," J. Comput. Phys. 100, 179 (1992).
[CrossRef]

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. D. Hammerich, G. Julicard, W. Karrlein, H. Dieter Meyer, N. Lipkin, O. Roncero, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrödinger equation," J. Comput. Phys. 94, 59 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

S. Leasure and R. E. Wyatt, "Floquet theory of the interaction of a molecule with a laser field: techniques and an application," Opt. Eng. 19, 46 (1980).
[CrossRef]

Opt. Lett. (1)

Phy. Rep. C (1)

Y. K. Ho, "The method of complex coordinate rotation and its application to atomic collision processes," Phy. Rep. C 99, 1 (1983).
[CrossRef]

Phys. Rev. A (2)

N. Moiseyev and H. J. Korsch, "Metastable quasienergy positions and widths for time-periodic Hamiltonians by the complex-coordinate method," Phys. Rev. A 41, 498 (1990); Isr. J. Chem. 30, 107 (1990).
[CrossRef] [PubMed]

C. Cerjan and R. Kosloff, "Efficient variable time-dependent scheme for intense field–atom interactions," Phys. Rev. A 47, 1852 (1993).
[CrossRef] [PubMed]

Other (2)

See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).

I. Vorobeichik, U. Peskin, and N. Moiseyev, "Propagation of light beam in optical fiber by the (t, t) method," Nonlinear Opt. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Index difference Δ(z) [defined in Eq. (52)] as a function of the fiber axis z for different values of the f0 parameter.

Fig. 2
Fig. 2

Spectra of the z-evolution operator as defined in Eq. (56) for different values of the field strength parameter f0 and the length period a [the definitions of f0 and a are given in Eqs. (52) and (53)]. The wavelength of the propagated ν = 2 light beam is λ = 900 nm. The index of refraction is defined in Eqs. (50) and (53) with the parameters n0 = 1.5, Δ0 = 0.031248 (as in Ref. 9), k0 = 10.472 μm−1, r0 = 3 μm. The spiral stands for nonconfined light beams, whereas the isolated points stand for the confined modes when f0 = 0 and a → 0 and for the decay modes when f0 ≠ 0 and a > 0.

Fig. 3
Fig. 3

Contour plots of the decay length L as defined in Eq. (24) (given on a logarithmic scale, that is, 2.00 in the plots stands for L = 102μm) as function of f0 and a [the definitions of f0 and a are given in Eqs. (52) and (53)].

Fig. 4
Fig. 4

As in Fig. 2 for λ = 500 nm,ν = 2, and the refractive-index parameters a = 57.1 μm, n0 = 1.5, and r0 = 3.5 μm.

Fig. 5
Fig. 5

Density plots of the irradiance of the three modes as a square-profile beam propagates through the optical fiber when f0 = 0.3 (the index-of-refraction parameters are defined in the caption of Fig. 4).

Equations (63)

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{ 2 x 2 + 2 y 2 + k 0 2 [ n 2 ( x , y , z ) n 0 2 1 ] } ψ ( x , y , z ) = 2 i k 0 ψ ( x , y , z ) z ,
t z , V ( x , y , z ) = k 0 2 [ n 2 ( x , y , z ) n 0 2 1 ] , Ĥ = 1 2 k 0 x , y 2 + V ( x , y , z ) ,
Ĥ ψ = i ψ z .
Ψ ( x , y , z ) = exp [ i Ĥ ( x , y ) ( z z 0 ) ] ψ ( x , y , z 0 ) ,
ψ β ( x , y , z = ) = [ 1 + G ( β ) V ( x , y ) ] ψ ( x , y , z 0 ) ,
G ( β ) = 1 β Ĥ ( x , y )
ψ ( x , y , z ) exp { i z 0 z Ĥ ( x , y , z ) d z + z 0 z d z z 0 z d z [ H ( x , y , z ) , H ( x , y , z ) ] } × ψ ( x , y , z 0 ) ,
E ( x , y , z ) = exp ( i k 0 z ) ( exp { i z 0 z Ĥ ( x , y , z ) d z + z 0 z d z z 0 z d z [ H ( x , y , z ) , H ( x , y , z ) ] } × E ( x , y , z = 0 ) ) .
n ( x , y , z ) = n ( x , y , z + a ) ,
ψ ( x , y , z ) = exp ( i β z ) Φ ( x , y , z ) ,
Φ ( x , y , z ) = Φ ( x , y , z + a ) = n = exp ( i 2 π a n z ) φ n ( x , y ) .
[ i z + H ( x ̂ , ŷ , ) ] ϕ ( x , y , z ) = β ϕ ( x , y , z )
( x , y ) φ = β φ ,
[ ( x , y ) n , n = 1 a 0 a d z exp ( i 2 π n z d ) × [ i z + H ( x , y , z ) ] exp ( + i 2 π n z a ) = ( 1 2 k 0 2 x 2 1 2 k 0 2 y 2 + 2 π n a ) δ n , n + k 0 2 a 0 a exp [ i ( 2 π a ) ( n n ) z ] × [ 1 n 2 ( x , y , z ) n 0 2 ] d z .
ψ ( x , y , z ) = exp ( i β z ) ϕ ( x , y ) ,
{ 1 2 k 0 2 x 2 1 2 k 0 2 y 2 + k 0 2 [ 1 n 2 ( x , y ) n 0 2 ] } ϕ ( x , y ) = β ϕ ( x , y ) .
lim x , y ϕ ( x , y ) = 0 ,
lim x ϕ ( x , y ) exp ( i 2 k 0 β x ) + S β ( y ) exp ( + i 2 k 0 β x ) .
S β ( y = ) = , S β ( x = ) =
β = β ν res , ν = 1 , 2 , , β ν res | β ν res | exp ( i φ ν res ) .
| ψ ν res ( x , y , z ) | 2 = exp [ 2 Im ( β ν res ) z ] | ϕ ν res ( x , y ) | 2 .
| ψ ν res ( x , y , z ) | 2 d x d y d z = 1 ,
ϕ ν res ( x , y )
lim x ϕ res ( x , y ) exp ( i 2 k 0 β res x ) = exp [ i 2 k 0 | β ν res | cos ( φ ν res ) x ] × exp [ + 2 k 0 | β ν res | sin ( φ ν res ) x ] .
L = 1 2 Im ( β res ) .
x = x e i θ , x , y = y e i θ , y ,
[ i z + H ( x e i θ , y e i θ , z ) ] ϕ θ res ( x , y , z ) = β res ϕ θ res ( x , y , z ) ,
lim x ϕ θ res ( x , y ) exp ( i 2 k 0 β res x ) = exp ( i β x ) exp ( α x ) ,
β = 2 k 0 | β ν res | cos ( θ φ θ ν res ) , α = 2 k 0 | β ν res | sin ( θ φ ν res ) .
lim x ϕ θ res ( x , y ) = 0 ,
( x , y , z ) ϕ ( x , y , z , z ) = i ϕ ( x , y , z , z ) z ,
( x , y , z ) = i z + Ĥ ( x , y , z ) ,
ψ ( x , y , z ) = ϕ ( x , y , z , z ) | z = z ,
E ( x , y , z ) = exp ( i k 0 z ) { exp [ i ̂ ( x , y , z ) z ] × E ( x , y , z = 0 ) } z = z .
ϕ ( x , y , z , z ) = η ( z ) χ ( x , y , z ) ,
η ( z ) = exp ( i β z )
( x , y , z ) χ ( x , y , z ) = β χ ( x , y , z ) .
{ exp ( i 2 π a n z ) ; n }
z = m τ ; m = 0 , 1 , , N ,
N τ = a
E ( x , y , z ) = exp ( i k 0 z ) m = 1 M Û [ m τ ( m 1 ) τ ] × E ( x , y , z = 0 ) ,
Û [ m τ ( m 1 ) τ ] = n = exp ( i 2 π n m τ / a ) A n , o ( x , y ) ,
A = exp [ i ( x , y ) τ ] 1 i τ ( x , y ) τ 2 2 2 ( x , y ) + ,
A 1 i τ ( x , y ) ,
A n , o 0
n = 1 , 0 , + 1 .
E ( x , y , z ) = exp ( i k 0 z ) m = 1 M { [ 1 i τ 1 , 0 ( x , y ) ] × exp ( i 2 π m τ / a ) + [ 1 i τ 0 , 0 ( x , y ) ] + [ 1 i τ + 1 , 0 ( x , y ) exp ( + i 2 π m τ / a ) ] } × E ( x , y , z = 0 ) .
n ( x , y , z ) = n ( r , z )
ψ ( r , ϕ , z ) = r 1 / 2 Φ υ ( r , z ) exp ( i ν ϕ ) , ν = 0 , 1 , 2 , ,
Ĥ Φ ν = i Φ ν z ,
Ĥ = 1 2 k 0 r 2 + ν 2 1 / 4 2 k 0 r 2 + k 0 2 { 1 [ n ( r , z ) n 0 ] 2 }
k 0 2 { 1 [ n ( r ) n 0 ] 2 } = k 0 Δ 0 2 Δ 0 1 [ 1 g ( r ) ] V ( r ) ,
g ( r ) = { ( r / r 0 ) 2 r r 0 1 r > r 0 .
Δ ( z ) = Δ 0 1 + 2 f 0 cos ( 2 π a z ) 1 + 2 f 0 Δ 0 cos ( 2 π a z ) .
k 0 2 { 1 [ n ( r , z ) n 0 ] 2 } = V ( r ) [ ( 1 + f 0 cos 2 π a z ) ] ,
Φ ν ( r , z ) = η ν ( z ) χ ν ( r , z )
η ν ( z ) = exp ( i β ν z ) , χ ν ( r , z ) = χ ν ( r , z + a ) .
U ̂ ( a 0 ) χ ( r , 0 ) = η ( a ) χ ( r , a ) ,
U ̂ ( a 0 ) = m = 1 M U [ m τ ( m 1 ) τ ] .
r = r exp ( i θ ) ,
0 r .
β n = β exp ( 2 i θ ) + 2 π n a , n = , , 1 , 0 , 1 , ,
η ( a ) = exp ( i β n a )

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