Abstract

The numerical scheme for solving the Helmholtz equation, based on the Lanczos orthogonalization scheme, is generalized so that it can be applied to media with space-dependent absorption or gain profiles.

© 1992 Optical Society of America

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References

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  1. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  2. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,”J. Opt. Soc. Am. A (to be published).
  3. R. P. Ratowsky, J. A. Fleck, Opt. Lett. 16, 787 (1991).
    [CrossRef] [PubMed]
  4. T. J. Park, J. C. Light, J. Chem. Phys. 85, 5870 (1986).
    [CrossRef]
  5. C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
    [CrossRef]
  6. See, for example, W. F. Ames, D. Lee, Appl. Num. Math. 3, 25 (1987).
    [CrossRef]
  7. N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.
  8. By factoring out the carrier exp(ikz) rather than exp(−ikz), we make it easier to select the exponentially decaying modes that correspond to propagation angles greater than 90 deg (see Ref. 2).
  9. A generalization for real nonsymmetric matrices appears in Lanczos’ original paper, C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).

1991 (2)

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, Opt. Lett. 16, 787 (1991).
[CrossRef] [PubMed]

1987 (1)

See, for example, W. F. Ames, D. Lee, Appl. Num. Math. 3, 25 (1987).
[CrossRef]

1986 (1)

T. J. Park, J. C. Light, J. Chem. Phys. 85, 5870 (1986).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1950 (1)

A generalization for real nonsymmetric matrices appears in Lanczos’ original paper, C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).

Ames, W. F.

See, for example, W. F. Ames, D. Lee, Appl. Num. Math. 3, 25 (1987).
[CrossRef]

Bisseling, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Cerjan, C.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Feit, M. D.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,”J. Opt. Soc. Am. A (to be published).

Fleck, J. A.

R. P. Ratowsky, J. A. Fleck, Opt. Lett. 16, 787 (1991).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,”J. Opt. Soc. Am. A (to be published).

Friesner, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Guldberg, A.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Hammerich, A.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Jolicard, G.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Karrlein, W.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Kosloff, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Lanczos, C.

A generalization for real nonsymmetric matrices appears in Lanczos’ original paper, C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).

Lee, D.

See, for example, W. F. Ames, D. Lee, Appl. Num. Math. 3, 25 (1987).
[CrossRef]

LeForestier, C.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Light, J. C.

T. J. Park, J. C. Light, J. Chem. Phys. 85, 5870 (1986).
[CrossRef]

Lipkin, N.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Meyer, H. D.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Park, T. J.

T. J. Park, J. C. Light, J. Chem. Phys. 85, 5870 (1986).
[CrossRef]

Ratowsky, R. P.

R. P. Ratowsky, J. A. Fleck, Opt. Lett. 16, 787 (1991).
[CrossRef] [PubMed]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,”J. Opt. Soc. Am. A (to be published).

Roncero, O.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

Sehmi, N. S.

N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.

Appl. Num. Math. (1)

See, for example, W. F. Ames, D. Lee, Appl. Num. Math. 3, 25 (1987).
[CrossRef]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

J. Chem. Phys. (1)

T. J. Park, J. C. Light, J. Chem. Phys. 85, 5870 (1986).
[CrossRef]

J. Comput. Phys. (1)

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991).
[CrossRef]

J. Res. Natl. Bur. Stand. (1)

A generalization for real nonsymmetric matrices appears in Lanczos’ original paper, C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).

Opt. Lett. (1)

Other (3)

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,”J. Opt. Soc. Am. A (to be published).

N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.

By factoring out the carrier exp(ikz) rather than exp(−ikz), we make it easier to select the exponentially decaying modes that correspond to propagation angles greater than 90 deg (see Ref. 2).

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Figures (2)

Fig. 1
Fig. 1

Comparison between Helmholtz and paraxial propagation in a quadratic refractive-index waveguide with a quadratic gain profile. Analytic and numerical solutions of the Helmholtz equation are superimposed and indistinguishable.

Fig. 2
Fig. 2

Comparison between Helmholtz and paraxial propagation in a quadratic refractive-index waveguide with a quadratic absorption profile. Analytic and numerical solutions of the Helmholtz equation are superimposed and indistinguishable.

Equations (15)

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2 Ψ x 2 + 2 Ψ y 2 + 2 Ψ z 2 + n 2 ( x , y ) ω 2 c 2 Ψ = 0 ,
2 ψ z 2 + 2 i k ψ z + 2 ψ + k 2 [ n 2 ( x , y ) n 0 2 1 ] ψ = 0 .
H = 2 + k 2 [ n 2 ( x , y ) n 0 2 1 ] ,
2 ψ z 2 + 2 i k ψ z + H ψ = 0 .
ψ ( z ) = exp { i z [ k k ( 1 + H / k 2 ) 1 / 2 ] } ψ ( 0 ) .
q n | q n = δ n n .
β n | q n + 1 = H | q n α n | q n β n 1 | q n 1 ,
β n q n + 1 | = q n | H α n q n | β n 1 q n 1 | ,
ψ ( Δ z ) = U 1 exp { i z [ k k ( 1 + β / k 2 ) 1 / 2 ] } U ψ ( 0 ) .
n 2 = n 1 2 ( 1 i ν ) 2 [ 1 2 Δ ( x x 0 ) 2 ] , x < x 0 , = n 0 2 = n 1 2 ( 1 2 Δ ) , x x 0 ,
ψ ( x , z ) = n = 0 N A n ψ n ( x ) exp { i k [ 1 ( 1 + 2 β n / k ) 1 / 2 ] z } ,
ψ n ( x ) = N n exp ( x 2 / 2 σ 2 ) H n ( x / σ ) ,
σ = [ x 0 ( 1 2 Δ ) 2 k Δ ( 1 i ν ) 2 ] 1 / 2 ,
β = Δ n 1 2 ( 1 i ν ) 2 n 0 ω c n 1 ( 1 i ν ) n 0 ( 2 Δ ) 1 / 2 x 0 ( n + 1 2 ) .
ψ ( x , 0 ) = n = 0 25 ψ ( x 1 ) ψ ( x ) ,

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