Abstract

A method is described for modeling two-dimensional reflecting structures based on a solution of the scalar Helmholtz equation. The equation is solved by use of an alternating-direction-implicit iterative method together with a semioptimum sequence of acceleration parameters similar to those introduced decades ago for the solution of elliptic equations with positive-definite operators. The resulting technique is efficient and simple to program, permits the simulation of complex structures with modest storage requirements, and is of very general applicability.

© 1994 Optical Society of America

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References

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  1. P. Kaczmarski, P. Lagasse, Electron. Lett. 24, 675 (1988).
    [CrossRef]
  2. C. H. Henry, IEEE J. Quantum Electron. 27, 523 (1991).
    [CrossRef]
  3. D. Marcuse, Bell Syst. Tech. J. 55, 1295 (1976).
  4. A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
    [CrossRef]
  5. E. I. Wachspress, J. Soc. Ind. Appl. Math. 10, 339 (1962).
    [CrossRef]
  6. A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
    [CrossRef]
  7. R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1963), Chap. 7 (especially p. 212).
  8. E. L. Wachspress, Iterative Methods for Large Linear Systems (Academic, San Diego, Calif., 1990), Chap. 15.
  9. W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
    [CrossRef]
  10. D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
    [CrossRef]

1991 (3)

C. H. Henry, IEEE J. Quantum Electron. 27, 523 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

1989 (1)

A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
[CrossRef]

1988 (1)

P. Kaczmarski, P. Lagasse, Electron. Lett. 24, 675 (1988).
[CrossRef]

1983 (1)

A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
[CrossRef]

1976 (1)

D. Marcuse, Bell Syst. Tech. J. 55, 1295 (1976).

1962 (1)

E. I. Wachspress, J. Soc. Ind. Appl. Math. 10, 339 (1962).
[CrossRef]

Bayliss, A.

A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
[CrossRef]

Chaudhuri, S. K.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

Chu, S. T.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

Goldstein, C. I.

A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
[CrossRef]

Goss, A.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

Hardy, A.

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
[CrossRef]

Henry, C. H.

C. H. Henry, IEEE J. Quantum Electron. 27, 523 (1991).
[CrossRef]

Huang, W. P.

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

Kaczmarski, P.

P. Kaczmarski, P. Lagasse, Electron. Lett. 24, 675 (1988).
[CrossRef]

Lagasse, P.

P. Kaczmarski, P. Lagasse, Electron. Lett. 24, 675 (1988).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 55, 1295 (1976).

Mehuys, D.

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

Parke, R.

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

Streifer, W.

A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
[CrossRef]

Turkel, E.

A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
[CrossRef]

Varga, R. S.

R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1963), Chap. 7 (especially p. 212).

Waarts, R. G.

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

Wachspress, E. I.

E. I. Wachspress, J. Soc. Ind. Appl. Math. 10, 339 (1962).
[CrossRef]

Wachspress, E. L.

E. L. Wachspress, Iterative Methods for Large Linear Systems (Academic, San Diego, Calif., 1990), Chap. 15.

Welch, D. F.

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 55, 1295 (1976).

Electron. Lett. (1)

P. Kaczmarski, P. Lagasse, Electron. Lett. 24, 675 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. H. Henry, IEEE J. Quantum Electron. 27, 523 (1991).
[CrossRef]

A. Hardy, D. F. Welch, W. Streifer, IEEE J. Quantum Electron. 25, 2096 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

W. P. Huang, S. T. Chu, A. Goss, S. K. Chaudhuri, IEEE Photon. Technol. Lett. 3, 524 (1991).
[CrossRef]

D. Mehuys, A. Hardy, D. F. Welch, R. G. Waarts, R. Parke, IEEE Photon. Technol. Lett. 3, 342 (1991).
[CrossRef]

J. Comp. Phys. (1)

A. Bayliss, C. I. Goldstein, E. Turkel, J. Comp. Phys. 49, 443 (1983).
[CrossRef]

J. Soc. Ind. Appl. Math. (1)

E. I. Wachspress, J. Soc. Ind. Appl. Math. 10, 339 (1962).
[CrossRef]

Other (2)

R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1963), Chap. 7 (especially p. 212).

E. L. Wachspress, Iterative Methods for Large Linear Systems (Academic, San Diego, Calif., 1990), Chap. 15.

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Figures (3)

Fig. 1
Fig. 1

Schematic illustrating a typical problem geometry for the ADI Helmholtz solution technique described in this Letter. An input (guided) mode enters at the lower boundary, scatters off the structure of interest, and is absorbed in the regions adjacent to the other three boundaries.

Fig. 2
Fig. 2

Geometry (not to scale) for the sample problem discussed in the text. A second-order grating has been etched into the waveguide as shown. Light scattered toward the substrate is reflected by the 10.5-period dielectric stack.

Fig. 3
Fig. 3

Intensity contours for the converged solution of the waveguide-with-grating problem shown in Fig. 2 and discussed in the text. The input guided mode was an approximate eigenmode of the waveguide as modified by the presence of the grating.

Equations (3)

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2 A + ( x , y ) A = 0 ,
( ω n + δ z 2 ) A n + ( 1 / 2 ) = ( ω n δ x 2 ) A n , ( ω n + δ x 2 ) A n + 1 = ( ω n δ z 2 ) A n + ( 1 / 2 ) ,
δ x 2 A | i , j A i + 1 , j + A i 1 , j 2 A i , j ( Δ x ) 2 + i , j 2 A i , j , δ z 2 A | i , j A i , j + 1 + A i , j 1 2 A i , j ( Δ z ) 2 + i , j 2 A i , j .

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