Abstract

A new beam-propagation method is presented whereby the exact scalar Helmholtz propagation operator is replaced by any one of a sequence of higher-order (n, n) Padé approximant operators. The resulting differential equation may then be discretized to obtain (in two dimensions) a matrix equation of bandwidth 2n + 1 that is solvable by using Standard implicit solution techniques. The final algorithm allows (for n = 2) accurate propagation at angles of greater than 55 deg from the propagation axis as well as propagation through materials with widely differing indices of refraction.

© 1992 Optical Society of America

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References

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  1. D. Yevick, M. Glasner, Electron. Lett. 25, 1611 (1989).
    [CrossRef]
  2. D. Yevick, M. Glasner, Opt. Lett. 15, 174 (1990).
    [CrossRef] [PubMed]
  3. J. Gerdes, R. Pregla, J. Opt. Soc. Am. B 8, 389 (1991).
    [CrossRef]
  4. R. P. Ratowsky, J. A. Fleck, Opt. Lett. 16, 787 (1991).
    [CrossRef] [PubMed]
  5. L. Halpern, L. N. Trefethen, J. Acoust. Soc. Am. 84, 1397 (1988).
    [CrossRef] [PubMed]
  6. B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
    [CrossRef]
  7. G. A. Baker, Essentials of Padé Approximants (Academic, New York, 1975).
  8. G. R. Hadley, Opt. Lett. 16, 624 (1991).
    [CrossRef] [PubMed]
  9. G. R. Hadley, IEEE J. Quantum Electron. 28, 363 (1992).
    [CrossRef]

1992 (2)

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

G. R. Hadley, IEEE J. Quantum Electron. 28, 363 (1992).
[CrossRef]

1991 (3)

1990 (1)

1989 (1)

D. Yevick, M. Glasner, Electron. Lett. 25, 1611 (1989).
[CrossRef]

1988 (1)

L. Halpern, L. N. Trefethen, J. Acoust. Soc. Am. 84, 1397 (1988).
[CrossRef] [PubMed]

Baker, G. A.

G. A. Baker, Essentials of Padé Approximants (Academic, New York, 1975).

Bardyszewski, W.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

Fleck, J. A.

Gerdes, J.

Glasner, M.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

D. Yevick, M. Glasner, Opt. Lett. 15, 174 (1990).
[CrossRef] [PubMed]

D. Yevick, M. Glasner, Electron. Lett. 25, 1611 (1989).
[CrossRef]

Hadley, G. R.

G. R. Hadley, IEEE J. Quantum Electron. 28, 363 (1992).
[CrossRef]

G. R. Hadley, Opt. Lett. 16, 624 (1991).
[CrossRef] [PubMed]

Halpern, L.

L. Halpern, L. N. Trefethen, J. Acoust. Soc. Am. 84, 1397 (1988).
[CrossRef] [PubMed]

Hermansson, B.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

Pregla, R.

Ratowsky, R. P.

Trefethen, L. N.

L. Halpern, L. N. Trefethen, J. Acoust. Soc. Am. 84, 1397 (1988).
[CrossRef] [PubMed]

Yevick, D.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

D. Yevick, M. Glasner, Opt. Lett. 15, 174 (1990).
[CrossRef] [PubMed]

D. Yevick, M. Glasner, Electron. Lett. 25, 1611 (1989).
[CrossRef]

Electron. Lett. (1)

D. Yevick, M. Glasner, Electron. Lett. 25, 1611 (1989).
[CrossRef]

IEEE J. Lightwave Technol. (1)

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, IEEE J. Lightwave Technol. 10, 772 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. R. Hadley, IEEE J. Quantum Electron. 28, 363 (1992).
[CrossRef]

J. Acoust. Soc. Am. (1)

L. Halpern, L. N. Trefethen, J. Acoust. Soc. Am. 84, 1397 (1988).
[CrossRef] [PubMed]

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

Opt. Lett. (3)

Other (1)

G. A. Baker, Essentials of Padé Approximants (Academic, New York, 1975).

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Figures (3)

Fig. 1
Fig. 1

Phase error incurred by the use of the operators shown for propagation of a plane wave at various angles with respect to the z axis. The curve labeled Padé (3, 3) is mostly obscured by that corresponding to the operator obtained by expanding the square root in Eq. (6) to 15th order in P.

Fig. 2
Fig. 2

Phase error incurred by the use of the operators shown for propagation of a plane wave along the z axis through media of differing dielectric constants. In each case the reference index is 3.5 (corresponding to a dielectric constant of 12.25).

Fig. 3
Fig. 3

Intensity profiles resulting from the propagation of an initial Gaussian beam with a 45-deg phase tilt a distance of 10 μm through a uniform medium. The beam was initially centered at zero, and the reference index employed was unity.

Tables (1)

Tables Icon

Table 1 Most Useful Low-Order Padé Approximants for the Helmholtz Operator in Terms of the Operator P Defined in Eq. (2)

Equations (13)

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H z i 2 k 2 H z 2 = i P 2 k H ,
P k 0 2 [ ( x ¯ ) 0 n ¯ 2 ] + 2 .
H z = i P 2 k 1 i 2 k z H .
z | n = i P 2 k 1 i 2 k z | n 1 .
H z = i N D H
H z = i ( P + k 2 k ) H ,
D ( H n + 1 H n ) = i Δ z 2 N ( H n + H n + 1 ) .
P H | i = 1 ( Δ x ) 2 ( v i H i + H i + 1 + H i 1 ) ,
v i k 0 2 ( Δ x ) 2 ( i 0 n ¯ 2 ) 2 .
P 2 k
P 2 k 1 + P 4 k 2
P 2 k + P 2 4 k 3 1 + 3 P 4 k 2 + P 2 16 k 4
P 2 k + P 2 2 k 3 + 3 P 3 32 k 5 1 + 5 P 4 k 2 + 3 P 2 8 k 4 + P 3 64 k 6

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