Abstract

We show that an appropriate variable transformation can redistribute the sample points for the field in the collocation method such that the numerical accuracy of the field-propagation algorithm improves significantly. We also present results for some specific transformations that improve the accuracy by more than 2 orders of magnitude.

© 1992 Optical Society of America

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References

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  1. M. D. Fiet, M. D. Fleck, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef]
  2. J. VanRoey, J. vander Donk, P. E. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).
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  3. L. Thylen, Opt. Quantum Electron. 15, 433 (1983).
    [CrossRef]
  4. D. Yevick, B. Hermansson, Opt. Lett. 11, 103 (1986).
    [CrossRef] [PubMed]
  5. M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
    [CrossRef]
  6. A. Sharma, S. Banerjee, Opt. Lett. 14, 96 (1989).
    [CrossRef] [PubMed]
  7. S. Banerjee, A. Sharma, J. Opt. Soc. Am. A 6, 1884 (1989).
    [CrossRef]
  8. A. Sharma, A. Taneja, in Digest of Meeting on Integrated Photonics Research (Optical Society of America, Washington, D.C., 1991), paper TuB4.
  9. A. Sharma, A. Taneja, Opt. Lett. 16, 1162 (1991).
    [CrossRef] [PubMed]

1991 (2)

M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
[CrossRef]

A. Sharma, A. Taneja, Opt. Lett. 16, 1162 (1991).
[CrossRef] [PubMed]

1989 (2)

1986 (1)

1983 (1)

L. Thylen, Opt. Quantum Electron. 15, 433 (1983).
[CrossRef]

1981 (1)

1978 (1)

Banerjee, S.

Fiet, M. D.

Fleck, J. A.

Fleck, M. D.

Glasner, M.

M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
[CrossRef]

Hermansson, B.

M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
[CrossRef]

D. Yevick, B. Hermansson, Opt. Lett. 11, 103 (1986).
[CrossRef] [PubMed]

Lagasse, P. E.

Sharma, A.

A. Sharma, A. Taneja, Opt. Lett. 16, 1162 (1991).
[CrossRef] [PubMed]

A. Sharma, S. Banerjee, Opt. Lett. 14, 96 (1989).
[CrossRef] [PubMed]

S. Banerjee, A. Sharma, J. Opt. Soc. Am. A 6, 1884 (1989).
[CrossRef]

A. Sharma, A. Taneja, in Digest of Meeting on Integrated Photonics Research (Optical Society of America, Washington, D.C., 1991), paper TuB4.

Taneja, A.

A. Sharma, A. Taneja, Opt. Lett. 16, 1162 (1991).
[CrossRef] [PubMed]

A. Sharma, A. Taneja, in Digest of Meeting on Integrated Photonics Research (Optical Society of America, Washington, D.C., 1991), paper TuB4.

Thylen, L.

L. Thylen, Opt. Quantum Electron. 15, 433 (1983).
[CrossRef]

vander Donk, J.

VanRoey, J.

Yevick, D.

M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
[CrossRef]

D. Yevick, B. Hermansson, Opt. Lett. 11, 103 (1986).
[CrossRef] [PubMed]

Appl. Opt. (1)

Electron. Lett. (1)

M. Glasner, D. Yevick, B. Hermansson, Electron. Lett. 27, 475 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (3)

Opt. Quantum Electron. (1)

L. Thylen, Opt. Quantum Electron. 15, 433 (1983).
[CrossRef]

Other (1)

A. Sharma, A. Taneja, in Digest of Meeting on Integrated Photonics Research (Optical Society of America, Washington, D.C., 1991), paper TuB4.

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Figures (3)

Fig. 1
Fig. 1

Error in the correlation factor between the calculated and exact fields after propagation through 100 μm. The solid curves are for the variable-transformed collocation method, and the dashed curves are for the normal collocation method. The transformation used is h(σ) = σ(1 + τσ2), with τ = 0.075 μm−2. The value of α is 0.75 μm−1, except for N ≥ 50 with Δz ≥ 1 μm, for which its value is 0.65 μm−1. Its value for the normal collocation is 0.245 μm−1. The error in BPM with 128 points and Δz = 2.5 μm is indicated in Fig. 2.

Fig. 2
Fig. 2

Same as Fig. 1, except that the transformation used is h(σ) = σ(1+ τσ4), with τ = 0.0002 μm−4. For N ≥ 50, α = 0.45 μm−1 with Δz < 1 μm and 0.35 μm−1 with Δz ≥ 1 μm, and for N ≥ 50, α = 0.35 μm−1 with Δz < 1 μm and 0.25 μm−1 with Δz ≥ 1 μm.

Fig. 3
Fig. 3

Incident field as a function of x, showing the distribution of collocation points for N = 20 in the transformed and normal collocation methods. The values of τ and α are the same as in Figs. 1 and 2.

Equations (10)

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2 E x 2 + 2 E z 2 + k 0 2 n 2 ( x , z ) E ( x , z ) = 0 ,
x = h ( σ ) , E ( x , z ) = h ( σ ) U ( σ , z ) ,
2 U z 2 + f ( σ ) 2 U σ 2 + [ g ( σ ) + k 0 2 n 2 ( σ , z ) ] U ( σ , z ) = 0 ,
d 2 U d z 2 + SU ( z ) = 0 ,
d 2 X d z 2 - 2 i k d X d z + ( S - k 2 I ) X ( z ) = 0 ,
d X d z = ( H 1 - H 2 ) X ( z ) / 2 i k ,
A = [ A i j : A i j = H j - 1 ( α σ i ) exp ( - ½ α 2 σ i 2 ) ] , D 1 = α 4 diag ( σ 1 2 , σ 2 2 , σ N 2 ) , D 2 = α 2 diag [ 1 , 3 , 5 , ( 2 N - 1 ) ] .
X ( z + Δ z ) = exp [ ( H 1 - H 2 ) Δ z / 2 i k ] X ( z ) ,
X ( z + Δ z ) = PQ ( z ) PX ( z ) + O [ ( Δ z ) 3 ] ,
n 2 ( x ) = n 2 2 + ( n 1 2 - n 2 2 ) sech 2 ( x / a ) ,

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