Abstract

The recent introduction of a fast Fourier transform– (FFT–) based method for calculating the Rayleigh–Sommerfeld full diffraction integral for tilted and offset planes permits high-speed evaluation of integrated optical systems. An important part of introducing a new calculational tool is its validation and an assessment of its limitations. The validity of the new FFT-based method was determined by comparison of that method with direct integration (DI) of the Rayleigh–Sommerfeld integral, a well-established method. Points of comparison were accuracy, computational speed, memory requirements of the host computer, and applicability to various optical modeling situations. The new FFT-based method is 228 times faster, yet requires 14 times more memory, than the DI method for a 500 µm by 500 µm real computational window.

© 2001 Optical Society of America

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References

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1998 (2)

1996 (1)

1993 (1)

1991 (1)

D. C. Bertilone, “On the exact Kirchhoff and Rayleigh–Sommerfeld theories for the focusing of an infinite scalar spherical wave-field,” Opt. Commun. 85, 153–158 (1991).
[CrossRef]

1990 (1)

1984 (1)

1982 (1)

R. Jozwicki, “The application of the vector analysis to the quasi-spherical wave scalar diffraction,” Optik 62, 231–247 (1982).

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

1964 (1)

Ball, M.

Bertilone, D. C.

D. C. Bertilone, “On the exact Kirchhoff and Rayleigh–Sommerfeld theories for the focusing of an infinite scalar spherical wave-field,” Opt. Commun. 85, 153–158 (1991).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Delen, N.

Fedor, A.

Feldman, M. R.

Flagello, D. G.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1983).

Hareb, S.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hooker, B.

Hooker, R. B.

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Hudson, J. A.

Jozwicki, R.

R. Jozwicki, “The application of the vector analysis to the quasi-spherical wave scalar diffraction,” Optik 62, 231–247 (1982).

Ju, T. H.

Lee, Y. C.

McCutchen, C. W.

Milster, T.

Morris, J. E.

Rosenbluth, A. E.

Sommargren, G. E.

Stirk, C. W.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1983).

Weaver, H. J.

Welch, W. H.

Wu, J. S.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Opt. Commun. (1)

D. C. Bertilone, “On the exact Kirchhoff and Rayleigh–Sommerfeld theories for the focusing of an infinite scalar spherical wave-field,” Opt. Commun. 85, 153–158 (1991).
[CrossRef]

Optik (1)

R. Jozwicki, “The application of the vector analysis to the quasi-spherical wave scalar diffraction,” Optik 62, 231–247 (1982).

Other (3)

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1983).

N. Delen, B. Hooker, “Optical field propagation between tilted or offset planes,” U.S. patent5,982,954 (9November1999).

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Figures (7)

Fig. 1
Fig. 1

Three views of the laser-to-fiber portion of the bidirectional link.

Fig. 2
Fig. 2

Intensity pattern generated at the fiber in Fig. 1 by the new FFT BM.

Fig. 3
Fig. 3

Intensity pattern generated at the fiber in Fig. 1 by the DI method.

Fig. 4
Fig. 4

Cross sections of the intensity patterns generated by the new FFT-based and the DI methods along the y direction at the fiber.

Fig. 5
Fig. 5

Cross sections of the intensity patterns generated by the new FFT-based and the DI methods along the x direction at the fiber.

Fig. 6
Fig. 6

Cost of the DI method relative to the FFT BM for several numbers of samples in one dimension.

Fig. 7
Fig. 7

Memory requirement for various numbers of samples in one dimension.

Tables (3)

Tables Icon

Table 1 Comparison of the Applicability Regions of the DI Method and the new FFT BM

Tables Icon

Table 2 Average CPU Time to Run Different-Sized Problems by the DI Method

Tables Icon

Table 3 Average CPU Time to Run Different-Sized Problems by the New FFT BM

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

s:=s+aik×bkj,
NOFnew FFT=2×log2 NFFT,NOFDI=NDI2.
NOFnew FFT=2×2×NFFTNFFT×log2NFFT2, NOFDI=NDI4.
cr=NDI42×(2×NFFTNFFT×log2NFFT/2=NDI42×NFFT2×log2NFFT.
cr=NDI232×2+log2NDI.
MRFFT=16×NFFT×NFFTbytes.
MRDI=8×NDI×NDI bytes.
NFFT=4×NDI.
MRFFT=256×NDI×NDI bytes, MRDI=8×NDI×NDI bytes,
MRFFT=32×MRDI.

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