Abstract

Computation of the readout signal of an optical disk involves Fourier transforms from the objective lens pupil to the disk and, after interaction with the disk, from the disk to the objective pupil. Traditionally, the complex two-dimensional Fourier transform is numerically evaluated as a two-dimensional fast Fourier transform. To obtain sufficient resolution in the involved planes, one must choose sampling grid sizes of typically 1024 × 1024 or higher, resulting in a substantial computation time if the calculation is to be repeated many times. Discussed is an alternative method for evaluating the Fourier transform, based on the chirp z transform, by which a considerable improvement in efficiency can be obtained without loss of accuracy.

© 2002 Optical Society of America

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References

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  1. A. Korpel, “Simplified diffraction theory of the video disk,” Appl. Opt. 17, 2037–2042 (1978).
    [CrossRef] [PubMed]
  2. J. Pasman, “Vector theory of diffraction” in Principles of Optical Disc SystemsE. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), pp. 88–124.
  3. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), pp. 393–399.
  4. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1989), pp. 452–457.
  5. H. H. Hopkins, “Diffraction theory of laser read-out systems for optical video discs,” J. Opt. Soc. Am. 69, 4–24 (1979).
    [CrossRef]

1979

1978

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), pp. 393–399.

Hopkins, H. H.

Korpel, A.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1989), pp. 452–457.

Pasman, J.

J. Pasman, “Vector theory of diffraction” in Principles of Optical Disc SystemsE. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), pp. 88–124.

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), pp. 393–399.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1989), pp. 452–457.

Appl. Opt.

J. Opt. Soc. Am.

Other

J. Pasman, “Vector theory of diffraction” in Principles of Optical Disc SystemsE. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), pp. 88–124.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), pp. 393–399.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1989), pp. 452–457.

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Figures (8)

Fig. 1
Fig. 1

Readout of an optical disk.

Fig. 2
Fig. 2

Sampling grids of the objective-collector and disk planes for method I. The dimensions are expressed in normalized units.

Fig. 3
Fig. 3

Matrices during the transform according to method II.

Fig. 4
Fig. 4

Sampling grids for method II.

Fig. 5
Fig. 5

Effect of application of a window on (a) the spot and (b) the transformed spot.

Fig. 6
Fig. 6

Disk geometry used for comparison of the two methods.

Fig. 7
Fig. 7

Data signal produced by method I for cases A and B.

Fig. 8
Fig. 8

(a) Results from methods I and II for case A. (b) Results from methods I and II for case B.

Equations (19)

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Fu, v =-+-+ fx, yexp-2πiux+vydxdy,
xm=-1+2m+1NR, m=0N-1, yn=-1+2n+1NR, n=0N-1, uk=-1+2k+1Mu0, k=0M-1, vl=-1+2l+1Mu0, l=0M-1.
Fuk, vl=4R2N2m=0N-1n=0N-1 fxm, yn×exp-2πiukxm+vlyn 
Fuk, vl=2RNm=0N-1 Axm, vlexp-2πiukxmk=0M-1,
Axm, vl=2RNn=0N-1 fxm, ynexp-2πivlynl=0M-1.
AkDFTf]k=n=0N-1 fnwnk, k=0M-1,
w=exp-2πi/N.
exp-2πivlyn=exp-2πi1-1N- 1M+1MNu0R ×exp-2πi1N-12u0RM l×exp-2πi1M-12u0RN n×exp-2πi4u0RMN ln.
M=N, 4u0RN=1.
b=cosϕb-i sinϕb,
ϕb=π1/N-1l,l=0N-1.
Ldisk=NL¯objλNA=N2RλNA,
RESdisk LdiskN=12RλNA.
AkCZTf]k=n=0N-1 fnaw-k-n, k=0  M-1,
exp-2πivlyn=exp-2πi1- 1N- 1M+1MNu0×exp-2πi1N-12u0M l×exp-2πi1M-12u0N n×exp-2πi4u0MN ln.
c=cosϕc-i sinϕc,
ϕc=2π 1N-12u0M l,l=0  M-1.
a=exp2πi1M-12u0N.
w=exp-2πi4u0MN.

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