Abstract

The diffractive processes within an optical system can be simulated by computer to compute the diffraction-altered electric-field distribution at the output of the system from the electric-field distribution at the input. In the paraxial approximation the system can be described by an ABCD ray matrix whose elements in turn can be used to simplify the computation such that only a single computational step is required. We describe two rearrangements of such computations that allow the simulation to be expressed in a linear systems formulation, in particular using the fast-Fourier-transform algorithm. We investigate the sampling requirements for the kernel-modifying function or chirp that arises. We also use the special properties of the chirp to determine the spreading imposed by the diffraction. This knowledge can be used to reduce the computation if only a limited region of either the input or the output is of interest.

© 1998 Optical Society of America

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References

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  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  2. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  3. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  4. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  5. P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  6. W. G. McKinley, H. T. Yura, S. G. Hanson, “Optical system defect propagation in ABCD systems,” Opt. Lett. 13, 333–335 (1988).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996).
  8. A. J. Lambert, “Optical image processing and its application,” Ph.D. dissertation (The University of New South Wales, Canberra, Australia, 1996).
  9. M. J. Beran, G. B. Perrant, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  10. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  11. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  12. I. W. Selesnick, C. S. Burrus, “Fast convolution and filterings,” in The Digital Signal Processing Handbook, V. K. Madisetti, D. B. Williams, eds. (IEEE/CRC Press, Boca Raton, Fla., 1998), Chap. 8.
  13. H. Stark, F. B. Tuteur, Modern Electrical Communications—Theory and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1979).
  14. A. J. Lambert, D. Fraser, “An implementation of the fast Fresnel transform,” in IVCNZ’95 Image and Vision Computing New Zealand (Industrial Research Ltd., Christchurch, New Zealand, 1995), pp. 143–148.

1991 (1)

1988 (1)

1987 (1)

1970 (1)

Belanger, P. A.

Beran, M. J.

M. J. Beran, G. B. Perrant, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Burrus, C. S.

I. W. Selesnick, C. S. Burrus, “Fast convolution and filterings,” in The Digital Signal Processing Handbook, V. K. Madisetti, D. B. Williams, eds. (IEEE/CRC Press, Boca Raton, Fla., 1998), Chap. 8.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Collins, S. A.

Fraser, D.

A. J. Lambert, D. Fraser, “An implementation of the fast Fresnel transform,” in IVCNZ’95 Image and Vision Computing New Zealand (Industrial Research Ltd., Christchurch, New Zealand, 1995), pp. 143–148.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

Hanson, S. G.

Lambert, A. J.

A. J. Lambert, D. Fraser, “An implementation of the fast Fresnel transform,” in IVCNZ’95 Image and Vision Computing New Zealand (Industrial Research Ltd., Christchurch, New Zealand, 1995), pp. 143–148.

A. J. Lambert, “Optical image processing and its application,” Ph.D. dissertation (The University of New South Wales, Canberra, Australia, 1996).

McKinley, W. G.

Perrant, G. B.

M. J. Beran, G. B. Perrant, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Selesnick, I. W.

I. W. Selesnick, C. S. Burrus, “Fast convolution and filterings,” in The Digital Signal Processing Handbook, V. K. Madisetti, D. B. Williams, eds. (IEEE/CRC Press, Boca Raton, Fla., 1998), Chap. 8.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Stark, H.

H. Stark, F. B. Tuteur, Modern Electrical Communications—Theory and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Tuteur, F. B.

H. Stark, F. B. Tuteur, Modern Electrical Communications—Theory and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Yura, H. T.

J. Opt. Soc. Am. (1)

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

Opt. Lett. (2)

Other (10)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

A. J. Lambert, “Optical image processing and its application,” Ph.D. dissertation (The University of New South Wales, Canberra, Australia, 1996).

M. J. Beran, G. B. Perrant, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

I. W. Selesnick, C. S. Burrus, “Fast convolution and filterings,” in The Digital Signal Processing Handbook, V. K. Madisetti, D. B. Williams, eds. (IEEE/CRC Press, Boca Raton, Fla., 1998), Chap. 8.

H. Stark, F. B. Tuteur, Modern Electrical Communications—Theory and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1979).

A. J. Lambert, D. Fraser, “An implementation of the fast Fresnel transform,” in IVCNZ’95 Image and Vision Computing New Zealand (Industrial Research Ltd., Christchurch, New Zealand, 1995), pp. 143–148.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (8)

Fig. 1
Fig. 1

To illustrate the effect of limiting the extent of the chirp, we show (a) an array of one-dimensional chirps with the aperture decreasing for each line in the image, and (b) the corresponding one-dimensional FFT of each case arranged on the corresponding line. Note that the smaller the aperture (toward the bottom), the more spread in the corresponding FFT that is due to the aperture and not the chirp.

Fig. 2
Fig. 2

Relationship between chirp extent (as set by an aperture) and spatial frequency bandwidth can be easily seen in this graph. The curves correspond to the highest-frequency occurrence of a fall in the magnitude of the FFT of an apertured chirp through a percentage threshold of the maximum value of the FFT magnitude. (The uppermost curve corresponds to the 10% threshold, and successively lower curves are for the 25%, 50%, and 90% thresholds, respectively.) The Fourier transform of the aperture controls the bandwidth at small aperture sizes, whereas for larger apertures the extent is set by the maximum spatial frequency of the chirp passed by the aperture.

Fig. 3
Fig. 3

Two-dimensional chirp, shown in (a) real and (b) imaginary components, exhibits a linear increase in spatial frequency with distance from the center. This chirp is sampled on 128 by 128 pixels and has a coefficient of 0.012 m-1, resulting in a frequency band limit that satisfies the sampling theorem. Should the chirp function be undersampled then spatial aliasing will occur, as shown in (c) real and (d) imaginary components of a two-dimensional chirp whose coefficient, 0.072 m-1, is three times the maximum specified by the sampled theorem, 0.024 m-1, for the same grid used in (a) and (b). This results in a three by three set of replicas or aliases of the central portion, which by itself is properly sampled in the portion specified by the boundary between the replicas. Direct simulation using this undersampled chirp will produce erroneous results.

Fig. 4
Fig. 4

To test the simulation of diffraction through an optical system defined by an ABCD ray matrix, we used this input image of 256 by 256 pixels.

Fig. 5
Fig. 5

Magnitude of the wave front emerging from an optical system in a plane that is defocused by 1 mm from the rear focal plane of a convex lens. This 256 by 256 pixel image is the center portion of the 512 by 512 pixel result generated with Eq. (7) with the test input shown in Fig. 4 (see the text for more details).

Fig. 6
Fig. 6

Magnitude of the wave front emerging from an optical system that amounts to having traversed 0.5 m of free space. This 384 by 384 pixel image is the central part of the result computed with Eq. (6) and the test input shown in Fig. 4.

Fig. 7
Fig. 7

One advantage in simulating the diffractive process is that the phase of an optical wave front is easily visualized. This 384 by 384 pixel image is the phase of the central part of the wave front whose magnitude is shown in Fig. 6. As the magnitude drops quickly to zero outside the central region, the apparent fluctuations in phase outside that region are insignificant.

Fig. 8
Fig. 8

Extent of the chirp used in the simulation defines the spread of a wave front. Here we illustrate the determination of (a) the extent of the result using this information with a given region of interest (ROI) of the input wave front and (b) the necessary region of the input given a ROI of the output. The gray circles indicate the extent of the chirp that causes the spreading. If the ROI in (a) is not surrounded by zeros, then only the inset rectangle of the output is valid, as is also the case for the nonshaded region of the output in (b).

Equations (11)

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U x = - j λ B exp - j π z λ -   U x 0 ×   exp - j π λ B Dx 2 - 2 xx 0 + Ax 0 2 d x 0
U x ,   y = - j λ B x B y exp - 2 j π z λ - -   U x 0 ,   y 0 × exp - j π λ B x D x x 2 - 2 xx 0 + A x x 0 2 × exp - j π λ B y D y y 2 - 2 yy 0 + A y y 0 2 d x 0 d y 0 ,
U Ax = - j λ B exp - j π z λ exp j β x 2 × -   U x 0 exp j α x - x 0 2 d x 0 ,
U - B λ 2 π   x = - j λ B   exp - j π z λ exp j β x 2 × -   U x 0 exp j α x 0 2 exp - jxx 0 d x 0 ,
q x ;   a = exp jax 2 .
q x ;   a = j π a   q u ; - 1 / 4 a .
U Ax = Kq x ;   β U x 0 q x 0 ;   α ,
U - B λ 2 π   x = Kq x ;   β U x 0 q x 0 ;   α ,
U Ax = Kq x ;   β - 1 U x 0 q x 0 ;   α .
δ / f f - 1 / f 0
1 f 0 1 .

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