Abstract

A set of generalized Dirac functions (Δ functions) is introduced which resemble the Dirac function (δ function) in that their masses equal unity. Different members out of this set are determined by different values of a certain parameter. The sampling theorem—which tells us that a band-limited function can be synthesized by a proper low-pass filtering of a sequence of equidistant δ functions having variable masses, these masses being equal to the corresponding sample values of the band-limited function—no longer holds, if the practically unrealizable δ functions are replaced by realized Δ functions. It must be replaced by a generalized sampling theorem, which tells us what relationship exists between the parameters of the Δ functions and the sample values of the function to be generated at the ouput of the low-pass filter. Once a set of Δ functions has been chosen, this relationship can be determined explicitly. An application to the synthesis of coherent optical fields by means of computer-generated transparencies is given.

© 1978 Optical Society of America

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References

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  1. A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968) pp. 119–128.
  2. B. R. Brown and A. W. Lohmann, "Complex spatial filtering with binary masks," Appl. Opt. 5, 967–969 (1966).
  3. J. J. Burch, "A computer algorithm for the synthesis of spatial frequency filters," Proc. IEEE 55, 599–601 (1967).
  4. A. W. Lohmann and D. P. Paris, "Binary Fraunhofer holograms, generated by computer," Appl. Opt. 6, 1739–1748 (1967).
  5. B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160–168 (1969).
  6. W. H. Lee, "Sampled Fourier transform hologram generated by computer," Appl. Opt. 9, 639–643 (1970).
  7. C. B. Burckhardt, "A simplification of Lee's method of generating holograms by computer," Appl. Opt. 9, 1949 (1970).
  8. W. H. Lee, "Binary synthetic holograms," Appl. Opt. 13, 1677–1682 (1974).
  9. J. P. Hugonin and P. Chavel, "A complement to the theory of Lohmann-type computer holograms," Opt. Commun. 16, 342–346 (1976).
  10. P. Chavel and J. P. Hugonin, "High quality computer holograms: The problem of phase representation," J. Opt. Soc. Am. 66, 989–996 (1976).
  11. B. Braunecker and R. Hauck, "Grey level on axis computer holograms for incoherent image processing," Opt. Commun. 20, 234–238 (1977).
  12. A. W. Lohmann, "Incoherent optical processing of complex data," Appl. Opt. 16, 261–263 (1977).
  13. V. Volterra, Theory of functionals and of integrals and integro-differential equations (Dover, New York, 1959).
  14. J. F. Barrett, "The use of functionals in the analysis of nonlinear physical systems," J. Electron. Control 15, 567–615 (1963).
  15. A. Halme, "Polynomial operators for nonlinear systems analysis," Act. Polytech. Math. 24, 1–64 (1972).
  16. L. A. Pipes, "The reversion method for solving nonlinear differential equations," J. App. Phys. 23, 202–207 (1952).

1977

B. Braunecker and R. Hauck, "Grey level on axis computer holograms for incoherent image processing," Opt. Commun. 20, 234–238 (1977).

A. W. Lohmann, "Incoherent optical processing of complex data," Appl. Opt. 16, 261–263 (1977).

1976

P. Chavel and J. P. Hugonin, "High quality computer holograms: The problem of phase representation," J. Opt. Soc. Am. 66, 989–996 (1976).

J. P. Hugonin and P. Chavel, "A complement to the theory of Lohmann-type computer holograms," Opt. Commun. 16, 342–346 (1976).

1974

1972

A. Halme, "Polynomial operators for nonlinear systems analysis," Act. Polytech. Math. 24, 1–64 (1972).

1970

1969

B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160–168 (1969).

1967

1966

1963

J. F. Barrett, "The use of functionals in the analysis of nonlinear physical systems," J. Electron. Control 15, 567–615 (1963).

1952

L. A. Pipes, "The reversion method for solving nonlinear differential equations," J. App. Phys. 23, 202–207 (1952).

Barrett, J. F.

J. F. Barrett, "The use of functionals in the analysis of nonlinear physical systems," J. Electron. Control 15, 567–615 (1963).

Braunecker, B.

B. Braunecker and R. Hauck, "Grey level on axis computer holograms for incoherent image processing," Opt. Commun. 20, 234–238 (1977).

Brown, B. R.

B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160–168 (1969).

B. R. Brown and A. W. Lohmann, "Complex spatial filtering with binary masks," Appl. Opt. 5, 967–969 (1966).

Burch, J. J.

J. J. Burch, "A computer algorithm for the synthesis of spatial frequency filters," Proc. IEEE 55, 599–601 (1967).

Burckhardt, C. B.

Chavel, P.

J. P. Hugonin and P. Chavel, "A complement to the theory of Lohmann-type computer holograms," Opt. Commun. 16, 342–346 (1976).

P. Chavel and J. P. Hugonin, "High quality computer holograms: The problem of phase representation," J. Opt. Soc. Am. 66, 989–996 (1976).

Halme, A.

A. Halme, "Polynomial operators for nonlinear systems analysis," Act. Polytech. Math. 24, 1–64 (1972).

Hauck, R.

B. Braunecker and R. Hauck, "Grey level on axis computer holograms for incoherent image processing," Opt. Commun. 20, 234–238 (1977).

Hugonin, J. P.

J. P. Hugonin and P. Chavel, "A complement to the theory of Lohmann-type computer holograms," Opt. Commun. 16, 342–346 (1976).

P. Chavel and J. P. Hugonin, "High quality computer holograms: The problem of phase representation," J. Opt. Soc. Am. 66, 989–996 (1976).

Lee, W. H.

Lohmann, A. W.

Papoulis, A.

A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968) pp. 119–128.

Paris, D. P.

Pipes, L. A.

L. A. Pipes, "The reversion method for solving nonlinear differential equations," J. App. Phys. 23, 202–207 (1952).

Volterra, V.

V. Volterra, Theory of functionals and of integrals and integro-differential equations (Dover, New York, 1959).

Act. Polytech. Math.

A. Halme, "Polynomial operators for nonlinear systems analysis," Act. Polytech. Math. 24, 1–64 (1972).

Appl. Opt.

IBM J. Res. Dev.

B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160–168 (1969).

J. App. Phys.

L. A. Pipes, "The reversion method for solving nonlinear differential equations," J. App. Phys. 23, 202–207 (1952).

J. Electron. Control

J. F. Barrett, "The use of functionals in the analysis of nonlinear physical systems," J. Electron. Control 15, 567–615 (1963).

J. Opt. Soc. Am.

Opt. Commun.

J. P. Hugonin and P. Chavel, "A complement to the theory of Lohmann-type computer holograms," Opt. Commun. 16, 342–346 (1976).

B. Braunecker and R. Hauck, "Grey level on axis computer holograms for incoherent image processing," Opt. Commun. 20, 234–238 (1977).

Other

V. Volterra, Theory of functionals and of integrals and integro-differential equations (Dover, New York, 1959).

A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968) pp. 119–128.

J. J. Burch, "A computer algorithm for the synthesis of spatial frequency filters," Proc. IEEE 55, 599–601 (1967).

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