Abstract

A new method of using an exponential filter to generate an intensity distribution in the complex plane is presented. This intensity, with the intensity on the real axis, allows the phase of the scattered field to be determined from its zero locations. It is shown that knowledge of all zero coordinates may not be required.

© 1981 Optical Society of America

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References

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  1. H. A. Ferwerda, in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
    [CrossRef]
  2. W. O. Saxton, “Computer techniques for image processing in electron microscopy,” in Advances in Electronics and Electron Physics, Supplement 10 (Academic, New York, 1978).
  3. G. Ross, M. A. Fiddy, M. Neito-Vesperinas, “The inverse scattering problem in structured determinations,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), Chap. 2.
    [CrossRef]
  4. E. L. O'Neill, A. Walther, Opt. Acta 10, 33 (1963).
    [CrossRef]
  5. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
    [CrossRef]
  6. G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
    [CrossRef]
  7. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  8. D. L. Misell, J. Phys. D. 6, L6 (1973).
    [CrossRef]
  9. H. A. Ferwerda, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).
  10. M. A. Fiddy, A. H. Greenaway, Nature (London) 276, 421 (1978).
    [CrossRef]
  11. R. P. Boas, Entire Functions (Academic, New York, 1954).
  12. M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).
  13. C. L. Mehta, Nuovo Cimento 36, 202 (1965).
    [CrossRef]
  14. D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 126 (1973).
    [CrossRef]
  15. M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).
  16. J. G. Walker, Opt. Acta 28, 735 (1981).
    [CrossRef]

1981 (1)

J. G. Walker, Opt. Acta 28, 735 (1981).
[CrossRef]

1980 (1)

G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
[CrossRef]

1978 (1)

M. A. Fiddy, A. H. Greenaway, Nature (London) 276, 421 (1978).
[CrossRef]

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

1973 (2)

1972 (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1965 (1)

C. L. Mehta, Nuovo Cimento 36, 202 (1965).
[CrossRef]

1963 (1)

E. L. O'Neill, A. Walther, Opt. Acta 10, 33 (1963).
[CrossRef]

Boas, R. P.

R. P. Boas, Entire Functions (Academic, New York, 1954).

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

Ferwerda, H. A.

H. A. Ferwerda, in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

H. A. Ferwerda, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

Fiddy, M. A.

G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
[CrossRef]

M. A. Fiddy, A. H. Greenaway, Nature (London) 276, 421 (1978).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

G. Ross, M. A. Fiddy, M. Neito-Vesperinas, “The inverse scattering problem in structured determinations,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), Chap. 2.
[CrossRef]

M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).

M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Greenaway, A. H.

M. A. Fiddy, A. H. Greenaway, Nature (London) 276, 421 (1978).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

Huiser, A. M. J.

M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).

Kohler, D.

Mandel, L.

Mehta, C. L.

C. L. Mehta, Nuovo Cimento 36, 202 (1965).
[CrossRef]

Misell, D. L.

D. L. Misell, J. Phys. D. 6, L6 (1973).
[CrossRef]

Moezzi, H.

G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
[CrossRef]

Neito-Vesperinas, M.

G. Ross, M. A. Fiddy, M. Neito-Vesperinas, “The inverse scattering problem in structured determinations,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), Chap. 2.
[CrossRef]

Nieto-Vesperinas, M.

M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).

M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

O'Neill, E. L.

E. L. O'Neill, A. Walther, Opt. Acta 10, 33 (1963).
[CrossRef]

Ross, G.

G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

G. Ross, M. A. Fiddy, M. Neito-Vesperinas, “The inverse scattering problem in structured determinations,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), Chap. 2.
[CrossRef]

M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

W. O. Saxton, “Computer techniques for image processing in electron microscopy,” in Advances in Electronics and Electron Physics, Supplement 10 (Academic, New York, 1978).

Walker, J. G.

J. G. Walker, Opt. Acta 28, 735 (1981).
[CrossRef]

Walther, A.

E. L. O'Neill, A. Walther, Opt. Acta 10, 33 (1963).
[CrossRef]

Wood, J. W.

M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

J. Opt. Soc. Am. (1)

J. Phys. D. (1)

D. L. Misell, J. Phys. D. 6, L6 (1973).
[CrossRef]

Nature (London) (1)

M. A. Fiddy, A. H. Greenaway, Nature (London) 276, 421 (1978).
[CrossRef]

Nuovo Cimento (1)

C. L. Mehta, Nuovo Cimento 36, 202 (1965).
[CrossRef]

Opt. Acta (3)

J. G. Walker, Opt. Acta 28, 735 (1981).
[CrossRef]

G. Ross, M. A. Fiddy, H. Moezzi, Opt. Acta 27, 1433 (1980).
[CrossRef]

E. L. O'Neill, A. Walther, Opt. Acta 10, 33 (1963).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Proc. R. Soc. London Ser. A (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, Proc. R. Soc. London Ser. A 350, 191 (1976).
[CrossRef]

Other (7)

H. A. Ferwerda, in Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

W. O. Saxton, “Computer techniques for image processing in electron microscopy,” in Advances in Electronics and Electron Physics, Supplement 10 (Academic, New York, 1978).

G. Ross, M. A. Fiddy, M. Neito-Vesperinas, “The inverse scattering problem in structured determinations,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), Chap. 2.
[CrossRef]

H. A. Ferwerda, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

R. P. Boas, Entire Functions (Academic, New York, 1954).

M. A. Fiddy, A. M. J. Huiser, M. Nieto-Vesperinas, G. Ross, Opt. Acta (to be published).

M. A. Fiddy, G. Ross, J. W. Wood, M. Nieto-Vesperinas, Optics in Four Dimensions—1980, AIP Conference Proceedings No. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981).

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Figures (2)

Fig. 1
Fig. 1

a, The intensity distribution on the real axis for a function having the complex zeros given in Table 1; b, the corresponding intensity on x + iy0 with y0 = 0.5; c, the zero distribution of a; d, the zero distribution of b; e, the common zeros from c and d; f, the Fourier transform of the F(x) generated from the zeros in e, i.e., the original source function.

Fig. 2
Fig. 2

a, The intensity on the real axis for a function having the complex zeros given in Table 1; b, the reconstructed intensity, IHy0(x); c, the zero distribution of Fh(x + iy0); d, the zeros in {h}, located after comparing the intensities in a and b, are shown flipped to their correct locations about the line y = y0; all other zeros remain at the same locations as in c.

Tables (1)

Tables Icon

Table 1 Determining the Zeros of F(x) from Two Intensitiesa

Equations (6)

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

F ( x + i y ) = | F ( z ) | exp [ i ϕ ( z ) ] = a a f ( t ) exp ( y t ) exp ( i x t ) d t = a a f ( t ) exp ( i z t ) d t = j ( 1 z / z j ) ,
ϕ ( x ) = x / π P log | F ( x ) | x ( x x ) d x ,
F H ( x ) = F ( x ) B k ( x ) = F ( x ) j = { k } ( x z j * x z j ) ;
F H ( x l ) = j l M ( j j l ) j M [ 1 l π 2 a / ( x j i | y j | ) ] ( 1 ) l ,
F H ( x + i y 0 ) = [ I ( x + i y 0 ) ] 1 / 2 exp [ i ϕ H ( x + i y 0 ) ] ,
F H ( x + i y 0 ) = F ( x + i y 0 ) j = { k } [ x ( z j * + i y 0 ) x ( z j i y 0 ) ] .

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