Abstract

The inverse source problem for monochromatic sources Re[ρ(<b>r</b>, ω)<i>e</i><sup><i>-iωt</i></sup>] to the scalar-wave equation is investigated. It is shown that a unique solution to the inverse source problem can be obtained by imposing the constraint that the solution minimize the source energy <i>E</i> = ∫ d<sup>3</sup><i>r</i>|ρ(<b>r</b>, ω)|<sup>2</sup>. For certain recording geometries the time derivative of the real image produced by a point-reference hologram is shown to be directly proportional to the time-reversed minimum energy source Re[ρ*<i>ME</i>(<b>r</b>, ω)<i>e</i><sup><i>-iωt</i></sup>] in the short-wavelength limit.

© 1982 Optical Society of America

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  1. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1967), p. 189.
  2. R. P. Porter, "Diffraction-limited, scalar image formation with holograms of arbitrary shape," J. Opt. Soc. Am. 60, 1051–1059 (1970); "Image formation with arbitrary holographic type surfaces," Phys. Lett. 29A, 193–194 (1969).
  3. N. N. Bojarski, "Inverse Scattering," Naval Air Systems Command Rep., contract N00019-73-C-0312, (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6. The form of Bojarski's integral equation employed in this paper is that given in Ref. 5.
  4. A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).
  5. N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).
  6. A. J. Devaney and G. C. Sherman, "Nonuniqueness in inverse source and scattering problems," IEEE Trans. Antennas Propag. (to be published).
  7. We use the definitions of the spherical harmonics and spherical Bessel and Hankel functions employed by A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I, Apps. BII and BIV.
  8. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 440.
  9. K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), p. 90.
  10. For a discussion on time reversal in holography see W. Lukosz, "Equivalent-lens theory of holographic imaging," J. Opt. Soc. Am. 58, 1084–1091 (1968).

1977 (1)

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

1973 (1)

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

1970 (1)

1968 (1)

Bleistein, N.

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

Bojarski, N. N.

N. N. Bojarski, "Inverse Scattering," Naval Air Systems Command Rep., contract N00019-73-C-0312, (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6. The form of Bojarski's integral equation employed in this paper is that given in Ref. 5.

Cohen, J.

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

Devaney, A. J.

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

A. J. Devaney and G. C. Sherman, "Nonuniqueness in inverse source and scattering problems," IEEE Trans. Antennas Propag. (to be published).

Gottfried, K.

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), p. 90.

Lukosz, W.

Porter, R. P.

Sherman, G. C.

A. J. Devaney and G. C. Sherman, "Nonuniqueness in inverse source and scattering problems," IEEE Trans. Antennas Propag. (to be published).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1967), p. 189.

Wolf, E.

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

J. Math. Phys. (1)

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

J. Opt. Soc. Am. (2)

Phys. Rev. D (1)

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

Other (6)

N. N. Bojarski, "Inverse Scattering," Naval Air Systems Command Rep., contract N00019-73-C-0312, (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6. The form of Bojarski's integral equation employed in this paper is that given in Ref. 5.

A. J. Devaney and G. C. Sherman, "Nonuniqueness in inverse source and scattering problems," IEEE Trans. Antennas Propag. (to be published).

We use the definitions of the spherical harmonics and spherical Bessel and Hankel functions employed by A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I, Apps. BII and BIV.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 440.

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), p. 90.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1967), p. 189.

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