Abstract

The inverse source problem for a monochromeatic source imbedded in a nonabsorbing inhomogeneous medium is investigated within the framework of the reduced scalar wave equation. The Porter-Bojarski integral equation previously formulated for sources imbedded in vacuum is generalized to this case, as are the class of nonradiating and minimum-energy sources considered in Part I [ J. Opt. Soc. Am. 72, 327 ( 1982)].

© 1985 Optical Society of America

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References

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  1. R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327–330 (1982).
    [CrossRef]
  2. A Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1967), p. 189.
  3. 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).
    [CrossRef]
  4. 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.
  5. This form of the integral equation was derived by N. Bleistein, J. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
    [CrossRef]
  6. C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, New York, 1969), Chap. III.
    [CrossRef]
  7. A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
    [CrossRef]
  8. A. J. Devaney, G. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
    [CrossRef]
  9. R. P. Porter, “Determination of structure of weak scatterers from holographic images,” Opt. Commun. 39, 362–365 (1981).
    [CrossRef]
  10. A. J. Devaney, “Inverse source and scattering problems in ultrasonics,” IEEE Trans. Sonics Ultrason. SU-30, 355–364 (1983).
    [CrossRef]
  11. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York1966), Chap. III.
  12. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I, Apps. BII and BIV.
  13. M. Fischer, K. J. Langenberg, “Limitations and defects of certain inverse scattering theories,” IEEE Trans. Antennas Propag. AP-32, 1080–1088 (1984).
    [CrossRef]
  14. I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
    [CrossRef]
  15. A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
    [CrossRef]

1985 (1)

1984 (1)

M. Fischer, K. J. Langenberg, “Limitations and defects of certain inverse scattering theories,” IEEE Trans. Antennas Propag. AP-32, 1080–1088 (1984).
[CrossRef]

1983 (1)

A. J. Devaney, “Inverse source and scattering problems in ultrasonics,” IEEE Trans. Sonics Ultrason. SU-30, 355–364 (1983).
[CrossRef]

1982 (2)

A. J. Devaney, G. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327–330 (1982).
[CrossRef]

1981 (1)

R. P. Porter, “Determination of structure of weak scatterers from holographic images,” Opt. Commun. 39, 362–365 (1981).
[CrossRef]

1979 (1)

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

1977 (1)

This form of the integral equation was derived by N. Bleistein, J. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

1973 (1)

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

1970 (1)

Bleistein, N.

This form of the integral equation was derived by N. Bleistein, J. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

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.

Cohen, J.

This form of the integral equation was derived by N. Bleistein, J. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York1966), Chap. III.

Devaney, A. J.

A. J. Devaney, “Inverse source and scattering problems in ultrasonics,” IEEE Trans. Sonics Ultrason. SU-30, 355–364 (1983).
[CrossRef]

A. J. Devaney, G. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327–330 (1982).
[CrossRef]

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

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

Fischer, M.

M. Fischer, K. J. Langenberg, “Limitations and defects of certain inverse scattering theories,” IEEE Trans. Antennas Propag. AP-32, 1080–1088 (1984).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York1966), Chap. III.

LaHaie, I. J.

Langenberg, K. J.

M. Fischer, K. J. Langenberg, “Limitations and defects of certain inverse scattering theories,” IEEE Trans. Antennas Propag. AP-32, 1080–1088 (1984).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I, Apps. BII and BIV.

Muller, C.

C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, New York, 1969), Chap. III.
[CrossRef]

Porter, R. P.

Sherman, G.

A. J. Devaney, G. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

Sommerfeld, A

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

Wolf, E.

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

IEEE Trans. Antennas Propag. (2)

A. J. Devaney, G. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. AP-30, 1034–1037 (1982).
[CrossRef]

M. Fischer, K. J. Langenberg, “Limitations and defects of certain inverse scattering theories,” IEEE Trans. Antennas Propag. AP-32, 1080–1088 (1984).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

A. J. Devaney, “Inverse source and scattering problems in ultrasonics,” IEEE Trans. Sonics Ultrason. SU-30, 355–364 (1983).
[CrossRef]

J. Math. Phys. (2)

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

This form of the integral equation was derived by N. Bleistein, J. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

R. P. Porter, “Determination of structure of weak scatterers from holographic images,” Opt. Commun. 39, 362–365 (1981).
[CrossRef]

Phys. Rev. D (1)

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

Other (5)

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York1966), Chap. III.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. I, Apps. BII and BIV.

C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, New York, 1969), Chap. III.
[CrossRef]

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.

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

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Equations (50)

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( 2 + k 2 ) ψ ( r , ω ) = ρ ( r , ω ) ,
τ d 3 r ρ ( r , ω ) Im G 0 ( r r ) = Θ ( r , ω ) ,
Θ ( r , ω ) = Σ d s [ ψ ( r , ω ) a Im G 0 ( r r ) Im G 0 ( r r ) a ( r , ω ) ] ,
G 0 ( r r ) = exp ( i k | r r | ) 4 π | r r | .
ψ ̂ ( r , ω ) = ψ ( r , ω ) ; a ψ ̂ ( r , ω ) = a ψ ( r , ω )
E = τ d 3 r | ρ ( r , ω ) | 2
[ 2 + k 2 n 2 ( r , ω ) ] ψ ( r , ω ) = ρ ( r , ω ) ,
[ 2 + k 2 n 2 ( r , ω ) ] G ( r , r ) = δ ( r r ) ,
G ( r , r ) = G ( r , r ) .
[ 2 + k 2 n 2 ( r , ω ) ] Im G ( r , r ) = 0 .
ψ ( r , ω ) 2 Im G ( r , r ) Im G ( r , r ) 2 ψ ( r , ω ) = ρ ( r , ω ) Im G ( r , r ) .
τ d 3 r ρ ( r , ω ) Im G ( r , r ) = Θ ( r , ω ) ,
Θ ( r , ω ) = Σ d s [ ψ ( r , ω ) a Im G ( r , r ) Im G ( r , r ) a ψ ( r , ω ) ] .
Σ d s [ ψ ( r , ω ) a G + ( r , r ) G + ( r , r ) a ψ ( r , ω ) ] = 0 ,
Θ ( r , ω ) = 1 2 i Γ * ( r , ω ) ,
Γ ( r , ω ) = Σ d s [ ψ * ( r , ω ) a G + ( r , r ) G + ( r , r ) a ψ * ( r , ω ) ]
2 i τ d 3 r ρ * ( r , ω ) Im G + ( r , r ) = Γ ( r , ω ) .
τ d 3 r ρ n.r. ( r , ω ) Im G ( r , r ) = 0 ,
ρ n.r. ( r , ω ) = [ 2 + k 2 n 2 ( r , ω ) Q ( r , ω )
V d 3 r d 3 r Im G ( r , r ) f ( r ) f * ( r ) 0
Im G ( r , r ) = n λ n ρ n ( r ) ρ n * ( r ) ,
V d 3 r Im G ( r , r ) ρ n ( r ) = λ n ρ n ( r ) .
V d 3 r ρ n ( r ) ρ n * ( r ) = δ n , n ,
G + ( r , r ) = ϕ n ( r ) ρ n * ( r ) ,
ϕ n ( r ) = τ d 3 r ρ n ( r ) G + ( r , r ) .
ρ ( r , ω ) = a n ρ n ( r ) ,
a n = d 3 r ρ ( r , ω ) ρ n * ( r ) .
ψ ( r , ω ) = τ d 3 r ρ ( r , ω ) G + ( r , r ) ,
ψ ( r , ω ) = a n φ n ( r ) ,
ψ n.r. ( r , ω ) = λ n = 0 a n ϕ n ( r )
ψ r ( r , ω ) = λ n > 0 a n ϕ n ( r ) .
n λ n a n ρ n ( r ) = Θ ( r , ω ) .
λ n a n = d 3 r Θ ( r , ω ) ρ n * ( r ) ,
ρ ( r , ω ) = ρ 1 ( r ) + ρ 2 ( r ) ,
ρ 1 ( r ) = λ n > 0 a n ρ n ( r )
τ d 3 r | ρ 1 ( r , ω ) + ρ 2 ( r , ω ) | 2 = τ d 3 r | ρ 1 | 2 + τ d 3 r | ρ 2 | 2 .
d 3 r | ρ 1 | 2 d 3 r | ρ ̂ | 2 ,
τ d 3 r Im G 0 ( r r ) ρ n ( r ) = λ n ρ n ( r ) .
ρ n ( r ) = { C l , m j l ( k r ) Y l m ( θ , ϕ ) , if r R 0 0 , otherwise ,
C l , m 1 [ d 3 r | j l ( k r ) Y l m ( θ , ϕ ) | 2 ] 1 / 2 = 1 [ d r r 2 j l 2 ( k r ) .
λ n = k C l , m 2 .
ρ m.e. ( r ) = { a l m C l , m j l ( k r ) Y l m ( θ , ϕ ) , if r R 0 0 , otherwise ,
a l m = 1 k C l , m 2 r R 0 d 3 r Θ ( r , ω ) j l ( k r ) Y l m * ( θ , ϕ ) .
Z τ d 3 r Γ ( r , ω ) ρ ( r , ω ) = Σ d s [ ψ * ( r , ω ) a ψ ( r , ω ) ψ ( r , ω ) a ψ * ( r , ω ) ] .
ψ ( r , ω ) = τ d 3 r G + ( r , r ) ρ ( r , ω ) .
ψ ( r , ω ) f ( r ̂ , ω ) e i k r r ,
Z = 2 i k 4 π d Ω | f ( r ̂ , ω ) | 2 ,
C d 3 r d 3 r Im G ( r , r ) ρ ( r , ω ) ρ * ( r , ω ) 0 .
C = τ d 3 r Θ ( r , ω ) ρ * ( r , ω ) = 1 2 i τ d 3 r Γ * ( r , ω ) ρ * ( r , ω ) = 1 2 i Z * .
C = k d Ω | f ( r ̂ , ω ) | 2 0 ,

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