Abstract

A general and practical method is developed for shape determination of specular reflectors from their scattered fields. Even though this problem has been of interest for many years, and even though a unique solution is believed possible, a general method has eluded us. The method presented here uses the observation that a class of incoming and outgoing waves that satisfy the boundary conditions of the object can be found. Holographic imaging of the scattered field is a practical approach for finding these eigenfunctions for the scatterer. As an example, we invert the scattered field from a sphere, precisely determining its radius.

© 1985 Optical Society of America

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References

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  1. W. A. Imbriale, R. Mittra, “The two-dimensional inverse scattering problem,” IEEE Trans. Antennas Propag. AP-18, 633 (1970).
    [CrossRef]
  2. W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
    [CrossRef]
  3. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805 (1965).
    [CrossRef]
  4. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417 (1969).
    [CrossRef]
  5. R. P. Porter, “Diffraction-limited scalar image formation with holograms of arbitrary shapes,” J. Opt. Soc. Am. 60, 1051 (1970).
    [CrossRef]
  6. R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. 29A, 193 (1969).
  7. R. P. Porter, A. J. Devaney, “Generalized holography and practical solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707 (1982).
    [CrossRef]
  8. R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327 (1982).
    [CrossRef]
  9. R. P. Porter, “Determination of structure of weak scatterers from holographic images,” Opt. Commun. 39, 362 (1981).
    [CrossRef]
  10. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11.3.
  11. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
    [CrossRef]
  12. D. J. N. Wall, “Methods of overcoming numerical instabilities associated with the T-matrix method,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), p. 269.

1982 (2)

1981 (1)

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

1971 (1)

W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
[CrossRef]

1970 (2)

W. A. Imbriale, R. Mittra, “The two-dimensional inverse scattering problem,” IEEE Trans. Antennas Propag. AP-18, 633 (1970).
[CrossRef]

R. P. Porter, “Diffraction-limited scalar image formation with holograms of arbitrary shapes,” J. Opt. Soc. Am. 60, 1051 (1970).
[CrossRef]

1969 (3)

R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. 29A, 193 (1969).

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[CrossRef]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805 (1965).
[CrossRef]

Boerner, W. M.

W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
[CrossRef]

Devaney, A. J.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11.3.

Hamid, M. A. K.

W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
[CrossRef]

Imbriale, W. A.

W. A. Imbriale, R. Mittra, “The two-dimensional inverse scattering problem,” IEEE Trans. Antennas Propag. AP-18, 633 (1970).
[CrossRef]

Mittra, R.

W. A. Imbriale, R. Mittra, “The two-dimensional inverse scattering problem,” IEEE Trans. Antennas Propag. AP-18, 633 (1970).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11.3.

Porter, R. P.

Vandenberghe, F. H.

W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
[CrossRef]

Wall, D. J. N.

D. J. N. Wall, “Methods of overcoming numerical instabilities associated with the T-matrix method,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), p. 269.

Waterman, P. C.

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805 (1965).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[CrossRef]

Can. J. Phys. (1)

W. M. Boerner, F. H. Vandenberghe, M. A. K. Hamid, “Determination of the electrical radius ka of a circular cylindrical scatterer from the scattered field,” Can. J. Phys. 49, 804 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. A. Imbriale, R. Mittra, “The two-dimensional inverse scattering problem,” IEEE Trans. Antennas Propag. AP-18, 633 (1970).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

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

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[CrossRef]

Phys. Lett. (1)

R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. 29A, 193 (1969).

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805 (1965).
[CrossRef]

Other (2)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 11.3.

D. J. N. Wall, “Methods of overcoming numerical instabilities associated with the T-matrix method,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), p. 269.

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Figures (4)

Fig. 1
Fig. 1

The object illumination and recording step are shown in (a). The object So is illuminated by the basis set, shown here as a plane wave, and the field and normal gradients are recorded on the holographic surface Sh. During the second or imaging step conjugate sources are established and the inward converging field is launched. The image field in the interior volume Vmeas is analyzed to find the T-matrix coefficients.

Fig. 2
Fig. 2

Reflecting sphere located inside the measure sphere R.

Fig. 3
Fig. 3

Surface identifier SUM for a sphere of radius ka = 10.

Fig. 4
Fig. 4

Surface identifier SUM for a sphere of radius ka = 30.

Equations (49)

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U m ( r ) = u m * ( r ) + s m u u m ( r )
s m n u = exp ( i θ m u ) δ m n .
SUM ( r ) = m | U m | 2 A m 2 ,
SUM ( r ) = m | n ̂ · U m | 2 A m 2 .
SUM ( r ) M | k r | 2 .
SUM ( r ) = m γ | U m | 2 A m 2 ,
Re ψ ν = ½ ψ ν + ½ ψ ν * ,
ψ I ( r ) = ν a ν Re ψ ν ( r ) .
G f ( r , r ) = i k 4 π ν ψ ν ( r ) Re ψ ν ( r ) , | r | > | r | .
ψ s r = ν f ν ψ ν ( r ) .
ψ ( r ) r out S h 0 r in S h } = ψ I ( r ) + S h { G f ( r , r ) ψ ( r ) ψ ( r ) G f ( r , r ) } · n ̂ d S ,
n ̂ · ψ ( r ) = α ν n ̂ · Re ψ ν ( r ) .
a ν = i γ k 4 π s o d S ψ ν ( r ) α γ n ̂ · Re ψ γ ( r )
a = i Q α ,
f = i Re Q α .
f = T a ,
T = Re ( Q ) ( Q ) 1 .
S = 1 + 2 T ,
ψ ( r ) = ½ ν γ [ a ν ψ ν * + S ν γ a γ ψ ν ] .
S = S ,
S * S = 1 .
S w m = exp ( i θ m ) w m ,
| Re Q + tan ( θ m / 2 ) Im Q | = 0 .
u m ( r ) = ν w ν m ψ ν ( r )
U m ( r ) = u m * ( r ) + exp ( i θ m ) u m ( r ) ,
exp ( i k 0 · r ) = n , m = 0 i n [ m ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] × P n m ( cos θ ) P n m ( cos θ o ) j n ( k r ) cos m ( ϕ ϕ o ) ,
m n = { 1 , m = 0 2 , m > 0 .
Re ψ ν ( r ) = Re ψ ( m , n , p ) ( r ) = [ m ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] 1 / 2 × P n m ( cos θ ) j n ( k r ) { cos m ϕ sin m ϕ .
a ν = i n [ m ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] 1 / 2 P n m ( cos θ o ) j n { cos m ϕ o sin m ϕ o .
V MEAS d V Re ψ ν ( r ) Re ψ ν ( r ) = 4 π δ ν γ 0 R d r r 2 j n 2 ( k r ) = σ n δ ν γ ,
δ ν γ = { 1 , ν = γ 0 , ν γ .
σ n = ( 2 π R / k 2 ) { [ 1 ( n + 1 / 2 ) 2 / k 2 R 2 ] 1 / 2 , n + 1 / 2 k R 0 , n + 1 / 2 > k R .
0 π 0 2 π d θ 0 d ϕ 0 sin θ o a ν a γ * = 4 π δ ν γ .
Ψ br ( r ) = V ρ * ( r ) K br ( r , r ) d V .
Ψ br ( r ) = S o n ̂ · Ψ * ( r ) K br ( r , r ) d S .
K br ( r , r ) = G f * ( r , r ) G f ( r , r ) ,
K br ( r , r ) = i k 2 π Re ψ ν ( r ) Re ψ ν ( r )
Ψ br ( r ) = 2 i ν γ α γ * ( Re Q ν γ ) [ Re ψ ν ( r ) ] ,
Ψ br ( r ) = 2 ν γ α γ * T ν γ * Re ψ ν ( r ) ,
V MEAS 0 π 0 2 π d V d θ o d ϕ sin θ o Ψ br ( r ) a γ Re ψ ν ( r ) = 8 π T ν γ * σ n ,
Ψ s ( r ) = n , m = 0 i n [ m ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] 1 / 2 × P n m ( cos θ o ) [ j n ( k a ) h n ( k a ) ] × [ cos m ϕ o ψ n , m , c ( r ) + sin m ϕ o ψ n , m , s ( r ) ] ,
Ψ br ( r ) = 2 n , m = 0 i n [ m ( 2 n + 1 ) ( n m ) ! ( n + m ) ! ] 1 / 2 × P n m ( cos θ o ) [ j n ( k a ) h n * ( k a ) ] × [ cos m ϕ o Re ψ n , m , c ( r ) + sin m ϕ o Re ψ n , m , s ( r ) ] .
T ν γ * σ n = [ j n ( k a ) h n * ( k a ) ] δ ν γ σ n ,
T ̂ ν γ * = { [ j n ( k a ) / h n * ( k a ) ] δ ν γ , n k a k R 0 , n > k a .
Ŝ = 1 + 2 T ̂ .
Ŝ ν ν = h n * ( k a ) / h n ( k a ) .
U ν ( r ) = w ν a ν [ ψ ν * ( r ) h n * ( k a ) h n ( k a ) ψ ν ( r ) ]
U n ( r ) = [ h n * ( k r ) h n * ( k a ) h n ( k a ) h n ( k r ) ] , n N .
SUM ( r ) = n = 0 N | h n * ( k r ) h n * ( k a ) h n ( k a ) h n ( k r ) | 2 .

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