Abstract

A fundamental result of scattering theory, the so-called optical theorem, applies to situations where the field incident on the scatterer is a monochromatic plane wave and the scatterer is deterministic. We present generalizations of the theorem to situations where either the incident field or the scatterer or both are spatially random. By using these generalizations we demonstrate the possibility of determining the structure of some random scatterers from the knowledge of the power absorbed from two plane waves incident on it.

© 1997 Optical Society of America

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  1. Although this term is frequently used by quantum physicists, it is hardly known to workers in classical optics. We will use instead the somewhat more descriptive term optical cross-section theorem.
  2. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). In the third paragraph on page 641 of this paper, there is an incorrect statement implying that the phase of the scattering amplitude can be determined from intensity correlation experiments pioneered by Hanbury Brown and Twiss.
    [CrossRef]
  3. E. Feenberg, “The scattering of slow electrons by neutral atoms,” Phys. Rev. 40, 40–54 (1932).
    [CrossRef]
  4. H. C. van der Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica (Utrecht) 15, 740–746 (1949).
    [CrossRef]
  5. D. S. Jones, “On the scattering cross section of an obstacle,” Phil. Mag. 46, 957–962 (1955).
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997), p. 659, Eq. (110).
  7. To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.
  8. See, for example, P. Roman, Advanced Quantum Theory (Addision-Wesley, Reading, Mass., 1965), Sec. 3-2.
  9. This point seems to have been overlooked by M. C. Li in his paper “Scattering initiated by two coherent beams” [Phys. Rev. A 9, 1635–1643 (1974)], in which the possibility of determining the imaginary part of the scattering amplitude from scattering of two monochromatic plane waves is discussed.
    [CrossRef]
  10. By a free field we mean a field that can be represented as a linear superposition of homogeneous plane-wave modes only. The properties of free fields appear to have been first systematically studied by G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968). and in J. Opt. Soc. Am. 59, 697–711 (1969). See also Manuel Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 2.7. General fields that propagate into the half-space z>0 include also inhomogeneous (evanescent) waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields encountered in practice, except in the immediate neighborhood of scattering bodies. An example is given in Ref. 11, Sec. 3.2.3.
    [CrossRef]
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Secs. 3.2.2 and 3.2.3.
  12. This theory applies to random fields that are statistically stationary, such as considered here. For fields of this kind the Fourier representation cannot be used, because the stationary fields do not die out with increasing time. For a discussion of coherence theory in the space-frequency domain, see Sec. 4.7 of Ref. 11.
  13. R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), p. 14.
  14. D. Rouseff, R. P. Porter, “Diffraction tomography with random media,” J. Acoust. Soc. Am. 89, 1599–1609 (1991).
    [CrossRef]
  15. J. Howard, “Laser probing of weakly scattering media,” J. Opt. Soc. Am. A 8, 1955–1963 (1991).
    [CrossRef]
  16. V. E. Kunitstyn, E. D. Tereshchenko, “Radio tomography of the ionosphere,” IEEE Antennas Propag. Mag. 34, 22–32 (1992).
    [CrossRef]
  17. D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
    [CrossRef]

1997 (1)

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

1992 (1)

V. E. Kunitstyn, E. D. Tereshchenko, “Radio tomography of the ionosphere,” IEEE Antennas Propag. Mag. 34, 22–32 (1992).
[CrossRef]

1991 (2)

D. Rouseff, R. P. Porter, “Diffraction tomography with random media,” J. Acoust. Soc. Am. 89, 1599–1609 (1991).
[CrossRef]

J. Howard, “Laser probing of weakly scattering media,” J. Opt. Soc. Am. A 8, 1955–1963 (1991).
[CrossRef]

1980 (1)

To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.

1976 (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). In the third paragraph on page 641 of this paper, there is an incorrect statement implying that the phase of the scattering amplitude can be determined from intensity correlation experiments pioneered by Hanbury Brown and Twiss.
[CrossRef]

1974 (1)

This point seems to have been overlooked by M. C. Li in his paper “Scattering initiated by two coherent beams” [Phys. Rev. A 9, 1635–1643 (1974)], in which the possibility of determining the imaginary part of the scattering amplitude from scattering of two monochromatic plane waves is discussed.
[CrossRef]

1968 (1)

By a free field we mean a field that can be represented as a linear superposition of homogeneous plane-wave modes only. The properties of free fields appear to have been first systematically studied by G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968). and in J. Opt. Soc. Am. 59, 697–711 (1969). See also Manuel Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 2.7. General fields that propagate into the half-space z>0 include also inhomogeneous (evanescent) waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields encountered in practice, except in the immediate neighborhood of scattering bodies. An example is given in Ref. 11, Sec. 3.2.3.
[CrossRef]

1955 (1)

D. S. Jones, “On the scattering cross section of an obstacle,” Phil. Mag. 46, 957–962 (1955).

1949 (1)

H. C. van der Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica (Utrecht) 15, 740–746 (1949).
[CrossRef]

1932 (1)

E. Feenberg, “The scattering of slow electrons by neutral atoms,” Phys. Rev. 40, 40–54 (1932).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997), p. 659, Eq. (110).

Feenberg, E.

E. Feenberg, “The scattering of slow electrons by neutral atoms,” Phys. Rev. 40, 40–54 (1932).
[CrossRef]

Fiddy, M. A.

To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.

Fischer, D. G.

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

Howard, J.

James, R. W.

R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), p. 14.

Jones, D. S.

D. S. Jones, “On the scattering cross section of an obstacle,” Phil. Mag. 46, 957–962 (1955).

Kunitstyn, V. E.

V. E. Kunitstyn, E. D. Tereshchenko, “Radio tomography of the ionosphere,” IEEE Antennas Propag. Mag. 34, 22–32 (1992).
[CrossRef]

Li, M. C.

This point seems to have been overlooked by M. C. Li in his paper “Scattering initiated by two coherent beams” [Phys. Rev. A 9, 1635–1643 (1974)], in which the possibility of determining the imaginary part of the scattering amplitude from scattering of two monochromatic plane waves is discussed.
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Secs. 3.2.2 and 3.2.3.

Newton, R. G.

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). In the third paragraph on page 641 of this paper, there is an incorrect statement implying that the phase of the scattering amplitude can be determined from intensity correlation experiments pioneered by Hanbury Brown and Twiss.
[CrossRef]

Nieto-Vesperinas, M.

To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.

Porter, R. P.

D. Rouseff, R. P. Porter, “Diffraction tomography with random media,” J. Acoust. Soc. Am. 89, 1599–1609 (1991).
[CrossRef]

Roman, P.

See, for example, P. Roman, Advanced Quantum Theory (Addision-Wesley, Reading, Mass., 1965), Sec. 3-2.

Ross, G.

To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.

Rouseff, D.

D. Rouseff, R. P. Porter, “Diffraction tomography with random media,” J. Acoust. Soc. Am. 89, 1599–1609 (1991).
[CrossRef]

Sherman, G. C.

By a free field we mean a field that can be represented as a linear superposition of homogeneous plane-wave modes only. The properties of free fields appear to have been first systematically studied by G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968). and in J. Opt. Soc. Am. 59, 697–711 (1969). See also Manuel Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 2.7. General fields that propagate into the half-space z>0 include also inhomogeneous (evanescent) waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields encountered in practice, except in the immediate neighborhood of scattering bodies. An example is given in Ref. 11, Sec. 3.2.3.
[CrossRef]

Tereshchenko, E. D.

V. E. Kunitstyn, E. D. Tereshchenko, “Radio tomography of the ionosphere,” IEEE Antennas Propag. Mag. 34, 22–32 (1992).
[CrossRef]

van der Hulst, H. C.

H. C. van der Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica (Utrecht) 15, 740–746 (1949).
[CrossRef]

Wolf, E.

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997), p. 659, Eq. (110).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Secs. 3.2.2 and 3.2.3.

Am. J. Phys. (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). In the third paragraph on page 641 of this paper, there is an incorrect statement implying that the phase of the scattering amplitude can be determined from intensity correlation experiments pioneered by Hanbury Brown and Twiss.
[CrossRef]

IEEE Antennas Propag. Mag. (1)

V. E. Kunitstyn, E. D. Tereshchenko, “Radio tomography of the ionosphere,” IEEE Antennas Propag. Mag. 34, 22–32 (1992).
[CrossRef]

J. Acoust. Soc. Am. (1)

D. Rouseff, R. P. Porter, “Diffraction tomography with random media,” J. Acoust. Soc. Am. 89, 1599–1609 (1991).
[CrossRef]

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

Opt. Commun. (1)

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

Optik (1)

To our knowledge, only one paper, by M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik 55, 165–171 (1980), has been published that attempts to generalize the theorem to partially coherent incident light. We are not aware of any publication that deals with its generalization to scattering on random media.

Phil. Mag. (1)

D. S. Jones, “On the scattering cross section of an obstacle,” Phil. Mag. 46, 957–962 (1955).

Phys. Rev. (1)

E. Feenberg, “The scattering of slow electrons by neutral atoms,” Phys. Rev. 40, 40–54 (1932).
[CrossRef]

Phys. Rev. A (1)

This point seems to have been overlooked by M. C. Li in his paper “Scattering initiated by two coherent beams” [Phys. Rev. A 9, 1635–1643 (1974)], in which the possibility of determining the imaginary part of the scattering amplitude from scattering of two monochromatic plane waves is discussed.
[CrossRef]

Phys. Rev. Lett. (1)

By a free field we mean a field that can be represented as a linear superposition of homogeneous plane-wave modes only. The properties of free fields appear to have been first systematically studied by G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968). and in J. Opt. Soc. Am. 59, 697–711 (1969). See also Manuel Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 2.7. General fields that propagate into the half-space z>0 include also inhomogeneous (evanescent) waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields encountered in practice, except in the immediate neighborhood of scattering bodies. An example is given in Ref. 11, Sec. 3.2.3.
[CrossRef]

Physica (Utrecht) (1)

H. C. van der Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica (Utrecht) 15, 740–746 (1949).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997), p. 659, Eq. (110).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Secs. 3.2.2 and 3.2.3.

This theory applies to random fields that are statistically stationary, such as considered here. For fields of this kind the Fourier representation cannot be used, because the stationary fields do not die out with increasing time. For a discussion of coherence theory in the space-frequency domain, see Sec. 4.7 of Ref. 11.

R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), p. 14.

See, for example, P. Roman, Advanced Quantum Theory (Addision-Wesley, Reading, Mass., 1965), Sec. 3-2.

Although this term is frequently used by quantum physicists, it is hardly known to workers in classical optics. We will use instead the somewhat more descriptive term optical cross-section theorem.

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Figures (3)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Illustration of the notation used in Section 4 in the analysis relating to structure determination of random scatterers from measurements of the power extinguished on scattering. u1 and u2 are unit vectors in directions of incidence of two monochromatic plane waves of amplitudes A1 and A2.

Fig. 3
Fig. 3

Illustrating the accessible Fourier components Γ˜η(K) of the function Γη(r) that characterizes the spatial fluctuations of the random medium, from measurements of the power extinguished on scattering of two monochromatic plane waves. (a) Ewald’s sphere of reflection (σ), (b) Ewald’s limiting sphere, generated by Ewald’s spheres of reflection σ1, σ2, for different directions of incidence, represented by vectors O1A, O2A,.

Equations (54)

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Ψ(i)(r, t)=ψ(i)(r, ω)exp(-iωt),
ψ(i)(r, ω)=exp(iku0·r).
k=ωc
F(r, ω)=k2η(r, ω)
=k24π[n2(r, ω)-1],
ψ(ru, u0; ω)=ψ(i)(u0; ω)+ψ(s)(ru, u0, ω),
ψ(s)(ru, u0, ω)=f(u, u0; ω) exp(ikr)r,
σ(u0; ω)=4πkIm f(u0, u0; ω),
σ(u0; ω)=4πkImf(u0, u0; ω).
f(u, u0)=n=1fn(u, u0),
f1(u, u0)=k2Vη(r1)exp[-ik(u-u0)·r1]d3r1,
f2(u, u0)=k4VVη(r1)η(r2)G(r1, r2)×exp[-ik(u·r1-u0·r2)]d3r1d3r2,
····
fn(u, u0)=k2nVVVη(r1)η(r2)η(rn)×G(r1, r2)G(r2, r3)G(rn-1, rn)×exp[-ik(u·r1-u0·rn)]×d3r1d3r2d3rn,
G(r1, r2)=exp(ik|r1-r2|)|r1-r2|
σ(u0)=n=1σn(u0),
σ1(u0)=4πk Im VC1(r1)d3r1,
σ2(u0)=4πk3 Im VC2(r1, r2)G(r1-r2)×exp[-iku0·(r1-r2)]d3r1d3r2,
····
σn(u0)=4πk2n-1 Im VVCn(r1, r2,rn)×G(r1-r2)G(r2-r3)G(rn-1-rn)×exp[-iku0·(r1-rn)]d3r1d3r2d3rn,
C1(r1)=η(r1)
Cn(r1, r2,rn)=η(r1)η(r2)η(rn)
ψ(i)(r)=|u|21a(u)exp(iku·r)d2u,
F(u)=|u|1a(u)f(u, u)d2u.
ψ(s)(ru)=F(u) exp(ikr)r,
P(e)=4πkIm |u|1a*(u)F(u)d2u.
P(e)=4πkIm |u|1|1|ua*(u)a(u)f(u, u)d2ud2u.
P(e)=4πkIm |u|1|u|1 A(u, u)f(u, u)d2ud2u.
A(u, u)=a*(u)a(u)
A(u, u)=|a(u0)|2[δ(2)(u-u0)δ(2)(u-u0)],
P(e)=|a(u0)|2 4πkIm f(u0, u0).
P(e)¯=4πkIm|u|1|u|1 A(u, u)×f(u, u)¯d2ud2u.
Im η(r)=0
η(r1)η(r2)=Γη(r1)δ(3)(r1-r2).
P(e)=4πk3|a|2 ImVVη(r1)η(r2)G(r1, r2)×exp[-iku0·(r1-r2)]d3r1d3r2.
P(e)=4πk4|a|2VVη(r1)η(r2) sin k|r1-r2|k|r1-r2|×cos[ku0·(r1-r2)]d3r1d3r2.
P(e)=4πk4|a|2Γ˜η(0),
Γ˜η(K)=VΓη(r)exp(iK·r)d3r
a(u)=A1δ(2)(u-u1)+A2δ(2)(u-u2),
P(e)=P11(e)+P22(e)+P12(e)+P21(e),
Pij(e)=4πkIm VVAi*Ajδ(2)(u-ui)δ(2)(u-uj)×f(u, u)d2ud2u=4πkIm Ai*Ajf(ui, uj).
Im Ai*Ajf(ui, uj)
=k4 Im VVAi*Ajη(r1)η(r2)G(r1, r2)×exp[-ik(ui·r1-uj·r2)]d3r1d3r2,
Im[A1*A2f(u1, u2)+A1A2*f(u2, u1)]=2k4 Im VVη(r1)η(r2)G(r1, r2)×Re{A1*A2 exp[ik(u1·r1-u2·r2)]}d3r1d3r2=2k5VV|A1A2|η(r1)η(r2) sin k|r1-r2|k|r1-r2|×cos[k(u1·r1-u2·r2)-θ]d3r1d3r2,
θ=arg(A1*A2).
Im[A1*A2f(u1, u2)+A1*A2f(u2, u1)]=2|A1A2|k5VΓη(r1)cos[k(u1-u2)·r1-θ]d3r1.
P12(e)+P21(e)=8πk4|A1A2|VΓη(r1)×cos[k(u1-u2)·r1-θ]d3r1,
P12(e)+P21(e)=4πk4|A1A2|{Γ˜η[k(u1-u2)]exp(-iθ)+Γ˜η[k(u2-u1)]exp(iθ)}.
P12(e)+P21(e)=Pint(θ).
Pint(θ1)=4πk4|A1A2|{Γ˜η[k(u1-u2)]exp(-iθ1)+Γ˜η[k(u2-u1)]exp(iθ1)},
Pint(θ2)=4πk4|A1A2|{Γ˜η[k(u1-u2)]exp(-iθ2)+Γ˜η[k(u2-u1)]exp(iθ2)}.
Γ˜η[k(u1-u2)]=-i8πk4|A1A2|sin(θ2-θ1)×[Pint(θ1)exp(iθ2)-Pint(θ2)exp(iθ1)].
Γ˜η[k(u1-u2)]=18πk4|A1A2|[Pint(0)+iPint(π/2)].
[Γη(r)]LP=1(2π)3|K|<2kΓ˜η(K)exp(iK·r)d3K,

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