Abstract

It is well known that scattering of a polychromatic plane wave by a random medium, i.e., by a medium whose refractive index varies randomly with position, may produce frequency shifts of spectral lines. It has been a common perception that a random medium is required for generation of such spectral shifts by scattering. In this Letter we show that such a phenomenon occurs even when the refractive index of the medium is a deterministic function of position. We also show that this phenomenon may be used to obtain information about the structure of a deterministic medium.

© 2012 Optical Society of America

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References

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  1. W. Friedrich, P. Knipping, and M. von Laue, in Sitzungs-berichte d. Bayer. Akademie der Wissenschaften(München, 1912), pp. 303–322.
  2. W. H. Bragg, Nature 90, 219 (1912).
    [CrossRef]
  3. W. L. Bragg, Nature 90, 410 (1912).
    [CrossRef]
  4. W. L. Bragg, Proc. Camb. Phil. Soc. 17, 43 (1913).
  5. P. P. Ewald, Phys. Z. 14, 465 (1913).
  6. P. P. Ewald, Z. Kristallogr. 56, 129 (1921).
  7. The classic papers on this subject are reprinted in [19].
  8. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
    [CrossRef]
  9. E. Wolf, J. T. Foley, and F. Gori, J. Opt. Soc. Am. A 6, 1142 (1989).
    [CrossRef]
  10. A non-rigorous treatment of color change produced by such objects is given in [20].
  11. E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
    [CrossRef]
  12. It was shown in [21] that spectral changes may take place in the neighborhood of phase singularities near the focus of a converging, spatially fully coherent polychromatic wave diffracted at an aperture.
  13. E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771(1996).
    [CrossRef]
  14. A. Dogariu and E. Wolf, Opt. Lett. 23, 1340 (1998).
    [CrossRef]
  15. D. Zhao, O. Korotkova, and E. Wolf, Opt. Lett. 32, 3483 (2007).
    [CrossRef]
  16. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  17. M. Born and E. Wolf, Priciples of Optics, 7th ed.(Cambridge University, 1999).
  18. In practice the spectral density has to be determined experimentally.
  19. G. E. Bacon, X-Ray and Neutron Diffraction (Pergamon, 1966).
  20. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, 1954).
  21. G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef]

2009 (1)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

2007 (1)

2002 (1)

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

1998 (1)

1996 (1)

E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771(1996).
[CrossRef]

1989 (1)

1986 (1)

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[CrossRef]

1921 (1)

P. P. Ewald, Z. Kristallogr. 56, 129 (1921).

1913 (2)

W. L. Bragg, Proc. Camb. Phil. Soc. 17, 43 (1913).

P. P. Ewald, Phys. Z. 14, 465 (1913).

1912 (2)

W. H. Bragg, Nature 90, 219 (1912).
[CrossRef]

W. L. Bragg, Nature 90, 410 (1912).
[CrossRef]

Bacon, G. E.

G. E. Bacon, X-Ray and Neutron Diffraction (Pergamon, 1966).

Born, M.

M. Born and E. Wolf, Priciples of Optics, 7th ed.(Cambridge University, 1999).

Bragg, W. H.

W. H. Bragg, Nature 90, 219 (1912).
[CrossRef]

Bragg, W. L.

W. L. Bragg, Proc. Camb. Phil. Soc. 17, 43 (1913).

W. L. Bragg, Nature 90, 410 (1912).
[CrossRef]

Dogariu, A.

Ewald, P. P.

P. P. Ewald, Z. Kristallogr. 56, 129 (1921).

P. P. Ewald, Phys. Z. 14, 465 (1913).

Fischer, D. G.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

Foley, J. T.

Friedrich, W.

W. Friedrich, P. Knipping, and M. von Laue, in Sitzungs-berichte d. Bayer. Akademie der Wissenschaften(München, 1912), pp. 303–322.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

Gori, F.

James, D. F. V.

E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771(1996).
[CrossRef]

Knipping, P.

W. Friedrich, P. Knipping, and M. von Laue, in Sitzungs-berichte d. Bayer. Akademie der Wissenschaften(München, 1912), pp. 303–322.

Korotkova, O.

Lahiri, M.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, 1954).

Shirai, T.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

von Laue, M.

W. Friedrich, P. Knipping, and M. von Laue, in Sitzungs-berichte d. Bayer. Akademie der Wissenschaften(München, 1912), pp. 303–322.

Wolf, E.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

D. Zhao, O. Korotkova, and E. Wolf, Opt. Lett. 32, 3483 (2007).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

A. Dogariu and E. Wolf, Opt. Lett. 23, 1340 (1998).
[CrossRef]

E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771(1996).
[CrossRef]

E. Wolf, J. T. Foley, and F. Gori, J. Opt. Soc. Am. A 6, 1142 (1989).
[CrossRef]

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

M. Born and E. Wolf, Priciples of Optics, 7th ed.(Cambridge University, 1999).

Zhao, D.

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

Nature (2)

W. H. Bragg, Nature 90, 219 (1912).
[CrossRef]

W. L. Bragg, Nature 90, 410 (1912).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (3)

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

Phys. Z. (1)

P. P. Ewald, Phys. Z. 14, 465 (1913).

Proc. Camb. Phil. Soc. (1)

W. L. Bragg, Proc. Camb. Phil. Soc. 17, 43 (1913).

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771(1996).
[CrossRef]

Z. Kristallogr. (1)

P. P. Ewald, Z. Kristallogr. 56, 129 (1921).

Other (9)

The classic papers on this subject are reprinted in [19].

W. Friedrich, P. Knipping, and M. von Laue, in Sitzungs-berichte d. Bayer. Akademie der Wissenschaften(München, 1912), pp. 303–322.

A non-rigorous treatment of color change produced by such objects is given in [20].

It was shown in [21] that spectral changes may take place in the neighborhood of phase singularities near the focus of a converging, spatially fully coherent polychromatic wave diffracted at an aperture.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

M. Born and E. Wolf, Priciples of Optics, 7th ed.(Cambridge University, 1999).

In practice the spectral density has to be determined experimentally.

G. E. Bacon, X-Ray and Neutron Diffraction (Pergamon, 1966).

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, 1954).

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Figures (2)

Fig. 1.
Fig. 1.

Illustration of the notations.

Fig. 2.
Fig. 2.

The dashed curve indicates the frequency dependence of the spectral density of the scattered field in the far zone, at an angle of scattering θ=24.9°. The solid curve shows the spectrum of the incident field. The parameters, characterizing incident beam and scattering potential, were chosen as follows: ω0=3.427×1015s1, Γ0=0.01ω0, λ0=550nm, σ=3λ0, and A2B/r2=1.

Equations (15)

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W(r1,r2;ω)=U*(r1;ω)U(r2;ω).
S(r,ω)|U(r,ω)|2=W(r,r;ω).
{U(in)(r,s0;ω)}={a(ω)exp(iks0·r)},
W(in)(r1,r2,s0;ω)=S(in)(ω)exp[iks0·(r2r1)],
F(r;ω)=14π(ωc)2[ϵ(r;ω)1],
Us0()(rs;ω)=a(ω)F˜(K;ω)eikrr,
F˜(K;ω)=DF(r;ω)exp[iK·r]d3r
K=ωc(ss0)
Ss0()(rs;ω)=S(in)(ω)r2|F˜(K;ω)|2.
S(in)(ω)=Bexp[(ωω0)22Γ02],
F(r;ω)=(ωc)2A(2πσ2)3/2exp[r22σ2],
F˜(K;ω)=(ωc)2Aexp[|K|2σ2/2].
F˜(K;ω)F˜(θ,ω)=(ωc)2Aexp[2(ωc)2σ2sin2θ2],
Ss0()(rs;ω)Ss0()(r,θ;ω)=A2Br2(ωc)4×exp[(ωω0)22Γ024(ωc)2σ2sin2θ2].
σ=c2ω0sin(θ/2){ln[(ω0c)41Ss0()(r,θ;ω0)]}1/2.

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