Abstract

In a recent paper [Phys. Rev. Lett. 88, 013901 (2002)] it was shown that when a convergent spatially coherent polychromatic wave is diffracted at an aperture, remarkable spectral changes take place on axis in the neighborhood of certain points near the geometrical focus. In particular, it was shown that the spectrum is redshifted at some points, blueshifted at others, and split into two lines elsewhere. In the present paper we extend the analysis and show that similar changes take place in the focal plane, in the neighborhood of the dark rings of the Airy pattern.

© 2002 Optical Society of America

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References

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  1. A pioneering paper on this subject is due to J. F. Nye, M. V. Berry, “Dislocation in wavetrains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. For a review of singular optics see, for example, M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.
  3. M. V. Berry, A. N. Wilson, “Black-and-white fringes and the colors of caustics,” Appl. Opt. 33, 4714–4718 (1994).
    [CrossRef] [PubMed]
  4. M. V. Berry, S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. U.S.A. 93, 2614–2619 (1996).
    [CrossRef] [PubMed]
  5. G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef] [PubMed]
  6. G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
  8. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  9. V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
    [CrossRef]
  10. The term “intensity” is used in optics with several different meanings. In this paper we mean by it the frequency-integrated spectrum, as indicated in Eq. (27).

2002 (2)

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

2001 (1)

V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
[CrossRef]

1996 (1)

M. V. Berry, S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. U.S.A. 93, 2614–2619 (1996).
[CrossRef] [PubMed]

1994 (1)

1974 (1)

A pioneering paper on this subject is due to J. F. Nye, M. V. Berry, “Dislocation in wavetrains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Berry, M. V.

M. V. Berry, S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. U.S.A. 93, 2614–2619 (1996).
[CrossRef] [PubMed]

M. V. Berry, A. N. Wilson, “Black-and-white fringes and the colors of caustics,” Appl. Opt. 33, 4714–4718 (1994).
[CrossRef] [PubMed]

A pioneering paper on this subject is due to J. F. Nye, M. V. Berry, “Dislocation in wavetrains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

Dogariu, A.

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Gbur, G.

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Klein, S.

M. V. Berry, S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. U.S.A. 93, 2614–2619 (1996).
[CrossRef] [PubMed]

Mandel, L.

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

Nye, J. F.

A pioneering paper on this subject is due to J. F. Nye, M. V. Berry, “Dislocation in wavetrains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Pas’ko, V. A.

V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
[CrossRef]

Popescu, G.

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Soskin, M. S.

V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
[CrossRef]

For a review of singular optics see, for example, M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.

Vasnetsov, M. V.

V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
[CrossRef]

For a review of singular optics see, for example, M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.

Visser, T. D.

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Wilson, A. N.

Wolf, E.

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

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

Appl. Opt. (1)

Opt. Commun. (1)

V. A. Pas’ko, M. S. Soskin, M. V. Vasnetsov, “Transversal optical vortex,” Opt. Commun. 198, 49–56 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. V. Berry, S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. U.S.A. 93, 2614–2619 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

A pioneering paper on this subject is due to J. F. Nye, M. V. Berry, “Dislocation in wavetrains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (4)

For a review of singular optics see, for example, M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.

The term “intensity” is used in optics with several different meanings. In this paper we mean by it the frequency-integrated spectrum, as indicated in Eq. (27).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

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

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Figures (9)

Fig. 1
Fig. 1

Notation relating to the focusing geometry. Here R=|r-r|.

Fig. 2
Fig. 2

Plot of the spectrum of the incident field for σ0/ω0=0.01.

Fig. 3
Fig. 3

Color-coded plot of the mean frequency ω¯ of the spectrum in the focal region as a function of u0, v0, for ω0=1015 s-1, σ0=1013 s-1, and N0=100. The color is more red or blue as the spectrum is more redshifted or blueshifted, respectively.

Fig. 4
Fig. 4

Depiction of the spectral changes on and about the axial zero u0=4π, for ω0=1015 s-1, σ0=1013 s-1, and N0=100, with δ=0.15. The peak values of each of the spectra are normalized to unity.

Fig. 5
Fig. 5

The standard deviation Δω of the spectrum about the frequency ω0, as a function of position u0, v0, with ω0=1015 s-1, σ0=1013 s-1, and N0=100.

Fig. 6
Fig. 6

The spectrum at various points around the first zero in the geometrical focal plane at u0=0, v0=3.83. (a) The geometry, (b)–(f) the changes in the spectrum at various positions around the circle in the u0, v0 plane of radius δ=0.04.

Fig. 7
Fig. 7

Color-coded plot of the mean frequency ω¯ of the spectrum in the geometrical focal plane, u0=0, for ω0=1015 s-1, σ0=1013 s-1, and N0=100.

Fig. 8
Fig. 8

Total intensity I(u0, 0) on axis in the focal region, with ω0=1015 s-1, σ0=1013 s-1, and N0=100. The inset shows an expanded view about the first two singular points. The dashed lines indicate the values predicted by Eq. (30).

Fig. 9
Fig. 9

The total intensity I(0, v0) in the focal plane, with ω0=1015 s-1, σ0=1013 s-1, and N0=100. The inset shows an expanded view about the first two singular points. The dashed lines indicate the values predicted by Eq. (32).

Equations (36)

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faλ,
U(r, ω)=-iλWU(0)(r, ω)exp(ik|r-r|)|r-r|d2r,
U(0)(r, ω)=A(ω)fexp(-ikf)
N=a2λf
U(r, ω)=-2πiN0ωω0A(ω)fexp(if2ωu0/a2ω0)×01J0(ωv0ξ/ω0)exp(-iωu0ξ2/2ω0)ξdξ,
N0=a2λ0f
u0=2πN0zf,
v0=2πN0ρa.
W(0)(r, r, ω)U(0)*(r, ω)U(0)(r, ω),
W(r1, r2, ω)=1λ2WW(0)(r1, r2, ω)×exp(-ik|r1-r1|)|r1-r1|exp(ik|r2-r2|)|r2-r2|×d2r1d2r2.
S(r, ω)W(r, r, ω)=1λ2WW(0)(r1, r2, ω)×exp(-ik|r-r1|)|r-r1|exp(ik|r-r2|)|r-r2|d2r1d2r2.
W(0)(r1, r2, ω)=|A(ω)|2f2.
S(r, ω)=M(r, ω)S(0)(ω),
S(0)(ω)|A(ω)|2f2
M(r, ω)(2πN0)2ωω0201J0(ωv0ξ/ω0)×exp[-iωu0ξ2/2ω0]ξdξ2.
S(0)(ω)=s0 exp[-(ω-ω0)2/2σ02].
ω¯(r)=ωS(r, ω)dωS(r, ω)dω.
M(u0, 0, ω)=(πN0)2ωω02sin(ωu0/4ω0)ωu0/4ω02.
Δω(r)=[ω-ω¯(r)]2S(r, ω)dωS(r, ω)dω1/2.
M(u0, 0, ω)M(u0, 0, ω0).
Δω(u0, 0)=σ0(nonsingularposition).
M(4πn, 0, ω)(πN0)222ω2sin(ωnπ/ω0)nπω=ω02(ω-ω0)2=πN0ω02(ω-ω0)2.
Δω(u0, 0)=3σ0(singularposition).
M(0, v0, ω)=(2πN0)2ωω02J1(ωv0/ω0)(ωv0/ω0)2,
u02+(v0-3.83)2=δ2.
S(4πn, 0, ω)πN0ω02(ω-ω0)2s0×exp[-(ω-ω0)2/2σ02].
I(u0, v0)0S(u0, v0, ω)dω.
I(4πn, 0)=12(2σ02)3/2πs0πN0ω02.
I(0, 0)=2πσ02(πN0)2s0,
I(4πn, 0)I(0, 0)=σ0ω02.
M(0, vn, ω)(2πN0)2ω02(J2[vn])2(ω-ω0)2,
I(0, vn)I(0, 0)=4σ02ω02(J2[vn])2.
Δuu4π-u±±4πσ0ω0.
Δzλ0=z4π-z±λ0±2fa2σ0ω0.
Δvv1-v±±3.83σ0ω0.
Δρλ0=±3.832πfaσ0ω0.

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