Abstract

It is shown that the recently discovered phenomenon of so-called spectral switches has a natural interpretation in the framework of singular optics with polychromatic light and that it should be regarded as being primarily a manifestation of diffraction-induced spectral changes rather than correlation-induced spectral changes as was suggested in the original papers [the first one appearing in Opt. Commun. 162, 57 (1999)] reporting this effect.

© 2002 Optical Society of America

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References

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  1. J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
    [Crossref]
  2. J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
    [Crossref]
  3. J. Pu, S. Nemoto, “Spectral changes and 1×N spectral switches in the diffraction of partially coherent light by an aperture,” J. Opt. Soc. Am. A 19, 339–344 (2002).
    [Crossref]
  4. H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A Pure Appl. Opt. 3, 296–299 (2001).
    [Crossref]
  5. H. C. Kandpal, A. S. Vaishya, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
    [Crossref]
  6. For a review of singular optics see M. S. Soskin, M. V. Vasnetov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.
  7. A pioneering paper in this field is due to J. F. Nye, M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [Crossref]
  8. 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]
  9. G. Gbur, T. D. Visser, E. Wolf, “Singular behavior of the spectrum in the neighborhood of focus,” J. Opt. Soc. Am. A 19, 1694–1700 (2002).
    [Crossref]
  10. G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [Crossref] [PubMed]
  11. In the present section we take the spectrum of the incident light to have a Lorenzian profile in order to compare our results more easily with those of Ref. 1. Later we will take the profile to be Gaussian in order to compare our results with those of Ref. 8.
  12. A. Gray, G. B. Mathews, T. M. MacRobert, A Treatise on Bessel Functions, 2nd ed. (MacMillan, London, 1922), p. 203, Eq. (49).
  13. J. Walker, The Analytical Theory of Light (Cambridge U. Press, Cambridge, UK, 1904), p. 147.
  14. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 8.8.1, Eqs. (11) and (21b). See also Ref. 12, p. 182, Eq. (10).
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions, 7th ed. (Cambridge U. Press, Cambridge, UK, 1944), Sec. 16.5.

2002 (5)

H. C. Kandpal, A. S. Vaishya, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
[Crossref]

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]

J. Pu, S. Nemoto, “Spectral changes and 1×N spectral switches in the diffraction of partially coherent light by an aperture,” J. Opt. Soc. Am. A 19, 339–344 (2002).
[Crossref]

G. Gbur, T. D. Visser, E. Wolf, “Singular behavior of the spectrum in the neighborhood of focus,” J. Opt. Soc. Am. A 19, 1694–1700 (2002).
[Crossref]

2001 (1)

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A Pure Appl. Opt. 3, 296–299 (2001).
[Crossref]

2000 (1)

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[Crossref]

1999 (1)

J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
[Crossref]

1974 (1)

A pioneering paper in this field is due to J. F. Nye, M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Berry, M. V.

A pioneering paper in this field is due to J. F. Nye, M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 8.8.1, Eqs. (11) and (21b). See also Ref. 12, p. 182, Eq. (10).

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]

G. Gbur, T. D. Visser, E. Wolf, “Singular behavior of the spectrum in the neighborhood of focus,” J. Opt. Soc. Am. A 19, 1694–1700 (2002).
[Crossref]

Gray, A.

A. Gray, G. B. Mathews, T. M. MacRobert, A Treatise on Bessel Functions, 2nd ed. (MacMillan, London, 1922), p. 203, Eq. (49).

Kandpal, H. C.

H. C. Kandpal, A. S. Vaishya, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
[Crossref]

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A Pure Appl. Opt. 3, 296–299 (2001).
[Crossref]

MacRobert, T. M.

A. Gray, G. B. Mathews, T. M. MacRobert, A Treatise on Bessel Functions, 2nd ed. (MacMillan, London, 1922), p. 203, Eq. (49).

Mathews, G. B.

A. Gray, G. B. Mathews, T. M. MacRobert, A Treatise on Bessel Functions, 2nd ed. (MacMillan, London, 1922), p. 203, Eq. (49).

Nemoto, S.

J. Pu, S. Nemoto, “Spectral changes and 1×N spectral switches in the diffraction of partially coherent light by an aperture,” J. Opt. Soc. Am. A 19, 339–344 (2002).
[Crossref]

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[Crossref]

J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
[Crossref]

Nye, J. F.

A pioneering paper in this field is due to J. F. Nye, M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Popescu, G.

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

Pu, J.

J. Pu, S. Nemoto, “Spectral changes and 1×N spectral switches in the diffraction of partially coherent light by an aperture,” J. Opt. Soc. Am. A 19, 339–344 (2002).
[Crossref]

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[Crossref]

J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
[Crossref]

Soskin, M. S.

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

Vaishya, A. S.

H. C. Kandpal, A. S. Vaishya, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
[Crossref]

Vasnetov, M. V.

For a review of singular optics see M. S. Soskin, M. V. Vasnetov, “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]

G. Gbur, T. D. Visser, E. Wolf, “Singular behavior of the spectrum in the neighborhood of focus,” J. Opt. Soc. Am. A 19, 1694–1700 (2002).
[Crossref]

Walker, J.

J. Walker, The Analytical Theory of Light (Cambridge U. Press, Cambridge, UK, 1904), p. 147.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 7th ed. (Cambridge U. Press, Cambridge, UK, 1944), Sec. 16.5.

Wolf, E.

G. Gbur, T. D. Visser, E. Wolf, “Singular behavior of the spectrum in the neighborhood of focus,” J. Opt. Soc. Am. A 19, 1694–1700 (2002).
[Crossref]

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 ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 8.8.1, Eqs. (11) and (21b). See also Ref. 12, p. 182, Eq. (10).

Zhang, H.

J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
[Crossref]

IEEE J. Quantum Electron. (2)

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[Crossref]

H. C. Kandpal, A. S. Vaishya, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
[Crossref]

J. Opt. A Pure Appl. Opt. (1)

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A Pure Appl. Opt. 3, 296–299 (2001).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Pu, H. Zhang, S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57–63 (1999).
[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. R. Soc. London, Ser. A (1)

A pioneering paper in this field is due to J. F. Nye, M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Other (6)

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

In the present section we take the spectrum of the incident light to have a Lorenzian profile in order to compare our results more easily with those of Ref. 1. Later we will take the profile to be Gaussian in order to compare our results with those of Ref. 8.

A. Gray, G. B. Mathews, T. M. MacRobert, A Treatise on Bessel Functions, 2nd ed. (MacMillan, London, 1922), p. 203, Eq. (49).

J. Walker, The Analytical Theory of Light (Cambridge U. Press, Cambridge, UK, 1904), p. 147.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 8.8.1, Eqs. (11) and (21b). See also Ref. 12, p. 182, Eq. (10).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 7th ed. (Cambridge U. Press, Cambridge, UK, 1944), Sec. 16.5.

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Figures (11)

Fig. 1
Fig. 1

On-axis spectrum showing the evolution of the spectrum, as a function of axial distance z, from a gradual spectral shift to a rapid spectral switch. The small solid circles in the plots indicate the positions of the maximum peak of the spectrum. The dotted curves are the original spectra. (Reproduced with permission from Ref. 1).

Fig. 2
Fig. 2

Basic geometry. The point PA is on the z axis a distance z from the origin. The off-axis point P is, in cylindrical polar coordinates, P=(ρ, ϕ, z).

Fig. 3
Fig. 3

Plots of the modifier function as a function of frequency for the cases (a) N0(z)=5.0 and (b) N0(z)=5.5.

Fig. 4
Fig. 4

Plots of the modifier function as a function of frequency for the cases (a) N0(z)=6.0 and (b) N0(z)=6.5.

Fig. 5
Fig. 5

Plots of M(z, ω) (thin curves), Si(ω)/S0 (medium-thickness curves), and S(z, ω)/S0 (thick curves) as a function of frequency. (a) N0(z)=7.0 and Γ/ω0=0.10; (b) N0(z)=7.5 and Γ/ω0=0.10.

Fig. 6
Fig. 6

Plots of M(z, ω) (thin curves), Si(ω)/S0 (medium-thickness curves), and S(z, ω)/S0 (thick curves) as a function of frequency. (a) N0(z)=8.0 and Γ/ω0=0.10; (b) N0(z)=8.5 and Γ/ω0=0.10.

Fig. 7
Fig. 7

Plot of the normalized frequency shift Ω as a function of the Fresnel number at line center N0(z) for the parameter values Γ=0.6×1015 rad/s and ω0=3.2×1015 rad/s.

Fig. 8
Fig. 8

Plot of the normalized spectrum S(i)(ω)/S0 as a function of ω/ω0 for σ/ω0=0.01.

Fig. 9
Fig. 9

Normalized spectrum S(P, ω)/S0 at the first axial zero at mean frequency ω0, i.e., at P[u1(ω0), 0], with σ/ω0=0.01.

Fig. 10
Fig. 10

Schematic illustration of the changes of the spectra along the loop [u(ω0)-u1(ω0)]2+ν2(ω0)=R2, with R=0.15, for selected values of the polar angle θ defined by Eq. (3.3).

Fig. 11
Fig. 11

Normalized spectra at the angles θ=0, 45°, 90°, 135°, and 180°, with σ/ω0=0.01.

Equations (21)

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Si(ω)=S0Γ2(ω-ω0)2+Γ2,
S(z, ω)=M(z, ω)Si(ω),
M(z, ω)=4 sin2[N(z, ω)π/2],
N(z, ω)=a2/λz=a2ω/(2πcz).
M(z, ω)=4 sin2[N0(z)π(ω/ω0)/2],
N0(z)N(z, ω0)=a2ω0/(2πcz).
N0(z)=2m(m=1, 2, 3,),
S(z, ω0)=0.
z=k04πm a2 k0=ω0c
Ω=(ωp-ω0)/ω0.
S(P, ω)=M(P, ω)Si(ω),
M(P, ω)=k2πzAexp[ik(ρ-ρ)2/2z]d2ρ2
u(w)=2πN(z, ω),v(ω)=2πN(z, ω)ρ/a,
M(P, ω)=W02[u(ω), v(ω)]+W12[u(ω), v(ω)],
W0(u, v)=V0(u, v)-cos[(u2+v2)/2u],
W1(u, v)=V1(u, v)-sin[(u2+v2)/2u].
Vn(u, v)=s=0(-1)s(u/v)2s+nJ2s+n(v),
Si(ω)=S0exp[-(ω-ω0)2/2σ2],
u(ω)um(ω)=4πm(m=1, 2, 3,).
[u(ω0)-u1(ω0)]2+v2(ω)=R2
cos θ=[u(ω0)-u1(ω0)]/R,sin θ=v(ω0)/R.

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