Abstract

Light propagation induces remarkable changes in the spectrum of focused diffracted beams. We show that spectral changes take place in the vicinity of phase singularities in the focal region of spatially coherent, polychromatic spherical waves of different Fresnel numbers. Instead of the Debye formulation, we use the Kirchhoff integral to evaluate the focal field accurately. We find that as a result of a decrease in the Fresnel number, some cylindrical spectral switches are geometrically transformed into conical spectral switches.

© 2004 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. L. Pan, B. Lü, “Spectral changes and spectral switches of partially coherent beams focused by an aperture lens,” J. Opt. Soc. Am. A 21, 140–148 (2004).
    [CrossRef]
  3. G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901–013904 (2002).
    [CrossRef] [PubMed]
  4. 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]
  5. J. T. Foley, E. Wolf, “Phenomenon of spectral switches as a new effect in singular optics with polychromatic light,” J. Opt. Soc. Am. A 19, 2510–2516 (2002).
    [CrossRef]
  6. S. A. Ponomarenko, E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
    [CrossRef]
  7. T. D. Visser, E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371–373 (2003).
    [CrossRef]
  8. D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
    [CrossRef]
  9. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [CrossRef]
  10. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  11. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  13. M. Martı́nez-Corral, V. Climent, “Focal switch: a new effect in low-Fresnel-number systems,” Appl. Opt. 35, 24–27 (1996).
    [CrossRef] [PubMed]
  14. C. J. Zapata-Rodrı́guez, M. Martı́nez-Corral, P. Andrés, A. Pons, “Inverse focal shift: a new effect in truncated cylindrical waves,” J. Mod. Opt. 46, 129–144 (1999).
    [CrossRef]
  15. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  16. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  18. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  19. E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Commun. 5, 264–266 (1980).
  20. C. J. Zapata-Rodrı́guez, P. Andrés, M. Martı́nez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
    [CrossRef]

2004 (1)

2003 (2)

T. D. Visser, E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371–373 (2003).
[CrossRef]

D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
[CrossRef]

2002 (4)

2000 (1)

1999 (2)

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]

C. J. Zapata-Rodrı́guez, M. Martı́nez-Corral, P. Andrés, A. Pons, “Inverse focal shift: a new effect in truncated cylindrical waves,” J. Mod. Opt. 46, 129–144 (1999).
[CrossRef]

1996 (1)

1984 (1)

1981 (4)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

1980 (1)

E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Commun. 5, 264–266 (1980).

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Andrés, P.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Chon, J. W. M.

D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
[CrossRef]

Climent, V.

Collett, E.

E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Commun. 5, 264–266 (1980).

Erkkila, J. H.

Foley, J. T.

Ganic, D.

D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
[CrossRef]

Gbur, G.

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901–013904 (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]

Gu, M.

D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
[CrossRef]

Li, Y.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Lü, B.

Marti´nez-Corral, M.

Muñoz-Escrivá, L.

Nemoto, S.

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]

Pan, L.

Ponomarenko, S. A.

Pons, A.

C. J. Zapata-Rodrı́guez, M. Martı́nez-Corral, P. Andrés, A. Pons, “Inverse focal shift: a new effect in truncated cylindrical waves,” J. Mod. Opt. 46, 129–144 (1999).
[CrossRef]

Pu, J.

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]

Rogers, M. E.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Visser, T. D.

T. D. Visser, E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371–373 (2003).
[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]

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901–013904 (2002).
[CrossRef] [PubMed]

Wolf, E.

T. D. Visser, E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371–373 (2003).
[CrossRef]

S. A. Ponomarenko, E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
[CrossRef]

J. T. Foley, E. Wolf, “Phenomenon of spectral switches as a new effect in singular optics with polychromatic light,” J. Opt. Soc. Am. A 19, 2510–2516 (2002).
[CrossRef]

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901–013904 (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]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Commun. 5, 264–266 (1980).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Zapata-Rodri´guez, C. J.

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]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. Ganic, J. W. M. Chon, M. Gu, “Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves,” Appl. Phys. Lett. 82, 1527–1528 (2003).
[CrossRef]

J. Mod. Opt. (1)

C. J. Zapata-Rodrı́guez, M. Martı́nez-Corral, P. Andrés, A. Pons, “Inverse focal shift: a new effect in truncated cylindrical waves,” J. Mod. Opt. 46, 129–144 (1999).
[CrossRef]

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

T. D. Visser, E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371–373 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Opt. Commun. (5)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[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]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Commun. 5, 264–266 (1980).

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximation,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901–013904 (2002).
[CrossRef] [PubMed]

Other (2)

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the optical setup.

Fig. 2
Fig. 2

Gaussian power spectrum of the incident radiation. The central value is given by ω0=1015 s-1 and the spectral width is σ=1013 s-1.

Fig. 3
Fig. 3

Gray-coded plot of the relative spectral shift δω/ω0 in the focal region as a function of normalized spatial variables u(ω0) and v(ω0). The spectral Gaussian parameters are given in Fig. 2.

Fig. 4
Fig. 4

Normalized power spectrum at transverse points u0=0 and (a) v0=1.220π, (b) v0=1.220π-Δ, and (c) v0=1.220π+Δ, where Δ=0.012π.

Fig. 5
Fig. 5

Relative spectral shift (a) in the transverse focal plane and (b) along the optical axis.

Fig. 6
Fig. 6

Gray-coded plots of the relative spectral shift in the focal region for (a) a high Fresnel number N0=100, some intermediate Fresnel numbers: (b) N0=10, (c) N0=5, (d) N0=2, (e) N0=1, and (f) a nearly flat beam characterized by a Fresnel number N0=0.01.

Equations (14)

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U(u, v)=B(u)exp ϕ(u, v)01J0(vρ)exp-i u2 ρ2ρdρ,
(u, v)=2πNz/f1+z/f, r/a1+z/f
B(u)=-2πiλaf21-u2πNA(ω)
N=a2/λf
N=ωω0 N0,
(u, v)=ωω0 (u0, v0),
B(u; ω)=B(u0; ω0)ω0A(ω0) ωA(ω);
S(u0, v0, ω)=|U(u, v; ω)|2=S(i)(ω)M(u0, v0, ω),
S(i)(ω)=|A(ω)|2f2,
S(i)(ω)=S0exp-(ω-ω0)22σ2,
(u, v)=2πN(z/f, r/a).
δωω0=ω¯-ω0ω0,
ω¯(u0, v0)=ωS(u0, v0, ω)dωS(u0, v0, ω)dω
δωω0max-δωω0min=0.0197,

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