Abstract

It is shown that when partially coherent polychromatic light is focused by a filter–lens system with chromatic aberration, a spectral shift exists in the focused field, and a spectral switch that is defined as a sharp transition of the spectral shift also takes place at some positions of the focused field. The influence of the chromatic aberration of the lens, the coherence of the partially coherent light in the filter (a circular aperture), the radius of the aperture, and the spectral width of the partially coherent light in the aperture on the spectral shift and the spectral switch are investigated in detail. The numerical results show that these parameters affect the spectral shift and the spectral switch significantly. Potential applications of the spectral shift and the spectral switch of the partially coherent light are discussed.

© 2004 Optical Society of America

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References

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  1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  2. E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
    [CrossRef] [PubMed]
  3. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
    [CrossRef]
  4. Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [CrossRef]
  5. J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” J. Opt. Soc. Am. A 8, 1099–1105 (1991).
    [CrossRef]
  6. C. Palma, G. Cincotti, “Spectral shifts of a partially coherent field after passing through a lens,” Opt. Lett. 22, 671–672 (1997).
    [CrossRef] [PubMed]
  7. C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
    [CrossRef]
  8. L. Wang, Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A Pure Appl. Opt. 3, 296–299 (2001).
    [CrossRef]
  13. H. C. Kandpal, S. Anand, J. S. Vaishya, “Experiment observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 38, 336–339 (2002).
    [CrossRef]
  14. S. Anand, B. K. Yadav, H. C. Kandpal, “Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A 19, 2223–2228 (2002).
    [CrossRef]
  15. L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
    [CrossRef]
  16. B. Lu, L. Pan, “Spectral switching of Gaussian-Schell model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340–344 (2002).
    [CrossRef]
  17. M. S. Soskin, M. V. Vasnetov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (Elesevier, Amsterdam, 2001), pp. 219–276.
  18. G. Gbur, T. D. Visser, E. Wolf, “Anomalous behavior of spectra near phase singularities of focused wave,” Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef]
  19. 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]
  20. 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]
  21. J. Pu, S. Nemoto, B. Lu, “The effect of spectral correlations on spectral switches in the diffraction of partially coherent light,” J. Opt. Soc. Am. A 20, 1933–1939 (2003).
    [CrossRef]
  22. J. Pu, S. Nemoto, “Spectral shifts of partially coherent beams focused by a lens with chromatic aberration,” Opt. Commun. 207, 1–5 (2002).
    [CrossRef]
  23. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
    [CrossRef]
  24. J. T. Foley, M. Wang, “A theoretical analysis of coherence-induced spectral shift experiments of Kandpal, Vaishya, and Joshi,” J. Res. Natl. Inst. Stand. Technol. 99, 267–280 (1994).
    [CrossRef]
  25. Z. L. Horvath, Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
    [CrossRef]
  26. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [CrossRef]

2003 (2)

L. Wang, Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003).
[CrossRef]

J. Pu, S. Nemoto, B. Lu, “The effect of spectral correlations on spectral switches in the diffraction of partially coherent light,” J. Opt. Soc. Am. A 20, 1933–1939 (2003).
[CrossRef]

2002 (8)

J. Pu, S. Nemoto, “Spectral shifts of partially coherent beams focused by a lens with chromatic aberration,” Opt. Commun. 207, 1–5 (2002).
[CrossRef]

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

S. Anand, B. K. Yadav, H. C. Kandpal, “Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A 19, 2223–2228 (2002).
[CrossRef]

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]

B. Lu, L. Pan, “Spectral switching of Gaussian-Schell model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340–344 (2002).
[CrossRef]

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

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]

2001 (2)

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

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (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]

1998 (1)

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

1997 (1)

1994 (2)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
[CrossRef]

J. T. Foley, M. Wang, “A theoretical analysis of coherence-induced spectral shift experiments of Kandpal, Vaishya, and Joshi,” J. Res. Natl. Inst. Stand. Technol. 99, 267–280 (1994).
[CrossRef]

1993 (1)

Z. L. Horvath, Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

1991 (1)

1989 (2)

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[CrossRef] [PubMed]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
[CrossRef]

1988 (2)

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Anand, S.

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

S. Anand, B. K. Yadav, H. C. Kandpal, “Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A 19, 2223–2228 (2002).
[CrossRef]

Bor, Z.

Z. L. Horvath, Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Cincotti, G.

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

C. Palma, G. Cincotti, “Spectral shifts of a partially coherent field after passing through a lens,” Opt. Lett. 22, 671–672 (1997).
[CrossRef] [PubMed]

Dacic, Z.

Foley, J. T.

Friberg, A. T.

Gbur, G.

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

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

Horvath, Z. L.

Z. L. Horvath, Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
[CrossRef]

Kandpal, H. C.

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

S. Anand, B. K. Yadav, H. C. Kandpal, “Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light,” J. Opt. Soc. Am. A 19, 2223–2228 (2002).
[CrossRef]

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

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
[CrossRef]

Lin, Q.

L. Wang, Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003).
[CrossRef]

Lu, B.

J. Pu, S. Nemoto, B. Lu, “The effect of spectral correlations on spectral switches in the diffraction of partially coherent light,” J. Opt. Soc. Am. A 20, 1933–1939 (2003).
[CrossRef]

B. Lu, L. Pan, “Spectral switching of Gaussian-Schell model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340–344 (2002).
[CrossRef]

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

Nemoto, S.

J. Pu, S. Nemoto, B. Lu, “The effect of spectral correlations on spectral switches in the diffraction of partially coherent light,” J. Opt. Soc. Am. A 20, 1933–1939 (2003).
[CrossRef]

J. Pu, S. Nemoto, “Spectral shifts of partially coherent beams focused by a lens with chromatic aberration,” Opt. Commun. 207, 1–5 (2002).
[CrossRef]

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]

Palma, C.

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

C. Palma, G. Cincotti, “Spectral shifts of a partially coherent field after passing through a lens,” Opt. Lett. 22, 671–672 (1997).
[CrossRef] [PubMed]

Pan, L.

B. Lu, L. Pan, “Spectral switching of Gaussian-Schell model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340–344 (2002).
[CrossRef]

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

Pu, J.

J. Pu, S. Nemoto, B. Lu, “The effect of spectral correlations on spectral switches in the diffraction of partially coherent light,” J. Opt. Soc. Am. A 20, 1933–1939 (2003).
[CrossRef]

J. Pu, S. Nemoto, “Spectral shifts of partially coherent beams focused by a lens with chromatic aberration,” Opt. Commun. 207, 1–5 (2002).
[CrossRef]

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.

M. S. Soskin, M. V. Vasnetov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (Elesevier, Amsterdam, 2001), pp. 219–276.

Turunen, J.

Vaishya, J. S.

H. C. Kandpal, S. Anand, J. S. Vaishya, “Experiment 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, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
[CrossRef]

Vasnetov, M. V.

M. S. Soskin, M. V. Vasnetov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (Elesevier, Amsterdam, 2001), pp. 219–276.

Visser, T. D.

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 wave,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

Wang, L.

L. Wang, Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003).
[CrossRef]

Wang, M.

J. T. Foley, M. Wang, “A theoretical analysis of coherence-induced spectral shift experiments of Kandpal, Vaishya, and Joshi,” J. Res. Natl. Inst. Stand. Technol. 99, 267–280 (1994).
[CrossRef]

Wolf, E.

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

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]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[CrossRef] [PubMed]

Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Yadav, B. K.

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. (6)

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

L. Wang, Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003).
[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]

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

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

B. Lu, L. Pan, “Spectral switching of Gaussian-Schell model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340–344 (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 (8)

J. Res. Natl. Inst. Stand. Technol. (1)

J. T. Foley, M. Wang, “A theoretical analysis of coherence-induced spectral shift experiments of Kandpal, Vaishya, and Joshi,” J. Res. Natl. Inst. Stand. Technol. 99, 267–280 (1994).
[CrossRef]

Opt. Commun. (4)

Z. L. Horvath, Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[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]

J. Pu, S. Nemoto, “Spectral shifts of partially coherent beams focused by a lens with chromatic aberration,” Opt. Commun. 207, 1–5 (2002).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–173 (1989).
[CrossRef]

Opt. Eng. (1)

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Correlation-induced spectral shifts in optical measurements,” Opt. Eng. 33, 1996–2012 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (3)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220–2223 (1989).
[CrossRef] [PubMed]

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

Other (1)

M. S. Soskin, M. V. Vasnetov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (Elesevier, Amsterdam, 2001), pp. 219–276.

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Figures (7)

Fig. 1
Fig. 1

Filter–lens system with chromatic aberration for focusing of partially coherent polychromatic light.

Fig. 2
Fig. 2

On-axis spectra in the focused field (at the position z2/f0=1.2) when the partially coherent light is focused by a lens with chromatic aberration. Dotted curves: (a) df/dω|0 =-2.5×10-18 m s (i.e., ζ=-0.126); (b) df/dω|0 =-5×10-18 m s (ζ=-0.252). The dashed curves represent the spectra in the case of no chromatic aberration (df/dω|0 =0), and the solid curves represent the incident spectrum of the light in the aperture, of Lorentzian type with central frequency ω0=7.57×1015 s-1, half-width at half-maximum Γ=0.6×1015 s-1, z1=2f0, a=0.5 mm, Δ0=1. The spectra are normalized to unity at maximum.

Fig. 3
Fig. 3

Normalized on-axis spectra showing the evolution from a gradual spectral shift to a rapid spectral switch. (a) z2/f0=1.21, (b) z2/f0=1.22, (c) z2/f0=1.23, (d) z2/f0=1.238, (e) z2/f0=1.24, (f) z2/f0=1.25, (g) z2/f0=1.26. The dotted curve is the spectrum in the aperture: ω0=7.57×1015 s-1, Γ=0.6×1015 s-1, df/dω|0 =-5×10-18 m s (ζ=-0.252); other parameters the same as in Fig. 2. Some special spectral parameters are listed in Table 1.

Fig. 4
Fig. 4

Spectral shifts from z2/f0=0.1 to z2/f0=1.5. Solid curve, df/dω|0 =-5×10-18 m s (i.e., ζ=-0.252); dotted curve, no chromatic aberration. The position at which the spectral shift experiences a sharp transition is a spectral switch. Δ0=1, other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Spectral shifts from z2/f0=0.1 to z2/f0=1.5. Δ0=2 (dotted curve), Δ0=1 (solid curve). df/dω|0 =-5×10-18 m s (i.e., ζ=-0.252). Other parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Variation of spectral shifts with the radius a of the aperture at three different chromatic aberrations. Solid curve, df/dω|0 =-5×10-18 m s (ζ=-0.252); dashed curve, df/dω|0 =-2.5×10-18 m s (ζ=-0.126); dotted curve, no chromatic aberration. Δ0=1 and the observation plane is located at z2/f0=1.2. Other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Variation of spectral shifts with the radius a of the aperture at three different spectral widths of the spectrum of the partially coherent light in the aperture. Solid curve, Γ=0.6×1015 s-1; dashed curve, Γ=0.4×1015 s-1; dotted curve, Γ=0.2×1015 s-1. The observation plane is located at z2/f0=1.2, and Δ0=1. The chromatic aberration df/dω|0 =-5×10-18 m s (ζ=-0.252), ω0=7.57×1015 s-1.

Tables (1)

Tables Icon

Table 1 Special Spectral Parameters Corresponding to Fig. 3

Equations (15)

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W(0)(ρ1, ρ2, z=0, ω)=S(0)(ω)exp-(ρ1-ρ2)22σ(ω)2,
f(λ)=f0+dfdλ0(λ-λ0)=f0-f0n-1dndλ0(λ-λ0),
f(ω)=f0+dfdω0(ω-ω0)=f01+ξωω0-1,
ξ=ω0f0dfdω0.
ABCD=1z201 10-1f(ω)1 1z101=1-z2f(ω)z1+z2-z1z2f(ω)-1f(ω)1-z1f(ω).
W(ρ1, ρ2, ω)=k2πB2W(0)(ρ1, ρ2, ω)×exp-ik2B [A(ρ12-ρ22)-2(ρ1ρ1-ρ2ρ2)+D(ρ12-ρ22)]dρ1dρ2,
S(0)(ω)=S0Γ2(ω-ω0)2+Γ2,
σ(ω)=σ0ω0ω,
Δ(ω)=σ(ω)a=Δ0ω0ω,
S(z2, ω)=S(0)(ω)ωω02z0f(ω)(z1+z2)f(ω)-z1z22×0101exp-r12+r222Δ(ω)2×exp-iπz0[f(ω)-z2](z1+z2)f(ω)-z1z2×ωω0 (r12-r22)I0r1r2Δ(ω)2r1r2dr1dr2.
r1=ρ1/a,r2=ρ2/a,
I0r1r2Δ(ω)2=12π02πexpr1r2Δ(ω)2cos(ϕ1-ϕ2)dϕ1
z0=a2/λ0,
δω=ωm-ω0,
δωω0=ωm-ω0ω0,

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