Abstract

Starting from the propagation law of partially coherent polychromatic light in the space-frequency domain, detailed numerical results and physical analysis are given to elucidate spectral changes and spectral switches at the geometrical focal plane of Gaussian Schell-model beams focused by an aperture lens. It is found that, in contrast to the aperture-induced spectral anomalies of spatially fully coherent polychromatic light, for partially coherent polychromatic light aperture diffraction plays an important role in spectral switching, but the truncation parameter, spectral correlation, and bandwidth all affect its spectral behavior.

© 2004 Optical Society of America

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References

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  1. M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.
  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]
  3. 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]
  4. S. A. Ponomarenko, E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. L. Pan, B. Lü, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
    [CrossRef]
  9. 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]
  10. H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A, Pure Appl. Opt. 3, 296–299 (2001).
    [CrossRef]
  11. H. C. Kandpal, S. Anand, J. 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]
  12. G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  13. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  14. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  15. J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
    [CrossRef]
  16. C. Palma, G. Cincotti, “Spectral shifts of a partially coherent field after passing through a lens,” Opt. Lett. 22, 671–672 (1997).
    [CrossRef] [PubMed]
  17. S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
    [CrossRef]

2002 (7)

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]

H. C. Kandpal, S. Anand, J. 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. 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]

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

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

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 (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]

S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
[CrossRef]

1997 (1)

1990 (1)

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[CrossRef]

1987 (1)

Anand, S.

H. C. Kandpal, S. Anand, J. 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]

Cincotti, G.

Dogariu, A.

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

Feng, S.

Foley, J. T.

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]

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[CrossRef]

Gbur, G.

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]

Hanson, S. G.

Kandpal, H. C.

H. C. Kandpal, S. Anand, J. 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]

Lü, B.

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

Mandel, L.

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

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]

Palma, C.

Pan, L.

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

Ponomarenko, S. A.

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.

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.

Vaishya, J. S.

H. C. Kandpal, S. Anand, J. 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]

Vasnetsov, M. V.

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]

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]

Winful, H. G.

Wolf, E.

Yura, H. T.

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. (3)

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]

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

H. C. Kandpal, S. Anand, J. 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 (5)

Opt. Commun. (2)

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[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]

Opt. Lett. (2)

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]

Other (2)

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

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, pp. 219–276.

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Figures (9)

Fig. 1
Fig. 1

Normalized intensity distributions at the geometrical focal plane of a diffracted GSM beam. The calculation parameters are given in the text. (a) β=1, (b) β=0.9, (c) β=0.5.

Fig. 2
Fig. 2

Normalized spectrum S(ω). (a) u=2.495, (b) u=2.500, (c) u=2.507, (d) u=2.515, (e) u=2.520. Dotted curves, original spectrum S(0)(ω)/S0; solid curves, spectrum S(ω).

Fig. 3
Fig. 3

Relative spectral shift δω/ω0 versus relative transversal position u. (a) β=1, (b) β=0.9.

Fig. 4
Fig. 4

Critical position uc of the first-, second-, and third-order spectral switches versus truncation parameter δ. (a) β=1, (b) β=0.9.

Fig. 5
Fig. 5

Normalized spectrum S(ω) of the second-order spectral switch. (a) β=1, (b) β=0.9, (c) β=0.85. Dotted curves, original spectrum S(0)(ω)/S0; solid curves, spectrum S(ω).

Fig. 6
Fig. 6

Critical position uc of the (a) first- and (b) second-order spectral switches versus spatial correlation β.

Fig. 7
Fig. 7

Normalized spectrum S(ω) of the second-order spectral switch. (a) σ0=0.10×1015 s-1, (b) σ0=0.08×1015s-1, (c) σ0=0.06×1015s-1, (d) σ0=0.04×1015s-1. Dotted curves, spectrum for β=1; solid curves, spectrum for β=0.9.

Fig. 8
Fig. 8

Critical position uc of the (a) first- and (b) second-order spectral switches versus spectral bandwidth σ0.

Fig. 9
Fig. 9

Relative spectral shift δω/ω0 versus relative transversal position, u. (a) N=0.2, (b) N=2, (c) N=10.

Tables (3)

Tables Icon

Table 1 Spectral Parameters Relating to Fig. 2

Tables Icon

Table 2 Spectral Parameters Relating to Fig. 5

Tables Icon

Table 3 Spectral Parameters Relating to Fig. 7

Equations (23)

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W(0)(x1, x2, z=0, ω)=S(0)(ω)exp-x12+x22w02×exp-(x1-x2)22σμ2
S(x, z, ω)=W(x, x, z, ω)=k2πB-aa-aaW(0)(x1, x2, z=0, ω)×exp-ik2B[A(x12-x22)-2x(x1-x2)]dx1dx2,
ABCD=1z0110-1/f1=1-zfz-1f1.
S(u, Δz, ω)=S(0)(ω)M(u, Δz, ω),
M(u, Δz, ω)=N1+Δz ωω0 12i πQ2×-δδ expQ1-(-1+β2)24Q2β4u2+iNπuu(1-β2+2Q2β2)(1+Δz)Q2β2 ωω0+N2π2u2Q2(1+Δz)2 ωω02H(u, u, ω)du,
H(u, u, ω)=erfi 12Q2β2(-u+uβ2+2Q2β2δ)-NπuQ2(1+Δz) ωω0-erfi 12Q2β2(-u+uβ2-2Q2β2δ)-NπuQ2(1+Δz) ωω0,
N=w02/λ0f(Fresnelnumber),
δ=a/w0(truncationparameter),
β=[1+(σμ/w0)-2]-1/2(spatialcorrelationparameter),
u=x/w0(relativetransversalcoordinate attheplanez=0),
u=x/w0(relativetransversalcoordinate atthezplane),
Δz=(z-f)/f(relativepropagationdistance),
Q1=-12-12β-2+iπNΔz1+Δz ωω0,
Q2=-12-12β-2-iπNΔz1+Δz ωω0,
S(0)(ω)=S0 exp-(ω-ω0)22σ02,
S(u, Δz, ω)=S0 exp-(ω-ω0)22σ02M(u, Δz, ω).
M(u, Δz, ω)=N1+Δz ωω0 π4Q˜1Q˜2×exp-2N2π2u2Q˜1Q˜2(1+Δz) ωω02×erf-Q˜1i(1+Δz)δ-Nπuω/ω0Q˜1(1+Δz)-erfQ˜1i(1+Δz)δ-Nπuω/ω0Q˜1(1+Δz)×erfQ˜2i(1+Δz)δ-Nπuω/ω0Q˜2(1+Δz)-erf-Q˜2i(1+Δz)δ-Nπuω/ω0Q˜2(1+Δz),
Q˜1=-1+iπNΔz1+Δz ωω0,
Q˜2=-1-iπNΔz1+Δz ωω0.
M(u, Δz=0, ω)=N ωω0 π2(1+β-2)×-δδ exp-2(u2-2iNπuuβ2ω/ω0+N2π2u2β2ω2/ω02)1+β2H(u, u, ω)du,
H(u, u, ω)=erfu(-1+β2)+δ(1+β2)+2iNπuβ2ω/ω0)2(1+β2)β-erfu(-1+β2)-δ(1+β2)+2iNπuβ2ω/ω0)2(1+β2)β.
M(u, Δz=0, ω)=N ωω0 π4 exp-2N2π2u2ωω02×erfδ+iNπu ωω0-erf-δ+iNπu ωω02.
I(u, Δz)=0S(u, Δz, ω)dω.

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