Abstract

Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical expressions of the spectrum of GSM beams propagating in dispersive media are derived, and the spectral properties are studied in detail. It is shown that, in comparison with propagation in free space and in turbulence, whether or not GSM beams satisfy the scaling law, the normalized spectrum of GSM beams in dispersive media changes on propagation in general, because the dispersive medium affects different spectral components differently. As compared with the free-space propagation, for the scaling-law GSM beams the dispersion results in spectrum change, and for the nonscaling-law GSM beams the dispersion gives rise to a further increase in spectral changes. The structure constant of the dispersive property of the media, the transverse coordinate of the observation point, the spatial correlation length of the source, and the propagation distance affect the spectral behavior of GSM beams; this effect is illustrated numerically.

© 2008 Optical Society of America

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References

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  1. E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
    [CrossRef] [PubMed]
  2. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
    [CrossRef]
  3. H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogenous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
    [CrossRef]
  4. C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378-383 (1998).
    [CrossRef]
  5. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
    [CrossRef]
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  7. Z. Dacic and 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]
  8. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205-1209 (1965).
    [CrossRef]

2006 (1)

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

2004 (1)

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogenous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[CrossRef]

1998 (1)

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

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

1995 (1)

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

1988 (1)

1986 (1)

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

1965 (1)

Cincotti, G.

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

Dacic, Z.

Guattari, G.

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

James, D. F. V.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Ji, X.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Lü, B.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Malitson, I. H.

Mandel, L.

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

Palma, C.

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

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogenous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[CrossRef]

Wolf, E.

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogenous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[CrossRef]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

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

Z. Dacic and 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 spectrum on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

Zhang, E.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogenous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

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

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Normalized spectra S ( ω ) of a scaling-law GSM beam propagating in dispersive media.

Fig. 2
Fig. 2

Relative spectral shift δ ω ω 0 versus transverse coordinate r in fused silica.

Fig. 3
Fig. 3

Normalized spectra S ( ω ) of a nonscaling-law GSM beam propagating in (a) free space and (b) fused silica.

Fig. 4
Fig. 4

Relative spectral shift δ ω ω 0 versus transverse coordinate r for different values of the spatial correlation length σ μ = 0.2 mm , 0.5 mm , and 2 mm in (a) free space and (b) fused silica at z = 1 m .

Fig. 5
Fig. 5

Variation of the normalized off-axis spectrum S ( ω ) with transverse coordinate r; the calculation parameters are seen in the text. (a) r = 4.6 mm , (b) r = 5 mm , (c) r = 5.036 mm , (d) r = 5.1 mm , (e) r = 5.5 mm . Dashed curve, original spectrum S ( 0 ) ( ω ) ; solid curve, off-axis spectrum S ( ω ) of the nonscaling-law GSM beam propagating in fused silica at transverse coordinate r.

Fig. 6
Fig. 6

Relative spectral shift δ ω ω 0 versus transverse coordinate r for different values of the spatial correlation length σ μ = 0.2 mm , 0.5 mm , and 2 mm in (a) free space and (b) fused silica at z = 5 m .

Fig. 7
Fig. 7

On-axis relative spectral shift δ ω ω 0 versus propagation distance z for different values of the spatial correlation length σ μ = 0.2 mm , 0.5 mm , and 2 mm in (a) free space and (b) fused silica.

Equations (15)

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W ( 0 ) ( r 1 , r 2 , 0 , ω ) = S ( 0 ) ( ω ) exp [ r 1 2 + r 2 2 w 0 2 ] exp [ ( r 1 r 2 ) 2 2 σ μ 2 ( ω ) ]
S ( r , z , ω ) = ( k ( ω ) 2 π z ) 2 W ( 0 ) ( r 1 , r 2 , 0 , ω ) exp { i k ( ω ) 2 z [ ( r 1 2 r 2 2 ) 2 r ( r 1 r 2 ) ] } d r 1 d r 2 ,
S ( r , z , ω ) = S ( 0 ) ( ω ) H ( z , ω ) exp [ 2 r 2 w 0 2 H ( z , ω ) ] ,
H ( z , ω ) = 1 + ( 2 c w 0 n ( ω ) ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ( ω ) ) z 2 .
S ( r , z , ω ) = S ( 0 ) ( ω ) H 0 ( z , ω ) exp [ 2 r 2 w 0 2 H 0 ( z , ω ) ] ,
H 0 ( z , ω ) = 1 + ( 2 c w 0 ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ( ω ) ) z 2 .
σ μ ( ω ) = w 0 γ w 0 4 ω 2 1 ,
H ( z , ω ) = 1 + 4 γ c 2 z 2 n 2 ( ω ) .
H ( z , ω ) = 1 + ( 2 c w 0 n ( ω ) ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 ,
H 0 ( z , ω ) = 1 + ( 2 c w 0 ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 .
S ( 0 ) ( ω ) = exp [ ( ω ω 0 ) 2 2 σ 0 2 ] .
S ( r , z , ω ) = 1 1 + 4 γ c 2 z 2 n 2 ( ω ) exp [ ( ω ω 0 ) 2 2 σ 0 2 ] exp [ 2 r 2 w 0 2 ( 1 + 4 γ c 2 z 2 n 2 ( ω ) ) ] .
S ( r , z , ω ) = 1 1 + ( 2 c w 0 n ( ω ) ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 exp [ ( ω ω 0 ) 2 2 σ 0 2 ] exp [ 2 r 2 w 0 2 [ 1 + ( 2 c w 0 n ( ω ) ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 ] ] .
S ( r , z , ω ) = 1 1 + ( 2 c w 0 ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 exp [ ( ω ω 0 ) 2 2 σ 0 2 ] exp [ 2 r 2 w 0 2 [ 1 + ( 2 c w 0 ω ) 2 ( 1 w 0 2 + 1 σ μ 2 ) z 2 ] ] .
n 2 ( λ ) = 1 + i = 1 3 B i 1 λ i 2 λ 2 ,

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