Abstract

Propagation of a partially coherent optical beam inside a linear, nondispersive, dielectric medium is studied, taking into account the vector nature of the electromagnetic field. Propagation-induced polarization changes are studied by using the Gaussian–Schell model for the cross-spectral-density tensor. The degree of polarization changes with propagation and also becomes nonuniform across the beam cross section. The extent of these changes depends on the coherence radius associated with the cross-correlation function. For optical beams with symmetric spectra, the bandwidth of the source spectra is found to play a relatively minor role.

© 2000 Optical Society of America

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References

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.
  2. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, New York, 1998).
  3. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 10.
  4. E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1373 (1986).
    [CrossRef] [PubMed]
  5. E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
    [CrossRef]
  6. A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
    [CrossRef]
  7. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]
  8. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–213 (1994).
    [CrossRef]
  9. R. Martinez-Herrero, P. M. Mejias, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [CrossRef] [PubMed]
  10. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  11. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
    [CrossRef]
  12. G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
    [CrossRef]
  13. G. Gbur, D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
    [CrossRef]
  14. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).
  15. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
    [CrossRef]
  16. A. Gamliel, G. P. Agrawal, “Spectrum-enhanced spreading of partially coherent beams,” Opt. Commun. 78, 203–207 (1990).
    [CrossRef]

2000 (1)

G. Gbur, D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
[CrossRef]

1999 (2)

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

1997 (1)

1996 (1)

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

1994 (2)

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–213 (1994).
[CrossRef]

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

1990 (1)

A. Gamliel, G. P. Agrawal, “Spectrum-enhanced spreading of partially coherent beams,” Opt. Commun. 78, 203–207 (1990).
[CrossRef]

1986 (1)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1373 (1986).
[CrossRef] [PubMed]

1981 (1)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

1973 (1)

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Agrawal, G. P.

A. Gamliel, G. P. Agrawal, “Spectrum-enhanced spreading of partially coherent beams,” Opt. Commun. 78, 203–207 (1990).
[CrossRef]

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 10.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, New York, 1998).

Carter, W. H.

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Gamliel, A.

A. Gamliel, G. P. Agrawal, “Spectrum-enhanced spreading of partially coherent beams,” Opt. Commun. 78, 203–207 (1990).
[CrossRef]

Gbur, G.

G. Gbur, D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Jaiswal, A. K.

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

James, D. F. V.

G. Gbur, D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
[CrossRef]

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

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–213 (1994).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

Martinez-Herrero, R.

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef] [PubMed]

Mehta, C. L.

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Mejias, P. M.

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef] [PubMed]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Movilla, J. M.

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef] [PubMed]

Piquero, G.

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Wolf, E.

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

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1373 (1986).
[CrossRef] [PubMed]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 10.

J. Mod. Opt. (1)

G. Gbur, D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
[CrossRef]

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

Nuovo Cimento B (1)

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far-field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Opt. Acta (1)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Opt. Commun. (3)

A. Gamliel, G. P. Agrawal, “Spectrum-enhanced spreading of partially coherent beams,” Opt. Commun. 78, 203–207 (1990).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–213 (1994).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Degree of polarization of nonuniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1373 (1986).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Rep. Prog. Phys. (1)

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

Other (4)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, New York, 1998).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 10.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, J. Eur. Opt. Soc. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Degree of polarization P plotted as a function of the ratio σc/σI for three values of R=(x2+y2)1/2/σI at z=Ld, for a partially coherent Gaussian beam. R=0 (solid curve) corresponds to the beam center. Values of other parameters are |γ0| =0.9, I2/I1=0.8, and σa/σI=2.

Fig. 2
Fig. 2

Degree of polarization P plotted as a function of propagation distance for two values of R=0 and R=1 with the choice σc/σI=0.5. R=0 (dotted curve) corresponds to beam center. Values of other parameters are same as in Fig. 1.

Fig. 3
Fig. 3

Same as in Fig. 1 except that |γ0| =1, I2/I1=1. The Gaussian input beam is completely polarized initially under these conditions. Not only does it become partially polarized when σc<σ0, but the degree of polarization also becomes nonuniform across the beam.

Fig. 4
Fig. 4

Effect of the spectral width of the source on beam polarization. Solid curves show the degree of polarization P plotted as a function of z/Ld for two values of R=0 and R=1, with the choice σc/σI=0.5, σa/σI=2, |γ0| =0.9, I2/I1=1, and δ=0.2. The case δ=0 (dotted curve) corresponding to a vanishingly narrow spectrum is included for comparison.

Equations (25)

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Γij(r1, r2, τ)=Ei*(r1, t)Ej*(r2, t+τ),
Jij(r)=Γij(r, r, 0),(i, j=x, y).
P(r)=1-4 detJ(trJ)21/2,
Wij(r1, r2, ω)=-Γij(r1, r2, τ)exp(iωτ)dτ.
Wij(r1, r2, ω)=K*(r1, ρ1, ω)K(r2, ρ2, ω)×Wijs(ρ1, ρ2, ω)dρ1dρ2,
K(r, ρ, ω)=k24π2-exp{ik[p(x-ξ)+q(y-η)+mz]}dpdq,
m=(1-p2-q2)1/2ifp2+q21i(p2+q2-1)1/2otherwise.
k(r, ρ, ω)k exp(ikz)2πizexpik2z [(x-ξ)2+(y-η)2].
 
Wxxs(ρ1, ρ2, ω)=S(ω)[Ix(ρ1)Ix(ρ2)]1/2γa(ρ1-ρ2),
Wyys(ρ1, ρ2, ω)=S(ω)[Iy(ρ1)Iy(ρ2)]1/2γa(ρ1-ρ2),
Wxys(ρ1, ρ2, ω)=S(ω)[Ix(ρ1)Iy(ρ2)]1/2γc(ρ1-ρ2).
Ix(ρ)=I1exp-|ρ|22σI2,Iy(ρ)=I2exp-|ρ|22σI2.
γa(ρ1-ρ2)=exp-|ρ1-ρ2|22σa2,
γc(ρ1-ρ2)=γ0exp-|ρ1-ρ2|22σc2.
P0=1-(1-|γ0|2) 4I1I2(I1+I2)21/2.
Jxx(r)=-I11+da2exp-x2+y22σI2(1+da2)S(ω)dω,
Jxy(r)=-γ0(I1I2)1/21+dc2exp-x2+y22σI2(1+dc2)S(ω)dω,
dμ(ω)=cz2nωσI21+4σI2σμ21/2(μ=a, c).
P(x, y)=|γ0|1+da21+dc2exp(x2+y2)(dc2-da2)2σI2(1+da2)(1+dc2),
S(ω)=12πΔω0exp-(ω-ω0)22Δω02.
Jxx(0, 0, z)I1da2(ω0)12π-(1+δf )2exp(-f2/2)df,
Jxx(0, 0, z)(1+δ2)I1Ld2z21+4σI2σa2.
P(0, 0, z)
1-4I1I2(I1+I2)1-|γ0|2(1+4σI2/σa2)2(1+4σI2/σc2)21/2.

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