Abstract

A generalization of the Stokes parameters of a random electromagnetic beam is introduced. Unlike the usual Stokes parameters, which depend on one spatial variable, the generalized Stokes parameters, depend on two spatial variables. They obey precise laws of propagation, both in free space and in any linear medium, whether deterministic or random. With the help of the generalized Stokes parameters, the changes in the ordinary Stokes parameters upon propagation can be determined. Numerical examples of such changes are presented. The generalized Stokes parameters contain information not only about the polarization properties of the beam but also about its coherence properties. We illustrate this fact by expressing the degree of coherence of the electromagnetic beam in terms of one of the generalized Stokes parameters.

© 2005 Optical Society of America

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References

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  1. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).
  2. M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, England, 1999).
    [CrossRef]
  3. D. F. V. James, J. Opt. Soc. Am. A 11, 1641 (1994).
    [CrossRef]
  4. G. P. Agrawal and E. Wolf, J. Opt. Soc. Am. A 17, 2019 (2000).
    [CrossRef]
  5. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
    [CrossRef]
  6. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  7. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef] [PubMed]
  8. H. Roychowdhury and E. Wolf, Opt. Commun. 266, 57 (2003). The sentence after Eq. (12) in that reference is misleading and should be replaced by “The other off-diagonal element can be determined in an analogous manner.”
    [CrossRef]
  9. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. (to be published).
  10. O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
    [CrossRef]
  11. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, Waves Random Media 14, 513 (2004).
    [CrossRef]
  12. O. Korotkova and E. Wolf, “Changes in state of polarization of a random electromagnetic beam propagating in free space,” submitted to Opt. Commun.
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995).
    [CrossRef]
  14. The spectral Stokes parameters are a refinement of the usual Stokes parameters, which are independent of both frequency and time (see Ref. 2, Sec. 10.9.3). The usual ones are essentially the integrals of the spectral Stokes parameters that we use taken over the bandwidth of the light, which is usually assumed to be quasi-monochromatic.
  15. Two-point Stokes parameters in the space–time domain rather than in the space–frequency domain were recently introduced somewhat formally in J. Ellis and A. Dogariu, Opt. Lett. 29, 536 (2004). Although it was not shown in that Letter, these generalized Stokes parameters obey, in free space, two wave equations; however, for propagation in inhomogeneous media they are governed by rather complicated laws. On the other hand, the generalized spectral Stokes parameters introduced in this Letter propagate according to relatively simple laws.
    [CrossRef] [PubMed]

2004 (3)

2003 (3)

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 266, 57 (2003). The sentence after Eq. (12) in that reference is misleading and should be replaced by “The other off-diagonal element can be determined in an analogous manner.”
[CrossRef]

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

2000 (1)

1994 (1)

Agrawal, G. P.

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, England, 1999).
[CrossRef]

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

Dogariu, A.

Ellis, J.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

James, D. F. V.

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, Waves Random Media 14, 513 (2004).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in state of polarization of a random electromagnetic beam propagating in free space,” submitted to Opt. Commun.

Mandel, L.

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

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. (to be published).

Roychowdhury, H.

H. Roychowdhury and E. Wolf, Opt. Commun. 266, 57 (2003). The sentence after Eq. (12) in that reference is misleading and should be replaced by “The other off-diagonal element can be determined in an analogous manner.”
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. (to be published).

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, Waves Random Media 14, 513 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

Wolf, E.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, Waves Random Media 14, 513 (2004).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 266, 57 (2003). The sentence after Eq. (12) in that reference is misleading and should be replaced by “The other off-diagonal element can be determined in an analogous manner.”
[CrossRef]

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

G. P. Agrawal and E. Wolf, J. Opt. Soc. Am. A 17, 2019 (2000).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. (to be published).

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

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, England, 1999).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in state of polarization of a random electromagnetic beam propagating in free space,” submitted to Opt. Commun.

J. Opt. A (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A 3, 1 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

H. Roychowdhury and E. Wolf, Opt. Commun. 266, 57 (2003). The sentence after Eq. (12) in that reference is misleading and should be replaced by “The other off-diagonal element can be determined in an analogous manner.”
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, Waves Random Media 14, 513 (2004).
[CrossRef]

Other (6)

O. Korotkova and E. Wolf, “Changes in state of polarization of a random electromagnetic beam propagating in free space,” submitted to Opt. Commun.

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

The spectral Stokes parameters are a refinement of the usual Stokes parameters, which are independent of both frequency and time (see Ref. 2, Sec. 10.9.3). The usual ones are essentially the integrals of the spectral Stokes parameters that we use taken over the bandwidth of the light, which is usually assumed to be quasi-monochromatic.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, England, 1999).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Changes of the Stokes parameters s0,s1,s2, and s3 of an electromagnetic Gaussian Schell-model beam5,11 on propagation in free space, calculated from Eq. (8). In the notation of Ref. 11 the parameters characterizing the source used here have the values ω=3×1015 Hz λ=0.6328 µm, Ax=1.5, Ay=1, δ=π/6, Bxy=0.35, σ=1 cm, δyy=0.2 mm, δxy=0.25 mm, and δxx=0.15 mm.

Fig. 3
Fig. 3

Same as Fig. 2 except that normalized Stokes parameters s1/s0, s2/s0, and s3/s0 are plotted rather than the actual ones.

Equations (22)

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s0r,ω=Ex*r,ωExr,ω+Ey*r,ωEyr,ω,
s1r,ω=Ex*r,ωExr,ω-Ey*r,ωEyr,ω,
s2r,ω=Ex*r,ωEyr,ω+Ey*r,ωExr,ω,
s3r,ω=iEy*r,ωExr,ω-Ex*r,ωEyr,ω,
S0r1,r2,ω=Ex*r1,ωExr2,ω+Ey*r1,ωEyr2,ω,
S1r1,r2,ω=Ex*r1,ωExr2,ω-Ey*r1,ωEyr2,ω,
S2r1,r2,ω=Ex*r1,ωEyr2,ω+Ey*r1,ωExr2,ω,
S3r1,r2,ω=iEy*r1,ωExr2,ω-Ex*r1,ωEyr2,ω.
Eir,ω=z=0Ei0ρ,ωGρ-ρ,z,ωd2ρ, i=x,y,
Gρ-ρ,z,ω=-ik2πzexpikρ-ρ/2z.
Ei*r1,ωEjr2,ω=z=0Ei0*ρ1,ωEj0ρ2,ω×Kρ1-ρ1,ρ2-ρ2,z,ωd2ρ1d2ρ2,    i=x,y, j=x,y,
Kρ1-ρ1,ρ2-ρ2,z,ω=G*ρ1-ρ1,z,ω×Gρ2-ρ2,z,ω.
Sαr1,r2,ω=z=0Sα0ρ1,ρ2,ωKρ1-ρ1,ρ2-ρ2,z,ωd2ρ1d2ρ2,    α=0,1,2,3.
sαr,ωSαr,r,ω=z=0Sα0ρ1,ρ2,ω×Kρ-ρ1,ρ-ρ2,z,ωd2ρ1d2ρ2.
Wr1,r2,ω=Wijr1,r2,ωEi*r1,ωEjr2,ω, i=x,y,j=x,y.
S0r1,r2,ω=Wxxr1,r2,ω+Wyyr1,r2,ω,
S1r1,r2,ω=Wxxr1,r2,ω-Wyyr1,r2,ω,
S2r1,r2,ω=Wxyr1,r2,ω-Wyxr1,r2,ω,
S3r1,r2,ω=iWyxr1,r2,ω-Wxyr1,r2,ω.
ηr1,r2,ω=Tr Wr1,r2,ωTr Wr1,r2,ωTr Wr2,r2,ω,
S0r1,r2,ω=Tr Wr1,r2,ω,
ηr1,r2,ω=S0r1,r2,ωS0r1,r1,ωS0r2,r2,ω.

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