Abstract

The polarization properties of vectorial coherent nonparaxial Gaussian beams are studied. It is shown that, when the source of a nonparaxial Gaussian beam is completely polarized, the degree of polarization of the propagation field maintains a constant value of 1. However, when the source is completely unpolarized, the degree of polarization does not maintain a constant value of 0. In the far field, vectorial nonparaxial Gaussian beams can become partially polarized, except on the propagation axis, and are spatially nonuniform across a typical cross section of the beam even though the source field is completely unpolarized.

© 2005 Optical Society of America

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References

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2003 (2)

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

K. Duan and B. Lu, Opt. Quantum Electron. 35, 179 (2003).
[CrossRef]

2001 (3)

P. Östlund and A. T. Friberg, Opt. Commun. 197, 1 (2001).
[CrossRef]

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

A. Ciattoni, B. Crosignani, and P. D. Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
[CrossRef]

2000 (2)

G. Gbur and D. F. V. James, J. Mod. Opt. 47, 1171 (2000).
[CrossRef]

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

1999 (1)

1998 (2)

1990 (1)

Agrawal, G. P.

An, Y.

Borghi, R.

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

Ciattoni, A.

Crosignani, B.

Duan, K.

K. Duan and B. Lu, Opt. Quantum Electron. 35, 179 (2003).
[CrossRef]

Friberg, A. T.

P. Östlund and A. T. Friberg, Opt. Commun. 197, 1 (2001).
[CrossRef]

Gbur, G.

G. Gbur and D. F. V. James, J. Mod. Opt. 47, 1171 (2000).
[CrossRef]

Gori, F.

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

F. Gori, Opt. Lett. 23, 241 (1998).
[CrossRef]

F. Gori, Opt. Lett. 23, 241 (1998).
[CrossRef]

James, D. F. V.

G. Gbur and D. F. V. James, J. Mod. Opt. 47, 1171 (2000).
[CrossRef]

Liang, C.

Lu, B.

K. Duan and B. Lu, Opt. Quantum Electron. 35, 179 (2003).
[CrossRef]

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: Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

Nemoto, S.

Östlund, P.

P. Östlund and A. T. Friberg, Opt. Commun. 197, 1 (2001).
[CrossRef]

Piquero, G.

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

Porto, P. D.

Santarsiero, M.

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

Simon, R.

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

Wolf, E.

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

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

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

Zeng, X.

Appl. Opt. (2)

J. Mod. Opt. (1)

G. Gbur and D. F. V. James, J. Mod. Opt. 47, 1171 (2000).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

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

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

Opt. Commun. (1)

P. Östlund and A. T. Friberg, Opt. Commun. 197, 1 (2001).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

K. Duan and B. Lu, Opt. Quantum Electron. 35, 179 (2003).
[CrossRef]

Phys. Lett. A (1)

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

Other (1)

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

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Figures (3)

Fig. 1
Fig. 1

Geometric scheme of the far-field components.

Fig. 2
Fig. 2

Degree of polarization Px,y,100zR of a vectorial nonparaxial Gaussian beam versus the f parameter for x=y=5λ, 10λ, and 20λ at the plane z=100zR.

Fig. 3
Fig. 3

Degree of polarization Px,y,100zR of a completely unpolarized vectorial nonparaxial Gaussian beam versus x/w100zR for f=0.7, 0.5, 0.3, and 0.01 at the plane z=100zR.

Equations (24)

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Exρ0,0;t=E0x exp-ρ022w02expiφxt-ωt,
Eyρ0,0;t=E0y exp-ρ022w02expiφyt-ωt,
Exρ,z;t=izE0xkf2expikr+φxt-ωtr2×exp-12f2ρ2r2,
Eyρ,z;t=izE0ykf2expikr+φyt-ωtr2×exp-12f2ρ2r2,
Epxρ,z;t=iE0xkf2expikz+φxt-ωtz×expik2zx2+y2exp-12f2ρ2z2,
Epyρ,z;t=iE0ykf2expikz+φyt-ωtz×expik2zx2+y2exp-12f2ρ2z2.
Ezxρ,z;t=-xzExρ,z;t,
Ezyρ,z;t=-yzEyρ,z;t,
E1ρ,z;t=iExρ,z;t+kEzxρ,z;t,
E2ρ,z;t=jEyρ,z;t+kEzyρ,z;t,
θ=arccosxyx2+z2y2+z21/2.
Pρ,z=1-4 Det Jρ,ρ,zTr Jρ,ρ,z21/2,
Jρ,ρ,z=E1*ρ,z;tE1ρ,z;tE1*ρ,z;tE2ρ,z;tE2*ρ,z;tE1ρ,z;tE2*ρ,z;tE2ρ,z;t,
Jpρ,ρ,z=Epx*ρ,z;tEpxρ,z;tEpx*ρ,z;tEpyρ,z;tEpy*ρ,z;tEpxρ,z;tEpy*ρ,z;tEpyρ,z;t.
Jρ,ρ,z=aEx*Ex+abEx*Ey+Ey*Excos θ+bEy*Ey cos2 θabEx*Ey sin θ+bEy*Ey sin θ cos θabEy*Ex sin θ+bEy*Ey sin θ cos θbEy*Ey sin2 θ,
Ex=izE0xkf21r2exp-12f2ρ2r2,
Ey=izE0ykf21r2exp-12f2ρ2r2,
a=z2+x2z2,
b=z2+y2z2.
Jρ,ρ,z=aEx*Ex+bEy*Ey cos2 θbEy*Ey sin θ cos θbEy*Ey sin θ cos θbEy*Ey sin2 θ.
Iρ,z=TrJρ,ρ,z=aEx*ρ,zExρ,z+bEy*ρ,zEyρ,z.
Jpρ,ρ,z=Epx*Epx00Epy*Epy,
Epxρ,z;t=iE0xkzf2exp-12f2ρ2z2,
Epyρ,z;t=iE0ykzf2exp-12f2ρ2z2.

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