Abstract

It is shown by use of a simple model that in general the state of polarization of a light beam generated by a partially coherent source changes as the beam propagates in free space.

© 1994 Optical Society of America

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References

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  1. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  2. E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  3. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  4. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  5. P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
    [CrossRef]
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, to be published).
  7. P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  8. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [CrossRef]
  9. The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].
  10. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 633.
  11. Note that the assumption of a completely unpolarized field is not incompatible with the assumption of partial spatial coherence. An example is the blackbody field, which is unpolarized and is partially coherent [see C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation, part III: cross spectral tensors,” Phys. Rev. 161, 1328–1334 (1967)].
  12. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  13. M. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
    [CrossRef]

1993 (1)

1987 (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

1986 (1)

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1982 (1)

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

1979 (1)

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1978 (1)

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

1973 (1)

The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Agrawal, G. P.

The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].

Collett, E.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Farina, J. D.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gori, F.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Guattari, G.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Jaiswal, A. K.

The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].

Kowarz, M.

Mandel, L.

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

Mehta, C. L.

The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Narducci, L. M.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Palma, C.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 633.

Simon, R.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Wolf, E.

M. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

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

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

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

Nuovo Cimento (2)

The influence of coherence on the degree of polarization was first considered for blackbody sources in A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento 15B, 295–307 (1973).This work is the subject of a forthcoming paper [D. F. V. James, “Polarization of light radiated by blackbody sources,” Opt. Commun. (to be published)].

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Opt. Acta (1)

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

Opt. Commun. (4)

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model beams,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Other (3)

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

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 633.

Note that the assumption of a completely unpolarized field is not incompatible with the assumption of partial spatial coherence. An example is the blackbody field, which is unpolarized and is partially coherent [see C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation, part III: cross spectral tensors,” Phys. Rev. 161, 1328–1334 (1967)].

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Figures (3)

Fig. 1
Fig. 1

Illustration of the notation used.

Fig. 2
Fig. 2

Change of the degree of polarization P(0, z,ω) at the center of the beam as it propagates along the z axis, with beam parameters α = 1.0 mm, ω = 3 × 1015 s−1 (λ = 6283 Å), l1 = 1.0 μm, l2 = 5.0 μm.

Fig. 3
Fig. 3

Far-zone degree of polarization, P(ρ, z,ω) (solid curve) and the normalized spectral distribution, S(ρ, z,ω)/S(0, z,ω) (dashed curve) plotted as functions of transverse distance from the z axis. The beam parameters are the same as in Fig. 2. Note that, in the model used, the degree of polarization and the spectral distribution are rotationally symmetric about the z axis.

Equations (20)

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W i j ( ρ a , ρ a , z a , ω ) = 1 2 π E i * ( ρ a , z a , t ) × E j ( ρ a , z a , t + τ ) exp ( i ω τ ) d τ ,
W i j ( ρ b , ρ b , z b , ω ) = W i j ( ρ a , ρ a , z a , ω ) × K * ( ρ b ρ a , z b z a , ω ) × K ( ρ b ρ a , z b z a ) d 2 ρ a d 2 ρ a .
K ( ρ b ρ a , z b z a , ω ) = i k 2 π exp [ i k ( z b z a ) ] z b z a × exp [ i k ( ρ b ρ a ) 2 2 ( z b z a ) ] ,
J i j ( ρ , z , ω ) W i j ( ρ , ρ , z , ω ) .
P ( ρ , z , ω ) = ( 1 4 Det [ J i j ( ρ , z , ω ) ] { Tr [ J i j ( ρ , z , ω ) ] } 2 ) 1 / 2 ,
W i j ( ρ a , ρ a , z a , ω ) = I ( ρ a , ρ a , ω ) × [ exp [ ( ρ a ρ a ) 2 2 l I 2 ] 0 0 exp [ ( ρ a ρ a ) 2 2 l 2 2 ] ] ,
J i j ( ρ a , z a , ω ) = I ( ρ a , ρ a , ω ) [ 1 0 0 1 ] ,
P ( ρ a , z a , ω ) = 0 .
I ( ρ a , ρ a , ω ) = S a ( ω ) exp ( | ρ a | 2 + | ρ a | 2 2 α 2 ) ,
J i j ( ρ b , z b , ω ) = S a ( ω ) [ A ( ρ b , z b , ω ) 0 0 B ( ρ b , z b , ω ) ] ,
P ( ρ b , z b , ω ) = | A ( ρ b , z b , ω ) B ( ρ b , z b , ω ) | A ( ρ b , z b , ω ) + B ( ρ b , z b , ω ) .
A ( ρ b , z b , ω ) = 1 β 1 2 ( z b ) exp [ | ρ b | 2 2 α 2 β 1 2 ( z b ) ] ,
B ( ρ b , z b , ω ) = 1 β 2 2 ( z b ) exp [ | ρ b | 2 2 α 2 β 2 2 ( z b ) ] ,
β i 2 ( z ) = 1 + z 2 4 k 2 α 4 + z 2 k 2 l i 2 α 2 , ( i = 1 , 2 ) .
J i j ( ω ) = W i j ( ρ , ρ , z , ω ) d 2 ρ ,
W i j ( ρ , ρ , z b , ω ) d 2 ρ = W i j ( ρ a , ρ a , z a , ω ) × K * ( ρ ρ a , z b z a , ω ) × K ( ρ ρ a , z b z a , ω ) d 2 ρ a d 2 ρ a d 2 ρ ,
K * ( ρ ρ a , z b z a , ω ) K ( ρ ρ a , z b z a , ω ) d 2 ρ = k 2 ( 2 π ) 2 1 ( z b z a ) 2 exp [ i k ( ρ ρ a ) 2 2 ( z b z a ) ] × exp [ i k ( ρ a ρ a ) 2 2 ( z b z a ) ] d 2 ρ = k 2 ( 2 π ) 2 1 ( z b z a ) 2 exp [ i k ( ρ a 2 ρ a 2 ) 2 ( z b z a ) ] × exp [ i k ( ρ a ρ a ) · ρ ( z b z a ) ] d 2 ρ .
K * ( ρ ρ a , z b z a , ω ) K ( ρ ρ a , z b z a , ω ) d 2 ρ = exp [ ik ( ρ a 2 ρ a 2 ) 2 ( z b z a ) ] 1 ( 2 π ) 2 × exp [ i ( ρ a ρ a ) · x ] d 2 x .
K * ( ρ ρ a , z b z a , ω ) K ( ρ ρ a , z b z a , ω ) d 2 ρ = δ ( 2 ) ( ρ a ρ a ) .
W i j ( ρ , ρ , z b , ω ) d 2 ρ = W i j ( ρ , ρ , z a , ω ) d 2 ρ ,

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