Abstract

The technique for generating the partially coherent and partially polarized source starting from the completely coherent and completely polarized laser source is proposed and analyzed. This technique differs from the known ones by the simplicity of its physical realization. The efficiency of the proposed technique is illustrated with the results of physical experiment in which an original technique for characterizing the coherence and polarization properties of the generated source is employed.

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References

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  1. A. S. Ostrovsky, P. Martínez-Vara, M. Á. Olvera-Santamaría, and G. Martínez-Niconoff, Vector coherence theory: An overview of basic concepts and definitions, in Recent Research Developments in Optics, S. G. Pandalai, ed., (Research Signpost, Kerala, India, 2009) Chap. 5.
  2. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screen and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
    [CrossRef]
  3. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
    [CrossRef]
  4. A. S. Ostrovsky, G. Martínez-Niconoff, V. Arrizón, P. Martínez-Vara, M. Á. Olvera-Santamaría, and C. Rickenstorff-Parrao, “Modulation of coherence and polarization using liquid crystal spatial light modulators,” Opt. Express 17(7), 5257–5264 (2009).
    [CrossRef] [PubMed]
  5. G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
    [CrossRef]
  6. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [CrossRef]
  7. H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003).
    [CrossRef]
  8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, UK, 2007).
  9. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [CrossRef]
  10. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21(11), 2205–2215 (2004).
    [CrossRef]
  11. M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008).
    [CrossRef]
  12. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
    [CrossRef] [PubMed]
  13. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24(4), 1063–1068 (2007).
    [CrossRef]
  14. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13(16), 6051–6060 (2005).
    [CrossRef] [PubMed]
  15. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
    [CrossRef]
  16. A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE Press, Bellingham, WA, USA, 2006).

2009 (1)

2008 (1)

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008).
[CrossRef]

2007 (1)

2005 (2)

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13(16), 6051–6060 (2005).
[CrossRef] [PubMed]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

2004 (2)

2003 (4)

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[CrossRef] [PubMed]

2002 (1)

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

1998 (1)

Alonso, M. A.

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008).
[CrossRef]

Arrizón, V.

Borghi, R.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Friberg, A. T.

Gori, F.

Goudail, F.

Guattari, G.

Korotkova, O.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

Luis, A.

Martínez-Niconoff, G.

Martínez-Vara, P.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Olvera-Santamaría, M. Á.

Ostrovsky, A. S.

Piquero, G.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Réfrégier, P.

Rickenstorff-Parrao, C.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003).
[CrossRef]

Santarsiero, M.

Satarsiero, M.

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Setälä, T.

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screen and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[CrossRef]

Simon, R.

Tervo, J.

Wolf, E.

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screen and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[CrossRef]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Other (3)

A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE Press, Bellingham, WA, USA, 2006).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, UK, 2007).

A. S. Ostrovsky, P. Martínez-Vara, M. Á. Olvera-Santamaría, and G. Martínez-Niconoff, Vector coherence theory: An overview of basic concepts and definitions, in Recent Research Developments in Optics, S. G. Pandalai, ed., (Research Signpost, Kerala, India, 2009) Chap. 5.

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Figures (4)

Fig. 1
Fig. 1

Schematic illustration of the technique for generating the partially coherent and partially polarized source: BS, beam splitter; PBS, polarizing beam splitter; M, mirror; GGP1, GGP2, rotating ground glass plates.

Fig. 2
Fig. 2

Schematic illustration of the technique for characterizing the generated secondary source: BS, beam splitter; M, mirror; TP, translating pinhole; P1, P2, polarizers; R1, R2; polarization rotators.

Fig. 3
Fig. 3

Experimental setup: L, laser; BE, beam expander; ZL, zoom-lens; PD, photodiode; the other abbreviations are just the same as in Figs. 1 and 2.

Fig. 4
Fig. 4

Results for the experimental characterization of generated sources for two different pairs of ground glass plates with the diffusion angles of 10° and 30°, and θ = 45°. Theoretical results are plotted by solid curves.

Equations (29)

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W ( x 1 , x 2 ) = [ E x * ( x 1 ) E x ( x 2 ) E x * ( x 1 ) E y ( x 2 ) E y * ( x 1 ) E x ( x 2 ) E y * ( x 1 ) E y ( x 2 ) ] ,
S ( x ) = Tr   W ( x , x ) ,
μ ( x 1 , x 2 ) = Tr   W ( x 1 , x 2 ) [ Tr   W ( x 1 , x 1 )   Tr   W ( x 2 , x 2 ) ] 1 / 2     ,
P ( x ) = [ 1 4 Det   W ( x , x ) [ Tr   W ( x , x ) ] 2 ] 1 / 2 .
W PS ( x 1 , x 2 ) = S 0 exp ( x 1 2 + x 2 2 4 ε 2 ) [ cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ] ,
T ( x ) = [ exp [ i φ 1 ( x ) ] 0 0 exp [ i φ 2 ( x ) ] ] ,
p [ φ 1 ( 2 ) ( x ) ] = 1 2 π   σ exp [ φ 1(2) 2 ( x ) 2 ]
φ 1 ( 2 ) ( x 1 ) φ 1 ( 2 ) ( x 2 ) = σ 2 exp ( ξ 2 2 )     .
p [ φ 1 ( x ) , φ 2 ( x ) ] = p [ φ 1 ( x ) ] p [ φ 2 ( x ) ] .
W SS ( x 1 , x 2 ) = T ( x 1 ) W PS ( x 1 , x 2 ) T ( x 2 ) ,
W SS ( x 1 , x 2 ) = S 0 e x p ( x 1 2 + x 2 2 4 ε 2 )
× [ exp { σ 2 [ 1 exp ( ξ 2 2 ) ] } cos 2 θ exp ( σ 2 )     sin θ cos θ exp ( σ 2 )     sin θ cos θ exp { σ 2 [ 1 exp ( ξ 2 2 ) ] } sin 2 θ ]     .
  exp ( σ 2 ) 0
  exp { σ 2 [ 1 exp ( ξ 2 2 ) ] } exp ( ξ 2 2 η 2 ) ,
W SS ( x 1 , x 2 ) = S 0 exp ( x 1 2 + x 2 2 4 ε 2 ) exp ( ξ 2 2 η 2 ) [ cos 2 θ 0 0 sin 2 θ ]     .
μ SS ( x 1 , x 2 ) = exp ( ξ 2 2 η 2 )     ,
P SS ( x ) = | 1 2 c o s 2 θ | .
S i j ( x ) = S i SS ( ξ 2 ) + S j SS ( ξ 2 ) + 2 | W i j SS ( ξ 2 , ξ 2 ) | cos [ k ξ z 0 x + α i j ( ξ 2 , ξ 2 ) ]     ( i , j = x , y ) ,
V i j ( ξ) = S i j max ( x ) S i j min ( x ) S i j max ( x ) + S i j min ( x )     .
|     W i j SS ( ξ)   | = 1 2 [ S i SS ( ξ 2 ) + S j SS ( ξ 2 ) ] V i j ( ξ)   .
exp [ ± ( x ) ] = 1 2 π   σ     -     exp [ ± ( x ) ] exp [ φ 2 ( x ) 2 ]   dφ   .
    -     exp ( π a 2 φ 2 ) exp ( ± i 2 πφ u )   dφ = 1 a exp ( π u 2 a 2 ) ,
exp [ ± ( x ) ] = exp ( σ 2 2 ) .
ψ( x 1 , x 2 ) = φ( x 2 ) φ( x 1 )   .
p [ ψ( x 1 , x 2 ) ] = 1 2 π β exp [ ψ 2 ( x 1 , x 2 ) 2 β 2 ] ,
β 2 = ψ 2 ( x 1 , x 2 ) = [ φ( x 2 ) φ( x 1 ) ] 2 .
β 2 = 2 σ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] .
exp {   i[φ( x 2 ) φ( x 1 )]   } = exp { σ 2 [ 1 exp ( ξ 2 2 ) ] } .
exp { i[φ 1 ( 2 ) ( x 2 ) φ 2 ( 1 ) ( x 1 ) ]   } = exp ( σ 2 )     .

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