Abstract

The sum of two uncorrelated and totally polarized lights with different coherence and polarization properties usually results in a partially polarized light. It is shown in this paper that the initial totally polarized lights can be recovered from the mixed partially polarized light. The proposed technique is based on coherence analysis and does not require the knowledge of the polarization states or the coherence properties of the initial perfectly polarized beams as long as these properties are different for the two waves. Some practical optical implementations of this technique are discussed on different illustrative applications.

© 2008 Optical Society of America

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References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  2. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137-1142 (2003).
    [CrossRef] [PubMed]
  3. 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, 2205-2215 (2004).
    [CrossRef]
  4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  5. 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, 78-84 (2003).
    [CrossRef]
  6. G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30, 120-122 (2005).
    [CrossRef] [PubMed]
  7. Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  8. Ph. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117-3119 (2005).
    [CrossRef] [PubMed]
  9. Ph. Réfrégier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. Am. A 23, 3036-3044 (2006).
    [CrossRef]
  10. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-670 (2006).
    [CrossRef] [PubMed]
  11. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
    [CrossRef] [PubMed]
  12. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669-2671 (2006).
    [CrossRef] [PubMed]
  13. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
    [CrossRef] [PubMed]
  14. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063-1068 (2007).
    [CrossRef]
  15. R. Martinez-Herrero and P. M. Mejias, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471-1473 (2007).
    [CrossRef] [PubMed]
  16. R. Martinez-Herrero and P. M. Mejias, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
    [CrossRef] [PubMed]
  17. I. Berezhnyy and A. Dogariu, “Polarimetric description of superposing random electromagnetic fields with different spectral composition,” J. Opt. Soc. Am. A 21, 218-222 (2004).
    [CrossRef]
  18. L. Mandel and E. Wolf, “Second-order coherence theory of scalar wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 160-170.
  19. J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116-156.
  20. E. Wolf, “Unified theory of polarization and coherence,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007), pp. 174-197.
  21. Ph. Réfrégier, “Fluctuations and covariance,” in Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004), pp. 28-32.
  22. E. Wolf, “Second-order coherence phenomena in the space-time domain,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007), pp. 31-58.
  23. Ph. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coherence,” Opt. Lett. 31, 1175-1177 (2006).
    [CrossRef] [PubMed]
  24. Ph. Réfrégier and A. Roueff, “Linear relations of partially polarized and coherent electromagnetic fields,” Opt. Lett. 31, 2827-2829 (2006).
    [CrossRef] [PubMed]
  25. L. Mandel and E. Wolf, “Second-order coherence theory of vector wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 342-349.
  26. S. Huard, “Propagation of states of polarization in optical devices,” in Polarization of Light (Wiley, 1997), pp. 86-130.
  27. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
    [CrossRef]

2007 (4)

2006 (6)

2005 (3)

2004 (3)

2003 (3)

1998 (1)

J. Opt. Soc. Am. A (5)

Opt. Express (2)

Opt. Lett. (12)

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328-330 (2004).
[CrossRef] [PubMed]

Ph. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117-3119 (2005).
[CrossRef] [PubMed]

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30, 120-122 (2005).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-670 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669-2671 (2006).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
[CrossRef] [PubMed]

R. Martinez-Herrero and P. M. Mejias, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471-1473 (2007).
[CrossRef] [PubMed]

R. Martinez-Herrero and P. M. Mejias, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
[CrossRef] [PubMed]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
[CrossRef]

Ph. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coherence,” Opt. Lett. 31, 1175-1177 (2006).
[CrossRef] [PubMed]

Ph. Réfrégier and A. Roueff, “Linear relations of partially polarized and coherent electromagnetic fields,” Opt. Lett. 31, 2827-2829 (2006).
[CrossRef] [PubMed]

Phys. Lett. A (1)

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

Other (7)

L. Mandel and E. Wolf, “Second-order coherence theory of scalar wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 160-170.

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116-156.

E. Wolf, “Unified theory of polarization and coherence,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007), pp. 174-197.

Ph. Réfrégier, “Fluctuations and covariance,” in Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004), pp. 28-32.

E. Wolf, “Second-order coherence phenomena in the space-time domain,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007), pp. 31-58.

L. Mandel and E. Wolf, “Second-order coherence theory of vector wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 342-349.

S. Huard, “Propagation of states of polarization in optical devices,” in Polarization of Light (Wiley, 1997), pp. 86-130.

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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of the mixing of two totally polarized lights. The two waves are assumed to have different polarization states and different standard degrees of coherence. The Jones matrices J 1 and J 2 are introduced to describe the action of the beam splitter.

Fig. 2
Fig. 2

Schematic optical implementation of the proposed separation technique. This polarization filtering technique allows one to recover optical fields that are proportional to E 1 ( r j , t ) and E 2 ( r j , t ) .

Fig. 3
Fig. 3

Schematic representation of the problem considered in the estimation of characteristics of a thick birefringent medium.

Fig. 4
Fig. 4

Physical illustration of a totally depolarized light with intrinsic degrees of coherence equal to μ ( τ ) .

Fig. 5
Fig. 5

Schematic representation of the problem considered in the time delay estimation of mixed fields.

Fig. 6
Fig. 6

The spectral filter allows one to modify the coherence properties of the beam E 1 ( t ) and thus to apply the separation technique proposed in this paper. The field [ h E 1 ] ( t ) corresponds to the convolution of h ( t ) with E 1 ( t ) .

Fig. 7
Fig. 7

Schematic illustration of a homodyne-like system. The signal φ ( t ) modulates the signal beam ( E S ( t ) ) , and a time delay η much larger than the coherence time is introduced on the reference beam ( E R ( t ) ) ; h ̂ ( ν ) corresponds to the transfer function of a nonsingular spectral filter h ( t ) to its impulse response and [ h ̂ ( ν ) ] 1 to its inverse. One thus has E R ( t ) = α R [ h E in ( t η ) ] and E S ( t ) = α S E in [ t φ ( t ) ] .

Equations (41)

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Γ ( r , t ) = ( E X ( r , t ) E X * ( r , t ) E X ( r , t ) E Y * ( r , t ) E Y ( r , t ) E X * ( r , t ) E Y ( r , t ) E Y * ( r , t ) ) ,
μ ( r 1 , r 2 , τ ) = E * ( r 1 , t ) E ( r 2 , t + τ ) I ( r 1 ) I ( r 2 ) ,
Ω ( r 1 , r 2 , τ ) = ( E X ( r 2 , t + τ ) E X * ( r 1 , t ) E X ( r 2 , t + τ ) E Y * ( r 1 , t ) E Y ( r 2 , t + τ ) E X * ( r 1 , t ) E Y ( r 2 , t + τ ) E Y * ( r 1 , t ) ) ,
E ( r j , t ) = E 1 ( r j , t ) e 1 ( r j ) + E 2 ( r j , t ) e 2 ( r j ) ,
μ i ( r 1 , r 2 , τ ) = E i * ( r 1 , t ) E i ( r 2 , t + τ ) I i ( r 1 ) I i ( r 2 ) ,
ϵ i ( r j , t ) = 1 I i ( r j ) E i ( r j , t ) , u i ( r j ) = I i ( r j ) e i ( r j ) .
u i ( r j ) = ( u i , X ( r j ) u i , Y ( r j ) ) , E ( r j , t ) = ( ϵ 1 ( r j , t ) ϵ 2 ( r j , t ) ) ,
E ( r j , t ) = U ( r j ) E ( r j , t ) ,
U ( r j ) = ( u 1 , X ( r j ) u 2 , X ( r j ) u 1 , Y ( r j ) u 2 , Y ( r j ) ) .
ϵ 1 * ( r j , t ) ϵ 2 ( r k , t + τ ) = 0
Γ ( r j ) = U ( r j ) U ( r j ) .
Ω ( r 1 , r 2 , τ ) = U ( r 2 ) D ( r 1 , r 2 , τ ) U ( r 1 ) ,
D ( r 1 , r 2 , τ ) = ( μ 1 ( r 1 , r 2 , τ ) 0 0 μ 2 ( r 1 , r 2 , τ ) ) .
M ( r 1 , r 2 , τ ) = Γ 1 2 ( r 2 ) Ω ( r 1 , r 2 , τ ) Γ 1 2 ( r 1 ) .
M ( r 1 , r 2 , τ ) = V ( r 2 ) D ( r 1 , r 2 , τ ) V ( r 1 ) ,
V ( r j ) = [ U ( r j ) U ( r j ) ] 1 2 U ( r j ) .
Λ ( r 1 , r 2 , τ ) = ( μ 1 ( r 1 , r 2 , τ ) 2 0 0 μ 2 ( r 1 , r 2 , τ ) 2 ) ,
M ( r 1 , r 2 , τ ) M ( r 1 , r 2 , τ ) = V ( r 2 ) Λ ( r 1 , r 2 , τ ) V ( r 2 ) ,
M ( r 1 , r 2 , τ ) M ( r 1 , r 2 , τ ) = V ( r 1 ) Λ ( r 1 , r 2 , τ ) V ( r 1 ) .
M ( r 1 , r 2 , τ ) M ( r 1 , r 2 , τ ) = N ( r 2 ) Λ ( r 1 , r 2 , τ ) N ( r 2 ) ,
M ( r 1 , r 2 , τ ) M ( r 1 , r 2 , τ ) = N ( r 1 ) Λ ( r 1 , r 2 , τ ) N ( r 1 ) ,
Ψ ( r j ) = ( e i ψ 1 ( r j ) 0 0 e i ψ 2 ( r j ) ) .
B ( r j ) = ( b 1 , X ( r j ) b 2 , X ( r j ) b 1 , Y ( r j ) b 2 , Y ( r j ) ) .
U ( r j ) = ( e i ψ 1 ( r j ) b 1 , X ( r j ) e i ψ 2 ( r j ) b 2 , X ( r j ) e i ψ 1 ( r j ) b 1 , Y ( r j ) e i ψ 2 ( r j ) b 2 , Y ( r j ) ) .
U ( r j ) = ( e i β 1 ( r j ) b 1 , X ( r j ) e i β 2 ( r j ) b 2 , X ( r j ) e i β 1 ( r j ) b 1 , Y ( r j ) e i β 2 ( r j ) b 2 , Y ( r j ) ) .
μ i ( τ ) = E i * ( t ) E i ( t + τ ) I i ,
E ( r 1 , t ) = E 1 ( t ) u 1 + E 2 ( t ) u 2 .
E ( r 2 , t ) = E 1 ( t η 1 ) u 1 + E 2 ( t η 2 ) u 2 ,
Γ ( r 1 ) = Γ ( r 2 ) = U ( I 0 0 0 I 0 ) U ,
U = ( u 1 , X u 2 , X u 1 , Y u 2 , Y ) ,
with u 1 = ( u 1 , X u 1 , Y ) , u 2 = ( u 2 , X u 2 , Y ) .
Ω ( r 1 , r 2 , τ ) = U ( E 1 ( t + τ η 1 ) E 1 * ( t ) E 1 ( t + τ η 1 ) E 2 * ( t ) E 2 ( t + τ η 2 ) E 1 * ( t ) E 2 ( t + τ η 2 ) E 2 * ( t ) ) U .
Ω ( r 1 , r 2 , τ ) = I 0 U ( μ ( τ η 1 ) 0 0 μ ( τ η 2 ) ) U .
μ ( τ ) = E * ( t ) E ( t + τ ) E ( t ) 2 = 0 if τ τ C .
Γ A = U Γ E U ,
Γ E = ( I E μ ( τ ) I E I E μ ( τ ) * I E I E I E ) ,
μ ( τ ) = E * ( t ) E ( t + τ ) I E I E .
Ω E ( η ) = ( μ ( η ) I E μ ( η + τ ) I E I E μ ( η τ ) * I E I E μ ( η ) I E ) ,
μ ( ξ ) = E * ( t ) E ( t + ξ ) I E , μ ( ξ ) = E * ( t ) E ( t + ξ ) I E .
Γ E = ( I E 0 0 I E )
Ω E ( η ) = ( μ ( η ) I E 0 0 μ ( η ) I E ) ,

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