Abstract

Two-path interference of transversal vectorial waves is embedded within a larger scheme: this is four-path interference between four scalar waves. This comprises previous approaches to coherence between vectorial waves and restores the equivalence between correlation-based coherence and visibility.

© 2010 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  2. J. W. Goodman, Statistical Optics (Wiley, 1985).
  3. A. Luis, An Overview of Coherence and Polarization Properties for Multicomponent Electromagnetic Waves (SPIE, 2008), pp. 171–188.
  4. B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
    [CrossRef]
  5. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  6. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
    [CrossRef] [PubMed]
  7. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005).
    [CrossRef] [PubMed]
  8. 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]
  9. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
    [CrossRef]
  10. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef] [PubMed]
  11. 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]
  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]
  13. E. Wolf, “Comment on ‘Complete electromagnetic coherence in the space-frequency domain',” Opt. Lett. 29, 1712–1712 (2004).
    [CrossRef] [PubMed]
  14. T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on `Complete electromagnetic coherence in the space-frequency domain,’” Opt. Lett. 29, 1713–1714 (2004).
    [CrossRef]
  15. P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
    [CrossRef] [PubMed]
  16. R. Martinez-Herrero and P. M. Mejias, “Maximizing Young’s fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009).
    [CrossRef] [PubMed]
  17. A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
    [CrossRef]
  18. P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
    [CrossRef]
  19. F. Herbut, “A quantum measure of coherence and incompatibility,” J. Phys. A 38, 2959–2974 (2005).
    [CrossRef]
  20. A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
    [CrossRef]
  21. A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
    [CrossRef] [PubMed]
  22. 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]
  23. 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]
  24. A. Luis, “Polarization ray picture of coherence for vectorial electromagnetic waves,” Phys. Rev. A 76, 043827 (2007).
    [CrossRef]

2009 (1)

2008 (3)

2007 (5)

2006 (2)

2005 (2)

2004 (3)

2003 (2)

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

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

2002 (2)

1963 (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Borghi, R.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

Goudail, F.

Herbut, F.

F. Herbut, “A quantum measure of coherence and incompatibility,” J. Phys. A 38, 2959–2974 (2005).
[CrossRef]

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Kutay, M. A.

Luis, A.

P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
[CrossRef]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[CrossRef]

A. Luis, “Polarization ray picture of coherence for vectorial electromagnetic waves,” Phys. Rev. A 76, 043827 (2007).
[CrossRef]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[CrossRef] [PubMed]

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]

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[CrossRef]

A. Luis, An Overview of Coherence and Polarization Properties for Multicomponent Electromagnetic Waves (SPIE, 2008), pp. 171–188.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Martinez-Herrero, R.

Mejias, P. M.

Ozaktas, H. M.

Réfrégier, P.

Santarsiero, M.

Setälä, T.

Tervo, J.

Wolf, E.

E. Wolf, “Comment on ‘Complete electromagnetic coherence in the space-frequency domain',” Opt. Lett. 29, 1712–1712 (2004).
[CrossRef] [PubMed]

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Yüksel, S.

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

F. Herbut, “A quantum measure of coherence and incompatibility,” J. Phys. A 38, 2959–2974 (2005).
[CrossRef]

J. Phys. A: Math. Gen. (1)

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (9)

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

E. Wolf, “Comment on ‘Complete electromagnetic coherence in the space-frequency domain',” Opt. Lett. 29, 1712–1712 (2004).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on `Complete electromagnetic coherence in the space-frequency domain,’” Opt. Lett. 29, 1713–1714 (2004).
[CrossRef]

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]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[CrossRef] [PubMed]

P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
[CrossRef] [PubMed]

Phys. Lett. (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Phys. Lett. A (1)

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

Phys. Rev. A (2)

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[CrossRef]

A. Luis, “Polarization ray picture of coherence for vectorial electromagnetic waves,” Phys. Rev. A 76, 043827 (2007).
[CrossRef]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Luis, An Overview of Coherence and Polarization Properties for Multicomponent Electromagnetic Waves (SPIE, 2008), pp. 171–188.

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Figures (4)

Fig. 1
Fig. 1

Illustration of the relation V 2 + W 2 = P 2 .

Fig. 2
Fig. 2

Scheme of a two-path Mach–Zehnder interferometer for two scalar waves.

Fig. 3
Fig. 3

Plot of P (dashed curve) and V (dotted curve) as functions of the intensity ratio r for μ = 0.7 (solid curve).

Fig. 4
Fig. 4

Scheme of a four-path Mach–Zehnder interferometer for two vectorial waves.

Equations (34)

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Γ = ( | E 1 | 2 E 1 E 2 * E 1 * E 2 | E 2 | 2 ) ,
I ( φ 1 , φ 2 ) = | E 1 e i φ 1 + E 2 e i φ 2 | 2 | E 1 | 2 + | E 2 | 2 ,
V ( E ) = I max I min I max + I min = 2 | E 1 E 2 * | | E 1 | 2 + | E 2 | 2 = 2 r 1 + r | μ | ,
μ ( E ) = E 1 E 2 * | E 1 | 2 | E 2 | 2 , r ( E ) = | E 1 | 2 | E 2 | 2 .
V 2 = 2 tr [ ( γ γ u ) 2 ] ,
γ = 1 tr Γ Γ , γ u = 1 | E 1 | 2 + | E 2 | 2 ( | E 1 | 2 0 0 | E 2 | 2 ) .
V 2 = 1 2 π 2 2 π d φ 1 d φ 2 [ I ( φ 1 , φ 2 ) 1 ] 2 .
P ( E ) = | λ 1 λ 2 | λ 1 + λ 2 = [ 1 4 det Γ ( tr Γ ) 2 ] 1 2 ,
P ( E ) = ( 1 r ) 2 + 4 | μ | 2 r 1 + r .
P 2 = 2 tr [ ( γ i ) 2 ] , i = 1 2 ( 1 0 0 1 ) ,
V 2 + W 2 = P 2 ,
W 2 ( E ) = 2 tr [ ( γ u i ) 2 ] = ( | E 1 | 2 | E 2 | 2 | E 1 | 2 + | E 2 | 2 ) 2 = ( 1 r 1 + r ) 2 .
U d Γ U d = ( λ 1 0 0 λ 2 ) ,
U 0 = 1 2 ( 1 1 1 1 ) ,
Γ ( A ) = U Γ ( E ) U = 1 2 ( λ 1 + λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 + λ 2 ) ,
A 1 = cos θ E 1 + sin θ E 2 , A 2 = sin θ E 1 + cos θ E 2 ,
V ( A ) = | sin ( 2 θ ) [ r ( E ) 1 ] + 2 μ ( E ) r ( E ) cos ( 2 θ ) | 1 + r ( E ) .
Γ = ( Γ 1 , 1 Γ 1 , 2 Γ 2 , 1 Γ 2 , 2 ) , Γ i , j = ( E i , x E j , x * E i , x E j , y * E i , y E j , x * E i , y E j , y * ) ,
V 2 ( E ) = 4 3 tr [ ( γ γ u ) 2 ] , P 2 ( E ) = 4 3 tr [ ( γ i ) 2 ] ,
W 2 ( E ) = 4 3 tr [ ( γ u i ) 2 ] ,
γ = 1 tr Γ Γ , i = 1 4 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ,
γ u = 1 tr Γ ( | E 1 , x | 2 0 0 0 0 | E 1 , y | 2 0 0 0 0 | E 2 , x | 2 0 0 0 0 | E 2 , y | 2 ) ,
E = ( E 1 E 2 E 3 E 4 ) = ( E 1 , x E 1 , y E 2 , x E 2 , y ) .
V 2 ( E ) = 4 3 1 ( 2 π ) 4 2 π d φ [ I ( φ ) 1 ] 2 = 4 3 j k | E j E k * | 2 ( l | E l | 2 ) 2 ,
I ( φ ) = | j , k E j , k e i φ j , k | 2 l , m | E l , m | 2 .
U 0 = 1 2 ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) .
P 2 ( E ) = 4 3 [ tr ( Γ 2 ) ( tr Γ ) 2 1 4 ] = 4 3 [ j λ j 2 ( k λ k ) 2 1 4 ] ,
Γ = U d ( λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) U d ,
E j E k * = λ U d ; 1 , j * U d ; 1 , k ,
V 2 = 4 3 j , k | A 1 , j A 2 , k * | 2 ( l , m | A l , m | 2 ) 2 = 4 3 tr ( Γ 1 , 2 Γ 2 , 1 ) ( tr Γ 1 , 1 + tr Γ 2 , 2 ) 2 ,
V 2 = 4 3 tr Γ 1 , 1 tr Γ 2 , 2 ( tr Γ 1 , 1 + tr Γ 2 , 2 ) 2 μ TSF 2 , μ TSF 2 = tr ( Γ 1 , 2 Γ 2 , 1 ) tr Γ 1 , 1 tr Γ 2 , 2 .
I ( φ 1 , φ 2 ) | E 1 e i φ 1 + E 2 e i φ 2 | 2 = | E 1 | 2 + | E 2 | 2 + 2 | E 1 E 2 | cos ( φ 2 φ 1 + δ ) ,
Γ j , k = E j E k , tr Γ j , k = E k E j ,
V = 2 tr Γ 1 , 1 tr Γ 2 , 2 tr Γ 1 , 1 + tr Γ 2 , 2 | μ K W | , μ K W = tr Γ 1 , 2 tr Γ 1 , 1 tr Γ 2 , 2 .

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