Abstract

We propose a method for modulation of coherence and polarization of electromagnetic fields, employing two crossed zero-twisted nematic liquid crystal spatial light modulators. In contrast to a similar technique analyzed by Shirai and Wolf [J. Opt. Soc. Am A, 21, 1907, (2004)] our method provides a wide range simultaneous modulation of coherence and polarization. The dependence of the obtained results on different definitions of electromagnetic coherence is considered.

© 2009 Optical Society of America

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References

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  1. J.W. Goodman, Statistical Optics (Wiley, New York, 1985).
  2. A.S. Ostrovsky, Coherent-Mode Representations in Optics, (SPIE Press, Bellingham, WA, 2006).
    [CrossRef]
  3. J. Turunen, A. Vasara, and A.T. Friberg, "Propagation invariance and self-imaging in variable-coherence optics," J. Opt. Soc. Am. A 8, 282-289 (1990).
    [CrossRef]
  4. A.S. Ostrovsky and E. Hernández-García, "Modulation of spatial coherence of optical field by means of liquid crystal light modulator," Revista Mexicana de Física 51, 442-446 (2005).
  5. T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
    [CrossRef]
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  7. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312,263-267 (2003).
    [CrossRef]
  8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, UK, 2007).
  9. A. Yariv and P. Yen, Optical Waves in Crystals, (Wiley, New York, 1984).
  10. K. Lu and B.E.A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
    [CrossRef]
  11. 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]
  12. P. Réfrégier and F. Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  13. 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]
  14. F. Gori, M. Santarsiero, and R. Borghi, "Maximizing Young’s fringe visibility through reversible optical transformations," Opt. Lett. 32, 588-590 (2007).
    [CrossRef] [PubMed]

2007

2005

A.S. Ostrovsky and E. Hernández-García, "Modulation of spatial coherence of optical field by means of liquid crystal light modulator," Revista Mexicana de Física 51, 442-446 (2005).

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

2004

2003

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

1990

K. Lu and B.E.A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

J. Turunen, A. Vasara, and A.T. Friberg, "Propagation invariance and self-imaging in variable-coherence optics," J. Opt. Soc. Am. A 8, 282-289 (1990).
[CrossRef]

Borghi, R.

Friberg, A.T.

Gori, F.

Goudail, F.

Hernández-García, E.

A.S. Ostrovsky and E. Hernández-García, "Modulation of spatial coherence of optical field by means of liquid crystal light modulator," Revista Mexicana de Física 51, 442-446 (2005).

Lu, K.

K. Lu and B.E.A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

Luis, A.

Ostrovsky, A.S.

A.S. Ostrovsky and E. Hernández-García, "Modulation of spatial coherence of optical field by means of liquid crystal light modulator," Revista Mexicana de Física 51, 442-446 (2005).

Réfrégier, P.

Saleh, B.E.A.

K. Lu and B.E.A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

Santarsiero, M.

Setälä, T.

Shirai, T.

Tervo, J.

Turunen, J.

Vasara, A.

Wolf, E.

J. Opt. Soc. Am. A

Opt. Eng.

K. Lu and B.E.A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

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

Revista Mexicana de Física

A.S. Ostrovsky and E. Hernández-García, "Modulation of spatial coherence of optical field by means of liquid crystal light modulator," Revista Mexicana de Física 51, 442-446 (2005).

Other

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

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

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

A. Yariv and P. Yen, Optical Waves in Crystals, (Wiley, New York, 1984).

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

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Figures (3)

Fig. 1.
Fig. 1.

Degree of coherence given by Eq. (16), σφ = 2 and γ = 1,2,3.

Fig. 2.
Fig. 2.

Degree of coherence given by Eq. (22), σφ = 2 and γ = 1,2,3.

Fig. 3.
Fig. 3.

Degree of coherence given by Eq. (22) (solid line) and degree of coherence given by Eq. (26) (dotted line), σφ =2 and γ = 2.

Equations (37)

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{ E ( x , v ) } = { E x ( x , v ) E y ( x , v ) } ,
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 2 ) E y ( x 2 ) ] ,
η ( x 1 , x 2 ) = Tr W ( x 1 , x 2 ) [ Tr W ( x 1 , x 2 ) Tr W ( x 1 , x 2 ) ] 1 / 2 ,
P ( x ) = ( 1 4 Det W ( x , x ) [ Tr W ( x , x ) ] 2 ) 1 / 2 ,
W out ( x 1 , x 2 ) = T ( x 1 ) W in ( x 1 , x 2 ) T ( x 2 ) ,
W in ( x 1 , x 2 ) = E 0 2 exp ( x 1 2 + x 2 2 4 ε 2 ) [ cos 2 θ cos θ sin θ sin θ cos θ sin 2 θ ] ,
T 1 ( x ) = [ 1 0 0 exp [ i β 1 ( x ) ] ] ,
β 1 = π d λ ( n e n o )
β 1 ( x ) = φ 0 + φ ( x ) ,
p [ φ ( x ) ] = 1 2 π σ φ exp ( φ 2 ( x ) 2 σ φ 2 ) ,
φ ( x 1 ) φ ( x 2 ) = σ φ 2 exp ( ξ 2 2 γ 2 ) ,
W out ( x 1 , x 2 ) = E 0 2 exp ( x 1 2 + x 2 2 4 ε 2 )
× [ cos 2 θ exp ( i φ 0 ) exp ( σ φ 2 2 ) sin θ cos θ exp ( i φ 0 ) exp ( σ φ 2 2 ) sin θ cos θ exp { σ φ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] } sin 2 θ ] .
η out ( ξ ) = cos 2 θ + exp { σ φ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] } sin 2 θ ,
P out ( x ) = { 1 [ 1 exp ( σ φ 2 ) ] sin 2 2 θ } 1 / 2 .
η out ( ξ ) = exp { σ φ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] } ,
η out ( ξ ) = 1 2 + 1 2 exp { σ φ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] } .
Δ η out = 0 η out 2 ( ξ ) ,
T 2 ( x ) = [ exp [ i β 2 ( x ) ] 0 0 1 ] ,
β 2 ( x ) = φ 0 φ ( x ) ,
T ( x ) = T 2 ( x ) T 1 ( x ) = exp ( i φ 0 ) [ exp [ ( x ) ] 0 0 exp [ ( x ) ] ] .
W out ( x 1 , x 2 ) = E 0 2 exp ( x 1 2 + x 2 2 4 ε 2 ) exp ( σ φ 2 )
× [ exp [ σ φ 2 exp ( ξ 2 2 γ 2 ) ] sin 2 θ exp [ σ φ 2 exp ( ξ 2 2 γ 2 ) ] cos θ sin θ exp [ σ φ 2 exp ( ξ 2 2 γ 2 ) ] sin θ cos θ exp [ σ φ 2 exp ( ξ 2 2 γ 2 ) ] cos 2 θ ] .
η out ( ξ ) = exp { σ φ 2 [ 1 exp ( ξ 2 2 γ 2 ) ] } ,
P out ( x ) = { 1 [ 1 exp ( 4 σ φ 2 ) ] sin 2 2 θ } 1 / 2 .
μ ( x 1 , x 2 ) = ( Tr [ W ( x 1 , x 2 ) W ( x 1 , x 2 ) Tr W ( x 1 , x 1 ) Tr W ( x 2 , x 2 ) ) 1 / 2 .
μ out ( ξ ) = exp ( σ φ 2 )
× { exp [ 2 σ φ 2 exp ( ξ 2 2 γ 2 ) ] ( sin 4 θ + cos 4 θ ) + 2 exp [ 2 σ φ 2 exp ( ξ 2 2 γ 2 ) ] sin 2 θ cos 2 θ } 1 2 ,
μ out ( ξ ) = 1 2 exp ( σ φ 2 ) { exp [ 2 σ φ 2 exp ( ξ 2 2 γ 2 ) ] + exp [ 2 σ φ 2 exp ( ξ 2 2 γ 2 ) ] } 1 / 2 .
exp [ ± i φ ( x ) ] = 1 2 π σ φ exp [ ± i φ ( x ) ] exp ( φ 2 ( x ) 2 σ φ 2 ) .
exp ( π a 2 φ 2 ) exp ( ± i 2 πφ u ) = 1 a exp ( π u 2 a 2 ) ,
exp [ ± i φ ( x ) ] = exp ( σ φ 2 2 ) .
ψ ( x 1 , x 2 ) = φ ( x 2 ) ± φ ( x 1 ) .
p [ ψ ( x 1 , x 2 ) ] = 1 2 π σ ψ exp ( ψ 2 ( x 1 , x 2 ) 2 σ ψ 2 ) ,
σ ψ 2 = ψ 2 ( x 1 , x 2 ) = [ φ ( x 2 ) ± φ ( x 1 ) ] 2 ,
σ ψ 2 = 2 σ φ 2 [ 1 ± exp ( ξ 2 2 γ 2 ) ] .
exp { + i [ φ ( x 2 ) ± φ ( x 1 ) ] } = exp { i [ φ ( x 2 ) ± φ ( x 1 ) ] } = exp { σ φ 2 [ 1 ± exp ( ξ 2 2 γ 2 ) ] . }

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