Abstract

We generate a stochastic electromagnetic beam (SEB) with complete controllable coherence, that is, the coherence degree can be controlled independently along two mutually perpendicular directions. We control the coherence of the SEB by adjusting the phase modulation magnitude applied onto two crossed phase only spatial light modulators. We measure the beam’s coherence properties using Young’s interference experiment, as well as the beam propagation factor. It is shown that the experimental results are consistent with our theoretical predictions.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  30. G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2009).

2015 (1)

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

2014 (1)

2013 (5)

2012 (3)

2011 (1)

2009 (4)

A. S. Ostrovsky, G. Martínez-Niconoff, V. Arrizón, P. Martínez-Vara, M. A. 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]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2009).

2007 (1)

2005 (2)

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]

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

2004 (1)

2003 (2)

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

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

2002 (3)

2000 (1)

1991 (1)

1963 (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5(3), 191–192 (1963).
[Crossref]

Arrizón, V.

Burger, L.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Cai, Y.

Chen, Z.

Chu, X.

Cui, S.

Davidson, F. M.

Forbes, A.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Friberg, A.

Friberg, A. T.

Gbur, G.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Goudail, F.

Ji, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5(3), 191–192 (1963).
[Crossref]

Korotkova, O.

Lajunen, H.

Li, Q.

Liang, C.

Litvin, I.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Liu, X.

Martínez-Niconoff, G.

Martínez-Vara, P.

Mei, Z.

Meng, P.

Ngcobo, S.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Olvera-Santamaría, M. A.

Ostrovsky, A. S.

Panezai, S.

Pu, J.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[Crossref]

Réfrégier, P.

Rickenstorff-Parrao, C.

Ricklin, J. C.

Rong, L.

Rosen, J.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Setala, T.

Setälä, T.

Shchepakina, E.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 25705 (2013).
[Crossref]

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[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 screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[Crossref] [PubMed]

Takeda, M.

Tervo, J.

Tong, Z.

Turunen, J.

Vasara, A.

Wang, D.

Wang, F.

Wang, Y.

Wolf, E.

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 screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[Crossref] [PubMed]

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

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref] [PubMed]

Zhang, B.

Zhang, L.

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Zhou, G.

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2009).

Appl. Opt. (1)

J. Opt. (2)

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 25705 (2013).
[Crossref]

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

J. Opt. A, Pure Appl. Opt. (4)

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]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2009).

J. Opt. Soc. Am. A (6)

Opt. Express (4)

Opt. Lett. (8)

Phys. Lett. (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5(3), 191–192 (1963).
[Crossref]

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)

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

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

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).

Supplementary Material (1)

NameDescription
» Visualization 1: AVI (7416 KB)      Dynamic pattern with modulation magnitude 0.5

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Figures (7)

Fig. 1
Fig. 1 An optical arrangement for producing SEB. L1, L2, lenses; HWP, half-wave-plate; NPBS, non-polarizing beam splitter; P1, P2, polarizers; SLM1, SLM2, spatial light modulators; PC1, PC2, computers.
Fig. 2
Fig. 2 One frame of dynamic patterns with modulation magnitude γ of (a) 0, (b) 0.2, (c) 0.5, (d) 1. (see Visualization 1).
Fig. 3
Fig. 3 (a) Filled circles and diamonds represent the measured μ x x and μ y y , respectively; Short dash line and solid line represent the fitting curve of μ x x and μ y y ; Dotted line represents the theoretical simulation of μ j j . (b) Measured M2 of the x-polarized light versus different modulation magnitude.
Fig. 4
Fig. 4 Coherence degree versus the distance of the two pinholes.
Fig. 5
Fig. 5 Experimental setup of modified Mach-Zehnder interferometer. PBS, polarizing beam-splitter. A1, A2, knife-edges. M1, M2, mirrors.
Fig. 6
Fig. 6 Coherence degree as a function of the modulation magnitude γ x when γ y is equal to (a) 0.4 and (b) 0.6, respectively.
Fig. 7
Fig. 7 M2 versus modulation magnitude γ x when γ y is equal to (a) 0, (b) 0.4, (c) 0.6, and (d) 1, respectively.

Equations (16)

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W ( r 1 , r 2 , ω ) = ( W x x ( r 1 , r 2 , ω ) W x y ( r 1 , r 2 , ω ) W y x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) ) ,
W j k ( r 1 , r 2 , ω ) = E j * ( r 1 , ω ) E k ( r 2 , ω ) ( j = x , y ; k = x , y ),
μ j k ( r 1 , r 2 ) = W j k ( r 1 , r 2 ) W j j ( r 1 ) W k k ( r 2 ) .
μ j k ( r 1 , r 2 ) = exp { i [ ϕ j ( r 1 ) ϕ k ( r 2 ) ] } .
W ( r 1 , r 2 ) = A 1 A 2 ( μ x x ( r 1 , r 2 ) μ x y ( r 1 , r 2 ) μ y x ( r 1 , r 2 ) μ y y ( r 1 , r 2 ) ) = A 1 A 2 ( exp { - i [ ϕ x ( r 1 ) ϕ x ( r 2 ) ] } exp { - i [ ϕ x ( r 1 ) ϕ y ( r 2 ) ] } exp { - i [ ϕ y ( r 1 ) ϕ x ( r 2 ) ] } exp { - i [ ϕ y ( r 1 ) ϕ y ( r 2 ) ] } ) .
p [ ϕ j ( r ) ] = { 1 2 π γ j ( r ) for π π γ j ( r ) < ϕ j ( r ) < π + π γ j ( r ) 0 otherwise , ( j = x , y )
exp [ i ϕ j ( r ) ] = exp [ i ϕ j ( r ) ] p [ ϕ j ( r ) ] d ϕ j = 1 2 π γ j ( r ) π π γ j ( r ) π + π γ j ( r ) exp [ i ϕ j ( r ) ] d ϕ j , = sin c [ γ j ( r ) ]
exp [ i ϕ j ( r ) ] = sin c [ γ j ( r ) ] ,
exp [ i ϕ j ( r 1 ) ] exp [ i ϕ k ( r 2 ) ] = { 1 for r 1 = r 2 and j = k exp [ -i ϕ j ( r 1 ) ] exp [ i ϕ k ( r 2 ) ] otherwise .
μ j k ( r 1 , r 2 ) = { 1 for r 1 = r 2 and j = k ( j = x , y ; k = x , y ) sin c [ γ j ( r 1 ) ] sin c[ γ k ( r 2 ) ] otherwise .
μ j k ( r 1 , r 2 ) = { 1 for r 1 = r 2 and j = k sin c [ γ j ] sin c [ γ k ] otherwise .
W ( r 1 , r 2 ) = A 1 A 2 ( [ sin c ( γ x ) ] 2 sin c ( γ x ) sinc ( γ y ) sin c ( γ y ) sin c ( γ x ) [ sin c ( γ y ) ] 2 ) ,
W ( r , r ) = A 1 A 2 ( 1 sin c ( γ x ) sin c ( γ y ) sin c ( γ y ) sin c ( γ x ) 1 )
η ( r 1 , r 2 ) = Tr W ( r 1 , r 2 ) [ Tr W ( r 1 , r 1 )Tr W ( r 2 , r 2 ) ] 1/2 = [ sin c(β x )] 2 +[ sin c(β y )] 2 2 .
P ( r ) = ( 1- 4det W ( r , r ) [ Tr W ( r , r ) ] 2 ) 1/2 = sin c(β x ) sin c(β y ) .
μ 12 = [ Tr ( W ( r 1 , r 2 ) W ( r 1 , r 2 ) ) I 1 I 2 ] 1 / 2 .

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