Abstract

The effect on the Stokes parameters of a Gaussian Schell model beam on propagation in free space is studied experimentally and results are matched with the theory [X. H. Zhao, et al. Opt. Express 17, 17888 (2009)] that in general the degree of polarization of a Gaussian Schell model beam doesn’t change on propagation if the three spectral correlation widths δxx, δyy, δxy are equal and the beam width parameters σx=σy. It is experimentally shown that all the four Stokes parameters at the center of the beam decrease on propagation while the magnitudes of the normalized Stokes parameters and the spectral degree of polarization at the center of the beam remain constant for different propagation distances.

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References

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  1. X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain the state of polarization on the propagation in free space,” Opt. Express17(20), 17888–17894 (2009).
    [CrossRef] [PubMed]
  2. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett.32(23), 3400–3401 (2007).
    [CrossRef] [PubMed]
  3. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
    [CrossRef] [PubMed]
  4. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A11(5), 1641–1643 (1994).
    [CrossRef]
  5. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
    [CrossRef]
  6. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
    [CrossRef]
  7. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
    [CrossRef]
  8. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
    [CrossRef]
  9. O. Korotkova, “Sufficient condition for polarization invariance of beams generated by quasi-homogeneous sources,” Opt. Lett.36(19), 3768–3770 (2011).
    [CrossRef] [PubMed]
  10. E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
    [CrossRef] [PubMed]
  11. J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
    [CrossRef] [PubMed]
  12. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  13. X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
    [CrossRef]
  14. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
    [CrossRef] [PubMed]
  15. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
    [CrossRef]
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  17. R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956).
    [CrossRef]

2011

O. Korotkova, “Sufficient condition for polarization invariance of beams generated by quasi-homogeneous sources,” Opt. Lett.36(19), 3768–3770 (2011).
[CrossRef] [PubMed]

2009

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain the state of polarization on the propagation in free space,” Opt. Express17(20), 17888–17894 (2009).
[CrossRef] [PubMed]

2007

2006

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

2005

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
[CrossRef]

2004

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
[CrossRef]

2003

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
[CrossRef]

2001

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

1994

1986

E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

1956

R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Brown, R. H.

R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956).
[CrossRef]

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

James, D. F. V.

Korotkova, O.

O. Korotkova, “Sufficient condition for polarization invariance of beams generated by quasi-homogeneous sources,” Opt. Lett.36(19), 3768–3770 (2011).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
[CrossRef]

Liu, C.

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain the state of polarization on the propagation in free space,” Opt. Express17(20), 17888–17894 (2009).
[CrossRef] [PubMed]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
[CrossRef]

Pu, J.

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
[CrossRef]

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

Sun, Y.

X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain the state of polarization on the propagation in free space,” Opt. Express17(20), 17888–17894 (2009).
[CrossRef] [PubMed]

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

Twiss, R.

R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956).
[CrossRef]

Wolf, E.

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett.32(23), 3400–3401 (2007).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett.30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
[CrossRef]

E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

Yao, Y.

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain the state of polarization on the propagation in free space,” Opt. Express17(20), 17888–17894 (2009).
[CrossRef] [PubMed]

Zhao, X.

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

Zhao, X. H.

J. Mod. Opt.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt.52, 1611–1618 (2005).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001).
[CrossRef]

J. Opt. Commun. Netw.

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw.1(4), 307–312 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Nature

R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956).
[CrossRef]

Opt. Lett.

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova, “Sufficient condition for polarization invariance of beams generated by quasi-homogeneous sources,” Opt. Lett.36(19), 3768–3770 (2011).
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett.31(14), 2097–2099 (2006).
[CrossRef] [PubMed]

Opt. Commun.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
[CrossRef]

Phys. Rev. Lett.

E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986).
[CrossRef] [PubMed]

Waves Random Media

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media14(4), 513–523 (2004).
[CrossRef]

Other

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).

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Figures (4)

Fig. 1
Fig. 1

Gaussian beam propagating in free space along the z-axis. The planes at ξ , z = 0) and ( r , z) are source plane and observation plane respectively. Here ξ and r are two dimensional vectors.

Fig. 2
Fig. 2

Schematics of the experimental setup. M, NDF, QWP, CCD refers to the mirror, Neutral density filter, Quarter wave plate and CCD camera respectively. P1 and P2 are polarizers. The curve in the inset shows the intensity profile of cross section of the beam in the plane at z = 0. Black dots represent the experimental values while red curve represents its Gaussian fit. The FWHM of the Gaussian fit (red curve) is 0.583 mm.

Fig. 3
Fig. 3

Experimental setup for determining the transverse coherence length in the source plane at z = 0. BS is Beam splitter, P1, P2 are polarizers, T1 and T2 are tips of two single mode fibers, APD refers to Avalanche photodiode, TAC is time to amplitude converter, MCA is multichannel analyzer. Curve in the inset shows the plot of coincidence counts with the displacement between the two fiber tips T1 and T2. Square dots represent the experimental values while the red curve represents its Gaussian fit. The FWHM of the Gaussian fit (red curve) is 0.613 mm.

Fig. 4
Fig. 4

(a) Variation in the magnitude of the Stokes parameters on propagation of the Gaussian Schell model beam in free space. Dots represent the experimentally calculated values while thick curves represent theoretical curves obtained from Eqs. (9) and (7) respectively. The experimental parameters A 0 x / A 0 y = ( I 0 x / I 0 y ) 1 / 2 = 1.45 , | B x y | = 0.91, arg ( B x y ) = π / 2 , σ = 0.248mm, ω = 3 × 10 15 sec 1 , c = 3 × 10 8 m / sec , δ x x = δ y y = δ x y = δ = 0.260mm are put in Eqs. (5) and (6) for getting the theoretically expected results. (b) Variation of normalized Stokes parameters S1/S0, S2/S0 and S3/S0 on propagation of the beam in free space. Dots represent the experimentally calculated values while thick curves represent theoretical curves.

Equations (8)

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S i ( 0 ) ( ξ , ω ) = | A 0 i | 2 exp ( | ξ | 2 2 σ i 2 ) ,
for i = j , B i j = 1 i j , | B i j | 1 } ,
W i j ( 0 ) ( ξ 1 , ξ 2 ; ω ) = S i ( 0 ) ( ξ 1 , ω ) S j ( 0 ) ( ξ 2 , ω ) μ i j ( 0 ) ( ξ 1 , ξ 2 , ω ) .
W i j ( r 1 , r 2 , z ; ω ) = A 0 i A 0 j B i j Δ i j 2 ( z ) exp ( ( | r 1 | 2 + | r 2 | 2 ) 8 σ 2 Δ i j 2 ( z ) ) exp ( | r 2 r 1 | 2 8 δ i j 2 Δ i j 2 ( z ) ) exp ( i k ( | r 2 | 2 | r 1 | 2 ) 2 Φ i j 2 ( z ) ) ,
Δ i j 2 ( z ) = 1 + 1 ( k σ ) 2 ( 1 4 σ 2 + 1 δ i j 2 ) Φ i j ( z ) = ( 1 + 1 Δ i j 2 ( z ) ) z } .
S 0 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) + W y y ( r 1 , r 2 , ω ) S 1 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) S 2 ( r 1 , r 2 , ω ) = W x y ( r 1 , r 2 , ω ) + W y x ( r 1 , r 2 , ω ) S 3 ( r 1 , r 2 , ω ) = i [ W x y ( r 1 , r 2 , ω ) W y x ( r 1 , r 2 , ω ) ] } .
P ( r , z ) = ( S 1 S 0 ) 2 + ( S 2 S 0 ) 2 + ( S 3 S 0 ) 2 .
S 0 = I ( 0 , 0 ) + I ( 0 , 90 ) S 1 = I ( 0 , 0 ) I ( 0 , 90 ) S 2 = I ( 0 , 45 ) I ( 0 , 135 ) S 3 = I ( 45 , 45 ) I ( 45 , 135 ) } .

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