Abstract

In the free space optical communication system with circle polarization shift keying (CPolSK) modulation, the changes of polarization state of light beam have significant influence on the system performance. Keeping the state of polarization (SOP) unchanged on propagation can reduce the bit error rate. Based on the unified theory of coherence and polarization, we derive the sufficient condition for Gaussian Schell-model (GSM) beam to keep the SOP unchanged. We found that when the three spectral correlation widths (δxx, δyy and δxy) equal to each other and σxy, the GSM beam maintains the SOP on propagation. This conclusion can be helpful for the design of the transmitter in the CPolSK system.

©2009 Optical Society of America

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References

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  1. X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle Polarization Shift Keying with Direct Detection for Free Space Optical Communication,” J. Opt. Netw. ((to be published).
  2. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
    [Crossref]
  3. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5–6), 263–267 (2003).
    [Crossref]
  4. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
    [Crossref]
  5. Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
    [Crossref]
  6. X. Du and D. Zhao, “Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere,” Opt. Commun. 281(24), 5968–5972 (2008).
    [Crossref]
  7. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
    [Crossref]
  8. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32(23), 3400–3401 (2007).
    [Crossref]
  9. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
  10. X. Du and D. Zhao, “Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation,” Opt. Express 16(20), 16172–16180 (2008).
    [Crossref]
  11. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
    [Crossref]
  12. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett.28(13), 1078–1080 (2003).
    [Crossref]
  13. M. Born and E. Wolf, Principles of Optics, 7th expanded ed.(Cambridge U. Press, Cambridge, UK1999).
  14. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  15. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
    [Crossref]

2008 (5)

Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
[Crossref]

X. Du and D. Zhao, “Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere,” Opt. Commun. 281(24), 5968–5972 (2008).
[Crossref]

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

X. Du and D. Zhao, “Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation,” Opt. Express 16(20), 16172–16180 (2008).
[Crossref]

2007 (1)

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

2005 (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

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

2003 (1)

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

1994 (1)

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed.(Cambridge U. Press, Cambridge, UK1999).

Du, X.

X. Du and D. Zhao, “Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation,” Opt. Express 16(20), 16172–16180 (2008).
[Crossref]

X. Du and D. Zhao, “Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere,” Opt. Commun. 281(24), 5968–5972 (2008).
[Crossref]

James, D. F. V.

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[Crossref]

Korotkova, O.

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

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[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. Netw. ((to be published).

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Salem, M.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref]

Sun, Y.

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle Polarization Shift Keying with Direct Detection for Free Space Optical Communication,” J. Opt. Netw. ((to be published).

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

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

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

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

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

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett.28(13), 1078–1080 (2003).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th expanded ed.(Cambridge U. Press, Cambridge, UK1999).

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

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. Netw. ((to be published).

Zhao, D.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
[Crossref]

X. Du and D. Zhao, “Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere,” Opt. Commun. 281(24), 5968–5972 (2008).
[Crossref]

X. Du and D. Zhao, “Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation,” Opt. Express 16(20), 16172–16180 (2008).
[Crossref]

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. Netw. ((to be published).

Zhu, Y.

Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
[Crossref]

J. Opt. Soc. Am. A (2)

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[Crossref]

Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944–1948 (2008).
[Crossref]

Opt. Commun. (4)

X. Du and D. Zhao, “Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere,” Opt. Commun. 281(24), 5968–5972 (2008).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Opt. Express (1)

X. Du and D. Zhao, “Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation,” Opt. Express 16(20), 16172–16180 (2008).
[Crossref]

Opt. Lett. (3)

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref]

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

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[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 (4)

X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle Polarization Shift Keying with Direct Detection for Free Space Optical Communication,” J. Opt. Netw. ((to be published).

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett.28(13), 1078–1080 (2003).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th expanded ed.(Cambridge U. Press, Cambridge, UK1999).

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

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Figures (2)

Fig. 1.
Fig. 1. Behaviors of normalized Stokes parameters and the DOP along the z-axis in situation 1. The source is assumed to be Gaussian Schell-model source with parameters Ax =1.5, Ay =1, Bxy =0.3exp(/6), λ=532nm, σ=1cm, ρ=0cm, δxx =δyy =0.25mm, (A)δxy =0.25mm, (B)δxy =0.30mm, (C)δxy =0.35mm, (D)δxy =0.40mm, (E)δxy =0.45mm.
Fig. 2.
Fig. 2. Changes of normalized Stokes parameters and DOP on propagation in situation 2. The source is assumed to be Gaussian Schell-model source with parameters Ax =1.5, Ay =1, Bxy =0.3exp(/6), λ=532nm, σ=1cm, ρ=0cm, δxx =δxy =0.25mm, (A)δyy =0.25mm, (B)δyy =0.22mm, (C)δyy =0.20mm, (D)δyy =0.17mm, (E)δyy =0.14mm.

Equations (29)

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W(ρ1,ρ2,z=0;ω)[Wij(ρ1,ρ2,z=0;ω)]
=[Ei*(ρ1,z=0;ω)Ej(ρ2,z=0;ω)] , (i=x,y;j=x,y)
Wij(ρ1,ρ2,z;ω)=Wij(ρ1,ρ2,z=0;ω)×K(ρ1ρ1,ρ2ρ2,z;ω)d2ρ1d2ρ2,
K(ρ1ρ1,ρ2ρ2,z;ω)=G* (ρ1ρ1,z;ω) G (ρ2ρ2,z;ω) ,
G(ρρ,z;ω)=ikexp(ikρρ2(2z))(2πz) .
s0(ρ,z;ω)=Wxx(ρ,ρ,z;ω)+Wyy(ρ,ρ,z;ω)
s1(ρ,z;ω)=Wxx(ρ,ρ,z;ω)Wyy(ρ,ρ,z;ω)
s2(ρ,z;ω)=Wxy(ρ,ρ,z;ω)+Wyx(ρ,ρ,z;ω)
s3(ρ,z;ω)=i[Wyx(ρ,ρ,z;ω)Wxy(ρ,ρ,z;ω)].
𝒫(ρ,z,ω)=s12+s22+s32s02=14DetW(ρ,ρ,z;ω)[TrW(ρ,ρ,z;ω)]2 ,
si(ρ,z;ω)=si(ρ,z;ω)s0(ρ,z;ω). (i=1,2,3)
Wij(ρ1,ρ2,z=0;ω)=AiAjBij×exp[(ρ124σi2+ρ224σj2)]×exp[(ρ2ρ1)22δij] .
Bij=1 (i=j) , Bij1 (ij) , Bij=Bji* , and δij=δji
s1(ρ,z=0;ω)=Wxx(ρ,ρ,z=0;ω)Wyy(ρ,ρ,z=0;ω)Wxx(ρ,ρ,z=0;ω)+Wyy(ρ,ρ,z=0;ω)=Ax2Ay2Ax2+Ay2 ,
s2 (ρ,z=0;ω) = Wxy(ρ,ρ,z=0;ω)Wyx(ρ,ρ,z=0;ω)Wxx(ρ,ρ,z=0;ω)+Wyy(ρ,ρ,z=0;ω) = AxAyBxy+AxAyByxAx2+Ay2
s3(ρ,z=0;ω)=i[Wyx(ρ,ρ,z=0;ω)Wxy(ρ,ρ,z=0;ω)]Wxx(ρ,ρ,z=0;ω)+Wyy(ρ,ρ,z=0;ω)=i[AxAyAyxAxAyBxy]Ax2+Ay2 ,
𝒫 (ρ,z=0;ω)(Ax2Ay2)2+4Bxy2Ax2Ay2(Ax2+Ay2)2.
Wij (ρ1,ρ2,z;ω) = AiAjBijΔij2(z) exp [(ρ1+ρ2)28σ2Δij2(z)]×exp[(ρ2ρ1)22Ωij2Δij2(z)]×exp[ik(ρ22ρ12)2Rij(z)],
Rij (z)=z[1+(kσΩijz)2],1Ωij2=14σ2+1δij2andΔij(z)=1+(zkσΩij)2.
Wij (ρ,ρ,z;ω)=AiAjBijΔij2(z)exp[ρ22σ2Δij2(z)].
s1 (ρ,z;ω)=Wxx(ρ,ρ,z;ω)Wyy(ρ,ρ,z;ω)Wxx(ρ,ρ,z;ω)+Wyy(ρ,ρ,z;ω)=Ax2Δxx2exp(ρ22σ2Δxx2)Ay2Δyy2exp(ρ22σ2Δyy2)Ax2Δxx2exp(ρ22σ2Δxx2)+Ay2Δyy2exp(ρ22σ2Δyy2),
s2 (ρ,z;ω)=Wxy(ρ,ρ,z;ω)+Wyx(ρ,ρ,z;ω)Wxx(ρ,ρ,z;ω)+Wyy(ρ,ρ,z;ω)=AxAyBxyΔxy2exp(ρ22σ2Δxy2)+AxAyByxΔyx2exp(ρ22σ2Δyx2)Ax2Δxx2exp(ρ22σ2Δxx2)+Ay2Δyy2exp(ρ22σ2Δyy2),
s3 (ρ,z;ω)=i[Wyx(ρ,ρ,z;ω)Wxy(ρ,ρ,z;ω)]Wxx(ρ,ρ,z;ω)+Wyy(ρ,ρ,z,ω)=i[AxAyByxΔyx2exp(ρ22σ2Δyx2)AxAyBxyΔxy2exp(ρ22σ2Δxy2)]Ax2Δxx2exp(ρ22σ2Δxx2)+Ay2Δyy2exp(ρ22σ2Δyy2),
𝒫(ρ,z;ω)=14[Ax2Ay2Δxx2Δyy2exp[ρ22σ2(1Δxx2+1Δyy2)]Ax2Ay2Bxy2Δxy2Δyx2exp[ρ22σ2(1Δxy2+1Δyx2)]][Ax2Δxx2exp[ρ22σ2Δxx2]+Ay2Δyy2exp[ρ22σ2Δyy2]]2 .
si(ρ,z=0;ω)=si(ρ,z;ω),i=1,2,3
δxx=δyy=δxy .
σx=σyandδxx=δyy=δxy ,
max{δxx,δyy}δxymin{δxxBxy,δyyBxy} .
s1(ρ,z;ω)=(Ax2Ay2) (Ax2+Ay2) .

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