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

1. Introduction

Propagation of light beam in free space involves many research fields such as satellite optical communication, deep space optical relay and astronomical observation. It has received special attention and extensive investigation. We proposed a polarization modulation scheme called circle polarization shift keying (CPolSK) in recent work [1]. For the free space optical communication system with CPolSK modulation, it is very important to study the changes of polarization properties on propagation. Traditionally, the polarization properties of the electromagnetic beams are considered to be unchanged as beams propagate in free space or in any linear medium. However, James demonstrated in 1994 that the degree of polarization (DOP) of a partially coherent light beam may experience changes on propagation, even in free space [2]. Later, Wolf proposed the unified theory of coherence and polarization [3]. Based on this theory, the generalized Stokes parameters were defined and used to study the propagation of light beam [46]. Meanwhile, the changes of the DOP of the partially polarized light on propagation in free space were studied [7]. Sufficient conditions were obtained for the DOP of a light beam to keep invariance throughout the far zone and across the source plane [810].

The researches of the light propagation mainly concentrate on the DOP recently. Studies of the state of polarization (SOP) are restricted to the expression for the SOP of the partially polarized light and the propagations of some particular examples [11]. According to the author’s knowledge, it has not been studied profoundly yet for the condition of polarization state invariance. Taking the Gaussian Schell-model (GSM), we calculate three normalized Stokes parameters based on the unified theory of coherence and polarization. Then the condition of polarization state invariance is obtained by theoretical analysis.

2. Theoretical derivation

Considering a stochastic, statistically stationary, electromagnetic beam generated by a source located in the plane z=0 (called the source plane), propagating close to the z direction and into the half-space z>0. The coherence and polarization properties of this beam can be characterized by the 2×2 cross-spectral density matrix (CSDM) [3]:

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)

where the asterisk denotes the complex conjugate and the angular brackets mean the ensemble average. x and y are two mutually orthogonal directions perpendicular to the beam axis. The elements of CSDM in the plane z=constant>0 can be obtained by the formula [12]:

Wij(ρ1,ρ2,z;ω)=Wij(ρ1,ρ2,z=0;ω)×K(ρ1ρ1,ρ2ρ2,z;ω)d2ρ1d2ρ2,

where

K(ρ1ρ1,ρ2ρ2,z;ω)=G*(ρ1ρ1,z;ω)G(ρ2ρ2,z;ω),
G(ρρ,z;ω)=ikexp(ikρρ2(2z))(2πz).

The Stokes parameters, which include the information of both the DOP and the SOP, can be expressed in terms of the elements of the CSDM by the formulas [4]:

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)].

Then, the degree of polarization can be expressed by [3]

𝒫(ρ,z,ω)=s12+s22+s32s02=14DetW(ρ,ρ,z;ω)[TrW(ρ,ρ,z;ω)]2,

where Det and Tr denote the determinant and trace of the matrix, respectively. It is helpful to define the normalized Stokes parameters as:

si(ρ,z;ω)=si(ρ,z;ω)s0(ρ,z;ω).(i=1,2,3)

Then the SOP can be expressed as a point s1,s2,s3 in the Poincaré sphere [13]. If the three normalized Stokes parameters don’t change, the SOP will keep invariance.

For the GSM beam, the CSDM in the source plane z=0 has elements [14]:

Wij(ρ1,ρ2,z=0;ω)=AiAjBij×exp[(ρ124σi2+ρ224σj2)]×exp[(ρ2ρ1)22δij].

The parameters Ai, Bij, σi, δij are independent of position and they satisfy the expressions [14]:

Bij=1(i=j),Bij1(ij),Bij=Bji*,andδij=δji

Assuming σ x=σ y=σ, the normalized Stokes parameters and the SOP in the source plane can be expressed as

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.

The elements of CSDM at a pair of points in any cross-sectional plane z=constant>0 are given by the expression [14]

Wij(ρ1,ρ2,z;ω)=AiAjBijΔij2(z)exp[(ρ1+ρ2)28σ2Δij2(z)]×exp[(ρ2ρ1)22Ωij2Δij2(z)]×exp[ik(ρ22ρ12)2Rij(z)],

where

Rij(z)=z[1+(kσΩijz)2],1Ωij2=14σ2+1δij2andΔij(z)=1+(zkσΩij)2.

In expressions above, k=2π/λ is the wave number of light beam. When the two points coincide, Eq. (12) simplifies to

Wij(ρ,ρ,z;ω)=AiAjBijΔij2(z)exp[ρ22σ2Δij2(z)].

Then the normalized Stokes parameters and the DOP in the plane z=constant>0 can be expressed as

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.

If the normalized Stokes parameters satisfy

si(ρ,z=0;ω)=si(ρ,z;ω),i=1,2,3

the SOP doesn’t change on propagation. Solving equations above, it is obtained

δxx=δyy=δxy.

From the derivation above, we conclude that when the parameters of GSM beam satisfy

σx=σyandδxx=δyy=δxy,

the normalized Stokes parameters will keep unchanged along the distance. So Eq. (19) is the sufficient condition for GSM beam to maintain the SOP on propagation in free space. Under this condition, there is no polarization state variation induced by the degree coherence and the spectral density. The use of light source under this condition is helpful for the simplification of polarization detection in CPolSK modulation system.

3. Analysis and discussion

In this section, propagations of GSM beams with different parameters are calculated. Changes of the SOP and the DOP along the distance z are obtained and the condition of polarization state invariance is exemplified.

For a GSM beam, the spectral correlation widths δij must correspond the realizable conditions as [15]

max{δxx,δyy}δxymin{δxxBxy,δyyBxy}.

Considering there are three variables involved in Eq. (18), we select two situations with different beam parameters. For the first situation, we suppose δxx=δyy and change the value of cross spectral correlation width δxy. Variations of normalized Stokes parameters and the DOP along with propagation distance are calculated. In situation 2, δxx=δxy are assigned and δyy is changed. The results of two situations are shown in Fig. 1 and Fig. 2, respectively.

 

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.

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Figure 1 shows the behaviors of three normalized Stokes parameters and the DOP as functions of the distance z along the z-axis (ρ=0) in situation 1. When the parameters of source satisfy Eq. (19) such as beam (A), the normalized Stokes parameters and DOP keep unchanged during the propagation. For the other beams, s2,s3 and DOP increase with the distance z. The significant differences between δxy and δxx lead to quicker increments. It should be noted that curves of five beams’ s1 are superposed and only one straight line can be seen in Fig. 1 (a). That is because when δxx=δyy the normalized Stokes parameter s1 in Eq. (15) can be simplified as

s1(ρ,z;ω)=(Ax2Ay2)(Ax2+Ay2).

Obviously, the result is independent of z and δxy.

The changes of both SOP and DOP of GSM beams along the z-axis in situation 2 are shown in Fig. 2. Satisfying the condition of polarization state invariance, both the SOP and the DOP of beam (A) don’t change on propagation. For the other beams, all the three normalized Stokes parameters and the DOP increase along the distance z.

The results show that the SOP and DOP keep unchanged for the beam satisfied Eq. (19). For the other beams, the normalized Stokes parameters and DOP increase along the distance z and they ultimately tend to certain values. These values are determined by parameters of GSM beams. Significant differences between the three spectral correlation widths induce noticeable differences between ultimate values and initial values. Because we have supposed two of the three spectral correlation widths are equal in calculations, results show monotone increasing trends of normalized Stokes parameters and DOP. If the three spectral correlation widths are mutually unequal to each other, downward trends can be obtained, such as some examples in [11].

 

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.

Download Full Size | PPT Slide | PDF

In the polarization modulation schemes such as CPolSK, the changes of polarization state on propagation induce the increase of bit error rate. If we design the light source according with Eq. (19), there will be no polarization state change even through long distance propagation. Consequently, the reliability of the polarization modulation system can be greatly improved. The results are helpful for the wide extensive use of the free space optical communication system in telecommunication links.

4. Conclusion

Based on the unified theory of coherence and polarization, we derive the sufficient condition for GSM beam to maintain the SOP on propagation in free space. The condition states that the three spectral correlation widths equal to each other and σxy. For GSM beams satisfied this condition, both the SOP and the DOP keep unchanged on propagation. If the condition is not satisfied, normalized Stokes parameters and the DOP experience variations and ultimately tend to certain values. These results are useful for the design of light source in the free space optical communication system with polarization modulation.

Acknowledgments

This work is financially supported by Guangdong Natural Science Foundation (8151805707000004) and Development Program for Outstanding Young Teachers in Harbin Institute of Technology (HITQNJS.2008.60). The authors appreciate the help of Key Laboratory of Network Oriented Intelligent Computation, HIT.

References and links

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]  

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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