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Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 16172–16180
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Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation

Xinyue Du and Daomu Zhao  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 16172-16180 (2008)
http://dx.doi.org/10.1364/OE.16.016172


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Abstract

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized are derived. We find that both the completely unpolarized stochastic electromagnetic Gaussian Schell-model beam and the completely polarized stochastic electromagnetic Gaussian Schell-model beam will keep their spectral degree of polarization or become partially polarized under different constraint conditions during their propagation in free space or through turbulent atmosphere. We give necessary theoretical explanation to the physical phenomena. They are considered as coherence-induced polarization changes and spectral density-induced polarization changes.

© 2008 Optical Society of America

1. Introduction

It has been known that polarization properties of stochastic electromagnetic beams, in particular, their spectral degree of polarization, can experience changes on propagation in free space [1

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

3

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

] and through turbulent atmosphere [4

4. 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]

6

6. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

]. We have also studied the changes in polarization of stochastic electromagnetic beams propagating through axially nonsymmetrical optical systems [7

7. X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711–2715 (2008). [CrossRef]

] and misaligned optical systems [8

8. X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773–779 (2008). [CrossRef]

].

Wolf has shown in his latest work that under sufficiency conditions the degree of polarization of a beam may be the same throughout the far zone and in the source plane [9

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

], and a light beam can be considered as the sum of a completely polarized and a completely unpolarized beam [10

10. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008). [CrossRef] [PubMed]

]. We are interested in these conclusions and put question to the propagation of completely unpolarized beams and completely polarized beams: will they keep their spectral degree of polarization or become partially polarized?

Recently the polarization invariance of the stochastic electromagnetic Gaussian Schell-model beam has been studied [11

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

], Salem and Wolf have also demonstrated that the coherence of the field in the source plane affects the polarization of the beam on propagation [12

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

]. But both of these investigations are performed under free space propagation condition and the stochastic electromagnetic beams are assumed to be generated by partially polarized sources.

In this paper we analyze the cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources, then the completely unpolarized source and the completely polarized source are given. Our purpose is to find out the key that leads the completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams to be partially polarized ones during their propagation not only in free space but also through turbulent atmosphere. Some typical examples are illustrated and necessary explanations are also given for the physical phenomena. All the work will be done within the framework of the paraxial approximation.

Fig. 1. Illustrating the notation.

2. Theoretical analysis

Let us consider a stochastic, statistically stationary, electromagnetic beam generated by a source located in the plane z=0 and propagating into the positive half-space z>0 (see Fig. 1). The second-order correlation properties of such a beam at a pair of points r 1, r 2 can be characterized by the 2×2 electric cross-spectral density matrix [13

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

]

W(r1,r2,ω)[Wij(r1,r2,ω)]=[Ei*(r1,ω)Ej(r2,ω)],(i=x,y;j=x,y),
(1)

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. E=(Ex, Ey) is a statistical ensemble of the fluctuating component of the transverse electric field.

In terms of the cross-spectral density matrix W ⃡(0) (ρ1,ρ2 ω) of the source, the cross-spectral density matrix of the beam at r 1≡(ρ 1, z) and r 2≡(ρ 2,z) in any plane z=const. can be obtained by the formula [14

14. 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, 1078–1080 (2003). [CrossRef] [PubMed]

]

W(ρ1,ρ2,z,ω)=W(0)(ρ1,ρ2,ω)K(ρ1ρ1,ρ2ρ2,z,ω)d2ρ1d2ρ2,
(2)

where

K(ρ1ρ1,ρ2ρ2,z,ω)=G*(ρ1ρ1,z,ω)G(ρ2ρ2,z,ω),
(3)

and G(ρ-ρ′, z,ω) is the Green’s function of the Helmholtz operator for paraxial propagation in free space. If the positive half-space is a medium, which is linear and random but static such as the turbulent atmosphere, Eq. (3) should be replaced by

Krm(ρ1ρ1,ρ2ρ2,z,ω)=Gm*(ρ1ρ1,z,ω)Gm(ρ2ρ2,z,ω)rm,
(4)

where 〈…rm〉 implies that the average is taken over the ensemble of the random medium.

To judge the completely unpolarized or completely polarized beams propagating in free space or through turbulent atmosphere, we should use the formula of the spectral degree of polarization at the point (ρ, z) [13

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

, 14

14. 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, 1078–1080 (2003). [CrossRef] [PubMed]

]:

P(ρ,ρ,z,ω)=14DetW(ρ,ρ,z,ω)[TrW(ρ,ρ,z,ω)]2,
(5)

where Det and Tr denote the determinant and the trace of the matrix, respectively.

For the completely unpolarized source, the cross-spectral density matrix can be expressed by [10

10. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008). [CrossRef] [PubMed]

]

W(0)(ρ1,ρ2,ω)=a(ρ1,ρ2,ω)[1001].
(6)

For the completely polarized source, the cross-spectral density matrix has the form [10

10. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008). [CrossRef] [PubMed]

]

W(0)(ρ1,ρ2,ω)=[ex*(ρ1,ω)ex(ρ2,ω)ex*(ρ1,ω)ey(ρ2,ω)ey*(ρ1,ω)ex(ρ2,ω)ey*(ρ1,ω)ey(ρ2,ω)],
(7)

where ex and ey are of the general forms of e 1 and e 2 which are “one point” constants shown in Ref. [15

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

], Eq. (6.3–12). The sources whose cross-spectral density matrixes are given by Eqs. (6) or (7) can be proved as completely unpolarized P(ρ′,ρ′, z=0,ω)=0 or completely polarized P(ρ′,ρ′, z=0,ω)=1 with the help of Eq. (5).

We now consider stochastic electromagnetic beams generated by electromagnetic Gaussian Schell-model sources. The elements of the cross-spectral density matrix for such sources are given by [3

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

5

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, 225–230 (2004). [CrossRef]

]

Wij(0)(ρ1,ρ2,ω)=AiAjBijexp(ρ124σi2ρ224σj2)exp(ρ2ρ122δij2),
(8)

where the coefficients Ai, Aj, Bij and the variances σi, σj, δij are independent of position but may depend on frequency (Ref. [15], Sec. 5.3.2), and Bii≡1. In order to calculate the spectral degree of polarization in the source plane, Eq. (8) can be simplified into one point (ρ′, z=0) as

Wij(0)(ρ,ρ,ω)=AiAjBijexp(ρ24σi2ρ24σj2).
(9)

For the completely unpolarized source, the parameters should be chosen as Ax=Ay=A, Bxy=0, σx=σy=σ, and the matrix is obtained as

W(0)(ρ,ρ,ω)=A2exp(ρ22σ2)[1001].
(10)

The spectral correlation width δxx and δyy can be selected freely. For the completely polarized source, we should choose |Bxy|=1 to make BxyByx=BxxByy. The spectral correlation width δij must correspond the realizable conditions as [16

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

]

max{δxx,δyy}δxymin{δxxBxy,δyyBxy},
(11)

so δxx, δyy and δxy should be equal. Other parameters can be selected freely. The matrix is then obtained as

W(0)(ρ,ρ,ω)=[AxAxBxxexp(ρ24σx2ρ24σx2)AxAyBxyexp(ρ24σx2ρ24σy2)AyAxByxexp(ρ24σy2ρ24σx2)AyAyByyexp(ρ24σy2ρ24σy2)].
(12)

The beam sources characterized by Eqs. (10) or (12) can also be proved as uniformly completely unpolarized or uniformly completely polarized with the help of Eq. (5).

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized have been derived, but what will happen on propagation? By using the tensor method, we have investigated the propagation of general stochastic electromagnetic Gaussian Schell-model beams through turbulent atmosphere in our recent work [6

6. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

] and obtained the following analytical expression for the elements of the cross-spectral density matrix as

Wij(ρ12,z,ω)=AiAjBij[Det(I¯+BP¯+B¯Mij1)]12exp{ik2ρ12T[(B¯1+P¯),
(B¯112P¯)T(B¯1+P¯+Mij1)1(B¯112P¯)]ρ12}
(13)

where k=2π/λ is the wave number, ρ 12 T=(ρ 1 T,ρ 2 T)=(x 1,y 1,x 2,y 2) is a four-dimensional vector in the output plane, and T means the matrix transpose operation. Mij -1 is a 4×4 complex matrix characterized by [6

6. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

8

8. X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773–779 (2008). [CrossRef]

]

Mij1=[i2kσi2ikδij20ikδij200i2kσi2ikδij20ikδij2ikδij20i2kσj2ikδij200ikδij20i2kσj2ikδij2].
(14)

Ī is a 4×4 unitary matrix, and

B¯=[zI00zI],P¯=2ikρ02[IIII],
(15)

where I is a 2×2 unitary matrix. ρ 0(0.545C 2 n k 2 z)-3/5 is the coherence length of a spherical wave propagating in the turbulent medium based on Kolmogorov spectrum model, and 2 n C is the structure parameter of the refractive index. If C 2 n=0, we can obtain that P̄=0 and Eq. (13) can be simplified as

Wij(ρ12,z,ω)=AiAjBij[Det(I¯+B¯Mij1)]12
×exp{ik2ρ12T[B¯1B¯1T(B¯1+Mij1)1B¯1]ρ12},
(16)

which represents the condition of propagating in free space.

With the help of Eqs. (16), (13) and (5), we can investigate the changes in the spectral degree of polarization P of stochastic electromagnetic Gaussian Schell-model beams generated by completely unpolarized or completely polarized sources on propagation in free space or through turbulent atmosphere.

3. Numerical calculations and discussions

In this section the criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation will be derived from both numerical calculation and theoretical analysis.

In Fig. 2, the source is assumed to be uniformly completely unpolarized electromagnetic Gaussian Schell-model source and the beam is propagating in free space. The profile of P across the plane y=0 is chosen, so it is a function of x-coordinate. We can find in Fig. 2(a) that the spectral degree of polarization at the initial plane has the same extreme value, zero. If δxx=δyy, P will not change when the beam propagates in free space as shown by the solid lines, so we can keep the completely unpolarized beam. This case that P dose not change is in accord with the condition derived in Ref. [11

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

]. If δxxδyy, P will have different distributions at different propagation distances but become changeless in the far-field. Under the condition of δxxδyy, the beam generated by a completely unpolarized source becomes partially polarized on propagation, and it can be considered as coherence-induced polarization changes [12

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

].

Fig. 2. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely unpolarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax=Ay=1, Bxy=0, σx σy,=1cm, δxx=2mm, and δyy is shown in the figure.

In Fig. 3, the beam generated by completely unpolarized electromagnetic Gaussian Schell-model source is passing through turbulent atmosphere. The distributions of P at the initial plane and the near-field shown in Fig. 3(a) and (b) are the same as those shown in Fig. 2(a) and (b), but in the far-field there will be obvious differences. It depends on the effect of the atmospheric turbulence, which occurs with the increase of the propagation distance. We find that, in the propagation environment of turbulent atmosphere, the criterion on δxx=δyy for keeping the completely unpolarized beam is the same as the condition of free space.

Fig.3. . s Fig. 2, but passing through the turbulent atmosphere with C 2 n=10-12 m-2/3.

We will give the necessary theoretical explanation to the physical phenomena shown in Figs. 2 and 3, especially under the condition of the beam propagating through turbulent atmosphere. For the completely unpolarized stochastic electromagnetic beam, if the parameters are chosen as Ax=Ay=A, Bxy=0, σx=σy, and δxx=δyy, we can find from Eq. (14) that Mxx -1=Myy -1, and if the function Fij(ρ 12, z, ω) is defined as

Fij(ρ12,z,ω)=[Det(I¯+BP¯+B¯Mij1)]12exp{ik2ρ12T[(B¯1+P¯),
(B¯112P¯)T(B¯1+P¯+Mij1)1(B¯112P¯)]ρ12}
(17)

it is obvious that Fxx(ρ 12,z,ω)=Fyy(ρ 12,z,ω)=F(ρ 12,z,ω), then Eq. (13) can be expressed briefly as

Wxx(ρ12,z,ω)=Wyy(ρ12,z,ω)=A2F(ρ12,z,ω),
Wxy(ρ12,z,ω)=Wyx(ρ12,z,ω)=0.
(18)

On substituting from Eq. (18) into the formula (5), we obtain

P(ρ,ρ,z,ω)=14DetW(ρ,ρ,z,ω)[TrW(ρ,ρ,z,ω)]2
=Wxx(ρ12,z,ω)Wyy(ρ12,z,ω)Wxx(ρ12,z,ω)+Wyy(ρ12,z,ω),
=0
(19)

where ρ 12 T=(ρ T, ρ T). The spectral degree of polarization is the same at every point, and the criterion δxx=δyy for keeping completely unpolarized stochastic electromagnetic beam is derived theoretically. The instance is the same when the beam propagates in free space described by Eq. (16).

Fig. 4. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely polarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax=2, Ay=1, Bxy=exp(/3), σx=1cm, δxx=δyy=δxy=2mm, and σy is shown in the figure.

In Fig. 4, the source is assumed to be uniformly completely polarized electromagnetic Gaussian Schell-model source and Fig. 4(a) shows that the spectral degree of polarization at the initial plane has the other extreme value, one. If σx=σy, P will not change when the beam propagates in free space, so we can keep the completely polarized beam. If σxσy, the beam will become partially polarized on propagation as shown in Fig. 4(b)–(d). It is known that the parameter σi represents the effective width of the spectral density, i.e. that [3

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

5

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, 225–230 (2004). [CrossRef]

]

Si(0)(ρ',ω)=Ai2exp(ρ'22σi2),(i=x,y).
(20)

The numerical examples indicate that spectral density properties of the electromagnetic field at the source plane affect the polarization properties of the beam on propagation. We may describe this phenomenon as spectral density-induced polarization changes.

In Fig. 5, we illustrate the beam generated by completely polarized electromagnetic Gaussian Schell-model source passing through turbulent atmosphere. The spectral density-induced polarization changes are also existent.

The theoretical explanation for the physical phenomena shown in Figs. 4 and 5 will also be given. For the completely polarized stochastic electromagnetic beam, if the parameters are chosen en as |Bxy|=1, δxx=δyy=δxy, and σx=σy, we can find from Eq. (14) that Mxx -1=Mxy -1=Myx -1=Myx -1=Myy -1, then Fxx(ρ 12,z,ω=Fyx(ρ 12,z,ω=Fyy(ρ 12,z,ω)=F(ρ 12,z,ω), Eq. (13) can be expressed briefly as

Wxx(ρ12,z,ω)=AxAxBxxF(ρ12,z,ω),
Wxy(ρ12,z,ω)=AxAyBxyF(ρ12,z,ω),
Wyx(ρ12,z,ω)=AyAxByxF(ρ12,z,ω),
Wyy(ρ12,z,ω)=AyAyByyF(ρ12,z,ω).
(21)

On substituting from Eq. (21) into the formula (5), we find that

DetW(ρ,ρ,z,ω)=Wxx(ρ12,z,ω)Wyy(ρ12,z,ω)Wxy(ρ12,z,ω)Wyx(ρ12,z,ω)
=Ax2Ay2(BxxByyBxyByx)F2(ρ12,z,ω)
=0
(22)

so the spectral degree of polarization

P(ρ,ρ,z,ω)=10[TrW(ρ,ρ,z,ω)]2.
=1
(23)

P is the same at every point, and the criterion σx=σy for keeping completely polarized stochastic electromagnetic beam is derived theoretically. The instance is also the same when the beam propagates in free space described by Eq. (16).

Fig. 5. As Fig. 4, but passing through turbulent atmosphere with C 2 n=10-12 m-2/3.

4. Conclusions

In conclusion, to judge the completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams propagating in free space or through turbulent atmosphere, we start with the analysis of cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources. We find that the completely unpolarized beams will keep their spectral degree of polarization on propagation if δxx=δyy in the source plane, and the completely polarized beams will keep their spectral degree of polarization on propagation if σx=σy in the source plane, otherwise both of them will become partially polarized. The phenomena are explained as coherence-induced polarization changes and spectral density-induced polarization changes. Finally, it is to be noted that these results may be important for many applications, such as tracking, remote sensing, and free-space optical communications.

Acknowledgments

This work was supported by the Program for New Century Excellent Talents in University (NCET-07-0760) and the National Natural Science Foundation of China under grant 60478041.

References and links

1.

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

2.

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

3.

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

4.

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]

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, 225–230 (2004). [CrossRef]

6.

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

7.

X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711–2715 (2008). [CrossRef]

8.

X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773–779 (2008). [CrossRef]

9.

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

10.

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008). [CrossRef] [PubMed]

11.

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

12.

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

13.

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

14.

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, 1078–1080 (2003). [CrossRef] [PubMed]

15.

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

16.

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

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(260.5430) Physical optics : Polarization

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: July 10, 2008
Revised Manuscript: August 16, 2008
Manuscript Accepted: September 19, 2008
Published: September 26, 2008

Citation
Xinyue Du and Daomu Zhao, "Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation," Opt. Express 16, 16172-16180 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16172


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References

  1. D. F. V. James, "Change of polarization of light beams on propagation in free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994). [CrossRef]
  2. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001). [CrossRef]
  3. O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005). [CrossRef]
  4. 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]
  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, 225-230 (2004). [CrossRef]
  6. X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007). [CrossRef] [PubMed]
  7. X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008). [CrossRef]
  8. X. Du and D. Zhao, "Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system," J. Opt. Soc. Am. A 25, 773-779 (2008). [CrossRef]
  9. E. Wolf, "Polarization invariance in beam propagation," Opt. Lett. 32, 3400-3401 (2007). [CrossRef] [PubMed]
  10. E. Wolf, "Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?" Opt. Lett. 33, 642-644 (2008). [CrossRef] [PubMed]
  11. D. Zhao and E. Wolf, "Light beams whose degree of polarization does not change on propagation," Opt. Commun. 281, 3067-3070 (2008). [CrossRef]
  12. M. Salem and E. Wolf, "Coherence-induced polarization changes in light beams," Opt. Lett. 33, 1180-1182 (2008). [CrossRef] [PubMed]
  13. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003). [CrossRef]
  14. 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, 1078-1080 (2003). [CrossRef] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
  16. H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005). [CrossRef]

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