## Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation

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

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. 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]

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]

## 2. Theoretical analysis

*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]

*x*and

*y*are two mutually orthogonal directions perpendicular to the beam axis.

*E*=(

*E*,

_{x}*E*) is a statistical ensemble of the fluctuating component of the transverse electric field.

_{y}*W*

^{⃡(0)}(

**ρ**′

_{1},

**ρ**′

_{2}

*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]

*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

*〉 implies that the average is taken over the ensemble of the random medium.*

_{rm}**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. 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]

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]

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]

*e*and

_{x}*e*are of the general forms of

_{y}*e*

_{1}and

*e*

_{2}which are “one point” constants shown in Ref. [15], 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).

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. 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]

*A*,

_{i}*A*,

_{j}*B*and the variances

_{ij}*σ*,

_{i}*σ*,

_{j}*δ*are independent of position but may depend on frequency (Ref. [15], Sec. 5.3.2), and

_{ij}*B*≡1. In order to calculate the spectral degree of polarization in the source plane, Eq. (8) can be simplified into one point (

_{ii}**ρ**′,

*z*=0) as

*A*=

_{x}*A*=

_{y}*A*,

*B*=0,

_{xy}*σ*=

_{x}*σ*=

_{y}*σ*, and the matrix is obtained as

*δ*and

_{xx}*δ*can be selected freely. For the completely polarized source, we should choose |

_{yy}*B*|=1 to make

_{xy}*B*=

_{xy}B_{yx}*B*. The spectral correlation width

_{xx}B_{yy}*δ*must correspond the realizable conditions as [16

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

*δ*,

_{xx}*δ*and

_{yy}*δ*should be equal. Other parameters can be selected freely. The matrix is then obtained as

_{xy}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]

*k*=2

*π*/

*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.

**M**′

_{ij}^{-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. 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]

**I**̄ is a 4×4 unitary matrix, and

**I**is a 2×2 unitary matrix.

*ρ*

_{0}(0.545

*C*

^{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}

*is the structure parameter of the refractive index. If*

_{n}C*C*

^{2}

*=0, we can obtain that*

_{n}**P**̄=0 and Eq. (13) can be simplified as

*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

*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

*δ*=

_{xx}*δ*,

_{yy}*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]

*δ*≠

_{xx}*δ*,

_{yy}*P*will have different distributions at different propagation distances but become changeless in the far-field. Under the condition of

*δ*≠

_{xx}*δ*, the beam generated by a completely unpolarized source becomes partially polarized on propagation, and it can be considered as coherence-induced polarization changes [12

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

*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

*δ*=

_{xx}*δ*for keeping the completely unpolarized beam is the same as the condition of free space.

_{yy}*A*=

_{x}*A*=

_{y}*A*,

*B*=0,

_{xy}*σ*=

_{x}*σ*, and

_{y}*δ*=

_{xx}*δ*, we can find from Eq. (14) that

_{yy}**M**′

_{xx}^{-1}=

**M**′

_{yy}^{-1}, and if the function

*F*(

_{ij}**ρ**

_{12},

*z*,

*ω*) is defined as

*F*(

_{xx}**ρ**

_{12},

*z*,

*ω*)=

*F*(

_{yy}**ρ**

_{12},

*z*,

*ω*)=

*F*(

**ρ**

_{12},

*z*,

*ω*), then Eq. (13) can be expressed briefly as

**ρ**

_{12}

*=(*

^{T}**ρ**

*,*

^{T}**ρ**

*). The spectral degree of polarization is the same at every point, and the criterion*

^{T}*δ*=

_{xx}*δ*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).

_{yy}*σ*=

_{x}*σ*,

_{y}*P*will not change when the beam propagates in free space, so we can keep the completely polarized beam. If

*σ*≠

_{x}*σ*, the beam will become partially polarized on propagation as shown in Fig. 4(b)–(d). It is known that the parameter

_{y}*σ*represents the effective width of the spectral density, i.e. that [3

_{i}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. 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]

*spectral density-induced polarization changes*.

*B*|=1,

_{xy}*δ*=

_{xx}*δ*=

_{yy}*δ*, and

_{xy}*σ*=

_{x}*σ*, we can find from Eq. (14) that

_{y}**M**′

_{xx}^{-1}=

**M**′

_{xy}^{-1}=

**M**′

_{yx}^{-1}=

**M**′

_{yx}^{-1}=

**M**′

_{yy}^{-1}, then

*F*(

_{xx}**ρ**

_{12},

*z*,

*ω*=

*F*(

_{yx}**ρ**

_{12},

*z*,

*ω*=

*F*(

_{yy}**ρ**

_{12},

*z*,

*ω*)=

*F*(

**ρ**

_{12},

*z*,

*ω*), Eq. (13) can be expressed briefly as

*P*is the same at every point, and the criterion

*σ*=

_{x}*σ*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).

_{y}## 4. Conclusions

*δ*=

_{xx}*δ*in the source plane, and the completely polarized beams will keep their spectral degree of polarization on propagation if

_{yy}*σ*=

_{x}*σ*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.

_{y}## Acknowledgments

## References and links

1. | D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A |

2. | F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A |

3. | O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. |

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. |

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. |

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 |

7. | X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. |

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 |

9. | E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. |

10. | E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. |

11. | D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. |

12. | M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. |

13. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

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. |

15. | L. Mandel and E. Wolf, |

16. | H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. |

**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

- 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]
- 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]
- 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]
- 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, 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]
- 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]
- X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008). [CrossRef]
- 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]
- E. Wolf, "Polarization invariance in beam propagation," Opt. Lett. 32, 3400-3401 (2007). [CrossRef] [PubMed]
- 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]
- D. Zhao and E. Wolf, "Light beams whose degree of polarization does not change on propagation," Opt. Commun. 281, 3067-3070 (2008). [CrossRef]
- M. Salem and E. Wolf, "Coherence-induced polarization changes in light beams," Opt. Lett. 33, 1180-1182 (2008). [CrossRef] [PubMed]
- E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 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, 1078-1080 (2003). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
- H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005). [CrossRef]

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