Abstract

Electromagnetic random beams with non-uniform source correlations have been recently shown to develop, on propagation in free space, the regions in transverse cross-sections where the degree and the state of polarization can significantly differ from those beyond that region. The size of the region and the values of polarimetric properties in it can be fully controlled from the source plane. In this paper the influence of a random isotropic medium on such beams is shown to suppress the effect in several ways, in particular by shifting the location of the region back to the axis.

©2012 Optical Society of America

1. Introduction

The possibility of prescribing the free-space evolution of random beams generated by spatially non-uniformly correlated sources has been recently illustrated in [1] on the basis of the general theory developed in [2]. Unlike the classic Gaussian Schell-model sources, for which the typical correlation size, say δ, is the same everywhere in the source plane, the sources in [1] can have a region where δ is greater than elsewhere. In this latter case the weaker diffraction of light from the region with larger δ and stronger diffraction from regions with smaller δ lead, on propagation at some distance, to redistribution of intensity benefiting the region with larger δ. Hence non-uniform source correlations can result in peculiar self-focusing effects, both on and off the beam axis. Moreover, if the intensity spike is set to be off the axis it is shown to remain at the same transverse location as the propagation distance increases. The intensity enhancement effect is generally suppressed if the beam propagates in the presence of a random isotropic medium, for example, the turbulent atmosphere [3]. It still can occur but both its strength and its displacement from the axis are reduced.

Recently the electromagnetic [EM] generalization of model in [1] was introduced [4]. Just like the intensity distribution in the scalar model, the polarization properties of electromagnetic non-uniformly correlated beams have been shown to form regions with much larger or much smaller values of polarimetric parameters compared to the values outside of that region. In this article we explore the behavior of EM non-uniformly correlated beams in greater detail and determine the effect of the isotropic and homogeneous atmospheric turbulence on polarimetric modulation.

2. Electromagnetic non-uniformly correlated beam propagation in atmosphere

Let us consider an electromagnetic non-uniformly correlated field propagating from the source plane z=0 into the half-space z>0 containing turbulent atmosphere. The cross- spectral density matrix of such a beam in the source plane has form [4]

Wαβ(0)(ρ1,ρ2,ω)=AαAβBαβexp(ρ12+ρ222σ02)exp{[(ρ1γα)2(ρ2γβ)2]2δαβ4},
where (α,β=x,y), ρ1 and ρ2 are two-dimensional position vectors in the source plane, Ax and Ay are the field amplitudes of the two mutually orthogonal components of the electric field, say Ex and Ey, σ0 is the r.m.s. source width, Bαβ=|Bαβ|eiφαβ is the single-point (complex) correlation coefficient between α and β (spanning x and y) components of the electric field, φαβ being its phase difference, and δαβ are the single-point correlation coefficient and the r.m.s. source correlation between Eα and Eβ field components, respectively, and γx and γy are real-valued, two-dimensional vectors describing off-axis shifts where the region with modulated spectral density and polarization is to be formed. In order to be a legitimate correlation function Wαβ(0) must have the integral representation of the form [2]
Wαβ(0)(ρ1,ρ2)=pαβHα(ρ1,v)Hβ(ρ2,v)d2v,
where star denotes complex conjugate, pαβ(v) is a nonnegative, Fourier-transformable function and Hα(ρ1,v)Hβ(ρ2,v) is an arbitrary kernel. In order for the model source in Eq. (1) to be written in the form of Eq. (2) we may set
pαβ(v)=Bαβkδαβ2/(2π)exp[k2δαβ4v2/4],
where k = 2π/λ is the wave number of light, λ being wavelength, and
Hj(ρ,v)=Ajexp[ρ2/(2σ02)]exp[ik(ργj)2v],(j=α,β).
The realizability and beam conditions for the source (1) have been derived in [4].

The paraxial form of the extended Huygens-Fresnel principle which describes the interaction of waves with random medium implies that the elements of the cross-spectral density matrix at two positions r1=(ρ1,z)and r2=(ρ2,z)in the same transverse plane of the half-space z>0 are related to those in the source plane as [5, 6]

Wαβ(ρ1,ρ2,z,ω)=(k/2πz)2d2ρ1d2ρ2Wαβ(0)(ρ1,ρ2,ω)×exp{ik[(ρ1ρ1)2(ρ2ρ2)2]/2z}exp[ϕ(ρ1,ρ1,z,ω)+ϕ(ρ2,ρ2,z,ω)]M.
Here ϕ denotes the complex phase perturbation due to the random medium and Mdenotes averaging over the ensemble of its realizations. Under the strong fluctuation condition of turbulence, the last term in the integrand of the right-hand side of Eq. (5) becomes [5]
exp[ϕ(ρ1,ρ1,z,ω)+ϕ(ρ2,ρ2,z,ω)]M=exp{(π2k2z/3)[(ρ1ρ2)2+(ρ1ρ2)(ρ1ρ2)+(ρ1ρ2)2]0κ3Φ(κ)dκ},
where Ф(к) is the spatial power spectrum of the turbulent refractive-index fluctuations.

On substituting from Eqs. (2) and (6) into Eq. (5) and interchanging the orders of the integrals, we obtain for the elements of the cross-spectral density matrix the formulas

Wαβ(ρ1,ρ2,z,ω)=[k2/(4π2z2)]pαβ(v)Hα(ρ1,v,z)Hβ(ρ2,v,z)dv,
Hα(ρ1,v,z)Hβ(ρ2,v,z)=d2ρ1d2ρ2Hα(ρ1,v)Hβ(ρ2,v)exp{ik[(ρ1ρ1)2(ρ2ρ2)2]/2z}×exp{(π2k2z/3)[(ρ1ρ2)2+(ρ1ρ2)(ρ1ρ2)+(ρ1ρ2)2]0κ3Φ(κ)dκ}.

On introducing new variables u=(ρ1+ρ2)/2,t=ρ1ρ2,u=(ρ1+ρ2)/2,t=ρ1ρ2 and substituting from Eq. (4) into Eq. (8) one finds, after long calculations, that

Hα(ρ1,v,z)Hβ(ρ2,v,z)=[4π2z2σ02AαAβ/k2w2(z,v)]×exp[2ikΓαβ+Γαβvikut2k2σ02(Γαβvt2z)2π2k2zt230κ3Φ(κ)dκ]×exp{[u2Γαβ+zv+ikσ02(12zv)(Γαβvt2z)+iπ2kz2t30κ3Φ(κ)dκ]2/w2(z,v)},
where Γαβ+=(γα+γβ)/2, Γαβ=γαγβ and

w2(z,v)=σ02(12zv)2+z2/(k2σ02)+(4π2z3/3)0κ3Φ(κ)dκ.

The elements of the matrix (7) can then be found on combination of Eqs. (9) and (10) with formula (3) and applying numerical integration. This article will be only concerned with evaluation of the distribution of spectral density (intensity at fixed frequency) and the degree of polarization which are related to the cross spectral density matrix by expressions

S(ρ,z)=TrW(ρ,z),
P(ρ,z)=14DetW(ρ,z)/[TrW(ρ,z)]2,
Tr and Det denoting the trace and the determinant of the matrix, respectively.

We will assume here that the turbulent atmosphere is isotropic, homogeneous and governed by statistics described by a generalized model for the power spectrum Ф(к) [7], in which the slope 11/3 of the conventional van Karman spectrum is generalized to an arbitrary parameter α, i.e.

Φα(κ)=A(α)C˜n2exp[(κ2/κm2)]/(κ2+κ02)α/2,0κ<,3<α<5,
where к0 = 2π/L0 with L0 and l0 being the outer and the inner scale of turbulence, κm=c(α)/l0,c(α)=[Γ(5α/2)A(α)2π/3]1/(α5). A(α)=Γ(α1)cos(απ/2)/(4π2) with Γ(x) being the Gamma function. The term C˜n2 in Eq. (13) is a generalized refractive-index structure parameter with units m3-α. With the power spectrum in Eq. (13) we have
0κ2Φ(κ)dκ=A(α)2(α2)C˜n2[κm2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ04α],
where β=2κ022κm2+ακm2 and Γ denotes the incomplete Gamma function.

3. Numerical examples

We will now illustrate the behavior of the spectral density and the degree of polarization of typical beams generated by source (1) on interaction with atmospheric turbulence (13) by a number of numerical examples. Unless specified in the captions, the source and the medium parameters are chosen as follows: k=107m1,L0=1m,l0=0.05m,|Bxy|=0.2,σ0=5mm, Ax=Ay=1,δxx=0.8σ0,δyy=δxy=0.9σ0,γx=(0.8σ0,0), γy=(0.9σ0,0),C˜n2=1012m3α.

The 2D and 3D transverse distribution of the beam’s spectral density at two selected distances z = 30 m and z = 1 km are shown in Figs. 1 , 2 , and 3 . One sees from Fig. 1 that, unlike the uniformly distributed beam, the non-uniformly distributed beam may generate two different spectral density peaks depending on the choices of the values of shifts at z = 30 m. Such trend diminishes at a larger distance, z = 1 km.

 

Fig. 1 The transverse distribution of the beam’s spectral density at the plane z = 30 m for α = 3.67, (a) γx = (0.8σ0,0), γy = (0.9σ0,0) and (b) γx = (0.8σ0,0), γy = (−0.9σ0,0); (c) and (d) are same as (a) and (b) but for z = 1 km.

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Fig. 2 There-dimensional transverse distributions of the beam’s spectral density at the plane z = 30 m and corresponding contour graphs for different parameters γx and γy, α = 3.67, δxx = 0.8σ0 and δyy = 0.9σ0 except for (f) δxx = δyy = 0.9σ0.

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Fig. 3 As Fig. 2 but at the plane z = 1 km.

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This effect can be more clearly seen from the 3D version in Figs. 2 and 3. Hence the control of the peak or even the distribution of the spectral density at certain transverse plane along the propagating path may be fulfilled by a careful selection of the shift values.

In Figs. 4 and 5 we show the shift of the intensity center of the beam at certain transverse plane on propagation and its spectral degree of polarization at the intensity center for selected source and turbulence parameters. One sees that although the spectral degree of polarization varies slightly with α for non-Kolmogorov turbulence, it alters significantly with the change in source correlation length δxy.

 

Fig. 4 The shift of the intensity center of the beam through non-Kolmogorov turbulence as a function of propagation distance z (a) for different α and (b) for α=3.67and different C˜n2.

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Fig. 5 The spectral degree of polarization at the intensity center of the beam with |Bxy| = 0.2 through non-Kolmogorov turbulence as a function of propagation distance z. (a) for different α with δxy = 0.9σ0; (b) for different δxy with α = 3.67.

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The number of turning points of the degree of polarization, which is a very special property of the non-uniformly distributed beams, increases and becomes more evident with the increase of the coherence length δxy. Unlike in the case of free space propagation [1], where the value of the degree of polarization saturates to certain level after propagating at long distances, it returns to its value at the intensity center at the source plane gradually resulting in one more turning point (compared to free space propagation) due to the uniformly correlated turbulence.

One more interesting presentation of the distribution of the spectral degree of polarization at z = 30m and z = 1 km in the three-dimensional version is shown in Figs. 6 and 7 . The maximum of the polarization profile is obvious in free space and depresses somewhat in non-Kolmogorov turbulence, with the largest depression corresponding to α = 3.05.

 

Fig. 6 There-dimensional transverse distributions of the degree of polarization of the beam at z = 30m with γx = (0.8σ0,-0.7σ0) and γy = (0.9σ0,-0.6σ0). (a) α = 3.05; (b) α = 3.67; (c) free space.

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Fig. 7 As Fig. 6 but at the plane z = 1 km.

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4. Summary

Thus we have illustrated that the EM non-uniformly correlated beams propagating in random isotropic media, such as atmosphere, at relatively small distances, can still form the regions in transverse cross-sections where the polarization properties are modulated differently from the rest of the cross-section. However such modulation is gradually suppressed by medium fluctuations, both in strength and in location, with respect to the optical axis. In the current paper we have also explored the new possibility of the EM non-uniformly correlated beams to form a controllable combination of the off-axis intensity and degree of polarization maxima, due to its vectorial nature.

Random EM beams carry huge potential for atmospheric applications since the combination of partial coherence and partial polarization, being two means of diversity, lead to significant decrease in scintillation and, hence, improve the receiver’s signal-to noise ratio. If, additionally, the strong self-interference effects leading to drastic intensity and polarization modulation are possible such beams become a unique class for atmospheric communication and sensing systems.

Acknowledgments

Z. Mei’s research is supported by Zhejiang Provincial Natural Science Foundation of China (Y6100605). O. Korotkova’s research is supported by grants N00189-12-T-0136 and FA9550-12-1-0449.

References and links

1. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]   [PubMed]  

2. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]  

3. Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef]   [PubMed]  

4. Z. Tong and O. Korotkova, “Electromagnetic non-uniformly correlated beams,” J. Opt. Soc. Am. A . in press).

5. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef]   [PubMed]  

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(25), 16909–16915 (2007). [CrossRef]   [PubMed]  

7. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]  

References

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  1. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  2. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  3. Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  4. Z. Tong and O. Korotkova, “Electromagnetic non-uniformly correlated beams,” J. Opt. Soc. Am. A. in press).
  5. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
    [Crossref] [PubMed]
  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(25), 16909–16915 (2007).
    [Crossref] [PubMed]
  7. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
    [Crossref]

2012 (1)

2011 (1)

2010 (1)

2009 (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

2008 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

2007 (1)

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Du, X.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Korotkova, O.

Lajunen, H.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Saastamoinen, T.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Shchepakina, E.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Tong, Z.

Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Electromagnetic non-uniformly correlated beams,” J. Opt. Soc. Am. A. in press).

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Zhao, D.

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

Z. Tong and O. Korotkova, “Electromagnetic non-uniformly correlated beams,” J. Opt. Soc. Am. A. in press).

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

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Figures (7)

Fig. 1
Fig. 1 The transverse distribution of the beam’s spectral density at the plane z = 30 m for α = 3.67, (a) γx = (0.8σ0,0), γy = (0.9σ0,0) and (b) γx = (0.8σ0,0), γy = (−0.9σ0,0); (c) and (d) are same as (a) and (b) but for z = 1 km.
Fig. 2
Fig. 2 There-dimensional transverse distributions of the beam’s spectral density at the plane z = 30 m and corresponding contour graphs for different parameters γx and γy, α = 3.67, δxx = 0.8σ0 and δyy = 0.9σ0 except for (f) δxx = δyy = 0.9σ0.
Fig. 3
Fig. 3 As Fig. 2 but at the plane z = 1 km.
Fig. 4
Fig. 4 The shift of the intensity center of the beam through non-Kolmogorov turbulence as a function of propagation distance z (a) for different α and (b) for α=3.67 and different C ˜ n 2 .
Fig. 5
Fig. 5 The spectral degree of polarization at the intensity center of the beam with |Bxy| = 0.2 through non-Kolmogorov turbulence as a function of propagation distance z. (a) for different α with δxy = 0.9σ0; (b) for different δxy with α = 3.67.
Fig. 6
Fig. 6 There-dimensional transverse distributions of the degree of polarization of the beam at z = 30m with γx = (0.8σ0,-0.7σ0) and γy = (0.9σ0,-0.6σ0). (a) α = 3.05; (b) α = 3.67; (c) free space.
Fig. 7
Fig. 7 As Fig. 6 but at the plane z = 1 km.

Equations (14)

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W αβ (0) ( ρ 1 , ρ 2 ,ω)= A α A β B αβ exp( ρ 1 2 + ρ 2 2 2 σ 0 2 )exp{ [ ( ρ 1 γ α ) 2 ( ρ 2 γ β ) 2 ] 2 δ αβ 4 },
W αβ (0) ( ρ 1 , ρ 2 )= p αβ H α ( ρ 1 ,v) H β ( ρ 2 ,v) d 2 v,
p αβ (v)= B αβ k δ αβ 2 /(2 π )exp[ k 2 δ αβ 4 v 2 /4 ],
H j ( ρ ,v)= A j exp[ ρ 2 /(2 σ 0 2 ) ]exp[ ik ( ρ γ j ) 2 v ], (j=α,β).
W αβ ( ρ 1 , ρ 2 ,z,ω)= (k/2πz) 2 d 2 ρ 1 d 2 ρ 2 W αβ (0) ( ρ 1 , ρ 2 ,ω) ×exp{ ik[ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ]/2z } exp[ ϕ ( ρ 1 , ρ 1 ,z,ω)+ϕ( ρ 2 , ρ 2 ,z,ω)] M .
exp[ ϕ ( ρ 1 , ρ 1 ,z,ω)+ϕ( ρ 2 , ρ 2 ,z,ω)] M = exp{ ( π 2 k 2 z/3)[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] 0 κ 3 Φ(κ)dκ },
W αβ ( ρ 1 , ρ 2 ,z,ω )=[ k 2 /(4 π 2 z 2 )] p αβ (v) H α ( ρ 1 ,v,z) H β ( ρ 2 ,v,z)dv ,
H α ( ρ 1 ,v,z) H β ( ρ 2 ,v,z)= d 2 ρ 1 d 2 ρ 2 H α ( ρ 1 ,v) H β ( ρ 2 ,v)exp{ ik[ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ]/2z } ×exp{ ( π 2 k 2 z/3)[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] 0 κ 3 Φ(κ)dκ }.
H α ( ρ 1 ,v,z) H β ( ρ 2 ,v,z)=[4 π 2 z 2 σ 0 2 A α A β / k 2 w 2 (z,v)] ×exp[ 2ik Γ αβ + Γ αβ v ikut 2 k 2 σ 0 2 ( Γ αβ v t 2z ) 2 π 2 k 2 z t 2 3 0 κ 3 Φ(κ)dκ ] ×exp{ [ u2 Γ αβ + zv+ik σ 0 2 (12zv)( Γ αβ v t 2z )+ i π 2 k z 2 t 3 0 κ 3 Φ(κ)dκ ] 2 / w 2 (z,v) },
w 2 (z,v)= σ 0 2 (12zv) 2 + z 2 /( k 2 σ 0 2 )+(4 π 2 z 3 /3) 0 κ 3 Φ(κ)dκ .
S(ρ,z)=Tr W (ρ,z),
P(ρ,z)= 14Det W (ρ,z)/ [Tr W (ρ,z)] 2 ,
Φ α (κ)=A(α) C ˜ n 2 exp[( κ 2 / κ m 2 )]/ ( κ 2 + κ 0 2 ) α/2 , 0κ<, 3<α<5,
0 κ 2 Φ(κ)dκ = A(α) 2(α2) C ˜ n 2 [ κ m 2α βexp( κ 0 2 κ m 2 )Γ(2 α 2 , κ 0 2 κ m 2 )2 κ 0 4α ],

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