Abstract

Analytic expression is derived for the cross-spectral density matrix of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere. The new formula can be used in study of the modulation in the spectral degree of polarization of the electromagnetic Gaussian Schell-model beam on propagation. We find that the spectral degree of polarization in the output plane can be directly controlled by the width of the slit aperture. The effect of polarization shaping is also illustrated by numerical examples.

©2009 Optical Society of America

1. Introduction

The propagation of light beams through the turbulent atmosphere is important in many applications such as tracking, remote sensing, and optical communication. After the unified theory of coherence and polarization presented by Wolf [1], the changes in the spectral density, the spectral degree of coherence, and the spectral degree of polarization of stochastic electromagnetic beams propagating in deterministic or random media can be determined by the 2×2 cross-spectral density matrix [2]. Since then the statistical properties of stochastic electromagnetic beams passing through the turbulent atmosphere have been extensively investigated [3–6]. We have also studied the propagation of stochastic anisotropic electromagnetic beams [7] and stochastic electromagnetic array beams [8] through the turbulent atmosphere.

In practical instance as well as optical communication through the turbulent atmosphere, apertures of different size or shape are usually present. The cosh-Gaussian beam diffracted by a circular aperture and passing through the turbulent atmosphere has been studied [9] and the effect of turbulence on the quality of the apertured beam has also been discussed [10]. The apertures may directly influence the propagation properties of light beams but the polarization modulation effect of truncated stochastic electromagnetic beams on propagation through the turbulent atmosphere has not, to the best of our knowledge, been previously adequately addressed.

In this paper, we introduce a slit aperture function with the help of the tensor method and study the propagation of an electromagnetic Gaussian Schell-model beam truncated by a slit aperture and passing through the turbulent atmosphere. The slit aperture is considered as a modulator and the changes in spectral degree of polarization are illustrated.

 

Fig. 1. Illustrating the notation.

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2. Theoretical analysis

Suppose the beam in the source plane is truncated by a hard-aperture of a slit shape as shown in Fig. 1. It is known that the aperture function can be approximated by a multi-Gaussian series [11], so the slit aperture function in x and y dimensions can be expressed as

Apx(x1)=n1=NNexp[(x1/βxn1)2]n1=NNexp(n12),andApy(y1)=1,

where N is the order of the multi-Gaussian series, βx is the spot size of the Gaussian function component. In two-dimensional case, the slit aperture function can be found by using tensor method as

Ap(ρ1)=Apx(x1)Apy(y1)=n1=NNexp[ik2(ρ1TI2ρ12ρ1TI1n1+n1TI0n1)]n1=NNexp(n1Tn1),

where

I0=[2ik000],I1=[2ikβx000],I2=[2ikβx2000],

k = 2π/λ, is the wave number, λ is the wavelength. The width of the slit aperture along x direction is 2σx with αx = x · ρT 1 = (x1, x2) and n T 1 = (n 1, 0), T denotes matrix transposition. Figure 2 shows that Eq. (2) is an appropriate approximate expression for the slit aperture function with N = 10 adopted in the numerical calculations.

 

Fig. 2. Three-dimensional distribution and corresponding gray-scale projection of the slit aperture function expressed by Eq. (2) with αx = 10 mm and N = 10.

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We have studied the propagation of stochastic electromagnetic beam through the turbulent atmosphere [7], if the beam in the source plane truncated by a slit aperture the cross-spectral density matrix in the output plane can be given by

Wij(ρ12,z,ω)=k24π2[Det(B̅)]1/2A̅p(ρ12)Wij(0)(ρ12,ω)
×exp[ik2(ρ12TB̅1ρ122ρ12TB̅1ρ12+ρ12TB̅1ρ12)],
×exp[ik2(ρ12TP̅ρ12ρ12TP̅ρ12+ρ12TP̅ρ12)]d4ρ12

where ρT 12 = (ρT 1, ρT 2) = (x1, y1, x2, y2) and ρ T 12 = ρT 12 = (ρ T 1, ρ T 2) = (x 1, y 1, x 2, y 2) are four-dimensional vectors, Det means the determinant,

B̅=[zI00zI],P̅=2ikρ02[IIII],

where I is a 2×2 unit 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 and C 2 n is the structure parameter of the refractive index. Āp (ρ12) = A * p (ρ1)Ap(ρ2) is a four-dimensional aperture function of the form

A̅p(ρ12)=n1=NNn2=NNexp[ik2(ρ12TI̅2ρ122ρ12I̅1n12+n12TI̅0n12)]n1=NNn2=NNexp(n12Tn12),

where

I̅0=[I000I0],I̅1=[I100I1],I̅2=[I200I2],

and n T 12 = (n T 1, n T 1) = (n 1, 0, n 2, 0)

Assume that a beam is generated by an electromagnetic Gaussian Schell-model source. The cross-spectral density matrix in the source plane is given in tensor form as [7]

Wij(0)(ρ12,ω)=AiAjBijexp(ik2ρ12TMij1ρ12),

where M-1 ij is a 4×4 complex matrix as

Mij1=[ik(12σi2+1δij2)Iik1δij2Iik1δij2Iik(12σj2+1δij2)I].

The coefficients Ai, Aj, Aj, Bij and the variances σi, σj, δij are independent of position but may depend on frequency.

On substituting from Eqs. (6) and (8) into Eq. (4), and after a vector integration, we obtain the analytic expression for the cross-spectral density matrix of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere as

Wij(ρ12,z,ω)=AiAjBij[Det(BI̅2+B̅Mij1+I̅+BP̅)]1/2n1=NNn2=NNexp(n12Tn12)
×exp{ik2ρ12T[(B̅1+P̅)(B̅112P̅)T(I̅2+Mij1+B̅1+P̅)1(B̅112P̅)]ρ12}
×n1=NNn2=NNexp[ikρ12T(B̅−112P̅)T(I̅2+Mij1+B̅1+P̅)1I̅1n12]
×exp{ik2n12T[I̅0I̅1T(I̅2+Mij1+B̅1+P̅)1I̅1]n12},

where I is a 4×4 unit matrix. If αx = ∞, Eq. (10) represents the cross-spectral density function of the beam passing through the turbulent atmosphere without aperture and reduces to the form

Wij(ρ12,z,ω)=AiAjBij[Det(B̅Mij1+I̅+BP̅)]1/2
×exp{ik2ρ12T[(B̅1+P̅)(B̅112P̅)T(Mij1+B̅1+P̅)1(B̅112P̅)]ρ12}.

(11) It is in agreement with Eq. (11) shown in Ref. [7].

3. Numerical calculations and discussions

In this section we are going to study the changes in the spectral degree of polarization of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere. The spectral degree of polarization at the point (ρ, z) with ρ T 12 = (ρ T, ρ T) is defined by the formula [1, 2]

P(ρ12,z,ω)=14DetW(ρ12,z,ω)[TrW(ρ12,z,ω)]2.

In Fig. 3, the stochastic electromagnetic beam is truncated by slit apertures of the same width, but the structure parameters are different. In the condition of the beam propagating through the turbulent atmosphere, the spectral degrees of polarization in the far field tend to an identical value, which is different from the condition of propagating in free space with C 2 n =0 and is independent of the local strength of atmospheric turbulence. The similar phenomenon has been shown in Fig. 4(b) of Ref. [7] for the stochastic anisotropic electromagnetic beam propagating in non-truncated case. Now the effect of slit aperture on the changes in polarization of the beam on propagation is additionally illustrated.

 

Fig. 3. Changes in the spectral degree of polarization along the z-axis of the stochastic electromagnetic beam truncated by the same slit aperture with αx = 10 mm and passing through the turbulent atmosphere with different C 2 n. The source is assumed to be Gaussian Schell-model source with: λ = 632.8 nm, Ax = 2, Ay = 1, Bxy = 0.2exp(/3), σx = 10 mm, σy = 20 mm, δxx = δyy = 2 mm and δxy = δyx = 3 mm.

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In Fig. 4, the stochastic electromagnetic beam is truncated by slit apertures of different widths and passing through the turbulent atmosphere with the same structure parameter. The dashed curve, dash-dot curve and dotted curve are calculated from Eqs. (10) and (12), and the solid curve is calculated from Eqs. (11) and (12). When the width is sufficiently large, it is similar to that in the limiting non-truncated case. The figure also shows that we can modulate the spectral degree of polarization in the output plane especially in the far-field by simply controlling the width of the slit aperture.

 

Fig. 4. Changes in the spectral degree of polarization along the z-axis of the stochastic electromagnetic beam truncated by slit apertures with different αx and passing through the turbulent atmosphere with C 2 n = 10-14 m-2/3. The parameters of the source are the same as Fig. 3.

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In Fig. 5, we choose the same structure parameter of the turbulent atmosphere and the same output plane of z = 1 km, but the beam is truncated by different slit apertures. When the width is large the distribution of the spectral degree of polarization is axially symmetrical shown in Fig. 5(a). As the width decreasing, the distribution of the spectral degree of polarization becomes more changeless in x direction shown in Fig. 5(b) and it can be regarded as uniformity in this direction shown in Fig. 5(c) when the width is sufficiently small. This phenomenon is induced by the diffraction effect of slit aperture and we consider it as a convenient method for shaping the spectral degree of polarization of the stochastic electromagnetic beam on propagation.

 

Fig. 5. Three-dimensional distributions of the spectral degree of polarization and corresponding gray-scale projections of the stochastic electromagnetic beam truncated by slit apertures with (a) αx = 50 mm, (b) αx =10 mm , (c) αx = 1 mm, and passing through the turbulent atmosphere with C 2 n = 10-14 m-2/3. The propagation distance z = 1 km and the parameters of the source are the same as Fig. 3.

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4. Conclusions

We conclude by saying that we have found a way of modulating the spectral degree of polarization of the stochastic electromagnetic beam on propagation through the turbulent atmosphere. The two-dimensional slit aperture function is given and the analytic expression for the cross-spectral density matrix is obtained. By virtue of numerical calculations, we have shown that the spectral degree of polarization of the electromagnetic Gaussian Schell-model beam in the output plane can be modulated by controlling the width of the slit aperture and the effect of polarization shaping is also illustrated.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (10874150 and 60478041) and the Program for New Century Excellent Talents in University (NCET-07-0760).

References and links

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

2. 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]  

3. 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]  

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 and E. Wolf, “Beam criterion for atmospheric propagation,” Opt. Lett. 32, 2137–2139 (2007). [CrossRef]   [PubMed]  

6. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007). [CrossRef]  

7. 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. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008). [CrossRef]   [PubMed]  

9. X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007). [CrossRef]  

10. X. Ji and G. Ji, “Effect of turbulence on the beam quality of apertured partially coherent beams,” J. Opt. Soc. Am. A 25, 1246–1252 (2008). [CrossRef]  

11. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897–1904 (2001). [CrossRef]  

References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [Crossref]
  2. 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]
  3. 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]
  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 and E. Wolf, “Beam criterion for atmospheric propagation,” Opt. Lett. 32, 2137–2139 (2007).
    [Crossref] [PubMed]
  6. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007).
    [Crossref]
  7. 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. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
    [Crossref] [PubMed]
  9. X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
    [Crossref]
  10. X. Ji and G. Ji, “Effect of turbulence on the beam quality of apertured partially coherent beams,” J. Opt. Soc. Am. A 25, 1246–1252 (2008).
    [Crossref]
  11. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897–1904 (2001).
    [Crossref]

2008 (2)

2007 (4)

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
[Crossref]

O. Korotkova and E. Wolf, “Beam criterion for atmospheric propagation,” Opt. Lett. 32, 2137–2139 (2007).
[Crossref] [PubMed]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007).
[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]

2005 (1)

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]

2004 (1)

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]

2003 (2)

2001 (1)

Chen, Z.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007).
[Crossref]

Chu, X.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
[Crossref]

Du, X.

Ji, G.

Ji, X.

Korotkova, O.

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
[Crossref]

Ponomarenko, S. A.

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]

Pu, J.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007).
[Crossref]

Roychowdhury, H.

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]

Salem, M.

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]

Tovar, A. A.

Wolf, E.

O. Korotkova and E. Wolf, “Beam criterion for atmospheric propagation,” Opt. Lett. 32, 2137–2139 (2007).
[Crossref] [PubMed]

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]

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]

Zhao, D.

Zhou, G.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
[Crossref]

Zhu, Y.

Appl. Phys. B (1)

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547–552 (2007).
[Crossref]

J. Mod. Opt. (1)

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]

J. Opt. A (1)

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A 9, 1123–1130 (2007).
[Crossref]

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

Opt. Commun. (1)

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]

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (1)

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

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Figures (5)

Fig. 1.
Fig. 1. Illustrating the notation.
Fig. 2.
Fig. 2. Three-dimensional distribution and corresponding gray-scale projection of the slit aperture function expressed by Eq. (2) with αx = 10 mm and N = 10.
Fig. 3.
Fig. 3. Changes in the spectral degree of polarization along the z-axis of the stochastic electromagnetic beam truncated by the same slit aperture with αx = 10 mm and passing through the turbulent atmosphere with different C 2 n . The source is assumed to be Gaussian Schell-model source with: λ = 632.8 nm, Ax = 2, Ay = 1, Bxy = 0.2exp(/3), σx = 10 mm, σy = 20 mm, δxx = δyy = 2 mm and δxy = δyx = 3 mm.
Fig. 4.
Fig. 4. Changes in the spectral degree of polarization along the z-axis of the stochastic electromagnetic beam truncated by slit apertures with different αx and passing through the turbulent atmosphere with C 2 n = 10-14 m-2/3. The parameters of the source are the same as Fig. 3.
Fig. 5.
Fig. 5. Three-dimensional distributions of the spectral degree of polarization and corresponding gray-scale projections of the stochastic electromagnetic beam truncated by slit apertures with (a) αx = 50 mm, (b) αx =10 mm , (c) αx = 1 mm, and passing through the turbulent atmosphere with C 2 n = 10-14 m-2/3. The propagation distance z = 1 km and the parameters of the source are the same as Fig. 3.

Equations (18)

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A px ( x 1 ) = n 1 = N N exp [ ( x 1 / β x n 1 ) 2 ] n 1 = N N exp ( n 1 2 ) , and A py ( y 1 ) = 1 ,
A p ( ρ 1 ) = A px ( x 1 ) A py ( y 1 ) = n 1 = N N exp [ ik 2 ( ρ 1 T I 2 ρ 1 2 ρ 1 T I 1 n 1 + n 1 T I 0 n 1 ) ] n 1 = N N exp ( n 1 T n 1 ) ,
I 0 = [ 2 ik 0 0 0 ] , I 1 = [ 2 ik β x 0 0 0 ] , I 2 = [ 2 ik β x 2 0 0 0 ] ,
W ij ( ρ 12 , z , ω ) = k 2 4 π 2 [ Det ( B ̅ ) ] 1 / 2 A ̅ p ( ρ 12 ) W ij ( 0 ) ( ρ 12 , ω )
× exp [ ik 2 ( ρ 12 T B ̅ 1 ρ 12 2 ρ 12 T B ̅ 1 ρ 12 + ρ 12 T B ̅ 1 ρ 12 ) ] ,
× exp [ ik 2 ( ρ 12 T P ̅ ρ 12 ρ 12 T P ̅ ρ 12 + ρ 12 T P ̅ ρ 12 ) ] d 4 ρ 12
B ̅ = [ z I 0 0 z I ] , P ̅ = 2 ik ρ 0 2 [ I I I I ] ,
A ̅ p ( ρ 12 ) = n 1 = N N n 2 = N N exp [ ik 2 ( ρ 12 T I ̅ 2 ρ 12 2 ρ 12 I ̅ 1 n 12 + n 12 T I ̅ 0 n 12 ) ] n 1 = N N n 2 = N N exp ( n 12 T n 12 ) ,
I ̅ 0 = [ I 0 0 0 I 0 ] , I ̅ 1 = [ I 1 0 0 I 1 ] , I ̅ 2 = [ I 2 0 0 I 2 ] ,
W ij ( 0 ) ( ρ 12 , ω ) = A i A j B ij exp ( ik 2 ρ 12 T M ij 1 ρ 12 ) ,
M ij 1 = [ i k ( 1 2 σ i 2 + 1 δ ij 2 ) I i k 1 δ ij 2 I i k 1 δ ij 2 I i k ( 1 2 σ j 2 + 1 δ ij 2 ) I ] .
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( BI ̅ 2 + B ̅ M ij 1 + I ̅ + BP ̅ ) ] 1 / 2 n 1 = N N n 2 = N N exp ( n 12 T n 12 )
× exp { ik 2 ρ 12 T [ ( B ̅ 1 + P ̅ ) ( B ̅ 1 1 2 P ̅ ) T ( I ̅ 2 + M ij 1 + B ̅ 1 + P ̅ ) 1 ( B ̅ 1 1 2 P ̅ ) ] ρ 12 }
× n 1 = N N n 2 = N N exp [ ik ρ 12 T ( B ̅ −1 1 2 P ̅ ) T ( I ̅ 2 + M ij 1 + B ̅ 1 + P ̅ ) 1 I ̅ 1 n 12 ]
× exp { ik 2 n 12 T [ I ̅ 0 I ̅ 1 T ( I ̅ 2 + M ij 1 + B ̅ 1 + P ̅ ) 1 I ̅ 1 ] n 12 } ,
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( B ̅ M ij 1 + I ̅ + BP ̅ ) ] 1 / 2
× exp { ik 2 ρ 12 T [ ( B ̅ 1 + P ̅ ) ( B ̅ 1 1 2 P ̅ ) T ( M ij 1 + B ̅ 1 + P ̅ ) 1 ( B ̅ 1 1 2 P ̅ ) ] ρ 12 } .
P ( ρ 12 , z , ω ) = 1 4 Det W ( ρ 12 , z , ω ) [ Tr W ( ρ 12 , z , ω ) ] 2 .

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