Abstract

An analytical model for electromagnetic twisted Gaussian Schell-model array (EM TGSMA) beams is introduced. We derive the analytical expression for the cross-spectral density matrix (CSDM) of such beam propagating in free space and investigate the spectral density, degree of polarization (DOP) and degree of coherence (DOC) in detail. Twist effects of the evolutions of DOC and DOP during propagation are also demonstrated. It is found that the twist effects of DOC and DOP can be influenced by the beam width and coherence length. In particular, senses of rotation can also be controlled by adjusting the twist strength. Furthermore, we find that twist phase always causes DOC to rotate in an opposite direction compared with that of DOP. Our results might be beneficial for free-space communications of the partially coherent beams endowed with twist.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Statistical properties of coherence and polarization of light waves have been treated separately for many years. Until 2003, Wolf developed the unified theory of coherence and polarization for random partially coherent electromagnetic beams [1, 2], which made it evident that coherence and polarization of light, whilst distinct phenomena, were just two related aspects of statistical optics. In particular, the spectral density, degree of coherence (DOC) and degree of polarization (DOP) of such an electromagnetic beam can be expressed by a 2 × 2 cross- spectral density matrix (CSDM). With the help of this theory, propagation statistical characteristics of light waves in different optical systems or media have been investigated in detail [3–12]. Coherent properties of the source can cause the spectrum of light change on propagation. The degree of polarization as beam propagates generally also changes, which is caused by the difference in the correlation coefficients between the x and y components of the electrical field vector at the two source points [13]. Besides, the shape and the orientation of the polarization ellipse of the polarization portion of the beam also change on propagation [14]. Light beams whose degree of polarization remains constant on propagation was also given, which corresponded to a linear polarization along the x direction [15].

Compared with the single beam, the coherent array beam has higher output power, and it has been used to multiple fields, including photonic lithography [16, 17], atomic capture and cooling [18], microfluidic sorting [19]. Besides, partially coherent beams endowed with twist have also aroused much attention from researchers [20–25]. Hence in the present paper, we extend the concept of partially coherent twisted array beams to the electromagnetic (vector) domain by using a generalized complex Gaussian representation for the CSDM. Moreover, propagation characteristics of such electromagnetic twisted Gaussian Schell-model array (EM TGSMA) beam in free space are analyzed. In particular, we demonstrated the twist effects of DOC and DOP during propagation in detail.

2. Propagation of EM TGSMA beam in free space

For a random statistically stationary electromagnetic beam propagating along the positive z-axis, the second-order correlation property of the light field in space-frequency can be presented by a 2 × 2 CSDM, as following [2]:

Wij(r1,r2,z,ω)=Ei*(r1,z,ω)Ej(r2,z,ω),(i,j=x,y),
where r1(x1,y1) and r2(x2,y2) are two arbitrary transverse position vectors; Ex and Ey denote two mutually orthogonal components of the random electric vector, perpendicular to the z-axis. The angular brackets stand for the ensemble average and the asterisk denotes the complex conjugate. For brevity, we ignore the dependence of the angular frequency.

Assume the field taken across the z=0 plane. When the beam axis is tilted, field distribution would become E(r,z,ω)exp [i(kxx+kyy)], where kx and ky are the transverse components of the mean wave vector kij. By giving an inclination kij(r0)=uij(y0,x0), with real uij, the corresponding element of CSDM Wij0(r1,r2,r0), is [26]

Wij(r1,r2,r0)=Wij0(r1r0,r2r0)exp (2πiυijr0),
with vector υij=uij2π(y2y1,x1x2), which denotes a rotated version of D=r1r2. Assuming an incoherent superposition of such mutually uncorrelated tilted fields Wij(r1,r2,r0), with a non-negative weight function p(r0), and introducing a variable change r0=rmr, where rm=(r1+r2)/2, a bona fide CSDM is produced, as following [27]:
Wij(r1,r2)=Rij(r1,r2)exp (2πiυijrm),
with function Rij referred as the reminder:
Rij(r1,r2)=p(r0)Wij0(r+D/2,rD/2)exp (2πiυijr)d2r.

For simplicity, we shall put p(r0)=1. With a Gaussian Schell-model array model Wij0, one can obtain a rotating EM Gaussian array pattern as following [28]:

Wij(r1,r2)=exp [(x1x2)22δ1ij2]exp[(y1y2)22δ2ij2]×Pn1=Pcos [C1ij(x1x2)]Qn2=Qcos[C2ij(y1y2)]×exp (iuijx1y2)exp (iuijx2y1).

In order to impose restriction on the amplitude of the source, we place a proper amplitude mask τ(r) as a spatial limitation of the actual source, as following [25]:

τ(r)=exp (x24σ1ij2)exp (y24σ2ij2),
here C1ij=2πn1R1/δ1ij, C2ij=2πn2R2/δ2ij, R1,R2 and δ1ij, δ2ij are coherence parameters, and uij denotes the twist strength. σ1ij and σ2ij are the beam widths of Gaussian intensity distribution. P=(N11)/2 and Q=(N21)/2, which N1 and N2 are positive integrals that determine the number of lobes of the array. Hence, the element of CSDM becomes:
Wij(r1,r2)=exp (x12+x224σ1ij2)exp (y12+y224σ2ij2)×exp [(x1x2)22δ1ij2]exp[(y1y2)22δ2ij2]×Pn1=Pcos [C1ij(x1x2)]Qn2=Qcos[C2ij(y1y2)]×exp (iuijx1y2)exp (iuijx2y1).

According to the extended Huygens-Fresnel principle in the paraxial form, the element of CSDM of the EM TGSMA beams during propagation can be presented as following [29]:

Wij(ρ1,ρ2,z)=Wij(r1,r2)Hz*(ρ1,r1)Hz(ρ2,r2)d2r1d2r2,
here the function Hz called the propagation kernal that characterizes propagation in a domain outside the source, given by the following form for free-space propagation:
Hz(ρ,r)=ik(2πz)1exp [ik(ρr)2/(2z)].

Substituting Eqs. (7) and (9) into Eq. (8), we obtain the element of CSDM in free space as following:

Wij(ρ1,ρ2,z)=Eijn1=PPQn2=Q(kπσ1ij)2AijBijDijz2exp [ik2z(ρ12ρ22)]×exp [k2σ1ij22z2(x1x2)2]×{exp[14Dij(2mij1++ikz(y1y2))2+14Aijγij++2+14Bijβij+2] +exp[14Dij(2mij1++ikz(y1y2))2+14Aijγij+2+14Bijβij+2]+exp[14Dij(2mij1+ikz(y1y2))2+14Aijγij+2+14Bijβij2]+exp[14Dij(2mij1+ikz(y1y2))2+14Aijγij2+14Bijβij2]},
setting:
Eij=1(i=j),Eij=0(ij);γij±=ik2z(y1+y2)mij2±+gij±4DijiC2ij;βij±=ik2z(x1+x2)+k2σ1ij2z2(x1x2)±iC1ij;mij1±=iuij4Bijβij±; mij2±=uijkσ1ij2z[βij±;2Bij(x1x2)];Aij=1α2ijuij2k2σ1ij44Bijz2+uij2σ1ij2214Dij(iuij2kσ1ij22Bijz+ikz)2;gij±=2(iuij2kσ1ij22Bijz+ikz)[2mij1±+ik(y1y2)/z];Bij=1α1ij+k2σ1ij22z2;Dij=12σ2ij2+uij24Bij;1α1ij=18σ1ij2+12δ1ij2;1α2ij=18σ2ij2+12δ2ij2.

Equations (10) and (11) can be considered as the general analytical expression of the CSDM of the beam generated by an EM TGSMA source. When N1=N2=1, it would reduce to the propagated CSDM for EM TGSM beams. Applying Eqs. (10) and (11), one may ready to study the statistical properties of such an EM TGSMA beam on propagation. Setting ρ1=ρ2=ρ, the spectral density can be defined as:

S(ρ,z)=TrW(ρ,ρ,z)=Wxx(ρ,ρ,z)+Wyy(ρ,ρ,z);
and the DOP of the beam with uncorrelated beam components can be calculated by the expression:
P(ρ,z)=|Wxx(ρ,ρ,z)Wyy(ρ,ρ,z)|Wxx(ρ,ρ,z)+Wyy(ρ,ρ,z).

The DOC at a pair of points ρ1 and ρ2 generally is given by:

μ(ρ1,ρ2,z)=Wxx(ρ1,ρ2,z)+Wyy(ρ1,ρ2,z)S(ρ1,z)S(ρ2,z).

3. Numerical results and discussions

In this section, we will demonstrate the evolution of statistical properties of EM TGSMA beam during propagation in free space. If no other explanation is made, those parameters are invariable in the following simulation: λ=632.8nm,R1=2R2=3mm,N1=2,N2=1.

 figure: Fig. 1

Fig. 1 Average intensity generated by an EM TGSMA source at several propagation distances in free space and setting: uxx = −uyy (mm−2); rows 1 (a1-a4) and 2 (b1-b4): σ1xx = σ1yy = 1mm, σ2xx = σ2yy = 0.3mm. δ1xx = δ1yy = 1mm, δ2xx = δ2yy = 0.3mm; rows 3 (c1-c4) and 4 (d1-d4): σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm,; σ2yy = δ2yy = 0.4mm.

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A stack of images taken in Fig. 1 illustrates the typical evolution of the transverse spectral density as an EM TGSMA beam propagates in free space. In rows 1 and 2 of Fig. 1, we show the intensity profiles with two different twist strength, uxx=uyy=3mm2 and uxx=uyy=2mm2 respectively. One can see that initial spots would progressively split into an array with propagation. Besides, the evolutions of intensity profiles during propagation are similar, both transformed from a horizontal ellipse to a vertical ellipse. Two other cases of beam propagation are given in rows 3 and 4 with uxx=uyy=3mm2 and uxx=uyy=3mm2 respectively. What need to emphasize that evolutions of spectral distribution in rows 1 and 2 are corresponding to the parameters setting for σ1xx/σ1yy=σ2xx/σ2yy and e˙lta1xx/δ1yy=δ2xx/δ2yy, and evolutions in rows 3 and 4 are for σ1xx/σ2yy=σ2xx/σ1yy and δ1xx/δ2yy=δ2xx/δ1yy.

 figure: Fig. 2

Fig. 2 DOP generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.

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Figure 2 presents the modulus of DOP of the field generated by the same source as in Fig. 1. In row 1 of Fig. 2, DOP transforms from a certain cross-like shape into two stable cross shapes arranged side by side in the far field without rotating. Then it remains shape-invariant (spreading with distance) as the beam propagates. When only changing twist strength, as seen in row 2, the evolution of DOP does not show the twist effect, either. Then in rows 3 and 4 of Fig. 2, we set σ1xx/σ2yy=σ2xx/σ1yy and δ1xx/δ2yy=δ2xx/δ1yy. It shows that DOP rotates counterclockwise around the beam center upon propagation in row 3, yet rotates clockwise in row 4, which are caused by the difference setting of twist strength. In addition, profiles of DOP in rows 3 and 4 rotate at the same pace during propagation.

 figure: Fig. 3

Fig. 3 DOC generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.

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In Fig. 3, we show the evolution of DOC of the field generated by the same source as in Fig. 2. As seen in rows 1 and 2, profiles of DOC transform from a horizontal ellipse to a vertical ellipse and have no twist effects. Then in rows 3 and 4, one can find that DOC rotates clockwise upon propagation in row 3, yet rotates counterclockwise in row 4.

Thus, one can conclude that twist effects of coherence distribution and polarization distribution during propagation can be influenced by the parameters σ1ij, σ2ij and δ1ij, δ2ij. Senses of rotation can also be controlled by adjusting the parameters of uij. In particular, for rows 1 and 2 of Figs. 2 and 3, coherent lengths for the element Wxx of CSDM are identical to the element Wyy, and beam widths are also the same. For uxx=3mm2, profiles of Wxx would rotate counterclockwise on propagation. Yet, the evolution of Wyy would rotate clockwise, for uyy=3mm2. Both rotate at a same pace. Hence, evolutions of the DOP and DOC for such EM beam propagating in free space do not show the twist effects. What’s more, an interesting feature from this rotating field is that the twist phase causes DOC to rotate in an opposite direction of DOP, as seen by comparing the rotation in Figs. 2 and 3.

 figure: Fig. 4

Fig. 4 The DOP on the axis of an EM TGSMA beam as a function of z for σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm, σ2yy = δ2yy = 0.4mm. (a) N1 = N2 = 2, (b) and (c) uxx = −uyy = −3mm−2.

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Let us now consider the on-axis behaviors of DOP of the EM TGSMA beam propagating in free space from the source plane, for selected values of the parameters. Figure 4(a) shows that for propagation in free space, DOP acquires a particular value after propagating a certain distance and it retains this value as the beam propagates further. Furthermore, twist strengths can also affect the behavior of beam statistics. The larger the value of uij, the greater the final stable value of the DOP. The evolutions of curves, corresponding to uxx=uyy=3mm2 and uxx=uyy=3mm2, are completely coincident. Figures 4(b) and 4(c) show the influence of N1 in the source plane on the DOP for EM TGSMA beam propagating in free space. When N1=N2 are odd, 3,5,7 or 9, curves of DOP present similar evolutionary. After two oscillations, the polarization value tends to zero. For the case where N1=N2 are even, 4,6,8 or 10, the polarization value tends to a certain value but not zero. Hence, the on-axis behavior of DOP during propagation can be beneficial to determine the number parity of such array.

It is also of interest to consider the transverse coherent distribution of such beam at distance z=5m and plots of this variation, as given in Fig. 5. One can find that twist strength can have effect on the coherent distribution, as shown in Fig. 5(a). Evolutions of curves, corresponding to uxx=uyy=3mm2 and uxx=uyy=3mm2 respectively, are also totally coincident. Coherence curve can present oscillation and as the propagation distance increases, the oscillation will gradually weaken. The decay rate of the case N1=3,5,7 or 9 is much greater than the decay rate of N1=2,4,6 or 10. For the case where N1=N2 are odd, 3,5,7 or 9, coherent curves present a Gaussian-like distribution. When N1=N2 are even, 4,6,8 or 10, DOP for EM TGSMA beams performs a central peak with two secondary peaks structure. It seems that the value of N1 (N1=N2) does not have a great impact on the trend of curves.

 figure: Fig. 5

Fig. 5 The DOC of EM TGSMA beam at z = 5m and yd = 0 with parameters as in Fig. 4.

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4. Conclusions

In summary, we introduced an analytic model for an EM TGSMA beam. We have derived the analytic expression for CSDM of such beam propagating in free space and investigated the spectral density, DOP and DOC in detail. Particularly, we investigated the twist effects of DOP and DOC of such beam propagating in free space. We have found that twist effects of polarization distribution and coherent distribution on propagation can be influenced by beam width and coherent length. Senses of rotation can be controlled by adjusting the parameters of uij. Moreover, the twist phase always causes DOC to rotate in an opposite direction compared with that of DOP. In addition, DOP on its axis during propagation would tend to a particular value. The larger the value of uij, the greater the final stable value of the DOP. Transverse coherent distribution at a certain distance are useful to determine the array’s parity. Typical behavior of DOP propagation on its axis can be beneficial to judge the parity of such array. Our results might be crucial for operation of communication, imaging, and sensing systems.

Funding

National Natural Science Foundation of China (NSFC) (11874321); Fundamental Research Funds for the Central Universities (2018FZA3005).

References

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. 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]  

4. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014). [CrossRef]  

5. M. Guo and D. Zhao, “Polarization properties of stochastic electromagnetic beams modulated by a wavefront-folding interferometer,” Opt. Express 26, 8581–8593 (2018). [CrossRef]   [PubMed]  

6. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008). [CrossRef]  

7. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21, 17512–17519 (2013). [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. Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22, 22534–22546 (2014). [CrossRef]   [PubMed]  

10. Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008). [CrossRef]  

11. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257–4262 (2009). [CrossRef]   [PubMed]  

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

13. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004). [CrossRef]  

14. 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]  

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

16. V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997). [CrossRef]  

17. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000). [CrossRef]   [PubMed]  

18. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003). [CrossRef]  

19. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003). [CrossRef]   [PubMed]  

20. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993). [CrossRef]  

21. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996). [CrossRef]  

22. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001). [CrossRef]  

23. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]  

24. L. Wan and D. Zhao, “Controllable rotating Gaussian Schell-model beams,” Opt. Lett. 44, 735–738 (2019). [CrossRef]   [PubMed]  

25. L. Wan and D. Zhao, “Twisted Gaussian Schell-model array beams,” Opt. Lett. 43, 3554–3557 (2018). [CrossRef]   [PubMed]  

26. A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994). [CrossRef]  

27. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018). [CrossRef]   [PubMed]  

28. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40, 5662–5665 (2015). [CrossRef]   [PubMed]  

29. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995). [CrossRef]  

References

  • View by:

  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. 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]
  4. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
    [Crossref]
  5. M. Guo and D. Zhao, “Polarization properties of stochastic electromagnetic beams modulated by a wavefront-folding interferometer,” Opt. Express 26, 8581–8593 (2018).
    [Crossref] [PubMed]
  6. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
    [Crossref]
  7. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21, 17512–17519 (2013).
    [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. Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22, 22534–22546 (2014).
    [Crossref] [PubMed]
  10. Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
    [Crossref]
  11. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257–4262 (2009).
    [Crossref] [PubMed]
  12. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
    [Crossref]
  13. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004).
    [Crossref]
  14. 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]
  15. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
    [Crossref]
  16. V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
    [Crossref]
  17. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
    [Crossref] [PubMed]
  18. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
    [Crossref]
  19. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
    [Crossref] [PubMed]
  20. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  21. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
    [Crossref]
  22. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
    [Crossref]
  23. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  24. L. Wan and D. Zhao, “Controllable rotating Gaussian Schell-model beams,” Opt. Lett. 44, 735–738 (2019).
    [Crossref] [PubMed]
  25. L. Wan and D. Zhao, “Twisted Gaussian Schell-model array beams,” Opt. Lett. 43, 3554–3557 (2018).
    [Crossref] [PubMed]
  26. A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
    [Crossref]
  27. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
    [Crossref] [PubMed]
  28. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40, 5662–5665 (2015).
    [Crossref] [PubMed]
  29. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
    [Crossref]

2019 (1)

2018 (3)

2015 (1)

2014 (2)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22, 22534–22546 (2014).
[Crossref] [PubMed]

2013 (1)

2009 (1)

2008 (4)

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

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

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]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[Crossref]

2007 (2)

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]

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

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2005 (1)

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]

2004 (1)

2003 (4)

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]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

2001 (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

2000 (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

1997 (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

1994 (1)

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

1993 (1)

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

Cai, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Chen, Z.

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

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

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Du, X.

Friberg, A. T.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Gori, F.

Guo, M.

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Korotkova, O.

Liu, L.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

Mao, Y.

Mei, Z.

Mukunda, N.

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

Pu, J.

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[Crossref]

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

Santarsiero, M.

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Shchepakina, E.

Shirai, T.

Simon, R.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Tervonen, E.

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Turunen, J.

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Wan, L.

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Wang, X.

Wolf, E.

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

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (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]

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

Zhang, Z.

Zhao, C.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Zhao, D.

Zhu, Y.

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

J. Appl. Phys. (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

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

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

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

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

Nature (2)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Opt. Commun. (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]

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

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Phys. Lett. A (2)

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

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

Phys. Rev. A (1)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Phys. Rev. E (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

Other (1)

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Average intensity generated by an EM TGSMA source at several propagation distances in free space and setting: uxx = −uyy (mm−2); rows 1 (a1-a4) and 2 (b1-b4): σ1xx = σ1yy = 1mm, σ2xx = σ2yy = 0.3mm. δ1xx = δ1yy = 1mm, δ2xx = δ2yy = 0.3mm; rows 3 (c1-c4) and 4 (d1-d4): σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm,; σ2yy = δ2yy = 0.4mm.
Fig. 2
Fig. 2 DOP generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.
Fig. 3
Fig. 3 DOC generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.
Fig. 4
Fig. 4 The DOP on the axis of an EM TGSMA beam as a function of z for σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm, σ2yy = δ2yy = 0.4mm. (a) N1 = N2 = 2, (b) and (c) uxx = −uyy = −3mm−2.
Fig. 5
Fig. 5 The DOC of EM TGSMA beam at z = 5m and yd = 0 with parameters as in Fig. 4.

Equations (14)

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W i j ( r 1 , r 2 , z , ω ) = E i * ( r 1 , z , ω ) E j ( r 2 , z , ω ) , ( i , j = x , y ) ,
W i j ( r 1 , r 2 , r 0 ) = W i j 0 ( r 1 r 0 , r 2 r 0 ) exp  ( 2 π i υ i j r 0 ) ,
W i j ( r 1 , r 2 ) = R i j ( r 1 , r 2 ) exp  ( 2 π i υ i j r m ) ,
R i j ( r 1 , r 2 ) = p ( r 0 ) W i j 0 ( r + D / 2 , r D / 2 ) exp  ( 2 π i υ i j r ) d 2 r .
W i j ( r 1 , r 2 ) = exp   [ ( x 1 x 2 ) 2 2 δ 1 i j 2 ] exp [ ( y 1 y 2 ) 2 2 δ 2 i j 2 ] × P n 1 = P cos   [ C 1 i j ( x 1 x 2 ) ] Q n 2 = Q cos [ C 2 i j ( y 1 y 2 ) ] × exp   ( i u i j x 1 y 2 ) exp   ( i u i j x 2 y 1 ) .
τ ( r ) = exp  ( x 2 4 σ 1 i j 2 ) exp  ( y 2 4 σ 2 i j 2 ) ,
W i j ( r 1 , r 2 ) = exp   ( x 1 2 + x 2 2 4 σ 1 i j 2 ) exp   ( y 1 2 + y 2 2 4 σ 2 i j 2 ) × exp   [ ( x 1 x 2 ) 2 2 δ 1 i j 2 ] exp [ ( y 1 y 2 ) 2 2 δ 2 i j 2 ] × P n 1 = P cos   [ C 1 i j ( x 1 x 2 ) ] Q n 2 = Q cos [ C 2 i j ( y 1 y 2 ) ] × exp   ( i u i j x 1 y 2 ) exp   ( i u i j x 2 y 1 ) .
W i j ( ρ 1 , ρ 2 , z ) = W i j ( r 1 , r 2 ) H z * ( ρ 1 , r 1 ) H z ( ρ 2 , r 2 ) d 2 r 1 d 2 r 2 ,
H z ( ρ , r ) = i k ( 2 π z ) 1 exp  [ i k ( ρ r ) 2 / ( 2 z ) ] .
W i j ( ρ 1 , ρ 2 , z ) = E i j n 1 = P P Q n 2 = Q ( k π σ 1 i j ) 2 A i j B i j D i j z 2 exp   [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] × exp   [ k 2 σ 1 i j 2 2 z 2 ( x 1 x 2 ) 2 ] × { exp [ 1 4 D i j ( 2 m i j 1 + + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + + 2 + 1 4 B i j β i j + 2 ]   + exp [ 1 4 D i j ( 2 m i j 1 + + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + 2 + 1 4 B i j β i j + 2 ] + exp [ 1 4 D i j ( 2 m i j 1 + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + 2 + 1 4 B i j β i j 2 ] + exp [ 1 4 D i j ( 2 m i j 1 + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j 2 + 1 4 B i j β i j 2 ] } ,
E i j = 1 ( i = j ) , E i j = 0 ( i j ) ; γ i j ± = i k 2 z ( y 1 + y 2 ) m i j 2 ± + g i j ± 4 D i j i C 2 i j ; β i j ± = i k 2 z ( x 1 + x 2 ) + k 2 σ 1 i j 2 z 2 ( x 1 x 2 ) ± i C 1 i j ; m i j 1 ± = i u i j 4 B i j β i j ± ;   m i j 2 ± = u i j k σ 1 i j 2 z [ β i j ± ; 2 B i j ( x 1 x 2 ) ] ; A i j = 1 α 2 i j u i j 2 k 2 σ 1 i j 4 4 B i j z 2 + u i j 2 σ 1 i j 2 2 1 4 D i j ( i u i j 2 k σ 1 i j 2 2 B i j z + i k z ) 2 ; g i j ± = 2 ( i u i j 2 k σ 1 i j 2 2 B i j z + i k z ) [ 2 m i j 1 ± + i k ( y 1 y 2 ) / z ] ; B i j = 1 α 1 i j + k 2 σ 1 i j 2 2 z 2 ; D i j = 1 2 σ 2 i j 2 + u i j 2 4 B i j ; 1 α 1 i j = 1 8 σ 1 i j 2 + 1 2 δ 1 i j 2 ; 1 α 2 i j = 1 8 σ 2 i j 2 + 1 2 δ 2 i j 2 .
S ( ρ , z ) = T r W ( ρ , ρ , z ) = W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) ;
P ( ρ , z ) = | W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) | W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) .
μ ( ρ 1 , ρ 2 , z ) = W x x ( ρ 1 , ρ 2 , z ) + W y y ( ρ 1 , ρ 2 , z ) S ( ρ 1 , z ) S ( ρ 2 , z ) .

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