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Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space

Yiming Dong, Fanlong Feng, Yahong Chen, Chengliang Zhao, and Yangjian Cai

Open Access Open Access

Abstract

Analytical nonparaxial propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integrals. Statistical properties, such as average intensity, degree of polarization and degree of coherence, of a nonparaxial cylindrical vector partially coherent field are illustrated numerically, and compared with that of a paraxial cylindrical vector partially coherent beam. It is found that the statistical properties of a nonparaxial cylindrical vector partially coherent field are much different from that of a paraxial cylindrical vector partially coherent beam, and are closely determined by the initial beam width and correlation coefficients. Our results will be useful for modulating the properties of a nonparaxial cylindrical vector partially coherent field.

©2012 Optical Society of America

1. Introduction

A vector laser beam can be classified as uniformly or non-uniformly polarized beam. Cylindrical vector beam is a typical vector beam with spatially non-uniform state of polarization (SOP) [1]. Radially or azimuthally polarized beam can be regarded as a special case of cylindrical vector beam. In the past decade, numerous efforts have been paid to generation and propagation of a cylindrical vector beam due to its unique properties under high-numerical-aperture focusing and important applications in particle trapping, optical data storage, material thermal processing, high-resolution imaging, electron acceleration, plasmonic focusing, laser machining, and free-space optical communications [1–20].

Most of previous literatures on cylindrical vector beam have been confined to coherent beam. Recently, more and more attention is being paid to vector partially coherent beam. Vector partially coherent beam with uniform SOP usually is named stochastic electromagnetic beam or partially coherent and partially polarized beam [21–23]. In the past ten years, generation and propagation of stochastic electromagnetic beam have been studied widely and it was found that stochastic electromagnetic beam has important application in free-space optical communications, active laser radar system, optical imaging, particle trapping, and optical scattering [21–40]. More recently, Dong et al. introduced a general theoretical model named cylindrical vector partially coherent LG beam to describe a vector partially coherent field with non-uniform SOP and studied its paraxial propagation properties in free space [41]. Wang et al. reported experimental generation of a partially coherent radially polarized beam [42], and found that we can shape its focused beam profile by varying its initial spatial coherence, which is useful for material thermal processing and particle trapping. Paraxial propagation properties of partially coherent radially polarized beam in turbulent atmosphere were reported in [43, 44].

When the beam width and the wavelength of a laser beam are comparable, the paraxial propagation theory is no longer valid. The beam emitted from semiconductor laser usually is nonparaxial. When a laser beam is focused by a thin lens with high-numerical-aperture, the focused beam also becomes nonparaxial. Nonparaxial field is commonly encountered in optical trapping and optical data storage. Thus it is of practical importance to study the nonparaxial propagation of beam [10, 45–52]. Deng explored the nonparaxial propagation properties of a radially polarized beam in free space [10]. Nonparaxial propagation of a scalar partially coherent beam and a vector partially coherent beam with uniform SOP were reported in [47–51]. Nonparaxial propagation of a scalar partially coherent beam in crystal was investigated in [52]. To our knowledge no results have been reported up until now on nonparaxial propagation of a cylindrical vector partially coherent LG beam, although paraxial propagation and experimental generation of scalar partially coherent LG beams carrying optical vortices have been investigated in detail [53–55]. In this paper, our aim is to explore the statistical properties of a nonparaxial cylindrical vector partially coherent LG field in free space. Analytical propagation formula is derived and some interesting results are found.

2. Nonparaxial propagation theory of a cylindrical vector partially coherent beam

Based on the unified theory of coherence and polarization, the second-order correlation properties of a nonparaxial vector partially coherent field, in space–frequency domain, can be characterized by the 3×3 cross-spectral density (CSD) matrix W↔(x1,y1,x2,y2,z) of the electric field, defined by the formula [21, 49, 52, 56]

W↔(x1,y1,x2,y2,z)=(Wxx(x1,y1,x2,y2,z)   Wxy(x1,y1,x2,y2,z)   Wxz(x1,y1,x2,y2,z)Wyx(x1,y1,x2,y2,z)   Wyy(x1,y1,x2,y2,z)   Wyz(x1,y1,x2,y2,z)Wzx(x1,y1,x2,y2,z)   Wzy(x1,y1,x2,y2,z)   Wzz(x1,y1,x2,y2,z)),
where
Wαβ(x1,y1,x2,y2,z)=〈Eα*(x1,y1,z)Eβ(x2,y2,z)〉,(α,β=x,y,z)
here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average, Ex, Ey and Ezdenote the components of the random electric vector along x, y and z directions, respectively .

The nonparaxial propagation of a vector coherent beam with source field E→(x0,y0,0)=Ex(x0,y0,0)e→x+Ey(x0,y0,0)e→yin free space can be studied with the help of the following vectorial Rayleigh-Sommerfeld diffraction integrals [57]

Eα(x,y,z)=−12π∬Eα(x0,y0,0)∂∂z[exp(ikR)R]dx0dy0, (α=x,y)
Ez(x,y,z)=12π∬{Ex(x0,y0,0)∂∂x[exp(ikR)R]+Ey(x0,y0,0)∂∂y[exp(ikR)R]}dx0dy0,
where R=(x−x0)2+(y−y0)2+z2, k=2π/λ is the wave number with λ being the wavelength. When R>>λ, the terms∂∂x[exp(ikR)R], ∂∂y[exp(ikR)R] and∂∂z[exp(ikR)R] in Eqs. (3) and (4) are approximated as

∂∂α[exp(ikR)R]=ikexp(ikR)R2(α−α0),   (α=x,y)
∂∂z[exp(ikR)R]=ikzexp(ikR)R2.

Applying Eqs. (1)-(6), we find that the nonparaxial propagation of the elements of the CSD matrix of a vector partially coherent beam can be studied with the help of the following generalized vectorial Rayleigh-Sommerfeld diffraction integrals [49]

Wαβ(x1,y1,x2,y2,z)=k2z24π2∬∬Wαβ(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22×dx10dy10dx20dy20, (α,β=x,y)
Wαz(x1,y1,x2,y2,z)=−k2z4π2∬∬[Wαx(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22(x2−x20)   +Wαy(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22(y2−y20)]dx10dy10dx20dy20,  (α=x,y)
Wzz(x1,y1,x2,y2,z)=k24π2∬∬[Wxx(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22   ×(x1−x10)(x2−x20)+Wxy(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22(x1−x10)(y2−y20)   +Wyx(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22(y1−y10)(x2−x20)   +Wyy(x10,y10,x20,y20,0)exp[ik(R1−R2)]R12R22(y1−y10)(y2−y20)]dx10dy10dx20dy20,
where

Wαβ(x10,y10,x20,y20,0)=〈Eα*(x10,y10,0)Eβ(x20,y20,0)〉.(α,β=x,y)

The electric vector field of a cylindrical vector coherent Laguerre-Gaussian (LG) beam at z = 0 is expressed in cylindrical coordinates as follows [4]

E→(r,ϕ,0)=exp(−r2w02)(2r2w02)(n±1)/2Lpn±1(2r2w02){cos(nϕ)e→ϕ∓sin(nϕ)e→r±sin(nϕ)e→ϕ+cos(nϕ)e→r},
where r and ϕ are the radial and azimuthal coordinates, Lpn±1 denotes the Laguerre polynomial with mode orders p and n±1, w0 is the beam width of the fundamental Gaussian mode. Under the condition ofp=0 and n=0, Eq. (11) reduces to the electric field of a radially or azimuthally polarized Gaussian beam.

In Cartesian coordinates, Eq. (11) can be expressed in the following alternative form [41]

E→(x0,y0,0)={Ex(x0,y0,0)e→x+Ey(x0,y0,0)e→yEy(x0,y0,0)e→x+Ex(x0,y0,0)e→y}  =exp(−x02+y02w02)(−1)p22p+n±1p!{12i∑m=0p∑s=0n±1is[1−(−1)s](pm)(n±1  s)H2m+n±1−s(2x0w0)H2p−2m+s(2y0w0)e→x12∑m=0p∑s=0n±1is[1+(−1)s](pm)(n±1  s)H2m+n±1−s(2x0w0)H2p−2m+s(2y0w0)e→x      +12∑m=0p∑s=0n±1is[1+(−1)s](pm)(n±1  s)H2m+n±1−s(2x0w0)H2p−2m+s(2y0w0)e→y+12i∑m=0p∑s=0n±1is[1−(−1)s](pm)(n±1  s)H2m+n±1−s(2x0w0)H2p−2m+s(2y0w0)e→y},
where Hm is the Hermite polynomial of order m, (pm) and (n±1  s) are binomial coefficients. Note the conversion of the source field expression of vector coherent LG beam from Laguerre polynomial into Hermite polynomial is to facilitate analytic derivations as shown later.

For the case of E→(x0,y0,0)=Ex(x0,y0,0)e→x+Ey(x0,y0,0)e→y, applying Eqs. (1), (2) and (10), we can express the CSD matrix of a cylindrical vector partially coherent beam generated by a Schell-model source as follows [41]

W↔(x10,y10,x20,y20,0)=(Wxx(x10,y10,x20,y20,0)   Wxy(x10,y10,x20,y20,0)   0Wyx(x10,y10,x20,y20,0)   Wyy(x10,y10,x20,y20,0)   0                0                                0                 0),
where
Wαβ(x10,y10,x20,y20,0)=Aαβ14exp[−x102+x202+y102+y202w02−(x10−x20)2+(y10−y20)22σαβ2]×124p+2n±2(p!)2∑m=0p∑s=0n±1∑l=0p∑h=0n±1is(−i)h[1+Cα(−1)s][1+Cβ(−1)h](pm)(pl)(n±1s)(n±1h) ×H2m+n±1−s(2x10w0)H2l+n±1−h(2x20w0)H2p−2m+s(2y10w0)H2p−2l+h(2y20w0),             (α,β=x,y)
Wyx(x10,y10,x20,y20,0)=[Wxy(x20,y20,x10,y10,0)]*, 
with
Aαβ={1           α=β=x,−iBxy     α=x, β=y1           α=β=y,,Cα={−1         α=x,1           α=y,
hereBxy=|Bxy|exp(iϕ) is the correlation coefficient between theEx and Ey field components, ϕ is the phase difference between the x-and y-components of the field and is removable in most case, σxx, σyyand σxydenote the r.m.s. widths of the auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively.

For the case of E→(x0,y0,0)=Ey(x0,y0,0)e→x+Ex(x0,y0,0)e→y, the elements of the CSD matrix of a cylindrical vector partially coherent LG beam are expressed as follows [41]

W1xx(x10,y10,x20,y20,0)=Wyy(x10,y10,x20,y20,0),W1yy(x10,y10,x20,y20,0)=Wxx(x10,y10,x20,y20,0),W1xy(x10,y10,x20,y20,0)=Wyx(x10,y10,x20,y20,0),W1xy(x10,y10,x20,y20,0)=Wxy(x10,y10,x20,y20,0),
where Wxx,Wxy,Wyx,Wyyare given by Eqs. (14) and (15).

Now we study the nonparaxial propagation of the cylindrical vector partially coherent LG beam in free space. After expansion, R1andR2can be approximated as [49]

Ri≈ri+xi02+yi02−2xixi0−2yiyi02ri,    (i=1,2)
where ri=(xi2+yi2+z2)1/2. Substituting Eqs. (14)-(16) and (18) into Eqs. (7)-(9), we obtain (after tedious integration and operation) the following expressions for the elements of the CSD matrix of the nonparaxial cylindrical vector partially coherent LG field at the output plane
Wαβ(x1,y1,x2,y2,z)=Aαβk2z2exp[ik(r1−r2)]16r12r22M1αβ125(2p+n±1)/2(p!)2(1−2M1αβw02)(n±1+2p)/2×exp{−k2(x12+y12)4M1αβr12−k24M2αβ[(x2r2−x12M1αβr1σαβ2)2+(y2r2−y12M1αβr1σαβ2)2]}×∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c1=02m+n±1−s ∑d1=0c1/2∑e1=0(2l+n±1−h)/2∑c2=02p−2m+s ∑d2=0c2/2∑e2=0(2p−2l+h)/2(2p−2m+sc2)(2m+n±1−sc1)×(pm)(pl)(n±1s)(n±1h)[1+Cα(−1)s][1+Cβ(−1)h](−1)d1+d2+e1+e2is(−i)h(2i)−(c1+c2−2d1−2d2+n±1+2p−2e1−2e2)                           ×(22w0)2p+n±1−2e2−2e1c1!d1!(c1−2d1)!c2!d2!(c2−2d2)!(2l+n±1−h)!e1!(2l+n±1−h−2e1)!(2p−2l+h)!e2!(2p−2l+h−2e2)!×1(M2αβ)c1+c2+n±1+2p−2d1−2d2−2e1−2e2+2[2σαβ2(w02M1αβ2−2M1αβ)1/2]c1+c2−2d1−2d2×Hc1−2d1+2l+n±1−h−2e1[k2M2αβ(x12M1αβr1σαβ2−x2r2)]H2m+n±1−s−c1[−ikx1r1(w02M1αβ2−2M1αβ)1/2]×Hc2−2d2+2p−2l+h−2e2[k2M2αβ(y12M1αβr1σαβ2−y2r2)]H2p−2m+s−c2[−iky1r1(w02M1αβ2−2M1αβ)1/2],   (α,β=x,y)
Wyx(x1,y1,x2,y2,z)=Wxy*(x2,y2,x1,y1,z), 
Wxz(x1,y1,x2,y2,z)=∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c1=02m+n±1−s∑d1=0c1/2∑e1=0(2l+n±1−h)/2∑c2=02p−2m+s ∑d2=0c2/2∑e2=0(2p−2l+h)/2Q1×[1−(−1)s]{Q3xxQ4xx[1−(−1)h]Hc2−2d2+2p−2l+h−2e2[k2M2xx(y12M1xxr1σxx2−y2r2)]−Q2Q5xy[1+(−1)h]Hc1−2d1+2l+n±1−h−2e1[k2M2xy(x12M1xyr1σxy2−x2r2)]},
Wyz(x1,y1,x2,y2,z)=∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c1=02m+n±1−s∑d1=0c1/2∑e1=0(2l+n±1−h)/2∑c2=02p−2m+s ∑d2=0c2/2∑e2=0(2p−2l+h)/2Q1[1+(−1)s]{Q2Q4xy[1−(−1)h]Hc2−2d2+2p−2l+h−2e2[k2M2xy(y12M1xyr1σxy2−y2r2)]+Q3yyQ5yy[1+(−1)h]Hc1−2d1+2l+n±1−h−2e1[k2M2yy(x12M1yyr1σyy2−x2r2)]},
Wzx(x1,y1,x2,y2,z)=Wxz*(x2,y2,x1,y1,z),
Wzy(x1,y1,x2,y2,z)=Wyz*(x2,y2,x1,y1,z),
Wzz(x1,y1,x2,y2,z)=Wzzxx(x1,y1,x2,y2,z)+Wzzxy(x1,y1,x2,y2,z)                           +Wzzyx(x1,y1,x2,y2,z)+Wzzyy(x1,y1,x2,y2,z), 
Wzzxx(x1,y1,x2,y2,z)=∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c1=0(2m+n±1−s)/2∑d1=0(2l+n±1−h)/2∑c2=02p−2m+s ∑d2=0c2/2∑e2=0(2p−2l+h)/2Q6xxQ7xx×Hc2−2d2+2p−2l+h−2e2[k2M2xx(y12M1xxr1σxx2−y2r2)]{x1∑f1=02m+n±1−s−2c1∑g1=0f1/2Q8xx(2m+n±1−s−2c1           f1)×H2m+n±1−s−2c1−f1(kx12M1xxr1)[2iM2xxx2Hf1−2g1+2l+n±1−h−2d1(k2M2xx(x12M1xxr1σxx2−x2r2))−Hf1−2g1+2l+n±1−h−2d1+1(k2M2xx(x12M1xxr1σxx2−x2r2))]−∑f2=02m+n±1−s−2c1+1∑g2=0f2/2Q9xx(2m+n±1−s−2c1+1            f2)×H2m+n±1−s−2c1+1−f2(kx12M1xxr1)[2iM2xxx2Hf2−2g2+2l+n±1−h−2d1(k2M2xx(x12M1xxr1σxx2−x2r2))−Hf2−2g2+2l+n±1−h−2d1+1(k2M2xx(x12M1xxr1σxx2−x2r2))]},                                                   
Wzzyy(x1,y1,x2,y2,z)=∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c2=02m+n±1−s∑d2=0c2/2∑e2=0(2l+n±1−h)/2∑c1=0(2p−2m+s)/2∑d1=0(2p−2l+h)/2Q6yy(2m+n±1−s        c2)×(2i)−(c2−2d2+n±1−2e2+4p−2m+s−2c1−2d1+2)(1−2w02M1yy)(2m+n±1−s)/2(22w0)n±1−2e2+4p−2m+s−2c1−2d1×(2l+n±1−h)!e2!(2l+n±1−h−2e2)!(2p−2m+s)!c1!(2p−2m+s−2c1)!(2p−2l+h)!d1!(2p−2l+h−2d1)!1(M1yy)2p−2m+s−2c1+3×Hc2−2d2+2l+n±1−h−2e2[k2M2yy(x12M1yyr1σyy2−x2r2)]H2m+n±1−s−c2[−ikx1(w02M1yy2r12−2M1yyr12)1/2]×{y1∑f1=02p−2m+s−2c1∑g1=0f1/2Q8yy(2p−2m+s−2c1          f1)H2p−2m+s−2c1−f1(ky12M1yyr1)[2iM2yyy2H2p−2l+h−2d2+f1−2g1(k2M2yy(y12M1yyr1σyy2−y2r2))−H2p−2l+h−2d1+f1−2g1+1(k2M2yy(y12M1yyr1σyy2−y2r2))]−∑f2=02p−2m+s−2c1+1∑g2=0f2/2Q9yy(2p−2m+s−2c1+1           f2)H2p−2m+s−2c1+1−f2(ky12M1yyr1)[2iM2yyy2H2p−2l+h−2d1+f2−2g2(k2M2yy(y12M1yyr1σyy2−y2r2))−H2p−2l+h−2d1+f2−2g2+1(k2M2yy(y12M1yyr1σyy2−y2r2))]}.   
Wzzyx(x1,y1,x2,y2,z)=[Wzzxy(x2,y2,x1,y1,z)]*, 
Wzzxy(x1,y1,x2,y2,z)=∑m=0p∑s=0n±1∑l=0p∑h=0n±1∑c1=0(2m+n±1−s)/2∑d1=0(2l+n±1−h)/2∑c2=02p−2m+s∑d2=0c2/2∑e2=0(2p−2l+h)/2Q5xyQ6xyQ7xy×{x1∑f1=02m+n±1−s−2c1∑g1=0f1/2Q8xy(2m+n±1−s−2c1            f1)H2m+n±1−s−2c1−f1(kx12M1xyr1)Hf1−2g1+2l+n±1−h−2d1[k2M2xy(x12M1xyr1σxy2−x2r2)]−∑f2=02m+n±1−s−2c1+1∑g2=0f2/2Q9xy(2m+n±1−s−2c1+1              f2)×H2m+n±1−s−2c1+1−f2(kx12M1xyr1)Hf2−2g2+2l+n±1−h−2d1[k2M2xy(x12M1xyr1σxy2−x2r2)]}, 
where
M1αβ=1/w02+1/(2σαβ2)−ik/(2r1),(α,β=x,y)
M2αβ=1/w02+1/(2σαβ2)+ik/(2r2)−1/(4M1αβσαβ4),(α,β=x,y)
Q1=−k2zexp[ik(r1−r2)]16r12r2225(2p+n±1)/2(p!)2(−1)d1+d2+e1+e2is(−i)h(2i)−(c1+c2−2d1−2d2+n±1+2p−2e1−2e2+1)×(22w0)n±1+2p−2e1−2e2(2m+n±1−s c1)(2p−2m+sc2)(pm)(pl)(n±1s)(n±1h)×c1!d1!(c1−2d1)!c2!d2!(c2−2d2)!(2l+n±1−h)!e1!(2l+n±1−h−2e1)!(2p−2l+h)!e2!(2p−2l+h−2e2)!,
Q2=iBxyM1xy1(M2xy)c1+c2−2d1−2d2+n±1+2p−2e1−2e2+3[2σxy2(w02M1xy2−2M1xy)1/2]c1+c2−2d1−2d2×(1−2M1xyw02)(n±1+2p)/2exp{−k2(x12+y12)4M1xyr12−k24M2xy[(x2r2−x12M1xyr1σxy2)2+(y2r2−y12M1xyr1σxy2)2]}×H2m+n±1−s−c1[−ikx1(w02M1xy2r12−2M1xyr12)1/2]H2p−2m+s−c2(−iky1r1(w02M1xy2−2M1xy)1/2),
Q3αα=1M1αα1(M2αα)c1+c2−2d1−2d2+n±1+2p−2e1−2e2+3[2σαα2(w02M1αα2−2M1αα)1/2]c1+c2−2d1−2d2×(w02M1αα−2w02M1αα)(2p+n±1)/2exp{−k2(x12+y12)4M1ααr12−k24M2αα[(x2r2−x12M1ααr1σαα2)2+(y2r2−y12M1ααr1σαα2)2]}×H2m+n±1−s−c1[−ikx1r1(w02M1αα2−2M1αα)1/2]H2p−2m+s−c2[−iky1r1(w02M1αα2−2M1αα)1/2],      (α=x,y)
Q4xα=2iM2xαx2Hc1−2d1+2l+n±1−h−2e1[k2M2xα(x12M1xαr1σxα2−x2r2)]−Hc1−2d1+2l+n±1−h−2e1+1[k2M2xα(x12M1xαr1σxα2−x2r2)],  (α=x,y)
Q5αy=2iM2αyy2Hc2−2d2+2p−2l+h−2e2[k2M2αy(y12M1αyr1σαy2−y2r2)]−Hc2−2d2+2p−2l+h−2e2+1[k2M2αy(y12M1αyr1σαy2−y2r2)],  (α=x,y)
Q6αβ=Aαβexp{−k2(x12+y12)4M1αβr12−k24M2αβ[(x2r2−x12M1αβr1σαβ2)2+(y2r2−y12M1αβr1σαβ2)2]}×k2exp[ik(r1−r2)]16r12r2225(2p+n±1)/2(p!)2(pm)(pl)(n±1s)(n±1h)is(−i)h(−1)c1+d1+d2+e2[1+Cα(−1)s]1+Cβ(−1)h2(−2c1+1)/2×1(M2αβ)n±1−2d1+c2−2d2+2p−2e2+3[2σαβ2(w02M1αβ2−2M1αβ)1/2]c2−2d2c2!d2!(c2−2d2)!,    (α,β=x,y)
Q7xα=(2p−2m+s       c2)(2i)−(2m+2(n±1)−s−2c1−2d1+c2−2d2+2p−2e2+2)(22w0)2m+2(n±1)−s−2c1−2d1+2p−2e2×(2m+n±1−s)!c1!(2m+n±1−s−2c1)!(2l+n±1−h)!d1!(2l+n±1−h−2d1)!(2p−2l+h)!e2!(2p−2l+h−2e2)!(1−2M1xαw02)(2p−2m+s)/2×1(M1xα)2m+n±1−s−2c1+3H2p−2m+s−c2[−iky1(w02M1xα2r12−2M1xαr12)1/2],    (α=x,y)
Q8αβ=2M1αβ(−1)g1(2i)−(f1−2g1−1)f1!g1!(f1−2g1)!1(M2αβ)f1−2g1(2iM1αβσαβ2)f1−2g1, (α,β=x,y)
Q9αβ=(−1)g2(2i)−(f2−2g2)f2!g2!(f2−2g2)!1(M2αβ)f2−2g2(2iM1αβσαβ2)f2−2g2. (α,β=x,y)
In above integration, we have used the following integral and expansion formulae [58, 59]
∫−∞∞exp[−(x−y)2]Hn(ax)dx=π(1−a2)n/2Hn(ay(1−a2)1/2),
∫−∞∞xnexp[−(x−β)2]dx=(2i)−nπHn(iβ),
Hn(x+y)=12n/2∑k=0n(nk)Hk(2x)Hn−k(2y),
Hn(x)=∑k=0n/2(−1)kn!k!(n−2k)!(2x)n−2k.
Under the condition of ri≈z+(xi2+yi2)/2z, Eqs. (19) and (20) reduces to the propagation formulae (Eqs. (15)-(18) of Ref [41].) for the elements of the2×2 CSD matrix of a paraxial cylindrical vector partially coherent LG beam in free space.

3. Statistical properties of a nonparaxial cylindrical vector partially coherent field

In this section, we study numerically the statistical properties of a nonparaxial cylindrical vector partially coherent LG field in free space by applying the formulae derived in section 2. In the following numerical examples, we set λ=632.8nm, Bxy=1, p=1, n±1=1, zr=πw02/λ.

The intensity of a nonparaxial cylindrical vector partially coherent LG field at the output plane in free space is expressed as

I(x,y,z)=Ix(x,y,z)+Iy(x,y,z)+Iz(x,y,z)             =Wxx(x,y,x,y,z)+Wyy(x,y,x,y,z)+Wzz(x,y,x,y,z),
where Wxx, Wyyand Wzzare given by Eqs. (19) and (25).

The degree of polarization of a nonparaxial cylindrical vector partially coherent LG field at the output plane can be expressed by the formula proposed by Ellis et al. [56]

P(x,y,z)=p1(x,y,z)−p2(x,y,z)p1(x,y,z)+p2(x,y,z)+p3(x,y,z),
wherep1(x,y,z),p2(x,y,z)andp3(x,y,z)are the three eigenvalues of the CSD matrix of a nonparaxial cylindrical vector partially coherent LG field, and they satisfy the relation p1(x,y,z)≥p2(x,y,z)≥p3(x,y,z). Note there is another definition of degree of polarization proposed by Setala et al. [60]. Both definitions of degree of polarization can be applied to study the polarization properties of a nonparaxial cylindrical vector partially coherent LG field. In this paper, we adopted the definition proposed by Ellis et al. for numerical calculation.

The spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field at a pair of points (x1,y1,z) and (x2,y2,z) is given by the formula [61]

μ(x1−x2,y1−y2,z)=TrW↔(x1,y1,x2,y2,z)TrW↔(x1,y1,x1,y1,z)TrW↔(x2,y2,x2,y2,z), 
where Tr stands for the trace of the CSD matrix.

Applying Eqs. (19), (25) and (45), we calculate in Figs. 1 -3 the normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances withρ=x2+y2for w0=10λ, w0=λ, and w0=0.5λ, respectively. For the convenience of comparison, the normalized intensity distribution Ip/Ipmaxand the corresponding cross line (y = x) calculated by the paraxial propagation formulae (Eqs. (15) and (18) of Ref [41].) are also shown in Figs. 1-3. The other parameters are chosen asσxx=σxy=σyy=λ. From Figs. 1-3, we see that the propagation properties of a cylindrical vector partially coherent LG beam are closely determined by the beam width w0, and the nonparaxial propagation properties are much different from the paraxial propagation properties. When w0is much larger than the wavelength λ(i.e., w0=10λ), there is almost no difference between the intensity distribution I/Imax calculated by the nonparaxial propagation formulae and the intensity distribution Ip/Ipmax calculated by the paraxial propagation formulae due to the fact that Iz/Imaxis extremely small compared to I/Imaxor (Ix+Iy)/Imaxand can be negligible (see Fig. 1), thus in this case the paraxial propagation formulae are enough to treat the propagation of the cylindrical vector partially coherent LG beam. With the decrease of w0, the difference between I/Imaxand Ip/Ipmax appears gradually (see Figs. 2 and 3). The intensity I/Imax calculated by the nonparaxial propagation formulae becomes of non-circular symmetry due to the fact that Iz/Imaxhas a non-circular intensity distribution and its contribution increases with the decrease of w0, while (Ix+Iy)/Imaxand Ip/Ipmaxcalculated by the nonparaxial propagation formulae and paraxial propagation formulae respectively still have circular symmetries. Thus, when w0 is comparable to λ, the nonparaxial propagation formulae should be applied to treat the propagation of a cylindrical vector partially coherent LG beam.

 figure: Fig. 1

Fig. 1 Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ=x2+y2and w0=10λ. Ip/Ipmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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 figure: Fig. 2

Fig. 2 Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ=x2+y2and w0=λ. Ip/Ipmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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 figure: Fig. 3

Fig. 3 Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances withρ=x2+y2andw0=0.5λ. Ip/Ipmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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To learn about the influence of correlation coefficients on the nonparaxial propagation properties of a cylindrical vector partially coherent LG beam, we calculate in Fig. 4 the normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at z=10zrwithρ=x2+y2 andw0=λfor different values of the correlation coefficients σxx,σxy,σyy. One finds that the nonparaxial propagation properties of a cylindrical vector partially coherent LG beam are also determined by its initial correlation coefficients. When w0 is comparable to λ, I/Imaxof the cylindrical vector partially coherent LG beam with large correlation coefficients σxx,σxy,σyy (i.e., high spectral degree of coherence) has a non-circular intensity distribution due to the contribution of the non-circular distribution of Iz/Imax. With the decrease of the correlation coefficients, I/Imaxgradually becomes of circular symmetry due to the fact that Iz/Imax gradually transforms from non-circular symmetry into circular symmetry. What’s more, the ratio of Iz/(Ix+Iy)also increases as the correlation coefficients decreases. Thus, one can shape the intensity distribution of a nonparaxial cylindrical vector LG field by modulating its initial spatial coherence, which is useful in some applications, such as material thermal processing and particle trapping. Note that in Fig. 4, we have considered the partially coherent cylindrical vector beam with correlation lengths being comparable to the wavelength or smaller than the wavelength. It is known from [62, 63] that when a partially coherent beam is focused by a thin lens with high numerical aperture, the correlation length of the focused beam can be comparable to the wavelength or smaller than the wavelength. Thus we may generate a nonparaxial cylindrical vector partially coherent field with correlation lengths being comparable to the wavelength or smaller than the wavelength by focusing a cylindrical vector partially coherent beam with a high numerical aperture thin lens.

 figure: Fig. 4

Fig. 4 Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, Iz/Imaxand the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at z=10zrwithρ=x2+y2andw0=λ for different values of the correlation coefficients σxx,σxy,σyy.

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Now we study the properties of the degree of polarization of a nonparaxial cylindrical vector partially coherent field on propagation in free space. We calculate in Fig. 5 the degree of polarization and the corresponding cross line (y = 0) of a cylindrical vector partially coherent LG beam in the source plane (z = 0). As shown in Fig. 5, the degree of polarization of a cylindrical vector partially coherent LG beam in the source plane equals 1 for all the points across the entire transverse plane. Furthermore, the degree of polarization in the source plane is independent of w0 and the correlation coefficients.

 figure: Fig. 5

Fig. 5 Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at the source plane (z = 0).

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We calculate in Figs. 6 -8 the degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances for w0=10λ, w0=λ, and w0=0.5λ, respectively. The other parameters are chosen as σxx=σxy=σyy=5λ. We find from Figs. 6-8 that the behavior of the degree of polarization of a cylindrical vector partially coherent beam on propagation is closely related with the beam width w0, and the evolution properties of the degree of polarization of a nonparaxial cylindrical vector partially coherent LG field are much different that of a paraxial cylindrical vector partially coherent LG beam. When w0is much larger than the wavelength λ(i.e., w0=10λ), the evolution properties of the degree of polarization (see Fig. 6) calculated by the nonparaxial propagation formulae are similar to that calculated by paraxial propagation formulae (see Fig. 3 of Ref [41].). The degree polarization varies on propagation, and one sees that a dip appears in the distribution of the degree of polarization, which means the cylindrical vector partially coherent LG beam was depolarized on propagation. The distribution of the degree of polarization on propagation has a circular symmetry. With the decrease of w0, the difference between the degree of polarization of a nonparaxial cylindrical vector partially coherent LG field and that of a paraxial cylindrical vector partially coherent LG beam appears. More dips appear in the distribution of the degree of polarization of a nonparaxial partially coherent LG beam, and the distribution of the degree of polarization gradually becomes of non-circular symmetry. The appearance of additional dips in Figs. 7 and 8 are due to the non-uniform contribution of the z-component field to the depolarization of the nonparaxial cylindrical vector partially coherent LG field on propagation. With the increase of the propagation distance, the additional dips disappear gradually, and in the far field, the additional dips disappear, and the degree of polarization recovers circular symmetry.

 figure: Fig. 6

Fig. 6 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w0=10λ and ρ=x2+y2.

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 figure: Fig. 7

Fig. 7 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with w0=λ andρ=x2+y2.

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 figure: Fig. 8

Fig. 8 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w0=0.5λ andρ=x2+y2.

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We calculate in Fig. 9 the degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at z=zrfor different values of the correlation coefficients σxx,σxy,σyy with w0=5λ. One finds from Fig. 9 that the behavior of the degree of polarization of a nonparaxial cylindrical vector partially coherent LG field is affected by the correlation coefficients. One can modulate the degree of polarization of a nonparaxial cylindrical vector partially coherent LG field by varying its initial correlation coefficients.

 figure: Fig. 9

Fig. 9 Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at z=zrfor different values of the correlation coefficients σxx,σxy,σyy.

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To learn about the evolution properties of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field, we calculate in Fig. 10 the modulus of the spectral degree of coherence between two transverse points (x1, 0) and (x2, 0) at several propagation distances for different values of w0with σxx=σxy=σyy=λ. Figure 11 shows the modulus of the spectral degree of coherence between two transverse points (x1, 0) and (x2, 0) at z=zrfor different values of the correlation coefficients σxx,σxy,σyywith w0=λ. It is clear that the spectral degree of coherence of a cylindrical vector partially coherent LG beam gradually becomes of non-Gaussian distribution on propagation, and the evolution properties of the spectral degree of coherence of nonparaxial cylindrical vector partially coherent LG field are much different from that of a paraxial cylindrical vector partially coherent LG beam (see Fig. 10), and the behavior of the spectral degree of coherence varies as the initial correlation coefficients vary (see Fig. 11).

 figure: Fig. 10

Fig. 10 Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1,0) and (x2,0) at several propagation distances for different values of w0.

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 figure: Fig. 11

Fig. 11 Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1, 0) and (x2, 0) at z=zrfor different values of the correlation coefficients σxx,σxy,σyy.

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4. Summary

We have derived analytical propagation formula for the CSD matrix of a nonparaxial cylindrical vector partially coherent LG field in free space by applying the generalized Raleigh-Sommerfeld diffraction integrals. As an application example, we have studied the statistical properties of paraxial and nonparaxial cylindrical vector partially coherent LG fields numerically and comparatively. Our numerical results show that the statistical properties of a nonparaxial cylindrical vector partially coherent LG field are closely determined by the initial beam width and correlation coefficients, and are much different from that of a paraxial beam. When the beam width is much larger than the wavelength, the results calculated by nonparaxial propagation formulae agree well with that calculated by paraxial propagation formulae, thus the paraxial propagation formulae are enough to treat the propagation of beam. When the beam width is comparable to the wavelength, significant differences between paraxial and nonparaxial results appear, and nonparaxial propagation formulae are required to treat the propagation of beam. By varying the initial correlation coefficients, one can modulate the statistical properties of a nonparaxial cylindrical vector partially coherent LG field, which will be useful in some applications, such as material thermal processing and particle trapping. Note that we have assumedR>>λ to obtain analytical nonparaxial propagation formula for the cylindrical vector partially coherent beam. With this assumption, the obtained nonparaxial propagation formula in this paper may break down in the near-field region, i.e., at the propagation distances of a few wavelengths from the source, because the evanescent waves at small distances from the source are excluded in our formula. To describe the nonparaxial fields in the near-field region, one can use the angular spectrum representation to derive analytical expression, and we leave this for future study.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 10904102&61008009, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Province under Grant No. 10KJB140011, the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant No. CXLX11_0064 and the National College Students Innovation Experiment Program under Grant No. 111028510.

References and links

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]  

2. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [PubMed]  

3. Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003). [CrossRef]  

4. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [CrossRef]  

5. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]   [PubMed]  

7. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001). [CrossRef]   [PubMed]  

8. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef]   [PubMed]  

9. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef]   [PubMed]  

10. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006). [CrossRef]  

11. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45(3), 470–479 (2006). [CrossRef]   [PubMed]  

12. B. J. Roxworthy and K. C. Toussaint Jr., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010). [CrossRef]  

13. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010). [CrossRef]   [PubMed]  

14. X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009). [CrossRef]  

15. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]  

16. R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010). [CrossRef]   [PubMed]  

17. K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006). [CrossRef]  

18. R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express 17(16), 13982–13988 (2009). [CrossRef]   [PubMed]  

19. G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004). [CrossRef]  

20. C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett. 32(24), 3543–3545 (2007). [CrossRef]   [PubMed]  

21. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]  

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

24. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). [CrossRef]   [PubMed]  

25. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]   [PubMed]  

26. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005). [CrossRef]  

27. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005). [CrossRef]  

28. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef]   [PubMed]  

29. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]  

30. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef]   [PubMed]  

31. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]  

32. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010). [CrossRef]  

33. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef]   [PubMed]  

34. G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011). [CrossRef]   [PubMed]  

35. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]   [PubMed]  

36. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). [CrossRef]  

37. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef]   [PubMed]  

38. L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011). [CrossRef]  

39. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011). [CrossRef]   [PubMed]  

40. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010). [CrossRef]  

41. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011). [CrossRef]   [PubMed]  

42. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]  

43. H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010). [CrossRef]  

44. H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009). [CrossRef]  

45. H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]  

46. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21(10), 2029–2037 (2004). [CrossRef]   [PubMed]  

47. K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004). [CrossRef]   [PubMed]  

48. Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. 29(23), 2710–2712 (2004). [CrossRef]   [PubMed]  

49. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004). [CrossRef]   [PubMed]  

50. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]  

51. L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011). [CrossRef]  

52. L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011). [CrossRef]   [PubMed]  

53. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003). [CrossRef]   [PubMed]  

54. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef]   [PubMed]  

55. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef]   [PubMed]  

56. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005). [CrossRef]  

57. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

58. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

59. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

60. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002). [CrossRef]   [PubMed]  

61. O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004). [CrossRef]   [PubMed]  

62. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008). [CrossRef]   [PubMed]  

63. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011). [CrossRef]  

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Figures (11)
Fig. 1
Fig. 1 Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =10λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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Fig. 2
Fig. 2 Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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Fig. 3
Fig. 3 Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =0.5λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

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Fig. 4
Fig. 4 Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at z=10 z r with ρ= x 2 + y 2 and w 0 =λ for different values of the correlation coefficients σ xx , σ xy , σ yy .

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Fig. 5
Fig. 5 Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at the source plane (z = 0).

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Fig. 6
Fig. 6 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =10λ and ρ= x 2 + y 2 .

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Fig. 7
Fig. 7 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with w 0 =λ and ρ= x 2 + y 2 .

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Fig. 8
Fig. 8 Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =0.5λ and ρ= x 2 + y 2 .

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Fig. 9
Fig. 9 Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

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Fig. 10
Fig. 10 Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1,0) and (x2,0) at several propagation distances for different values of w 0 .

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Fig. 11
Fig. 11 Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1, 0) and (x2, 0) at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

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Equations (47)

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W ↔ ( x 1 , y 1 , x 2 , y 2 ,z )=( W xx ( x 1 , y 1 , x 2 , y 2 ,z )    W xy ( x 1 , y 1 , x 2 , y 2 ,z )    W xz ( x 1 , y 1 , x 2 , y 2 ,z ) W yx ( x 1 , y 1 , x 2 , y 2 ,z )    W yy ( x 1 , y 1 , x 2 , y 2 ,z )    W yz ( x 1 , y 1 , x 2 , y 2 ,z ) W zx ( x 1 , y 1 , x 2 , y 2 ,z )    W zy ( x 1 , y 1 , x 2 , y 2 ,z )    W zz ( x 1 , y 1 , x 2 , y 2 ,z ) ),
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )=〈 E α * ( x 1 , y 1 ,z ) E β ( x 2 , y 2 ,z ) 〉,( α,β=x,y,z )
E α (x,y,z)=− 1 2π ∬ E α ( x 0 , y 0 ,0) ∂ ∂z [ exp( ikR ) R ]d x 0 d y 0 , (α=x,y)
E z ( x,y,z )= 1 2π ∬ { E x ( x 0 , y 0 ,0 ) ∂ ∂x [ exp( ikR ) R ] + E y ( x 0 , y 0 ,0 ) ∂ ∂y [ exp( ikR ) R ] }d x 0 d y 0 ,
∂ ∂α [ exp( ikR ) R ]= ikexp( ikR ) R 2 ( α− α 0 ),   (α=x,y)
∂ ∂z [ exp( ikR ) R ]= ikzexp( ikR ) R 2 .
W αβ ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 z 2 4 π 2 ∬ ∬ W αβ ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ×d x 10 d y 10 d x 20 d y 20 , (α,β=x,y)
W αz ( x 1 , y 1 , x 2 , y 2 ,z)=− k 2 z 4 π 2 ∬ ∬ [ W αx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ( x 2 − x 20 )    + W αy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ( y 2 − y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,  (α=x,y)
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 4 π 2 ∬ ∬ [ W xx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2    ×( x 1 − x 10 )( x 2 − x 20 )+ W xy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ( x 1 − x 10 )( y 2 − y 20 )    + W yx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ( y 1 − y 10 )( x 2 − x 20 )    + W yy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 − R 2 ) ] R 1 2 R 2 2 ( y 1 − y 10 )( y 2 − y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )=〈 E α * ( x 10 , y 10 ,0 ) E β ( x 20 , y 20 ,0 ) 〉. ( α,β=x,y )
E → ( r,ϕ,0 )=exp( − r 2 w 0 2 ) ( 2 r 2 w 0 2 ) ( n±1 )/2 L p n±1 ( 2 r 2 w 0 2 ){ cos( nϕ ) e → ϕ ∓sin( nϕ ) e → r ±sin( nϕ ) e → ϕ +cos( nϕ ) e → r },
E → ( x 0 , y 0 ,0 )={ E x ( x 0 , y 0 ,0 ) e → x + E y ( x 0 , y 0 ,0 ) e → y E y ( x 0 , y 0 ,0 ) e → x + E x ( x 0 , y 0 ,0 ) e → y }   =exp( − x 0 2 + y 0 2 w 0 2 ) (−1) p 2 2p+n±1 p! { 1 2i ∑ m=0 p ∑ s=0 n±1 i s [ 1− (−1) s ]( p m )( n±1   s ) H 2m+n±1−s ( 2 x 0 w 0 ) H 2p−2m+s ( 2 y 0 w 0 ) e → x 1 2 ∑ m=0 p ∑ s=0 n±1 i s [ 1+ (−1) s ]( p m )( n±1   s ) H 2m+n±1−s ( 2 x 0 w 0 ) H 2p−2m+s ( 2 y 0 w 0 ) e → x        + 1 2 ∑ m=0 p ∑ s=0 n±1 i s [ 1+ (−1) s ]( p m )( n±1   s ) H 2m+n±1−s ( 2 x 0 w 0 ) H 2p−2m+s ( 2 y 0 w 0 ) e → y + 1 2i ∑ m=0 p ∑ s=0 n±1 i s [ 1− (−1) s ]( p m )( n±1   s ) H 2m+n±1−s ( 2 x 0 w 0 ) H 2p−2m+s ( 2 y 0 w 0 ) e → y },
W ↔ ( x 10 , y 10 , x 20 , y 20 ,0 )=( W xx ( x 10 , y 10 , x 20 , y 20 ,0 )    W xy ( x 10 , y 10 , x 20 , y 20 ,0 )   0 W yx ( x 10 , y 10 , x 20 , y 20 ,0 )    W yy ( x 10 , y 10 , x 20 , y 20 ,0 )   0                 0                                0                 0 ),
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )= A αβ 1 4 exp[ − x 10 2 + x 20 2 + y 10 2 + y 20 2 w 0 2 − ( x 10 − x 20 ) 2 + ( y 10 − y 20 ) 2 2 σ αβ 2 ] × 1 2 4p+2n±2 (p!) 2 ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 i s (−i) h [ 1+ C α (−1) s ][ 1+ C β (−1) h ]( p m )( p l )( n±1 s )( n±1 h )  × H 2m+n±1−s ( 2 x 10 w 0 ) H 2l+n±1−h ( 2 x 20 w 0 ) H 2p−2m+s ( 2 y 10 w 0 ) H 2p−2l+h ( 2 y 20 w 0 ),             (α,β=x,y)
W yx ( x 10 , y 10 , x 20 , y 20 ,0 )= [ W xy ( x 20 , y 20 , x 10 , y 10 ,0 ) ] * , 
A αβ ={ 1           α=β=x, −i B xy      α=x, β=y 1           α=β=y, , C α ={ −1         α=x, 1           α=y,
W 1xx ( x 10 , y 10 , x 20 , y 20 ,0 )= W yy ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1yy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W yx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xy ( x 10 , y 10 , x 20 , y 20 ,0 ),
R i ≈ r i + x i0 2 + y i0 2 −2 x i x i0 −2 y i y i0 2 r i ,    (i=1,2)
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )= A αβ k 2 z 2 exp[ ik( r 1 − r 2 ) ] 16 r 1 2 r 2 2 M 1αβ 1 2 5( 2p+n±1 )/2 (p!) 2 ( 1− 2 M 1αβ w 0 2 ) ( n±1+2p )/2 ×exp{ − k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 − k 2 4 M 2αβ [ ( x 2 r 2 − x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 − y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 1 =0 2m+n±1−s   ∑ d 1 =0 c 1 /2 ∑ e 1 =0 (2l+n±1−h)/2 ∑ c 2 =0 2p−2m+s   ∑ d 2 =0 c 2 /2 ∑ e 2 =0 (2p−2l+h)/2 ( 2p−2m+s c 2 )( 2m+n±1−s c 1 ) ×( p m )( p l )( n±1 s )( n±1 h )[ 1+ C α (−1) s ][ 1+ C β (−1) h ] (−1) d 1 + d 2 + e 1 + e 2 i s (−i) h (2i) −( c 1 + c 2 −2 d 1 −2 d 2 +n±1+2p−2 e 1 −2 e 2 )                             × ( 2 2 w 0 ) 2p+n±1−2 e 2 −2 e 1 c 1 ! d 1 !( c 1 −2 d 1 )! c 2 ! d 2 !( c 2 −2 d 2 )! (2l+n±1−h)! e 1 !(2l+n±1−h−2 e 1 )! (2p−2l+h)! e 2 !(2p−2l+h−2 e 2 )! × 1 ( M 2αβ ) c 1 + c 2 +n±1+2p−2 d 1 −2 d 2 −2 e 1 −2 e 2 +2 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 −2 M 1αβ ) 1/2 ] c 1 + c 2 −2 d 1 −2 d 2 × H c 1 −2 d 1 +2l+n±1−h−2 e 1 [ k 2 M 2αβ ( x 1 2 M 1αβ r 1 σ αβ 2 − x 2 r 2 ) ] H 2m+n±1−s− c 1 [ −ik x 1 r 1 ( w 0 2 M 1αβ 2 −2 M 1αβ ) 1/2 ] × H c 2 −2 d 2 +2p−2l+h−2 e 2 [ k 2 M 2αβ ( y 1 2 M 1αβ r 1 σ αβ 2 − y 2 r 2 ) ] H 2p−2m+s− c 2 [ −ik y 1 r 1 ( w 0 2 M 1αβ 2 −2 M 1αβ ) 1/2 ],   (α,β=x,y)
W yx ( x 1 , y 1 , x 2 , y 2 ,z)= W xy * ( x 2 , y 2 , x 1 , y 1 ,z), 
W xz ( x 1 , y 1 , x 2 , y 2 ,z)= ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 1 =0 2m+n±1−s ∑ d 1 =0 c 1 /2 ∑ e 1 =0 ( 2l+n±1−h ) /2 ∑ c 2 =0 2p−2m+s   ∑ d 2 =0 c 2 /2 ∑ e 2 =0 (2p−2l+h)/2 Q 1 ×[ 1− (−1) s ]{ Q 3xx Q 4xx [ 1− (−1) h ] H c 2 −2 d 2 +2p−2l+h−2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 − y 2 r 2 ) ] − Q 2 Q 5xy [ 1+ (−1) h ] H c 1 −2 d 1 +2l+n±1−h−2 e 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 − x 2 r 2 ) ] },
W yz ( x 1 , y 1 , x 2 , y 2 ,z)= ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 1 =0 2m+n±1−s ∑ d 1 =0 c 1 /2 ∑ e 1 =0 ( 2l+n±1−h ) /2 ∑ c 2 =0 2p−2m+s   ∑ d 2 =0 c 2 /2 ∑ e 2 =0 (2p−2l+h)/2 Q 1 [ 1+ (−1) s ]{ Q 2 Q 4xy [ 1− (−1) h ] H c 2 −2 d 2 +2p−2l+h−2 e 2 [ k 2 M 2xy ( y 1 2 M 1xy r 1 σ xy 2 − y 2 r 2 ) ] + Q 3yy Q 5yy [ 1+ (−1) h ] H c 1 −2 d 1 +2l+n±1−h−2 e 1 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 − x 2 r 2 ) ] },
W zx ( x 1 , y 1 , x 2 , y 2 ,z)= W xz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zy ( x 1 , y 1 , x 2 , y 2 ,z)= W yz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzxy ( x 1 , y 1 , x 2 , y 2 ,z)                            + W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzyy ( x 1 , y 1 , x 2 , y 2 ,z), 
W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)= ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 1 =0 ( 2m+n±1−s )/2 ∑ d 1 =0 ( 2l+n±1−h )/2 ∑ c 2 =0 2p−2m+s   ∑ d 2 =0 c 2 /2 ∑ e 2 =0 (2p−2l+h)/2 Q 6xx Q 7xx × H c 2 −2 d 2 +2p−2l+h−2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 − y 2 r 2 ) ]{ x 1 ∑ f 1 =0 2m+n±1−s−2 c 1 ∑ g 1 =0 f 1 /2 Q 8xx ( 2m+n±1−s−2 c 1             f 1 ) × H 2m+n±1−s−2 c 1 − f 1 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 1 −2 g 1 +2l+n±1−h−2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 − x 2 r 2 ) ) − H f 1 −2 g 1 +2l+n±1−h−2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 − x 2 r 2 ) ) ]− ∑ f 2 =0 2m+n±1−s−2 c 1 +1 ∑ g 2 =0 f 2 /2 Q 9xx ( 2m+n±1−s−2 c 1 +1              f 2 ) × H 2m+n±1−s−2 c 1 +1− f 2 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 2 −2 g 2 +2l+n±1−h−2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 − x 2 r 2 ) ) − H f 2 −2 g 2 +2l+n±1−h−2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 − x 2 r 2 ) ) ] },                                                   
W zzyy ( x 1 , y 1 , x 2 , y 2 ,z)= ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 2 =0 2m+n±1−s ∑ d 2 =0 c 2 /2 ∑ e 2 =0 ( 2l+n±1−h ) /2 ∑ c 1 =0 ( 2p−2m+s )/2 ∑ d 1 =0 ( 2p−2l+h )/2 Q 6yy ( 2m+n±1−s          c 2 ) × ( 2i ) −( c 2 −2 d 2 +n±1−2 e 2 +4p−2m+s−2 c 1 −2 d 1 +2) ( 1− 2 w 0 2 M 1yy ) ( 2m+n±1−s ) /2 ( 2 2 w 0 ) n±1−2 e 2 +4p−2m+s−2 c 1 −2 d 1 × ( 2l+n±1−h )! e 2 !(2l+n±1−h−2 e 2 )! ( 2p−2m+s )! c 1 !( 2p−2m+s−2 c 1 )! ( 2p−2l+h )! d 1 !( 2p−2l+h−2 d 1 )! 1 ( M 1yy ) 2p−2m+s−2 c 1 +3 × H c 2 −2 d 2 +2l+n±1−h−2 e 2 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 − x 2 r 2 ) ] H 2m+n±1−s− c 2 [ −ik x 1 ( w 0 2 M 1yy 2 r 1 2 −2 M 1yy r 1 2 ) 1/2 ] ×{ y 1 ∑ f 1 =0 2p−2m+s−2 c 1 ∑ g 1 =0 f 1 /2 Q 8yy ( 2p−2m+s−2 c 1            f 1 ) H 2p−2m+s−2 c 1 − f 1 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p−2l+h−2 d 2 + f 1 −2 g 1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 − y 2 r 2 ) ) − H 2p−2l+h−2 d 1 + f 1 −2 g 1 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 − y 2 r 2 ) ) ] − ∑ f 2 =0 2p−2m+s−2 c 1 +1 ∑ g 2 =0 f 2 /2 Q 9yy ( 2p−2m+s−2 c 1 +1             f 2 ) H 2p−2m+s−2 c 1 +1− f 2 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p−2l+h−2 d 1 + f 2 −2 g 2 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 − y 2 r 2 ) ) − H 2p−2l+h−2 d 1 + f 2 −2 g 2 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 − y 2 r 2 ) ) ] }.   
W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)= [ W zzxy ( x 2 , y 2 , x 1 , y 1 ,z) ] * , 
W zzxy ( x 1 , y 1 , x 2 , y 2 ,z)= ∑ m=0 p ∑ s=0 n±1 ∑ l=0 p ∑ h=0 n±1 ∑ c 1 =0 ( 2m+n±1−s )/2 ∑ d 1 =0 ( 2l+n±1−h )/2 ∑ c 2 =0 2p−2m+s ∑ d 2 =0 c 2 /2 ∑ e 2 =0 ( 2p−2l+h ) /2 Q 5xy Q 6xy Q 7xy ×{ x 1 ∑ f 1 =0 2m+n±1−s−2 c 1 ∑ g 1 =0 f 1 /2 Q 8xy ( 2m+n±1−s−2 c 1              f 1 ) H 2m+n±1−s−2 c 1 − f 1 ( k x 1 2 M 1xy r 1 ) H f 1 −2 g 1 +2l+n±1−h−2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 − x 2 r 2 ) ]− ∑ f 2 =0 2m+n±1−s−2 c 1 +1 ∑ g 2 =0 f 2 /2 Q 9xy ( 2m+n±1−s−2 c 1 +1                f 2 ) × H 2m+n±1−s−2 c 1 +1− f 2 ( k x 1 2 M 1xy r 1 ) H f 2 −2 g 2 +2l+n±1−h−2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 − x 2 r 2 ) ] }, 
M 1αβ =1/ w 0 2 +1/(2 σ αβ 2 )−ik/(2 r 1 ), (α,β=x,y)
M 2αβ =1/ w 0 2 +1/(2 σ αβ 2 )+ik/(2 r 2 )−1/(4 M 1αβ σ αβ 4 ), (α,β=x,y)
Q 1 =− k 2 zexp[ ik( r 1 − r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 ) /2 (p!) 2 (−1) d 1 + d 2 + e 1 + e 2 i s (−i) h ( 2i ) −( c 1 + c 2 −2 d 1 −2 d 2 +n±1+2p−2 e 1 −2 e 2 +1) × ( 2 2 w 0 ) n±1+2p−2 e 1 −2 e 2 ( 2m+n±1−s   c 1 )( 2p−2m+s c 2 )( p m )( p l )( n±1 s )( n±1 h ) × c 1 ! d 1 !( c 1 −2 d 1 )! c 2 ! d 2 !( c 2 −2 d 2 )! ( 2l+n±1−h )! e 1 !(2l+n±1−h−2 e 1 )! (2p−2l+h)! e 2 !(2p−2l+h−2 e 2 )! ,
Q 2 = i B xy M 1xy 1 ( M 2xy ) c 1 + c 2 −2 d 1 −2 d 2 +n±1+2p−2 e 1 −2 e 2 +3 [ 2 σ xy 2 ( w 0 2 M 1xy 2 −2 M 1xy ) 1/2 ] c 1 + c 2 −2 d 1 −2 d 2 × ( 1− 2 M 1xy w 0 2 ) ( n±1+2p )/2 exp{ − k 2 ( x 1 2 + y 1 2 ) 4 M 1xy r 1 2 − k 2 4 M 2xy [ ( x 2 r 2 − x 1 2 M 1xy r 1 σ xy 2 ) 2 + ( y 2 r 2 − y 1 2 M 1xy r 1 σ xy 2 ) 2 ] } × H 2m+n±1−s− c 1 [ −ik x 1 ( w 0 2 M 1xy 2 r 1 2 −2 M 1xy r 1 2 ) 1/2 ] H 2p−2m+s− c 2 ( −ik y 1 r 1 ( w 0 2 M 1xy 2 −2 M 1xy ) 1/2 ),
Q 3αα = 1 M 1αα 1 ( M 2αα ) c 1 + c 2 −2 d 1 −2 d 2 +n±1+2p−2 e 1 −2 e 2 +3 [ 2 σ αα 2 ( w 0 2 M 1αα 2 −2 M 1αα ) 1/2 ] c 1 + c 2 −2 d 1 −2 d 2 × ( w 0 2 M 1αα −2 w 0 2 M 1αα ) ( 2p+n±1 ) /2 exp{ − k 2 ( x 1 2 + y 1 2 ) 4 M 1αα r 1 2 − k 2 4 M 2αα [ ( x 2 r 2 − x 1 2 M 1αα r 1 σ αα 2 ) 2 + ( y 2 r 2 − y 1 2 M 1αα r 1 σ αα 2 ) 2 ] } × H 2m+n±1−s− c 1 [ −ik x 1 r 1 ( w 0 2 M 1αα 2 −2 M 1αα ) 1/2 ] H 2p−2m+s− c 2 [ −ik y 1 r 1 ( w 0 2 M 1αα 2 −2 M 1αα ) 1/2 ],      (α=x,y)
Q 4xα =2i M 2xα x 2 H c 1 −2 d 1 +2l+n±1−h−2 e 1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 − x 2 r 2 ) ] − H c 1 −2 d 1 +2l+n±1−h−2 e 1 +1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 − x 2 r 2 ) ],  (α=x,y)
Q 5αy =2i M 2αy y 2 H c 2 −2 d 2 +2p−2l+h−2 e 2 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 − y 2 r 2 ) ] − H c 2 −2 d 2 +2p−2l+h−2 e 2 +1 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 − y 2 r 2 ) ],  (α=x,y)
Q 6αβ = A αβ exp{ − k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 − k 2 4 M 2αβ [ ( x 2 r 2 − x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 − y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × k 2 exp[ ik( r 1 − r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 )/2 (p!) 2 ( p m )( p l )( n±1 s )( n±1 h ) i s (−i) h (−1) c 1 + d 1 + d 2 + e 2 [ 1+ C α (−1) s ] 1+ C β (−1) h 2 ( −2 c 1 +1 )/2 × 1 ( M 2αβ ) n±1−2 d 1 + c 2 −2 d 2 +2p−2 e 2 +3 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 −2 M 1αβ ) 1/2 ] c 2 −2 d 2 c 2 ! d 2 !( c 2 −2 d 2 )! ,    (α,β=x,y)
Q 7xα =( 2p−2m+s         c 2 ) ( 2i ) −( 2m+2( n±1 )−s−2 c 1 −2 d 1 + c 2 −2 d 2 +2p−2 e 2 +2 ) ( 2 2 w 0 ) 2m+2( n±1 )−s−2 c 1 −2 d 1 +2p−2 e 2 × ( 2m+n±1−s )! c 1 !( 2m+n±1−s−2 c 1 )! ( 2l+n±1−h )! d 1 !( 2l+n±1−h−2 d 1 )! (2p−2l+h)! e 2 !(2p−2l+h−2 e 2 )! ( 1− 2 M 1xα w 0 2 ) ( 2p−2m+s )/2 × 1 ( M 1xα ) 2m+n±1−s−2 c 1 +3 H 2p−2m+s− c 2 [ −ik y 1 ( w 0 2 M 1xα 2 r 1 2 −2 M 1xα r 1 2 ) 1/2 ],    (α=x,y)
Q 8αβ = 2 M 1αβ (−1) g 1 (2i) −( f 1 −2 g 1 −1) f 1 ! g 1 !( f 1 −2 g 1 )! 1 ( M 2αβ ) f 1 −2 g 1 ( 2 i M 1αβ σ αβ 2 ) f 1 −2 g 1 , (α,β=x,y)
Q 9αβ = (−1) g 2 (2i) −( f 2 −2 g 2 ) f 2 ! g 2 !( f 2 −2 g 2 )! 1 ( M 2αβ ) f 2 −2 g 2 ( 2 i M 1αβ σ αβ 2 ) f 2 −2 g 2 . (α,β=x,y)
∫ −∞ ∞ exp[ − ( x−y ) 2 ] H n ( ax )dx= π ( 1− a 2 ) n/2 H n ( ay ( 1− a 2 ) 1/2 ),
∫ −∞ ∞ x n exp[ − ( x−β ) 2 ] dx= ( 2i ) −n π H n ( iβ ),
H n ( x+y )= 1 2 n/2 ∑ k=0 n ( n k ) H k ( 2 x ) H n−k ( 2 y ),
H n ( x )= ∑ k=0 n/2 ( −1 ) k n! k!( n−2k )! ( 2x ) n−2k .
I(x,y,z)= I x (x,y,z)+ I y (x,y,z)+ I z (x,y,z)              = W xx (x,y,x,y,z)+ W yy (x,y,x,y,z)+ W zz (x,y,x,y,z),
P(x,y,z)= p 1 (x,y,z)− p 2 (x,y,z) p 1 (x,y,z)+ p 2 (x,y,z)+ p 3 (x,y,z) ,
μ( x 1 − x 2 , y 1 − y 2 ,z)= Tr W ↔ ( x 1 , y 1 , x 2 , y 2 ,z) Tr W ↔ ( x 1 , y 1 , x 1 , y 1 ,z) Tr W ↔ ( x 2 , y 2 , x 2 , y 2 ,z) , 
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