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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 15908–15927
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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  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 15908-15927 (2012)
http://dx.doi.org/10.1364/OE.20.015908


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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 OSA

1. Introduction

When the beam width and the wavelength of a laser beam are comparable, the paraxial propagation theory is no longer valid.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

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

, 45

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

52

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]

]. Deng explored the nonparaxial propagation properties of a radially polarized beam in free space [10

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

]. Nonparaxial propagation of a scalar partially coherent beam and a vector partially coherent beam with uniform SOP were reported in [47

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

51

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]

]. Nonparaxial propagation of a scalar partially coherent beam in crystal was investigated in [52

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]

]. 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

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]

55

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]

]. 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

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

, 49

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

, 52

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]

, 56

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]

]
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)),
(1)
where
Wαβ(x1,y1,x2,y2,z)=Eα*(x1,y1,z)Eβ(x2,y2,z),(α,β=x,y,z)
(2)
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)ex+Ey(x0,y0,0)eyin free space can be studied with the help of the following vectorial Rayleigh-Sommerfeld diffraction integrals [57

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

]
Eα(x,y,z)=12πEα(x0,y0,0)z[exp(ikR)R]dx0dy0,(α=x,y)
(3)
Ez(x,y,z)=12π{Ex(x0,y0,0)x[exp(ikR)R]+Ey(x0,y0,0)y[exp(ikR)R]}dx0dy0,
(4)
where R=(xx0)2+(yy0)2+z2, k=2π/λ is the wave number with λ being the wavelength. When R>>λ, the termsx[exp(ikR)R], y[exp(ikR)R] andz[exp(ikR)R] in Eqs. (3) and (4) are approximated as

α[exp(ikR)R]=ikexp(ikR)R2(αα0),(α=x,y)
(5)
z[exp(ikR)R]=ikzexp(ikR)R2.
(6)

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

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

]
Wαβ(x1,y1,x2,y2,z)=k2z24π2Wαβ(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22×dx10dy10dx20dy20,(α,β=x,y)
(7)
Wαz(x1,y1,x2,y2,z)=k2z4π2[Wαx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(x2x20)+Wαy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y2y20)]dx10dy10dx20dy20,(α=x,y)
(8)
Wzz(x1,y1,x2,y2,z)=k24π2[Wxx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22×(x1x10)(x2x20)+Wxy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(x1x10)(y2y20)+Wyx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y1y10)(x2x20)+Wyy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y1y10)(y2y20)]dx10dy10dx20dy20,
(9)
where

Wαβ(x10,y10,x20,y20,0)=Eα*(x10,y10,0)Eβ(x20,y20,0).(α,β=x,y)
(10)

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

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]

]
E(r,ϕ,0)=exp(r2w02)(2r2w02)(n±1)/2Lpn±1(2r2w02){cos(nϕ)eϕsin(nϕ)er±sin(nϕ)eϕ+cos(nϕ)er},
(11)
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

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]

]
E(x0,y0,0)={Ex(x0,y0,0)ex+Ey(x0,y0,0)eyEy(x0,y0,0)ex+Ex(x0,y0,0)ey}=exp(x02+y02w02)(1)p22p+n±1p!{12im=0ps=0n±1is[1(1)s](pm)(n±1s)H2m+n±1s(2x0w0)H2p2m+s(2y0w0)ex12m=0ps=0n±1is[1+(1)s](pm)(n±1s)H2m+n±1s(2x0w0)H2p2m+s(2y0w0)ex+12m=0ps=0n±1is[1+(1)s](pm)(n±1s)H2m+n±1s(2x0w0)H2p2m+s(2y0w0)ey+12im=0ps=0n±1is[1(1)s](pm)(n±1s)H2m+n±1s(2x0w0)H2p2m+s(2y0w0)ey},
(12)
where Hm is the Hermite polynomial of order m, (pm) and (n±1s) 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)ex+Ey(x0,y0,0)ey, 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

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]

]
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)0000),
(13)
where
Wαβ(x10,y10,x20,y20,0)=Aαβ14exp[x102+x202+y102+y202w02(x10x20)2+(y10y20)22σαβ2]×124p+2n±2(p!)2m=0ps=0n±1l=0ph=0n±1is(i)h[1+Cα(1)s][1+Cβ(1)h](pm)(pl)(n±1s)(n±1h)×H2m+n±1s(2x10w0)H2l+n±1h(2x20w0)H2p2m+s(2y10w0)H2p2l+h(2y20w0),(α,β=x,y)
(14)
Wyx(x10,y10,x20,y20,0)=[Wxy(x20,y20,x10,y10,0)]*,
(15)
with
Aαβ={1α=β=x,iBxyα=x,β=y1α=β=y,,Cα={1α=x,1α=y,
(16)
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)ex+Ex(x0,y0,0)ey, the elements of the CSD matrix of a cylindrical vector partially coherent LG beam are expressed as follows [41

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]

]
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),
(17)
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

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

]
Riri+xi02+yi022xixi02yiyi02ri,(i=1,2)
(18)
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(r1r2)]16r12r22M1αβ125(2p+n±1)/2(p!)2(12M1αβw02)(n±1+2p)/2×exp{k2(x12+y12)4M1αβr12k24M2αβ[(x2r2x12M1αβr1σαβ2)2+(y2r2y12M1αβr1σαβ2)2]}×m=0ps=0n±1l=0ph=0n±1c1=02m+n±1sd1=0c1/2e1=0(2l+n±1h)/2c2=02p2m+sd2=0c2/2e2=0(2p2l+h)/2(2p2m+sc2)(2m+n±1sc1)×(pm)(pl)(n±1s)(n±1h)[1+Cα(1)s][1+Cβ(1)h](1)d1+d2+e1+e2is(i)h(2i)(c1+c22d12d2+n±1+2p2e12e2)×(22w0)2p+n±12e22e1c1!d1!(c12d1)!c2!d2!(c22d2)!(2l+n±1h)!e1!(2l+n±1h2e1)!(2p2l+h)!e2!(2p2l+h2e2)!×1(M2αβ)c1+c2+n±1+2p2d12d22e12e2+2[2σαβ2(w02M1αβ22M1αβ)1/2]c1+c22d12d2×Hc12d1+2l+n±1h2e1[k2M2αβ(x12M1αβr1σαβ2x2r2)]H2m+n±1sc1[ikx1r1(w02M1αβ22M1αβ)1/2]×Hc22d2+2p2l+h2e2[k2M2αβ(y12M1αβr1σαβ2y2r2)]H2p2m+sc2[iky1r1(w02M1αβ22M1αβ)1/2],(α,β=x,y)
(19)
Wyx(x1,y1,x2,y2,z)=Wxy*(x2,y2,x1,y1,z),
(20)
Wxz(x1,y1,x2,y2,z)=m=0ps=0n±1l=0ph=0n±1c1=02m+n±1sd1=0c1/2e1=0(2l+n±1h)/2c2=02p2m+sd2=0c2/2e2=0(2p2l+h)/2Q1×[1(1)s]{Q3xxQ4xx[1(1)h]Hc22d2+2p2l+h2e2[k2M2xx(y12M1xxr1σxx2y2r2)]Q2Q5xy[1+(1)h]Hc12d1+2l+n±1h2e1[k2M2xy(x12M1xyr1σxy2x2r2)]},
(21)
Wyz(x1,y1,x2,y2,z)=m=0ps=0n±1l=0ph=0n±1c1=02m+n±1sd1=0c1/2e1=0(2l+n±1h)/2c2=02p2m+sd2=0c2/2e2=0(2p2l+h)/2Q1[1+(1)s]{Q2Q4xy[1(1)h]Hc22d2+2p2l+h2e2[k2M2xy(y12M1xyr1σxy2y2r2)]+Q3yyQ5yy[1+(1)h]Hc12d1+2l+n±1h2e1[k2M2yy(x12M1yyr1σyy2x2r2)]},
(22)
Wzx(x1,y1,x2,y2,z)=Wxz*(x2,y2,x1,y1,z),
(23)
Wzy(x1,y1,x2,y2,z)=Wyz*(x2,y2,x1,y1,z),
(24)
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),
(25)
Wzzxx(x1,y1,x2,y2,z)=m=0ps=0n±1l=0ph=0n±1c1=0(2m+n±1s)/2d1=0(2l+n±1h)/2c2=02p2m+sd2=0c2/2e2=0(2p2l+h)/2Q6xxQ7xx×Hc22d2+2p2l+h2e2[k2M2xx(y12M1xxr1σxx2y2r2)]{x1f1=02m+n±1s2c1g1=0f1/2Q8xx(2m+n±1s2c1f1)×H2m+n±1s2c1f1(kx12M1xxr1)[2iM2xxx2Hf12g1+2l+n±1h2d1(k2M2xx(x12M1xxr1σxx2x2r2))Hf12g1+2l+n±1h2d1+1(k2M2xx(x12M1xxr1σxx2x2r2))]f2=02m+n±1s2c1+1g2=0f2/2Q9xx(2m+n±1s2c1+1f2)×H2m+n±1s2c1+1f2(kx12M1xxr1)[2iM2xxx2Hf22g2+2l+n±1h2d1(k2M2xx(x12M1xxr1σxx2x2r2))Hf22g2+2l+n±1h2d1+1(k2M2xx(x12M1xxr1σxx2x2r2))]},
(26)
Wzzyy(x1,y1,x2,y2,z)=m=0ps=0n±1l=0ph=0n±1c2=02m+n±1sd2=0c2/2e2=0(2l+n±1h)/2c1=0(2p2m+s)/2d1=0(2p2l+h)/2Q6yy(2m+n±1sc2)×(2i)(c22d2+n±12e2+4p2m+s2c12d1+2)(12w02M1yy)(2m+n±1s)/2(22w0)n±12e2+4p2m+s2c12d1×(2l+n±1h)!e2!(2l+n±1h2e2)!(2p2m+s)!c1!(2p2m+s2c1)!(2p2l+h)!d1!(2p2l+h2d1)!1(M1yy)2p2m+s2c1+3×Hc22d2+2l+n±1h2e2[k2M2yy(x12M1yyr1σyy2x2r2)]H2m+n±1sc2[ikx1(w02M1yy2r122M1yyr12)1/2]×{y1f1=02p2m+s2c1g1=0f1/2Q8yy(2p2m+s2c1f1)H2p2m+s2c1f1(ky12M1yyr1)[2iM2yyy2H2p2l+h2d2+f12g1(k2M2yy(y12M1yyr1σyy2y2r2))H2p2l+h2d1+f12g1+1(k2M2yy(y12M1yyr1σyy2y2r2))]f2=02p2m+s2c1+1g2=0f2/2Q9yy(2p2m+s2c1+1f2)H2p2m+s2c1+1f2(ky12M1yyr1)[2iM2yyy2H2p2l+h2d1+f22g2(k2M2yy(y12M1yyr1σyy2y2r2))H2p2l+h2d1+f22g2+1(k2M2yy(y12M1yyr1σyy2y2r2))]}.
(27)
Wzzyx(x1,y1,x2,y2,z)=[Wzzxy(x2,y2,x1,y1,z)]*,
(28)
Wzzxy(x1,y1,x2,y2,z)=m=0ps=0n±1l=0ph=0n±1c1=0(2m+n±1s)/2d1=0(2l+n±1h)/2c2=02p2m+sd2=0c2/2e2=0(2p2l+h)/2Q5xyQ6xyQ7xy×{x1f1=02m+n±1s2c1g1=0f1/2Q8xy(2m+n±1s2c1f1)H2m+n±1s2c1f1(kx12M1xyr1)Hf12g1+2l+n±1h2d1[k2M2xy(x12M1xyr1σxy2x2r2)]f2=02m+n±1s2c1+1g2=0f2/2Q9xy(2m+n±1s2c1+1f2)×H2m+n±1s2c1+1f2(kx12M1xyr1)Hf22g2+2l+n±1h2d1[k2M2xy(x12M1xyr1σxy2x2r2)]},
(29)
where
M1αβ=1/w02+1/(2σαβ2)ik/(2r1),(α,β=x,y)
(30)
M2αβ=1/w02+1/(2σαβ2)+ik/(2r2)1/(4M1αβσαβ4),(α,β=x,y)
(31)
Q1=k2zexp[ik(r1r2)]16r12r2225(2p+n±1)/2(p!)2(1)d1+d2+e1+e2is(i)h(2i)(c1+c22d12d2+n±1+2p2e12e2+1)×(22w0)n±1+2p2e12e2(2m+n±1sc1)(2p2m+sc2)(pm)(pl)(n±1s)(n±1h)×c1!d1!(c12d1)!c2!d2!(c22d2)!(2l+n±1h)!e1!(2l+n±1h2e1)!(2p2l+h)!e2!(2p2l+h2e2)!,
(32)
Q2=iBxyM1xy1(M2xy)c1+c22d12d2+n±1+2p2e12e2+3[2σxy2(w02M1xy22M1xy)1/2]c1+c22d12d2×(12M1xyw02)(n±1+2p)/2exp{k2(x12+y12)4M1xyr12k24M2xy[(x2r2x12M1xyr1σxy2)2+(y2r2y12M1xyr1σxy2)2]}×H2m+n±1sc1[ikx1(w02M1xy2r122M1xyr12)1/2]H2p2m+sc2(iky1r1(w02M1xy22M1xy)1/2),
(33)
Q3αα=1M1αα1(M2αα)c1+c22d12d2+n±1+2p2e12e2+3[2σαα2(w02M1αα22M1αα)1/2]c1+c22d12d2×(w02M1αα2w02M1αα)(2p+n±1)/2exp{k2(x12+y12)4M1ααr12k24M2αα[(x2r2x12M1ααr1σαα2)2+(y2r2y12M1ααr1σαα2)2]}×H2m+n±1sc1[ikx1r1(w02M1αα22M1αα)1/2]H2p2m+sc2[iky1r1(w02M1αα22M1αα)1/2],(α=x,y)
(34)
Q4xα=2iM2xαx2Hc12d1+2l+n±1h2e1[k2M2xα(x12M1xαr1σxα2x2r2)]Hc12d1+2l+n±1h2e1+1[k2M2xα(x12M1xαr1σxα2x2r2)],(α=x,y)
(35)
Q5αy=2iM2αyy2Hc22d2+2p2l+h2e2[k2M2αy(y12M1αyr1σαy2y2r2)]Hc22d2+2p2l+h2e2+1[k2M2αy(y12M1αyr1σαy2y2r2)],(α=x,y)
(36)
Q6αβ=Aαβexp{k2(x12+y12)4M1αβr12k24M2αβ[(x2r2x12M1αβr1σαβ2)2+(y2r2y12M1αβr1σαβ2)2]}×k2exp[ik(r1r2)]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±12d1+c22d2+2p2e2+3[2σαβ2(w02M1αβ22M1αβ)1/2]c22d2c2!d2!(c22d2)!,(α,β=x,y)
(37)
Q7xα=(2p2m+sc2)(2i)(2m+2(n±1)s2c12d1+c22d2+2p2e2+2)(22w0)2m+2(n±1)s2c12d1+2p2e2×(2m+n±1s)!c1!(2m+n±1s2c1)!(2l+n±1h)!d1!(2l+n±1h2d1)!(2p2l+h)!e2!(2p2l+h2e2)!(12M1xαw02)(2p2m+s)/2×1(M1xα)2m+n±1s2c1+3H2p2m+sc2[iky1(w02M1xα2r122M1xαr12)1/2],(α=x,y)
(38)
Q8αβ=2M1αβ(1)g1(2i)(f12g11)f1!g1!(f12g1)!1(M2αβ)f12g1(2iM1αβσαβ2)f12g1,(α,β=x,y)
(39)
Q9αβ=(1)g2(2i)(f22g2)f2!g2!(f22g2)!1(M2αβ)f22g2(2iM1αβσαβ2)f22g2.(α,β=x,y)
(40)
In above integration, we have used the following integral and expansion formulae [58

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

, 59

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

]
exp[(xy)2]Hn(ax)dx=π(1a2)n/2Hn(ay(1a2)1/2),
(41)
xnexp[(xβ)2]dx=(2i)nπHn(iβ),
(42)
Hn(x+y)=12n/2k=0n(nk)Hk(2x)Hnk(2y),
(43)
Hn(x)=k=0n/2(1)kn!k!(n2k)!(2x)n2k.
(44)
Under the condition of riz+(xi2+yi2)/2z, Eqs. (19) and (20) reduces to the propagation formulae (Eqs. (15)-(18) of Ref [41

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]

].) 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),
(45)
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

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]

]
P(x,y,z)=p1(x,y,z)p2(x,y,z)p1(x,y,z)+p2(x,y,z)+p3(x,y,z),
(46)
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

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]

]. 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

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]

]
μ(x1x2,y1y2,z)=TrW(x1,y1,x2,y2,z)TrW(x1,y1,x1,y1,z)TrW(x2,y2,x2,y2,z),
(47)
where Tr stands for the trace of the CSD matrix.

Applying Eqs. (19), (25) and (45), we calculate in Figs. 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.
-3
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.
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.
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

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]

].) 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.

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
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.
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

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

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]

] 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.

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
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).
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.

We calculate in 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.
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.

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
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.
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
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.
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).

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.

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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]

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(140.3300) Lasers and laser optics : Laser beam shaping
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: April 24, 2012
Revised Manuscript: June 12, 2012
Manuscript Accepted: June 14, 2012
Published: June 28, 2012

Citation
Yiming Dong, Fanlong Feng, Yahong Chen, Chengliang Zhao, and Yangjian Cai, "Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space," Opt. Express 20, 15908-15927 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15908


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