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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13312–13325
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Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal

Lina Zhang and Yangjian Cai  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13312-13325 (2011)
http://dx.doi.org/10.1364/OE.19.013312


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Abstract

Nonparaxial propagation theory of coherent beams in a uniaxial crystal is extended to the partially coherent case. An analytical formula for the 3 × 3 cross-spectral density matrix of a nonparaxial Gaussian Schell-model (GSM) beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. Statistical properties, such as the spectral intensity and the degree of polarization, of a nonparaxial GSM beam in a uniaxial crystal are studied numerically. It is found that the statistical properties of a nonparaxial GSM beam are closely determined by its initial beam parameters and the parameters of the crystal. Uniaxial crystal can be used to modulate the spectral density and degree of polarization of a nonparaxial partially coherent beam. Our results may be useful in some applications, such as optical trapping and nonlinear optics, where a light beam with special beam profile and polarization is required.

© 2011 OSA

1. Introduction

Partially coherent beams have important applications in free-space optical communications, remote sensing, LIDAR systems, optical trapping, optical imaging, optical projection, nonlinear optics, optical projection, laser scanning and inertial confinement fusion [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

14

14. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

]. Gaussian Schell-model (GSM) beam is a typical partially coherent beam, whose spectral density and spectral degree of coherence have Gaussian shapes [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

,15

15. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]

17

17. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

]. By scattering a coherent laser beam from a rotating grounded glass, then transforming the spectral density distribution of the scattered light into Gaussian profile with a Gaussian amplitude filter, a GSM beam can be generated [18

18. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]

]. GSM beams can also be generated with synthetic acousto-optic holograms [19

19. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

]. A more general GSM can possess a twist phase [20

20. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]

]. Paraxial propagation properties of a GSM beam with or without twist phase in free space or in turbulent atmosphere have been investigated in detail in the past decades [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

7

7. G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]

,16

16. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990). [CrossRef]

,17

17. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

,20

20. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]

30

30. F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]

].

2. Nonparaxial theory of a partially coherent beam propagating in a uniaxial crystal orthogonal to the optical axis

The geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis is shown in Fig. 1
Fig. 1 Geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis.
. We assume that a partially coherent beam, which is linearly polarized in the x-direction, is incident on a uniaxial crystal at the plane z = 0. The optical axis of the crystal coincides with the x-axis, and the dielectric tensor of the crystal can be expressed as

ε=(ne2000no2000no2),
(1)

where no and ne are the ordinary and extraordinary refractive index, respectively.

Propagation of a monochromatic light of frequency ω in linear media is described by the following equation [49

49. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

]:

2E(E)+k02εE=0,
(2)

where k0=ω/c, ε is the relative dielectric tensor, and E is the complex amplitude of the electric field. According to [52

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]

], an exact solution of Eq. (2) can be expressed as follows:

E(r,z)=d2kexp(ikr)exp(ikezz)(E˜x(k)kxkyk02no2kx2E˜x(k)kezkxk02no2kx2E˜x(k))             +d2kexp(ikr)exp(ikozz)(0kxkyk02no2kx2E˜x(k)+E˜y(k)kykoz[kxkyk02no2kx2E˜x(k)+E˜y(k)]),   
(3)

where k=kxe˜x+kye˜y, r=xe˜x+ye˜y,E˜s(k) is the two dimensional Fourier transform of the transverse part of the electric field at z = 0 given by

E˜s(k)=12π2d2rexp(ikr)Es(r,0),         (s=x,y)
(4)

and

koz(k)=(k02no2k2)1/2,kez(k)=[k02ne2(ne2/no2)kx2ky2]1/2.     
(5)

Within the validity of the paraxial approximation, Eq. (3) reduces to [52

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]

]

Ep(r,z)=d2kexp(ikr)exp(ik0nez)exp(ine2kx2+no2ky22k0neno2z)(E˜x(k)00)               +d2kexp(ikr)exp(ik0noz)exp(ikx2+ky22k0noz)(0E˜y(k)0).
(6)

The elements of Ep(r,z)can be expressed in the following alternative form:

Epx(r,z)=k0no2πizd2r0Γe(r0,r,0)Ex(r0,0), Epy(r,z)=k0no2πizd2r0Γo(r0,0)Ey(r0,r,0),Epz(r,z)   =0,           
(7)

with

Γe(r0,r,0)=exp(ik0nez)exp{k02izne[no2(xx0)2+ne2(yy0)2]},
(8)
Γo(r0,r,0)=exp(ik0noz)exp{k0no2iz[(xx0)2+(yy0)2]}.
(9)

To describe the nonparaxial field in a uniaxial crystal, Eq. (3) can be expressed as the sum of the paraxial field and a nonparaxial correction term [52

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]

]

E(r,z)=exp(ik0nez)d2kexp(ikr)×exp(ine2kx2+no2ky22k0neno2z)(E˜x(k)kxkyk02no2E˜x(k)nekxk0no2E˜x(k))               +exp(ik0noz)d2kexp(ikr)exp(ikx2+ky22k0noz)(0kxkyk02no2E˜x(k)+E˜y(k)kyk0noE˜y(k)).
(10)

The elements of E(r,z)can be expressed in the following alternative form:

Ex(r,z)=k0no2πizd2r0Γe(r0,r,0)Ex(r0,0),
(11)
Ey(r,z)=ik0no2πz3d2r0(xx0)(yy0)[Γe(r0,r,0)Γo(r0,r,0)]Ex(r0,0)               +k0no2πizd2r0Γo(r0,r,0)Ey(r0,0),
(12)
Ez(r,z)   =ik0no2πz2d2r0(xx0)Γe(r0,r,0)Ex(r0,0)               +ik0no2πz2d2r0(yy0)Γo(r0,r,0)Ey(r0,0).
(13)

Now we extend the nonparaxial propagation theory of coherent beams in a uniaxial crystal to the partially coherent case. The second-order statistics of a nonparaxial partially coherent beam can be characterized by the3×3cross-spectral density (CSD) matrix W(r1,r2,z) defined as [42

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

,43

43. K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B 22(8), 1585–1593 (2005). [CrossRef]

,57

57. 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]

]

W(r1,r2,z)=(Wxx(r1,r2,z)Wxy(r1,r2,z)Wxz(r1,r2,z)Wxy*(r1,r2,z)Wyy(r1,r2,z)Wyz(r1,r2,z)Wxz*(r1,r2,z)Wyz*(r1,r2,z)Wzz(r1,r2,z)),
(14)

where

Wαβ(r1,r2,z)=Eα*(r1,z)Eβ(r2,z),(α,β=x,y,z).
(15)

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.

Applying Eqs. (11)-(15), we can express the elements of the 3×3 CSD matrix of a nonparaxial partially coherent beam in a uniaxial crystal as follows:

Wxx(r1,r2,z)=k02no24π2z2d2r10d2r20ΠeeW0xx(r10,r20,0),
(16)
Wyy(r1,r2,z)=k02no24π2z4d2r10d2r20(ΠooΠoe*)W0xy(r10,r20,0)(x1x10)(y1y10)+k02no24π2z6d2r10d2r20(ΠeeΠoeΠoe*+Πoo)W0xx(r10,r20,0)(x1x10)(y1y10)×(x2x20)(y2y20)+k02no24π2z4d2r10d2r20(ΠooΠoe)W0yx(r10,r20,0)×(x2x20)(y2y20)+k02no24π2z2d2r10d2r20ΠooW0yy(r10,r20,0),
(17)
Wzz(r1,r2,z)=k02no24π2z4d2r10d2r20ΠeeW0xx(r10,r20,0)(x1x10)(x2x20)+k02no24π2z4d2r10d2r20Πoe*W0xy(r10,r20,0)(x1x10)(y2y20)+k02no24π2z4d2r10d2r20ΠoeW0yx(r10,r20,0)(y1y10)(x2x20)+k02no24π2z4d2r10d2r20ΠooW0yy(r10,r20,0)(y1y10)(y2y20)
(18)
Wxy(r1,r2,z)=k02no24π2z4d2r10d2r20(Πoe*Πee)W0xx(r10,r20,0)×(x2x20)(y2y20)k02no24π2z2d2r10d2r20Πoe*W0xy(r10,r20,0),
(19)
Wxz(r1,r2,z)=k02no24π2z3d2r10d2r20ΠeeW0xx(r10,r20,0)(x2x20)k02no24π2z3d2r10d2r20Πoe*W0xy(r10,r20,0)(y2y20),
(20)
Wyz(r1,r2,z)=k02no24π2z5d2r10d2r20(ΠeeΠoe)W0xx(r10,r20,0)(x1x10)×(y1y10)(x2x20)+k02no24π2z5d2r10d2r20(Πoe*Πoo)W0xy(r10,r20,0)×(x1x10)(y1y10)(y2y20)k02no24π2z3d2r10d2r20ΠoeW0yx(r10,r20,0)×(x2x20)k02no24π2z3d2r10d2r20ΠooW0yy(r10,r20,0)(y2y20),
(21)

where

Πee=Γe(r10,r1,0)Γe*(r20,r2,0)=exp{k02izne[no2(x1x10)2+ne2(y1y10)2]}         ×exp{k02izne[no2(x2x20)2+ne2(y2y20)2]},
(22)
Πoo=Γo(r10,r1,0)Γo*(r20,r2,0)=exp{k0no2iz[(x1x10)2+(y1y10)2]}         ×exp{k0no2iz[(x2x20)2+(y2y20)2]},
(23)
Πoe=Γo(r10,r1,0)Γe*(r20,r2,0)=exp{k0no2iz[(x1x10)2+(y1y10)2]}         ×exp{k02izne[no2(x2x20)2+ne2(y2y20)2]+ik0(none)z}.
(24)

Equations (16)-(24) are the nonparaxial propagation equations for a partially coherent beam propagating in a uniaxial crystal orthogonal to the optical axis.

3. Statistical properties of a nonparaxial GSM beam propagating in a uniaxial crystal orthogonal to the optical axis

In this section, as an application example of the formulae derived in section 2, we study the statistical properties of a nonparaxial GSM beam propagating in a uniaxial crystal orthogonal to the optical axis. We assume that a GSM beam, which is linearly polarized in the x-direction, is incident on a uniaxial crystal at the plane z = 0. The CSD matrix of the initial linearly polarized GSM beam at z = 0 is expressed as [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

]

W0(r10,r20,0)=(W0xx(r10,r20,0)00000000),
(25)

where

W0xx(r10,r20,0)=exp(x102+y102+x202+y2024σI2(x10x20)2+(y10y20)22σg2),
(26)

where σIand σgare the transverse beam width and coherence width, respectively.

Substituting Eq. (26) into Eqs. (16)-(21), we obtain (after tedious integration and operation)

Wxx(r1,r2,z)=aoe2HxooHyeeAooAeeBooooBeeee,
(27)
Wyy(r1,r2,z)=aoe2ExooooEyeeeeHxooHyeez4AooAeeBooooBeeee+aoe2FxoeooGyoeeeEoe*LxoeooJyoeeez4AooAeeBoeooBoeee                       +aoe2GxoooeFyeeoeEoeLyeeoeJxoooez4AoeBoooeBeeoeaoe2ExoeoeEyoeoeHxoeHyoez4AoeBoeoe,
(28)
Wzz(r1,r2,z)=aoe2ExooooHxooHyeez2AooAeeBooooBeeee,
(29)
Wxy(r1,r2,z)=aoe2CxooooCyeeeeHxooHyeez2AooAeeBooooBeeeeaoe2CxoeooCyoeeeEoe*LxoeooJyoeeez2AooAeeBoeooBoeee,
(30)
Wxz(r1,r2,z)=aoe2CxooooHxooHyeezAooAeeBooooBeeee,
(31)
Wyz(r1,r2,z)=aoe2ExooooDyHxooHyeez3AooAeeBooooBeeee+aoe2GxoooeDyoeEoeLyeeoeJxoooez3AoeBoooeBeeoe,
(32)

where

aμγ=k0nμnγ2izne, Aμγ=14σI2+12σg2+aμγ, Bμγτν=14σI2+12σg2aμγ14Aτνσg4,
Csμγτν=aμγs2Bμγτνaτνs12AτνBμγτνσg2+s2,  Dy=y1aeey1Aeeaee2AeeBeeeeσg2(y12Aeeσg2y2),
Dyoe=y1aeey1Aeeaee2AoeBeeoeσg2(noy12neAoeσg2y2), Eoe=exp[ik0(none)z],
Esμγτν=s1s2aτνs1s2Aτν+14AτνBμγτνσg2+aμγ22AτνBμγτν2σg2(s12Aτνσg2s2)2                                                                       +aμγBμγτν(s12Aτνσg2s2)(aτνs1Aτνs1s22Aτνσg2),
Fsμγτν=s1s2aτνs1s2Aτν+14AτνBμγτνσg2+aμγ22AτνBμγτν2σg2(nos12neAτνσg2s2)2                                                                     +aμγBμγτν(nos12neAτνσg2s2)(aτνσ1Aτνs1s22Aτνσg2),
Gsμγτν=s1s2aτνs1s2Aτν+14AτνBμγτνσg2+aμγ22AτνBμγτν2σg2(nes12noAτνσg2s2)2                                                                     +aμγBμγτν(nes12noAτνσg2s2)(aτνs1Aτνs1s22Aτνσg2),
Hsμγ=exp[aμγs12+aμγs22+aμγ2s12Aμγ+aμγ2Bμγμγ(s2s12Aμγσg2)2],
Lsμγτν=exp[aτνs12+aμγs22+aτν2s12Aτν+aμγ2Bμγτν(s2nos12neAτνσg2)2],
Jsμγτν=exp[aτνs12+aμγs22+aτν2s12Aτν+aμγ2Bμγτν(s2nes12noAτνσg2)2], (s=x      or    y;   μ,γ,τ,ν=o      or    e)
(33)

In the derivation of above formulae, we have used the following integral formula:

(ax2+bx+c)exp(px2+qx)dx=14p2(2ap+aq22bpq+4cp2)πpexp(q24p)             (p<0).
(34)

Equations (27)-(32) are the analytical expressions for the elements of the CSD matrix of a nonparaxial GSM beam propagating in a uniaxial crystal orthogonal to the optical axis, and they are the main analytical results of present paper. Within the validity of the paraxial approximation, we can easily obtain the following analytical expressions for the elements of the CSD matrix of a paraxial GSM beam propagating in a uniaxial crystal orthogonal to the optical axis

Wαβ(r1,r2,z)={aoe2HxooHyeeAooAeeBooooBeeeeα=β=x0otherwise.
(35)

The total intensity distribution of the nonparaxial GSM beam in a uniaxial crystal is expressed as

I(r,z)=Ix(r,z)+Iy(r,z)+Iz(r,z)          =Wxx(r,r,z)+Wyy(r,r,z)+Wzz(r,r,z),
(36)

where Ix(r,z), Iy(r,z)and Iz(r,z) are the intensity distributions of the x, y and z components of the field, respectively. For a paraxial GSM beam in a uniaxial crystal, its intensity distribution is expressed asI(r,z)=Ix(r,z). The degree of polarization of a nonparaxial GSM beam is defined as [57

57. 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]

]

P(r,z)=32Wxx(r,r,z)2+Wyy(r,r,z)2+Wzz(r,r,z)2[Wxx(r,r,z)+Wyy(r,r,z)+Wzz(r,r,z)]212   .
(37)

The beam is regarded as a completely polarized beam if P=1 and as a completely unpolarized beam ifP=0. Note there is another definition of degree of polarization as shown in [58

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

]. Both definitions of degree of polarization can be used to characterize the polarization properties of a paraxial beam. The definition given by Eq. (37) is more suitable to describe the polarization properties of a nonparaxial beam. In the following numerical examples, the propagation distance is normalized to the Rayleigh distance z/zr         (zr=πσI2/λ).

We calculate in Fig. 1 the normalized intensity distribution (contour graph) of a paraxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σg with λ=0.5μm, no=2, ne=1.25no, σI=10λ. One finds from Fig. 1 that the paraxial propagation properties of a GSM beam in uniaxial crystal are much different from its propagation properties in free space. In free space, the beam spot of the paraxial GSM beam on propagation is always of circular symmetry and the beam spot spreads more rapidly as the initial coherence width σgdecreases [15

15. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]

19

19. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

]. In the uniaxial crystal, the beam spot of the GSM beam on propagation gradually becomes of elliptical symmetry due to anisotropic diffraction. As the initial coherence width σg decreases, the conversion from a circular beam spot to an elliptical beam spot becomes more quickly and the beam spot also spreads more rapidly. Thus the uniaxial crystal provides a convenient way to transform a stigmatic GSM beam to an astigmatic GSM beam, which is useful in nonlinear optics and free-space optical communications [4

4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]

,11

11. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]

].

Fig. 1 Normalized intensity distribution (contour graph) of a paraxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σg.

To learn about the polarization properties of a nonparaxial GSM beam in a uniaxial crystal, we calculate in Fig. 4
Fig. 4 Distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σg
the distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σgwith λ=0.5μm, no=2, ne=1.25no, σI=0.5λ. Figure 5
Fig. 5 Distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the ratio of extraordinary index to ordinary refractive index ne/no
shows the distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the ratio of extraordinary index to ordinary refractive index ne/nowith λ=0.5μm, σI=0.5λ, σg=0.5λ. One can see from Fig. 4 and Fig. 5 that the evolution properties of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal are much different from that of a paraxial linearly polarized GSM beam. For a paraxial linearly polarized GSM beam, it remains linearly polarized on propagation in the uniaxial crystal (i.e., the degree of polarization equals to 1 on propagation). For a nonparaxial GSM beam, it becomes nonuniformly polarized and varies on propagation in the uniaxial crystal (i.e., the GSM beam is depolarized and becomes partially polarized). Furthermore, the distribution of the degree of polarization is determined by σgand ne/no. Thus, one comes to the conclusion that we can modulate the polarization structure of a nonparaxial GSM beam by a uniaxial crystal through controlling the parameters of the crystal and the initial parameters of the initial beam.

4. Summary

We have introduced the nonparaxial propagation theory of a partially coherent beam in a uniaxial crystal, and we have obtained the analytical nonparaxial formula for a GSM beam propagating in a uniaxial crystal orthogonal to the optical axis. The statistical properties of a nonparaxial GSM beam in a uniaxial crystal have been studied numerically. Our numerical results show that the nonparaxial properties of a GSM beam in a uniaxial crystal are much different from its paraxial properties, and are closely determined by the parameters of the crystal and the parameters of the initial beam. Our results show that we can modulate the intensity distribution and polarization properties of a nonparaxial GSM beam by the uniaxial crystal, which will be useful in some applications, such as optical trapping and nonlinear optics, where light beam with special beam profile and polarization is required.

Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation 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 and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References and links

1.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

2.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).

3.

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]

4.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]

5.

G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24(3), 745–752 (2007). [CrossRef] [PubMed]

6.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004). [CrossRef]

7.

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]

8.

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975). [CrossRef]

9.

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004). [CrossRef]

10.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005). [CrossRef] [PubMed]

11.

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]

12.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]

13.

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

14.

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

15.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]

16.

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990). [CrossRef]

17.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

18.

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]

19.

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

20.

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]

21.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]

22.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001). [CrossRef] [PubMed]

23.

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]

24.

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef] [PubMed]

25.

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [CrossRef] [PubMed]

26.

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]

27.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007). [CrossRef]

28.

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef] [PubMed]

29.

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011). [CrossRef]

30.

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]

31.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]

32.

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

33.

A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9(5), 765–774 (1992). [CrossRef]

34.

G. P. Agrawal and D. N. Pattanayak, “Gaussain beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575–578 (1979). [CrossRef]

35.

S. R. S. Seshadri, “Virtual source for the Bessel-Gaussian beam,” Opt. Lett. 27, 988–1000 (2002).

36.

K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11(13), 1474–1480 (2003). [CrossRef] [PubMed]

37.

G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008). [CrossRef] [PubMed]

38.

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26(11), 2044–2049 (2009). [CrossRef]

39.

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:. [CrossRef]

40.

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

41.

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]

42.

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

43.

K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B 22(8), 1585–1593 (2005). [CrossRef]

44.

F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A 27(5), 1120–1126 (2010). [CrossRef] [PubMed]

45.

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

46.

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009). [CrossRef]

47.

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009). [CrossRef] [PubMed]

48.

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:. [CrossRef]

49.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

50.

J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66(8), 780–788 (1976). [CrossRef]

51.

J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18(7), 1662–1669 (2001). [CrossRef]

52.

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]

53.

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004). [CrossRef]

54.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008). [CrossRef]

55.

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008). [CrossRef]

56.

B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009). [CrossRef] [PubMed]

57.

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]

58.

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

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: May 20, 2011
Revised Manuscript: June 14, 2011
Manuscript Accepted: June 15, 2011
Published: June 24, 2011

Citation
Lina Zhang and Yangjian Cai, "Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal," Opt. Express 19, 13312-13325 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13312


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).
  3. Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]
  4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]
  5. G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24(3), 745–752 (2007). [CrossRef] [PubMed]
  6. O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004). [CrossRef]
  7. G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]
  8. D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975). [CrossRef]
  9. M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004). [CrossRef]
  10. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005). [CrossRef] [PubMed]
  11. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
  12. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]
  13. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
  14. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]
  15. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]
  16. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990). [CrossRef]
  17. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
  18. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]
  19. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]
  20. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]
  21. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]
  22. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001). [CrossRef] [PubMed]
  23. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]
  24. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef] [PubMed]
  25. Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [CrossRef] [PubMed]
  26. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]
  27. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007). [CrossRef]
  28. F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef] [PubMed]
  29. G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011). [CrossRef]
  30. F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]
  31. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]
  32. H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]
  33. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9(5), 765–774 (1992). [CrossRef]
  34. G. P. Agrawal and D. N. Pattanayak, “Gaussain beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575–578 (1979). [CrossRef]
  35. S. R. S. Seshadri, “Virtual source for the Bessel-Gaussian beam,” Opt. Lett. 27, 988–1000 (2002).
  36. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11(13), 1474–1480 (2003). [CrossRef] [PubMed]
  37. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008). [CrossRef] [PubMed]
  38. D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26(11), 2044–2049 (2009). [CrossRef]
  39. L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:. [CrossRef]
  40. K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004). [CrossRef] [PubMed]
  41. 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]
  42. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004). [CrossRef] [PubMed]
  43. K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B 22(8), 1585–1593 (2005). [CrossRef]
  44. F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A 27(5), 1120–1126 (2010). [CrossRef] [PubMed]
  45. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]
  46. D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009). [CrossRef]
  47. D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009). [CrossRef] [PubMed]
  48. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:. [CrossRef]
  49. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  50. J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66(8), 780–788 (1976). [CrossRef]
  51. J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18(7), 1662–1669 (2001). [CrossRef]
  52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]
  53. B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004). [CrossRef]
  54. D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008). [CrossRef]
  55. D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008). [CrossRef]
  56. B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009). [CrossRef] [PubMed]
  57. 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]
  58. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

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