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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 15254–15267
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An alternative theoretical model for an anomalous hollow beam

Yangjian Cai, Zhaoying Wang, and Qiang Lin  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 15254-15267 (2008)
http://dx.doi.org/10.1364/OE.16.015254


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Abstract

An alternative and convenient theoretical model is proposed to describe a flexible anomalous hollow beam of elliptical symmetry with an elliptical solid core, which was observed in experiment recently (Phys. Rev. Lett, 94 (2005) 134802). In this model, the electric field of anomalous hollow beam is expressed as a finite sum of elliptical Gaussian modes. Flat-topped beams, dark hollow beams and Gaussian beams are special cases of our model. Analytical propagation formulae for coherent and partially coherent anomalous hollow beams passing through astigmatic ABCD optical systems are derived. Some numerical examples are calculated to show the propagation and focusing properties of coherent and partially coherent anomalous hollow beams.

© 2008 Optical Society of America

1. Introduction

Beam combination is a subject of current interest for some practical applications, where laser beams with high power or special beam profiles are required. A variety of laser beams, e.g., laser array beams, dark hollow beams, flat-topped beams and general-type beams have been developed through beam combination, and have found application in atomic physics, high-power laser systems, free-space optical communications and inertial confinement fusion [1

1. C. Palma, “Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams,” Appl. Opt. 36, 1116–1120 (1997). [CrossRef] [PubMed]

11

11. C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006). [CrossRef] [PubMed]

].

Over the past decade, conventional dark-hollow beams with zero central intensity have been widely investigated and have found wide applications in atomic physics, free space optical communications, binary optics, optical trapping of particles and medical sciences [6

6. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398–1407 (2006). [CrossRef]

9

9. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008). [CrossRef]

, 12

12. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., (North-Holland, Amsterdam, 2003), pp.119–204.

25

25. Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A 25, 1497–1503 (2008). [CrossRef]

]. Recently, Wu et al. observed an anomalous hollow electron beam of elliptical symmetry with an elliptical solid core in experiment [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

]. Anomalous hollow beam provides a unique model system for studying the transverse instabilities, and it can be used for studying the linear and nonlinear particle dynamics in the storage ring [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

]. Up to now, only one approximate model was recently proposed by Cai to describe an anomalous hollow beam [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

], and the propagation formulae of coherent and partially coherent anomalous hollow beams passing through paraxial stigmatic (i.e., symmetric) ABCD optical systems have been derived in [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

, 28

28. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).

]. While we note that this approximate model is not convenient for controlling the dark size, the relative peak value and beam spot size of the solid core, and in this model the ellipticity (ratio of the beam width along the long axis to that along the short axis) and the orientation angle (angle between the long axis to the horizontal axis x) of the out elliptical ring are the same with those of the elliptical solid core, but in practical case, they can be different as shown in [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

]. The main new results of present paper is that we propose an alternative and convenient model to describe a flexible anomalous hollow beam through beam combination, which is more close to experimental results reported in [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

]. In our model, the dark size, the relative peak value, beam spot size of the solid core, the ellipticity and the orientation angle of the out elliptical ring and the elliptical solid core can be controlled conveniently by controlling the parameters of the beam. What’s more, propagation formulae of coherent and partially coherent anomalous hollow beams passing through paraxial astigmatic (i.e., non-symmetric) ABCD optical systems are derived. Some numerical examples are given.

2. An alternative model for an anomalous hollow beam

In this section, we propose an alternative and convenient model to describe a flexible anomalous hollow beam of elliptical symmetry with an elliptical solid core, which is more close to the experimental results reported in [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

].

We express the electric field of an anomalous hollow beam at z = 0 as combination of a series of elliptical Gaussian modes as follows

EN(x,y,0)=n=1N(1)n1N(Nn)[exp(x2w0xxn22xyw0xyn2y2w0yyn2)
exp(x2w0xxnp22xyw0xynp2y2w0yynp2)αexp(x2w1xxβ22xyw1xyβ2y2w1yyβ2)],
(1)

where (Nn) denotes a binomial coefficient, and

1w0xxn2=ncos2θw0x2+nsin2θw0y2,1w0yyn2=ncos2θw0y2+nsin2θw0x2,1w0xyn2=nsin2θ2w0x2nsin2θ2w0y2,1w0xxnp2=ncos2θp2w0x2+nsin2θp2w0y2,1w0yynp2=ncos2θp2w0y2+nsin2θp2w0x2,1w0xynp2=nsin2θ2p2w0x2nsin2θ2p2w0y2,1w1xxβ2=cos2ϕβ2w1x2+sin2ϕβ2w1y2,1w1yyβ2=cos2ϕβ2w1y2+sin2ϕβ2w1x2,1w1xyβ2=sin2ϕ2β2w1x2sin2ϕ2β2w1y2,
(2)

w 0x and x 0y are the beam widths along the long axis and short axis of the fundamental elliptical Gaussian mode for constructing the out ring of the elliptical anomalous hollow beam, respectively, and θ is the orientation angle between the long axis of the out elliptical ring and the horizontal axis x. w1x and w1y are the beam widths along the long axis and short axis of the fundamental elliptical Gaussian mode for constructing the elliptical solid core of the elliptical anomalous hollow beam, respectively, and ϕ is the orientation angle between the long axis of the elliptical solid core and the horizontal axis x. We call N the beam order of the anomalous hollow beam mainly for controlling the dark size of the anomalous hollow beam and relative peak value of the solid core. p is a parameter mainly for controlling the dark size and the relative peak value of the solid core and satisfy 0<p<1. α is a parameter mainly for controlling the relative peak value of the solid core and satisfy α > 0. β is a parameter mainly for controlling the dark size and the beam spot size of the solid core and satisfy β > 0. When α = 0 and p = 0, Eq. (1) reduces to the expression for the electric filed of an elliptical flat-topped beam [4, 5]. When α = 0, Eq. (1) reduces to the expression for the electric filed of a controllable elliptical dark hollow beam [7]. When p = 1, Eq. (1) reduces to the expression for the electric filed of an elliptical Gaussian beam. Thus, with suitable beam parameters w0x, w0y, w1x, w1y, N, p, α and β, Eq. (1) provides an alternative and convenient model for describing an anomalous hollow beam with controllable beam properties (i.e., beam spot size, orientation angle, dark size, relative peak value of the solid core and ellipticity) as shown in Fig. 1. From Eq. (1) and Fig. 1, we can find the effective beam widths of the out ring of the elliptical anomalous hollow beam are determined by N, w0x and w0y together (i.e., the first term and second term of Eq. (1) mainly determine the out ring), and the effective beam widths of the solid core are determined by β, w1x and w1y together (i.e., the third term of Eq. (1) mainly determines the solid core). For suitable value of β, we can choose the values of w1x and w1y to be larger than w0x and w0y, but of course we can’t choose the values of w1x and w1y arbitrary large as shown in Fig. 2.

Fig. 1. Normalized intensity (contour graph) of an anomalous hollow beam and corresponding cross line (y = 0) for different values of θ, ϕ, N, p, α and β with w 0x = 2mm, w 0y = 1mm, w 1x = 3mm, w 1y = 1mm (a) θ = ϕ = π/4, N = 5, p = 0.8, α = 0.4, β = 0.5, (b) θ = ϕ = π/4, N = 10, p = 0.8, α = 0.4, β = 0.5, (c) θ = 0, ϕ = π/10, N = 10, p = 0.8,α = 0.4, β = 0.5, (d) θ = 0, ϕ = π/10, N = 10, p = 0.5, α = 0.4, β = 0.5, (e) θ = 0, ϕ = π/10, N = 10, p = 0.8, α0.35, β = 0.5, (f) θ = 0, ϕ = π/10, N = 10, p = 0.8, α = 0.4, β = 0.2
Fig. 2. Normalized intensity (contour graph) of an anomalous hollow beam and corresponding cross line (y = 0) for different values of w 1x and w 1y with w 0x = 2mm w 0y = 1mm, θ = 0, ϕ = 0, N = 10, p = 0.8, α = 0.35 and β = 0.5 (a) w 1x = 2mm and w 1y = 1mm, (b) w 1x = 3mm, and w 1y = 1mm, (c) w 1x = 10mm and w 1y = 1mm

Note in the approximate proposed in [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

], for fixed values of w 0x and w 0y, the dark size, the relative peak value and beam spot size of the solid core are fixed. By controlling the values of w 0x, w 0y and α, we can control the ellipticity and the orientation angle of the out elliptical ring and the elliptical solid core, but the ellipticity and the orientation angle of the out elliptical ring are the same with those of the elliptical solid core in any case. Thus the alternative model proposed in present paper is more suitable and flexible than the model in [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

] to describe an anomalous hollow beam reported in [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

]. Eq. (1) is the main new result of this paper.

After some operation, we can express Eq. (1) in the following tensor form

EN(r1,0)=n=1N(1)n1N(Nn)[exp(ik2r1TQ1n1r1)exp(ik2r1TQ1np1r1)αexp(ik2r1TQ1β1r1)],
(3)

where k = 2π/λ is the wave number, λ is the wavelength of the beam, r 1 is the position vector given by r T 1=(x y), Q -1 1 is a 2 × 2 matrix called the complex curvature tensor for an elliptical Gaussian beam [29

29. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.

, 30

30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67 (1990).

]. In our case, Q -1 1n, Q -1 1np and Q -1 are given by|

Q1n1=(2ikw0xxn22ikw0xyn22ikw0xyn22ikw0yyn2),Q1np1=(2ikw0xxnp22ikw0xynp22ikw0xynp22ikw0yynp2),Q1β1=(2ikw1xxβ22ikw1xyβ22ikw1xyβ22ikw1yyβ2).
(4)

The introduction of the complex curvature tensor allows us to treat the propagation of an anomalous hollow beam conveniently through some vector integration and tensor operation (as shown later).

3. Paraxial propagation of an anomalous hollow beam through ABCD optical systems

In [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

], we have derived the propagation formula for an anomalous hollow beam passing through paraxial stigmatic ABCD optical systems based on the proposed theoretical model. In this section, we study the propagation of a flexible anomalous hollow beam through astigmatic ABCD optical systems. Within the validity of the paraxial approximation, propagation of a coherent laser beam through an astigmatic ABCD optical system can be studied with the help of the following generalized Collins formula [16

16. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058–1065 (2004). [CrossRef]

, 30

30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67 (1990).

, 31

31. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]

]

E(ρ1,l)=iλ[det(B)]12E(r1,0)exp[ik2(r1TB1Ar12r1TB1ρ1+ρ1TDB1ρ1)]dr1,
(5)

where det stands for the determinant of a matrix, E(r 1, 0) and E(ρ1, l) are the electric fields of the laser beam in the source plane (z = 0) and the output plane (z = l), respectively. ρ T 1=(ρ 1x ρ 1y) with ρ 1 being the position vectors in the output planes. k = 2π/λ is the wave number, λ is the wavelength of light. l is the axial distance from the input plane to the output plane. A,B,C and Dare the 2 × 2 sub-matrices of the astigmatic optical system [29

29. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.

, 30

30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67 (1990).

] and satisfy the following Luneburg relations that describe the symplecticity of an astigmatic optical system [32

32. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).

]

(B1A)T=B1A,(B1)T=(CDB1A),(DB1)T=DB1.
(6)

Substituting Eq. (3) into Eq. (5), we obtain (after some vector integration and tensor operation) the following propagation formula for an anomalous hollow beam through an astigmatic ABCD optical system

EN(ρ1,l)=n=1N(1)n1N(Nn)[1[det(A+BQ1n1)]12exp(ik2ρ1TQ2n1ρ1)
1[det(A+BQ1np1)]12exp(ik2ρ1TQ2np1ρ1)α1[det(A+BQ1β1)]12exp(ik2ρ1TQ2β1ρ1),
(7)

Q2n1=(C+DQ1n1)(A+BQ1n1)1,Q2np1=(C+DQ1np1)(A+BQ1np1)1,
Q2β1=(C+DQ1β1)(A+BQ1β1)1.
(8)
Fig. 3. Normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances (a) z = 0, (b) z = 0.3m, (c) z = 1m, (d) z = 2m, (e) z = 5m, (f) z = 15m

As a numerical example, we calculate the propagation properties of an anomalous hollow beam in free space by using the derived propagation formula. The elements of the transfer matrix for free space of distance z read as A=I,B=z I,C = 0I,D=Iwith I being a 2 × 2unit matrix. Figure 3 shows the normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances with w 0x = 1mm, w 0y = 0.5mm, w 1x = 1mm, w 1y = 0.5mm, θ = ϕ = 0, N = 3, p = 0.8, α = 0.2, β = 0.5 and λ = 632.8nm. One sees from Fig. 3 that as the propagation distance z increases, the initial beam profile gradually disappears, i.e., the dark region disappears, the central intensity increases gradually and the beam profile becomes non-elliptical symmetry. In the far field, the anomalous hollow beam retains its elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot (see Fig. 3 (f)), and the long axis and short axis of the elliptical beam spot in far field has interchanged their positions compared to the elliptical beam spot in near field (see Fig. 3 (a)). The interesting propagation properties of anomalous hollow beams are caused by the fact that an anomalous hollow beam is not a pure mode, but a combination of elliptical Gaussian modes, and these different modes will overlap and interfere in propagation. The propagation properties of anomalous hollow beam in free space in this paper are consistent with those in Ref. [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

] where another theoretical model for anomalous hollow beam was introduced. So the mathematical model and the analytical propagation formulae in this paper provide a reliable and convenient way for studying the properties anomalous hollow beams.

3. Partially coherent an anomalous hollow beam and its propagation

A partially coherent beam is characterized by the second-order correlation (at plane z) [34

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

], Γ(x 1,y 1,x 2,y 2,z)=〈E(x 1,y 1,z)E* (x 2,y 2,z)〉, where 〈 〉 denotes the ensemble average and * denotes the complex conjugate. The intensity distribution of a partially coherent beam is given by I(x,y,z)=Γ(x,y,x,y,z). For a partially coherent beam generated by an Schell-model source (at z = 0), the second-order correlation at z = 0 can be expressed in the following well-known form [34

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

]

Γ(x1,y1,x2,y2,0)=I(x1,y1,0)I(x2,y2,0)g(x1x2,y1y2),
(9)

where g(x 1-x 2,y 1-y 2) is the spectral degree of coherence given by

g(x1x2,y1y2)=exp[(x1x2)22σg02(y1y2)22σg02],
(10)

where σ g0 is called the transverse coherence width.

If we assume that the intensity distribution of the Schell-model source can be represented by I(x,y,0)=|E(x,y,0)|2, where E(x,y,0) is given by Eq. (1), after some operation, we can express the second-order correlation of a partially coherent anomalous hollow beam at z = 0 in following tensor form:

ΓN(r˜,0)=n=1Nm=1N(1)n+mN2(Nn)(Nm)[exp(ik2r˜TM111r˜)exp(ik2r˜TM121r˜)αexp(ik2r˜TM131r˜)
exp(ik2r˜TM141r˜)+exp(ik2r˜TM151r˜)+αexp(ik2r˜TM161r˜)αexp(ik2r˜TM171r˜)+αexp(ik2r˜TM181r˜)+α2exp(ik2r˜TM191r˜)],
(11)

where r̂ T=(r T 1 r 2 T)=(x 1 y 1 x 2 y 2) with r 1 and r 2 being the two arbitrary position vectors in the source plane z = 0 and

M111=(2ikw0xxn2ikσg022ikw0xyn2ikσg0202ikw0xyn22ikw0yyn2ikσg020ikσg02ikσg0202ikw0xxm2ikσg022ikw0xym20ikσg022ikw0xym22ikw0yym2ikσg02),
M121=(2ikw0xxn2ikσg022ikw0xyn2ikσg0202ikw0xyn22ikw0yyn2ikσg020ikσg02ikσg0202ikw0xxmp2ikσg022ikw0xymp20ikσg022ikw0xymp22ikw0yymp2ikσg02),
M131=(2ikw0xxn2ikσg022ikw0xyn2ikσg0202ikw0xyn22ikw0yyn2ikσg020ikσg02ikσg0202ikw1xxβ2ikσg022ikw1xyβ20ikσg022ikw1xyβ22ikw1yyβ2ikσg02),
M141=(2ikw0xxnp2ikσg022ikw0xynp2ikσg0202ikw0xynp22ikw0yynp2ikσg020ikσg02ikσg0202ikw0xxm2ikσg022ikw0xym20ikσg022ikw0xym22ikw0yym2ikσg02),
M151=(2ikw0xxnp2ikσg022ikw0xynp2ikσg0202ikw0xynp22ikw0yynp2ikσg020ikσg02ikσg0202ikw0xxmp2ikσg022ikw0xymp20ikσg022ikw0xymp22ikw0yymp2ikσg02),
M161=(2ikw0xxnp2ikσg022ikw0xynp2ikσg0202ikw0xynp22ikw0yynp2ikσg020ikσg02ikσg0202ikw1xxβ2ikσg022ikw1xyβ20ikσg022ikw1xyβ22ikw1yyβ2ikσg02),
M171=(2ikw1xxβ2ikσg022ikw1xyβ2ikσg0202ikw1xyβ22ikw1yyβ2ikσg020ikσg02ikσg0202ikw0xxm2ikσg022ikw0xym20ikσg022ikw0xym22ikw0yym2ikσg02),
M181=(2ikw1xxβ2ikσg022ikw1xyβ2ikσg0202ikw1xyβ22ikw1yyβ2ikσg020ikσg02ikσg0202ikw0xxmp2ikσg022ikw0xymp20ikσg022ikw0xymp22ikw0yymp2ikσg02),
M191=(2ikw1xxβ2ikσg022ikw1xyβ2ikσg0202ikw1xyβ22ikw1yyβ2ikσg020ikσg02ikσg0202ikw1xxβ2ikσg022ikw1xyβ20ikσg022ikw1xyβ22ikw1yyβ2ikσg02),
(12)

with M -1 being the partially coherent complex curvature tensor [37

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

].

Within the validity of the paraxial approximation, propagation of a partially coherent beam through an astigmatic ABCD optical system can be studied with the help of the following generalized Collins formula [37

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

]

Γ(ρ˜,l)=k24π2[det(B˜)]12Γ(r˜,0)exp[ik2(r˜TB˜1A˜r˜2r˜TB˜1ρ˜+ρ˜TD˜B˜1ρ˜)]dr˜,
(13)

where Γ(r̂, 0) and Γ(ρ̂, l) are second-order correlation of a partially coherent beam in the source (z = 0) and output planes (z=l), d r̂=d r 1 d r 2, ρ T=(ρ T 1 ρ T 2) and

A˜=(A0I0IA*),B˜=(B0I0IB*),C˜=(C0I0IC*),D˜=(D0I0ID*),
(14)

A,B,C and D are the 2 × 2 sub-matrices of the astigmatic optical system, and, Â,,Ĉ and D also satisfy the following Luneburg relations [37

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

]

(B˜1A˜)T=B˜1A˜,(B˜1)T=(C˜D˜B˜1A˜),(D˜B˜1)T=D˜B˜1.
(15)

“*” denotes the complex conjugate. We note that in Ref. [37

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

] A,B,C and D are assumed to be real quantities implying that the “*” is not needed anywhere in Eq. (14). However, for a general optical system with loss or gain (e.g. dispersive media, a Gaussian aperture, helical gas lenses, etc.)A,B,C and D take complex values and “*” is then required in Eq. (14).

Substituting Eq. (11) into Eq. (13), we obtain (after some vector integration and tensor operation) the following propagation formula for a partially coherent anomalous hollow beam through an astigmatic ABCD optical system

ΓN(ρ˜,l)=n=1Nm=1N(1)n+mN2(Nn)(Nm)[S1exp(ik2ρ˜TM211ρ˜)S2exp(ik2ρ˜TM221ρ˜)αS3exp(ik2ρ˜TM231ρ˜)
S4exp(ik2ρ˜TM241ρ˜)+S5exp(ik2ρ˜TM251ρ˜)+αS6exp(ik2ρ˜TM261ρ˜)αS7exp(ik2ρ˜TM271ρ˜)
+αS8exp(ik2ρ˜TM281ρ˜)+α2S9exp(ik2ρ˜TM291ρ˜)],
(16)

where

Si=1[det(A˜+B˜M1i1)]12,(i=1,2,3,4,5,6,7,8,9)
(17)
M2i1=(C˜+D˜M1i1)(A˜+B˜M1i1)1,(i=1,2,3,4,5,6,7,8,9)
(18)

Eq.(18) is called the tensor ABCD law for a partially coherent beam [37

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

]. Eqs. (1), (3), (7), (11) and (16) are the main analytical results in this paper, they provide a convenient and reliable way for characterizing coherent and partially coherent anomalous hollow beams and for studying their transformation and propagation properties.

Note that under the condition of σg 0- > ∞, a partially coherent anomalous hollow beam becomes a coherent anomalous hollow beam and Eq. (16) reduces to the expression for the intensity distribution of a coherent anomalous hollow beam after propagation when ρ 1=ρ 2. But we can’t obtain the electric field (Eq. (7)) of a coherent anomalous hollow beam after propagation directly from Eq. (16). So it is necessary to describe coherent and partially coherent anomalous hollow beam separately. In some applications, coherent anomalous hollow beam are required, and it is sufficient and convenient for us to calculate the electric field of anomalous hollow beam with Eq. (7), calculation of the second-order correlation of anomalous hollow beam with Eq. (16) will make the problem more complicated. In other applications, partially coherent anomalous hollow beams are required, and we have to calculate its second-order correlation with Eq. (16).

Fig. 4. Normalized 3D-intensity distribution of a partially coherent anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances (a) z = 0, (b) z = 0.3m, (c) z = 1m, (d) z = 2m, (e) z = 5m, (f) z = 15m

Now we apply Eqs. (16)(18) to study the propagation properties of a partially coherent anomalous hollow beam in free space. The elements of the transfer matrix for free space of distance z read as

A˜=(I0I0II),B=(zI0I0IzI),C=(0I0I0I0I),D=(I0I0II).
(19)

Substituting Eq. (19) into Eqs. (16)(18), we calculate in Fig. 4 the normalized 3D-intensity distribution of a partially coherent anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances with w 0x = 1mm, w 0y = 0.5mm, w 1x = 1mm, w 1y = 0.5mm, θ = ϕ = 0, N = 3, p = 0.8, α = 0.2, β = 0.5, λ = 632.8nm and σ g0 = 0.5mm. One sees from Figs. 4(a)–(c) that the beam profile of a partially coherent anomalous hollow beam also becomes non-elliptical symmetry at intermediate propagation distances, which is similar to that of a coherent anomalous hollow beam (see Figs. 3 (a)–(c)). In the far field, however, it is interesting to find that the partially coherent anomalous hollow beam gradually converses into a Gaussian beam (see Figs. 4 (d)–(f)), which is much different from that of a coherent anomalous hollow beam (see Figs. 3 (d)–(f)). This interesting phenomenon can be explained as follows. Partially coherent anomalous hollow beam can be regarded as a combination of a series of partially coherent modes with the same initial transverse coherence width σ g0. Different modes or different points across the beam section interfere during propagation. As the initial transverse coherence width decreases, the coherence of all modes at the source plane decreases, then the interference effect between different modes on propagation decreases, which leads to the disappearance of out small ring around the main Gaussian peak in intensity distribution of the far field. Note the intensity distribution of the partially coherent anomalous hollow beam at the source plane is independent of its initial transverse coherence width. The phenomenon that decreasing the spatial coherence can lead to the disappearance of interference pattern was demonstrated in experiment recently in [48

48. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33,1389–1391(2008). [CrossRef] [PubMed]

]. We also may say that decreasing the initial spatial coherence can shape the intensity distribution of partially coherent anomalous hollow beam in the far field. Using spatial coherence for shaping the intensity distribution of partially coherent beam was reported recently both theoretically and experimentally [49

49. F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008). [CrossRef]

, 50]. One also finds from Figs. 3 and 4 that the beam spot of a partially coherent anomalous hollow beam spreads more rapidly than that of a coherent anomalous hollow beam as expected [34

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

]. The propagation properties of a partially coherent anomalous hollow beam in free space are consistent with those in [28

28. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).

].

Fig. 5. Normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) at geometrical focal plane for different values of the coherence width (a) σ g = 0 (coherent case), (b) σ g = 2mm, (c) σ g = 0.5mm, (d) σ g = 0.1mm, (e) cross lines

As another numerical example, we study the focusing properties of an anomalous hollow beam focused by a thin lens. Assume an anomalous hollow beam is focused by an thin lens (with focal length f) that is located at z = 0, and the output plane is located at z = f (geometrical focal plane). The elements of the transfer matrix of the optical system between the source plane (z = 0) and output plane is expressed as follows

A˜=(0I0I0I0I),B=(fI0I0IfI),C=((1f)I0I0I(1f)I),D=(I0I0II).
(20)

Substituting Eq. (20) into Eqs. (16)(18), we calculate in Fig. 5 the normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) at the geometrical focal plane (z = f) for different values of the initial coherence width σ g0. with w 0y1mm, w 0y = 0.5mm, w 1x = 1mm, w 1y = 0.5mm, θ = ϕ = 0, N = 3, p = 0.8, α = 0.2, β = 0.5, λ = 632.8nm, f = 50mm, and σ g0 = 0.5mm. It is clear from Fig. 5 that the intensity distribution of an anomalous hollow beam at the geometrical focal plane is also closely controlled by its initial coherence. For a coherent anomalous hollow beam (σ g0=Infinity), the focused beam profiel is of elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot (see Fig. 5(a)), which is similar to the far field beam profile of a coherent anomalous hollow beam in free space. For a partially coherent anomalous hollow beam, the focused beam profile gradually becomes a circular Gaussian distribution as the initial coherence decreases (see Fig. 5(b)–(d)). Physical reason is the same as given for Fig. 4. One also finds from Fig. 5 (e) that the focused beam spot size decreases as the initial coherence width increases, which means that an anomalous hollow beam with higher initial coherence can be focused more tightly, which is consistent with the focusing properties of a partially coherent Gaussian beam [34

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

]. Our results also are consistent with those in [28

28. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).

]. From above discussions, we find it is necessary to take the coherence of anomalous hollow beams into consideration in some practical cases.

5. Conclusion

Acknowledgments

Y. Cai gratefully acknowledges the support from the Alexander von Humboldt Foundation. Q. Lin gratefully acknowledges the supports from the Ministry of Science and Technology of China (grant no. 2006CB921403 & 2006AA06A204).

References and links

1.

C. Palma, “Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams,” Appl. Opt. 36, 1116–1120 (1997). [CrossRef] [PubMed]

2.

J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation constans for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998). [CrossRef]

3.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev . 7, 216–220 (2000). [CrossRef]

4.

Y. Li, “Light beam with flat-topped profiles,” Opt. Lett . 27, 1007–1009 (2002). [CrossRef]

5.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profile,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004). [CrossRef]

6.

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398–1407 (2006). [CrossRef]

7.

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006). [CrossRef] [PubMed]

8.

Z. Mei and D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A 23, 919–925 (2006). [CrossRef]

9.

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008). [CrossRef]

10.

Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006). [CrossRef]

11.

C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006). [CrossRef] [PubMed]

12.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., (North-Holland, Amsterdam, 2003), pp.119–204.

13.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997). [CrossRef]

14.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000). [CrossRef]

15.

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084–1086 (2003). [CrossRef] [PubMed]

16.

Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058–1065 (2004). [CrossRef]

17.

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44, 7187–7190 (2005). [CrossRef] [PubMed]

18.

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, 202–202 (2008). [CrossRef]

19.

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008). [CrossRef]

20.

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076–2078 (2007). [CrossRef] [PubMed]

21.

Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun . 281, 508–514 (2008). [CrossRef]

22.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934(2008). [CrossRef]

23.

G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417–6424 (2008). [CrossRef] [PubMed]

24.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008). [CrossRef]

25.

Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A 25, 1497–1503 (2008). [CrossRef]

26.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . 94, 134802 (2005). [CrossRef] [PubMed]

27.

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . 32, 3179–3181 (2007). [CrossRef] [PubMed]

28.

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).

29.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.

30.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67 (1990).

31.

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]

32.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).

33.

S. Wang and L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (North-Holland, 1988), pp. 279.

34.

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

35.

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, 1057–1060 (1984). [CrossRef]

36.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002). [CrossRef]

37.

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

38.

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

39.

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004). [CrossRef] [PubMed]

40.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005). [CrossRef]

41.

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798–1804 (2005). [CrossRef]

42.

F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999–7004 (2006). [CrossRef] [PubMed]

43.

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

44.

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

45.

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

46.

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, 1937–1944 (2007). [CrossRef]

47.

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007). [CrossRef]

48.

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33,1389–1391(2008). [CrossRef] [PubMed]

49.

F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008). [CrossRef]

50.

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett . 33, 1795–1797 (2008). [CrossRef] [PubMed]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(140.3430) Lasers and laser optics : Laser theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: June 17, 2008
Revised Manuscript: September 2, 2008
Manuscript Accepted: September 7, 2008
Published: September 12, 2008

Citation
Yangjian Cai, Zhaoying Wang, and Qiang Lin, "An alternative theoretical model for an anomalous hollow beam," Opt. Express 16, 15254-15267 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15254


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References

  1. C. Palma, "Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams," Appl. Opt. 36, 1116-1120 (1997). [CrossRef] [PubMed]
  2. J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, "Beam propagation constans for a radial laser array," Appl. Opt. 37, 1045-1048 (1998). [CrossRef]
  3. N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, "Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target," Opt. Rev. 7, 216-220 (2000). [CrossRef]
  4. Y. Li, "Light beam with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002). [CrossRef]
  5. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profile," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004). [CrossRef]
  6. Y. Cai and L. Zhang, "Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties," J. Opt. Soc. Am. B 23, 1398-1407 (2006). [CrossRef]
  7. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]
  8. Z. Mei and D. Zhao, "Controllable elliptical dark-hollow beams," J. Opt. Soc. Am. A 23, 919-925 (2006). [CrossRef]
  9. H. T. Eyyuboðlu, "Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence," Opt. Laser Technol. 40, 156-166 (2008). [CrossRef]
  10. Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006). [CrossRef]
  11. C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006). [CrossRef] [PubMed]
  12. J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics, Vol. 44, E. Wolf, ed., (North-Holland, Amsterdam, 2003), pp.119-204.
  13. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997). [CrossRef]
  14. J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000). [CrossRef]
  15. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beam and its propagation," Opt. Lett. 28, 1084-1086 (2003). [CrossRef] [PubMed]
  16. Y. Cai and Q. Lin, "Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems," J. Opt. Soc. Am. A 21, 1058-1065 (2004). [CrossRef]
  17. D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, "Far-field intensity distribution and M2 factor of hollow Gaussian beams," Appl. Opt. 44, 7187-7190 (2005). [CrossRef] [PubMed]
  18. 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, 202-202 (2008). [CrossRef]
  19. D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am. B 25, 83-87 (2008). [CrossRef]
  20. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076-2078 (2007). [CrossRef] [PubMed]
  21. Y. Zhang, "Generation of thin and hollow beams by the axicon with a large open angle," Opt. Commun. 281, 508-514 (2008). [CrossRef]
  22. R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma," Opt. Commun. 281, 923-934 (2008). [CrossRef]
  23. G. Wu, Q. Lou, and J. Zhou, "Analytical vectorial structure of hollow Gaussian beams in the far field," Opt. Express 16, 6417-6424 (2008). [CrossRef] [PubMed]
  24. T. Wang, J. Pu, and Z. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008). [CrossRef]
  25. Y. Cai, H. T. Eyyuboðlu, and Y. Baykal, "Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere," J. Opt. Soc. Am. A 25, 1497-1503 (2008). [CrossRef]
  26. Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005). [CrossRef] [PubMed]
  27. Y. Cai, "Model for an anomalous hollow beam and its paraxial propagation," Opt. Lett. 32, 3179-3181 (2007). [CrossRef] [PubMed]
  28. Y. Cai and F. Wang, "Partially coherent anomalous hollow beam and its paraxial propagation," Phys. Lett. A 372, 4654-4660 (2008).
  29. J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247-304.
  30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).
  31. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970). [CrossRef]
  32. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).
  33. S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E. Wolf, ed. (North-Holland, 1988), pp. 279.
  34. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  35. 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, 1057-1060 (1984). [CrossRef]
  36. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002). [CrossRef]
  37. Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian Schell-model beams," Opt. Lett. 27, 216-218 (2002). [CrossRef]
  38. Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005). [CrossRef]
  39. Y. Cai and S. Zhu, "Ghost interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004). [CrossRef] [PubMed]
  40. Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phy. Lett. 86, 021112 (2005). [CrossRef]
  41. Y. Cai and S. Zhu, "Coincidence fractional Fourier transform with partially coherent light radiation," J. Opt. Soc. Am. A 22, 1798-1804 (2005). [CrossRef]
  42. F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006). [CrossRef] [PubMed]
  43. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]
  44. Y. Cai and L. Hu, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system," Opt. Lett. 31, 685-687 (2006). [CrossRef] [PubMed]
  45. Y. Cai and U. Peschel, "Second-harmonic generation by an astigmatic partially coherent beam," Opt. Express 15, 15480-15492 (2007). [CrossRef] [PubMed]
  46. 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, 1937-1944 (2007). [CrossRef]
  47. X. Lü and Y. Cai, "Partially coherent circular and elliptical dark hollow beams and their paraxial propagations," Phys. Lett. A 369, 157-166 (2007). [CrossRef]
  48. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, "Generation of a high-quality partially coherent dark hollow beam with a multimode fiber," Opt. Lett. 33, 1389-1391(2008). [CrossRef] [PubMed]
  49. F. V. Dijk, G. Gbur, and T. D. Visser, "Shaping the focal intensity distribution using spatial coherence," J. Opt. Soc. Am. A 25, 575-581 (2008). [CrossRef]
  50. F. Wang and Y. Cai, "Experimental generation of a partially coherent flat-topped beam," Opt. Lett. 33, 1795-1797 (2008). [CrossRef] [PubMed]

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