Abstract

Partially coherent standard and elegant Laguerre-Gaussian (LG) beams of all orders are introduced as a natural extension of coherent standard and elegant LG beams to the stochastic domain. By expanding the LG modes into a finite sum of Hermite-Gaussian modes, the analytical formulae are obtained for the cross-spectral densities of partially coherent standard and elegant LG beams in the source plane and after passing through paraxial ABCD optical system, based on the generalized Collins integral formula. A comparative study of the propagation properties of the partially coherent standard and elegant LG beams in free space is carried out via a set of numerical examples. Our results indicate that the intensity and spreading properties of partially coherent standard and elegant LG beams are closely related to their initial coherence states, and are very different from the corresponding results for the coherent standard and elegant LG beams. In particular, an elegant LG beam spreads slower than a standard LG beam, while this advantage disappears when their initial coherences are very small. Our results may find applications in connection with laser beam shaping, singular optics and astrophysical measurements of angular momentum of radiation.

© 2009 OSA

1. Introduction

Standard Laguerre-Gaussian (LG) beams are commonly encountered in laser optics, material processing and atomic optics [1]. Such beams were paid enormous attention during the past decades, and various aspects relating to their generation, propagation and applications were profoundly treated in the literature [223]. A standard LG beam can be generated by spatial light modulators [2], mode selection [3], cooperative frequency locking in a helium–neon laser [4,5], in fiber-coupled laser diode end-pumped lasers [6,7], through the conversion of Hermite-Gaussian beams by an astigmatic mode coverter [8], or through the computer hologram [9]. Standard LG beams have found wide applications in free space optical communications [10,11], atom trapping [12], atom guiding [13], atom interferometers [14], electron acceleration [15] and nanoparticles trapping [16]. Paraxial and non-paraxial propagation properties of standard LG beams have been studied in details [1725]

Standard Hermite-Gaussian (HG) and LG modes are eigenmodes of the paraxial wave equation. The Gaussian part of the standard HG or LG modes has a complex argument, but the Hermite or Laguerre part is purely real. Siegman was the first to introduce new HG solutions named elegant HG modes that satisfy the paraxial wave equation but have a more symmetrical form [26]. Later, Takenaka et al. proposed the elegant LG beam as an extension of standard LG beam [2729]. Relationship between elegant LG and Bessel-Gauss beams was studied in [30]. Elegant LG beams were also proposed as a tool for describing axisymmetric flattened Gaussian beams [31]. Paraxial and non-paraxial propagation of elegant LG beams have been carried out in [3237].

Coherence is one important properties of a laser beam. Laser beams with low coherence named partially coherent beams have advantages over coherent beam in many applications, such as free-space optical communication [38], inertial confinement fusion [39], harmonic generation [40,41], optical trapping [42], optical projection [43], photography [44], optical imaging [4547].

In the past decades, most research on partially coherent beams is devoted to Gaussian Schell-model beams whose intensity distribution and degree of coherence are Gaussian functions. Recently, more and more attention is being paid to Schell-model beams with special profiles [42,4863], such as partially coherent Hermite-Gaussian beams, partially coherent dark hollow beams, partially coherent flat-topped beams, partially coherent vortex beams, partially coherent cosh-Gaussian, cos-Gaussian and cosine-Gaussian beam, partially coherent Laguerre-Gaussian (0,1) beams, due to their important potential applications in various fields, including free-space optical communication, optical trapping and singular optics.

Several attempts were previously made in the literature to extend the coherent LG beams to stochastic domain. Ponamorenko [58] used the LG modes for construction, by means of the coherent mode decomposition [59], of the partially coherent fields carrying separable vortexes. This has led to the class of stochastic fields with the degree of coherence in a form of a modified Bessel function. Later this class of beams was further generalized to dark and anti-dark beams with a degree of coherence being a linear combination of Bessel and modified Bessel beams [60].

An alternative way of obtaining a class of partially coherent beams on the basis of the LG modes is to assume that the cross-spectral density function has the Gaussian Schell-model form [59], in which the intensity part is based on the LG modes but the degree of coherence can be chosen simply Gaussian. To our knowledge, only few papers were dedicated to beams belonging to this class, e.g [6163]. These studies, however, were aimed at investigating issues like dynamics of optical singularities and of radial polarization and discussed only particular representatives of this rich beam class. No studies were performed so far on partially coherent elegant LG beams.

To our knowledge, up to now, partially coherent standard or elegant LG beam of all orders has not been formulated. Thus it is of practical importance to formulate partially standard or elegant LG beam of all orders and study its propagation properties. Partially coherent vortex beams [61,62] and partially coherent LG(0,1) beams [63] can be regarded as special cases of partially coherent LG beams. Recent research has shown that partially coherent beams with special profiles have advantages over corresponding coherent beams for reducing turbulence-induced intensity fading in laser communication systems [50,52,5557,62], and for optical trapping [42]. We expect to apply partially coherent standard or elegant LG beam of all orders to free-space optical communication and optical trapping.

The goal of this paper is to establish the unified theoretical model for both standard and elegant stochastic LG beams of all orders (with LG intensity distribution and Gaussian degree of coherence). The paper is organized as follows. We will first derive the formulas for the cross-spectral density function of the new class of beams in the source plane and in a transverse plane after passing through a linear optical ABCD system (Section 2). We will then carefully study, via numerous examples, the free-space propagation of standard and elegant partially coherent LG beams, which is just a particular case of the general passage of a beam through the ABCD system (Section 3). Finally, the summary of our results will be given (Section 4).

2. Theory

Let us begin by recalling that the electric field of a standard or elegant LG beam in the plane of the source, z = 0, is expressed as follows [135]

Epl(r,φ;0)=(qrω0)lLpl(q2r2ω02)exp(r2ω02)exp(ilφ),
where r and φ are the radial and azimuthal (angle) coordinates, Lpldenotes the Laguerre polynomial with mode orders p and l, ω0 is the beam width of the fundamental Gaussian mode. For q=2, Eq. (1) reduces to the electric field of a standard LG beam; for q=1, it gives the electric field of an elegant LG beam; also for p=0 and l=0, Eq. (1) degenerates to the electric field of a fundamental Gaussian beam.

By use of the following relation between an LG mode and an HG mode [64]

eilφρlLpl(ρ2)=(1)p22p+lp!m=0pn=0lin(pm)(ln)H2m+ln(x)H2p2m+n(y),
withH(x) being the Hermite polynomial,(pm)and (ln)being binomial coefficients, Eq. (1) can be expressed in following alternative form in Cartesian coordinates

Epl(x,y;0)=(1)p22p+lp!m=0ps=0lis(pm)(ls)H2m+ls(qxω0)H2p2m+s(qyω0)exp(x2+y2ω02).

We will now extend standard and elegant LG beams to the partially coherent case. The second-order statistical properties of a partially coherent beam are generally characterized by the cross-spectral density W(x1,y1,x2,y2;z)=E*(x1,y1;z)E(x2,y2;z), where denotes the ensemble average and “*” is the complex conjugate [59]. The intensity distribution, of a partially coherent beam at any position (x, y) in plane z, z ≥ 0, can be determined from the relation I(x,y;z)=W(x,y,x,y;z). For a partially coherent beam generated by a Schell-model source (at z = 0), the cross-spectral density can be expressed in the following well-known form [59]

W(x1,y1,x2,y2;0)=I(x1,y1;0)I(x2,y2;0)g(x1x2;y1y2;0),
where g(x1x2;y1y2;0) is the spectral degree of coherence which we will assume to have Gaussian profile, i.e.
g(x1x2;y1y2;0)=exp[(x1x2)2+(y1y2)22σg2],
whereσgis the transverse coherence width. We now assume that the intensity distribution of our the Gaussian Schell-model source can be represented by a standard or elegant LG beam. ThenI(xi,yi;0)can be written as

I(x,y;0)=|(1)p22p+lp!m=0ps=0lis(pm)(ls)H2m+ls(qxω0)H2p2m+s(qyω0)exp(x2+y2ω02)|2.

Substituting Eq. (6) into Eq. (4), we can express the cross-spectral density of a partially coherent standard or elegant LG beam as follows

W(x1,y1,x2,y2;0)=124p+2l(p!)2m=0pn=0lh=0ps=0l(in)*is(pm)(ln)(ph)(ls)                               ×H2m+ln(qx1ω0)H2h+ls(qx2ω0)H2p2m+n(qy1ω0)H2p2h+s(qy2ω0)                               ×exp(x12+y12+x22+y22ω02)exp((x1x2)2+(y1y2)22σg2).

Under the condition of σg, Eq. (7) reduces to the expression for a coherent standard or elegant LG beam. Under the condition of p=0 and l=0, Eq. (7) reduces to the expression for a partially coherent Gaussian Schell-model beam [6570]. By transforming Eq. (1) into Eq. (3) with the help of Eq. (2) and expressing the cross-spectral density of a partially coherent standard or elegant LG beam of all orders in the form of Eq. (7) in the Cartesian coordinates, we can obtain analytical propagation formula for the cross-spectral density and analytical expression for the effective beam size of a partially coherent standard or elegant LG beam in an easy way as shown later. In the cylindrical coordinates, it is very difficult for us to obtain analytical propagation formula of a partially coherent standard or elegant LG beam.

Now we study the propagation of beams generated by a partially coherent standard or elegant LG source (7) through a paraxial ABCD optical system. Within the validity of the paraxial approximation, the propagation of the cross-spectral density of a partially coherent beam through an aligned ABCD optical system in free space can be studied with the help of the following generalized Collins formula [49,71]

W(u1,v1,u2,v2,z)=(1λ|B|)2W(x1,y1,x2,y2;0)          ×exp[ik2B*(A*x122x1u1+D*u12)ik2B*(A*y122y1v1+D*v12)]          ×exp[ik2B(Ax222x2u2+Du22)+ik2B(Ay222y2v2+Dy22)]dx1dx2dy1dy2,
wherexi,yi and ui,vi are the position coordinates in the input and output planes, A,B,C and D are the transfer matrix elements of optical system, k=2π/λ is the wave number withλbeing the wavelength.

Substituting from Eq. (7) into Eq. (8), we obtain the formula

W(u1,v1,u2,v2,z)=(1λ|B|)2124p+2l(p!)2m=0pn=0lh=0ps=0l(in)*is(pm)(ln)(ph)(ls)                           H2m+ln(qx1ω0)H2h+ls(qx2ω0)exp(x12+x22ω02)exp((x1x2)22σg2)                             ×exp[ik2B*(A*x122x1u1+D*u12)+ik2B(Ax222x2u2+Du22)]dx1dx2                         H2p2m+n(qy1ω0)H2p2h+s(qy2ω0)exp(y12+y22ω02)exp((y1y2)22σg2)                           ×exp[ik2B*(A*y122y1v1+D*v12)+ik2B(Ay222y2v2+Dy22)]dy1dy2.
After integration over x1,x2,y1,y2, Eq. (9) becomes
W(u1,u2,v1,v2,z)=(1λ|B|)2124p+2l(p!)2π2M1M2(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l                        ×exp[ikD*2B*u12+ikD2Bu22)]exp(k2u224M1B2)exp[k24M2(u1B*u22M1σg2B)2]                        ×exp[ikD*2B*v12+ikD2Bv22)]exp(k2v224M1B2)exp[k24M2(v1B*v22M1σg2B)2]                        ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2](in)*is(pm)(ln)                        ×(ph)(ls)(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2(2m+ln)!c1!(2m+ln2c1)!                        ×d!c2!(d2c2)!(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2i)2c1+2c2+2e1+2e2dd12pl                        ×(1M2)d+d12c12c22e12e2(2qω0)2c12e1(2q2σg2M12ω02q2M1)d+d12c22e2                        ×H2h+lsd(iqku22BM12ω02q2M1)H2m+ln+d2c12c2(ku24M1M2σg2Bku12M2B*)                        ×H2p2h+sd1(iqkv22BM12ω02q2M1)H2p2m+n+d12e12e2(kv24M1M2σg2Bkv12M2B*),
with M1=1/ω02+1/(2σg2)ikA/(2B),M2=1/ω02+1/(2σg2)+ikA*/(2B*)1/(4M1σg4). In above derivations, we have used the following integral and expansion formulae [72,73]

exp[(xy)2]Hn(ax)dx=π(1a2)n2Hn(ay(1a2)1/2)
xnexp[(xβ)2]dx=(2i)nπHn(iβ),
Hn(x+y)=12n/2k=0n(nk)Hk(2x)Hnk(2y),
Hn(x1)=m=0[n/2](1)mn!m!(n2m)!(2x1)n2m.

Thus, Eq. (10) is the analytical formulae for the cross-spectral density of a partially coherent standard or elegant LG beam passing a paraxial ABCD optical system. The intensity of the partially coherent standard or elegant LG beam at the output plane can be obtained by setting, in Eq. (10), u1=u2andv1=v2.

The effective beam size is a useful parameter for characterizing the spreading properties of a beam. According to [74], by use of twice the variance of x or y, the effective beam size of a partially coherent standard or elegant LG beam at plane z is defined as

Wsz(z)=2s2I(x,y,z)dxdyI(x,y,z)dxdy      (s=x,y) .

On substituting from Eq. (10) into Eq. (15), we obtain the following expression for the effective beam size of a partially coherent standard or elegant LG beam after propagation

Wxz=Wyz=A1(z)A2(z)
where
A1(z)=(1λ|B|)2124p+2l(p!)2π3M1M2M3(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l         ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2]f1=0[(2h+lsd)/2]         ×f2=0[(2m+ln+d2c12c2)/2]g1=0[(2p2h+sd1)/2]g2=0[(2p2m+n+d12e12c2)/2](in)*is(pm)(ln)(ph)         ×(ls)(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2+f1+f2+g1+g2(2m+ln)!c1!(2m+ln2c1)!d!c2!(d2c2)!         ×(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2m+ln+d2c12c2)!f2!(2m+ln+d2c12c22f2)!         ×(2h+lsd)!f1!(2h+lsd2f1)!(2p2m+n+d12e12e2)!g2!(2p2m+n+d12e12e22g2)!(2p2h+sd1)!g1!(2p2h+sd12g1)!         ×(2i)4c1+4c2+4e1+4e22f12f22g12g2dd16p3l2(1M2)d+d12c12c22e12e2(2qω0)2c12e1         ×(2q2σg2M12ω02q2M1)d+d12c22e2(2iqk2BM12ω02q2M1)2h+lsd2f1         ×(k2M1M2σg2BkM2B*)2m+ln+d2c12c22f2(2iqk2BM12ω02q2M1)2p2h+sd12g1         ×(k2M1M2σg2BkM2B*)2p2m+n+d12e12e22g2(1M3)2p+lc1c2f1f2e1e2g1g2+1         ×H2h+2m+2lsn2c12c22f12f2+2(0)H4p2h2m+s+n2e12e22g12g2(0),                                       
and
A2(z)=(1λ|B|)2124p+2l(p!)2π3M1M2M3(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l               ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2]f1=0[(2h+lsd)/2]            ×f2=0[(2m+ln+d2c12c2)/2]g1=0[(2p2h+sd1)/2]g2=0[(2p2m+n+d12e12c2)/2](in)*is(pm)(ln)(ph)(ls)           ×(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2+f1+f2+g1+g2(2m+ln)!c1!(2m+ln2c1)!d!c2!(d2c2)!           ×(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2m+ln+d2c12c2)!f2!(2m+ln+d2c12c22f2)!           ×(2h+lsd)!f1!(2h+lsd2f1)!(2p2m+n+d12e12e2)!g2!(2p2m+n+d12e12e22g2)!(2p2h+sd1)!g1!(2p2h+sd12g1)!           ×(2i)4c1+4c2+4e1+4e2+2g1+2g2+2f1+2f2dd16p3l(1M2)d+d12c12c22e12e2(2qω0)2c12e1           ×(2q2σg2M12ω02q2M1)d+d12c22e2(2iqk2BM12ω02q2M1)2h+lsd2f1          ×(k2M1M2σg2BkM2B*)2m+ln+d2c12c22f2(2iqk2BM12ω02q2M1)2p2h+sd12g1          ×(k2M1M2σg2BkM2B*)2p2m+n+d12e12e22g2(1M3)2p+le1e2g1g2c1c2f1f2          ×H2h+2m+2lsn2c12c22f12f2(0)H4p2h2m+s+n2e12e22g12g2(0),                                             
with M3=k24M1B2+k24M2(1B*12M1σg2B)2+ikD2BikD*2B*. Equations (10) and (16)-(18) are the main analytical results of the paper. Although Eqs. (10) and (16)-(18) involve up to 14 nested sums of binomial coefficients and Hermite function, it takes only several minutes to calculate each line of Figs. 1 -4 . If we want to calculate each line of Figs. 1-4 by Eqs. (7) and (8) through direct numerical integration, it is very time consuming due to four integrals, usually its take several hours or more to do this. More comparisons between calculation by analytical formula and by direct numerical calculation can be found in [75].

 

Fig. 1 Normalized intensity distribution (cross line v = 0) of a partially coherent standard LG beam for different values of the initial coherence width σg with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

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Fig. 4 Effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l and coherence widthσg

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3. Propagation properties of partially coherent standard and elegant LG beams in free space

In this section we will carry out a comparative study of the properties of partially coherent standard and elegant LG beams propagating in free space by using the formulae derived in Section 2.

The ray transfer matrix relating to free-space propagation between the source plane (z = 0) and output plane (z ≥ 0) takes the form

(ABCD)=(1z01).

Substituting from Eq. (19) into Eq. (10), we calculate in Fig. 1 the normalized intensity distribution (cross line v = 0) of a partially coherent standard LG beam for different values of the initial coherence width σg at several propagation distances in free space with p=1, l=1,ω0=2mmand λ=632.8nm. One finds from Fig. 1 that the propagation properties of a partially coherent LG beam are very different from those of a coherent standard LG beam. For a coherent standard LG beam, its initial source beam profile remains invariant on propagation although its beam spot increases, the result being in good aggreement with previous results of [125]. For a partially coherent LG beam, its propagation properties are closely related to its initial degree of coherence. The initial source beam profile of a partially coherent standard LG beam does not remain invariant on propagation, but gradually disappears on propagation and eventually takes a Gaussian shape. As the initial degree of coherence decreases, the transition from a standard LG beam into a Gaussian beam occurs sooner and the beam spreads more rapidly.

For a convenient comparison, we calculate in Fig. 2 the normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σg at several propagation distances in free space with p=1, l=1,ω0=2mmand λ=632.8nm. We find from Fig. 2 that the propagation properties of an elegant LG beam are also largely determined by its initial degree of coherence. The beam profile of a coherent elegant LG beam transforms into a dark hollow beam profile in the far field, as expected [2737]. Similar to a partially coherent standard LG beam, the beam profile of a partially coherent elegant LG beam also gradually disappears on propagation and eventually takes a Gaussian shape. As the initial coherence decreases, the conversion from an elegant LG beam into a Gaussian beam occurs more quickly and the beam spreads more rapidly. From Figs. 1 and 2, one comes to the conclusion that by degrading the coherence of a standard or elegant LG beam it possible to perform beam shaping in the far field.

 

Fig. 2 Normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σg with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

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To learn about the spreading properties of the partially coherent standard and elegant LG beams on propagation in free space, we calculate in Fig. 3 the effective beam sizes of partially coherent standard and elegant LG beams versus the propagation distance z in free space for different values of the initial coherence width with p=1, l=1and λ=632.8nm. For the convenience of comparison, we have set ω0=10mmfor a partially coherent elegant LG beam and ω0=5.773mm for a partially coherent standard LG beam, so that they have the same effective beam sizes in the source plane (z = 0). One finds from Fig. 3 that the spreading properties of standard and elegant LG beams are also closely related to their initial coherence widths. A coherent standard LG beam spreads more rapidly than a partially coherent elegant LG beam (see Fig. 3(a)), as expected [2737]. When the coherence width is large, a partially coherent elegant LG beam still spreads slower than a partially coherent standard LG beam (see Fig. 3(b) and (c)). When the coherence width is small, there is no distinct difference between the spreading properties of a partially coherent standard LG beam and that of a partially coherent elegant LG beam (see Fig. 3(d)).

 

Fig. 3 Effective beam sizes of partially coherent standard LG beam and elegant LG beam versus the propagation distance z in free space for different values of the initial coherence width

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In Fig. 4, we calculate the effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l. For the convenience of comparison, in Fig. 4(a)-(c), we have chosen ω0=10.0mm for p=1, l=1,ω0=10.54mm for p=2, l=1,ω0=10.80mm for p=3,l=1,ω0=8.165mm for p=1, l=2,ω0=6.90mm for p=1, l=3, respectively, so that all partially coherent elegant LG beams have the same effective beam sizes on the source plane. In Fig. 4 (d)-(f), we have chosen ω0=10.0mm for p=1, l=1,ω0=8.165mm for p=2, l=1,ω0=7.07mm for p=3, l=1,ω0=8.944mm for p=1, l=2,ω0=8.166mm for p=1, l=3 respectively. It is evident from Fig. 4 that the mode orders p and l of partially coherent standard and elegant LG beams affect their spreading properties strongly when the initial degree of coherence is high. Both partially coherent standard and elegant LG beams spread more rapidly as their mode orders p and l increase when the initial degree of coherence is high (see Fig. 4(a), (b), (d) and (e))). When the initial coherence is small, the partially coherent LG beams, both standard and elegant, with different mode orders exhibit almost the same spreading features (see Fig. 4 (c) and (f)).

4. Summary

We have proposed theoretical model to describe partially coherent standard and elegant LG beams, and have derived the analytical formulae for the cross-spectral densities of such beams propagating through paraxial ABCD optical systems. By numerical examples, we have studied the intensity and spreading properties of partially coherent standard and elegant LG beams in free space, comparatively. We have found that the properties of standard and elegant LG beams on free-space propagation are much different from those pertaining to coherent standard and elegant LG beams. As a general rule, a partially coherent elegant LG beam spreads more slowly in free space than a partially coherent standard LG beam. The advantage of a partially coherent elegant LG beam over a partially coherent standard LG beam disappears when the initial coherence in the source plane is very low. Thus, coherent or partially coherent elegant LG beams have some advantages over the corresponding standard LG beams, the result which can be employed in applications, such as free-space optical communications and remote sensing.

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 and the Natural Science of Jiangsu Province under Grant No. BK2009114. O. Korotkova’s research is funded by the AFOSR (grant FA 95500810102).

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11. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 ( 2009). [CrossRef]   [PubMed]  

12. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 ( 1997). [CrossRef]  

13. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 ( 2000). [CrossRef]  

14. Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 ( 2009). [CrossRef]  

15. X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 ( 2008). [CrossRef]  

16. D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. 30(22), 3039–3041 ( 2005). [CrossRef]   [PubMed]  

17. J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 ( 2001).

18. V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 ( 2000). [CrossRef]  

19. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 ( 2000). [CrossRef]   [PubMed]  

20. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 ( 2002). [CrossRef]   [PubMed]  

21. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 ( 2002). [CrossRef]  

22. S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 ( 2003). [CrossRef]  

23. Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 ( 2006). [CrossRef]  

24. G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 ( 2008). [CrossRef]  

25. C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express 17(5), 3690–3697 ( 2009). [CrossRef]   [PubMed]  

26. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 ( 1973). [CrossRef]  

27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 ( 1985). [CrossRef]  

28. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 ( 1986). [CrossRef]  

29. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 ( 1998). [CrossRef]  

30. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 ( 2001). [CrossRef]  

31. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 ( 2001). [CrossRef]  

32. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 ( 2004). [CrossRef]   [PubMed]  

33. Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 ( 2004). [CrossRef]  

34. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 ( 2004). [CrossRef]  

35. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 ( 2004). [CrossRef]  

36. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 ( 2008). [CrossRef]   [PubMed]  

37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 ( 2009). [CrossRef]   [PubMed]  

38. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 ( 2002). [CrossRef]  

39. 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]  

40. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 ( 1987). [CrossRef]   [PubMed]  

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

42. 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]  

43. G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 ( 1983).

44. C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. 38(32), 6687–6691 ( 1999). [CrossRef]   [PubMed]  

45. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 ( 2004). [CrossRef]   [PubMed]  

46. 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]  

47. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 ( 2009). [CrossRef]   [PubMed]  

48. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 ( 2005). [CrossRef]  

49. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 ( 2007). [CrossRef]  

50. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 ( 2008). [CrossRef]  

51. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 ( 2009). [CrossRef]  

52. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 ( 2007). [CrossRef]  

53. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 ( 2008). [CrossRef]   [PubMed]  

54. 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(12), 1389–1391 ( 2008). [CrossRef]   [PubMed]  

55. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 ( 2008). [CrossRef]  

56. M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 ( 2008). [CrossRef]  

57. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 ( 2009). [CrossRef]   [PubMed]  

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

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

60. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 ( 2007). [CrossRef]   [PubMed]  

61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 ( 2009). [CrossRef]  

62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 ( 2008). [CrossRef]  

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 ( 2009). [CrossRef]  

64. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 ( 1993).

65. 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]  

66. P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 ( 1979). [CrossRef]  

67. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 ( 1980). [CrossRef]  

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

69. 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]  

70. 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]  

71. 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]  

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

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

74. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 ( 1980). [CrossRef]   [PubMed]  

75. 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]  

References

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  12. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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  13. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
    [CrossRef]
  14. Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009).
    [CrossRef]
  15. X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
    [CrossRef]
  16. D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. 30(22), 3039–3041 (2005).
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  17. J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).
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    [CrossRef]
  19. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000).
    [CrossRef] [PubMed]
  20. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002).
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  21. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002).
    [CrossRef]
  22. S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
    [CrossRef]
  23. Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006).
    [CrossRef]
  24. G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008).
    [CrossRef]
  25. C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express 17(5), 3690–3697 (2009).
    [CrossRef] [PubMed]
  26. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973).
    [CrossRef]
  27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985).
    [CrossRef]
  28. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986).
    [CrossRef]
  29. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
    [CrossRef]
  30. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001).
    [CrossRef]
  31. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001).
    [CrossRef]
  32. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004).
    [CrossRef] [PubMed]
  33. Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
    [CrossRef]
  34. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
    [CrossRef]
  35. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
    [CrossRef]
  36. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
    [CrossRef] [PubMed]
  37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
    [CrossRef] [PubMed]
  38. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
    [CrossRef]
  39. 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]
  40. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
    [CrossRef] [PubMed]
  41. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
    [CrossRef] [PubMed]
  42. 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]
  43. G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).
  44. C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. 38(32), 6687–6691 (1999).
    [CrossRef] [PubMed]
  45. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
    [CrossRef] [PubMed]
  46. 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]
  47. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
    [CrossRef] [PubMed]
  48. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
    [CrossRef]
  49. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
    [CrossRef]
  50. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008).
    [CrossRef]
  51. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
    [CrossRef]
  52. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
    [CrossRef]
  53. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
    [CrossRef] [PubMed]
  54. 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(12), 1389–1391 (2008).
    [CrossRef] [PubMed]
  55. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
    [CrossRef]
  56. M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
    [CrossRef]
  57. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 (2009).
    [CrossRef] [PubMed]
  58. S. A. Ponamorenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. 18(1), 150–156 (2001).
    [CrossRef]
  59. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  60. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 (2007).
    [CrossRef] [PubMed]
  61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
    [CrossRef]
  62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
    [CrossRef]
  63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
    [CrossRef]
  64. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).
  65. 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]
  66. P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
    [CrossRef]
  67. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
    [CrossRef]
  68. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
    [CrossRef]
  69. 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]
  70. 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]
  71. 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]
  72. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  73. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
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2009 (10)

Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009).
[CrossRef]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
[CrossRef]

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]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express 17(5), 3690–3697 (2009).
[CrossRef] [PubMed]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[CrossRef]

G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 (2009).
[CrossRef] [PubMed]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[CrossRef] [PubMed]

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

2008 (10)

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(12), 1389–1391 (2008).
[CrossRef] [PubMed]

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
[CrossRef] [PubMed]

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008).
[CrossRef]

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[CrossRef]

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008).
[CrossRef]

2007 (5)

2006 (2)

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]

Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006).
[CrossRef]

2005 (4)

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]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13(13), 4952–4962 (2005).
[CrossRef] [PubMed]

D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. 30(22), 3039–3041 (2005).
[CrossRef] [PubMed]

2004 (5)

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

2003 (1)

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
[CrossRef]

2002 (5)

2001 (5)

M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001).
[CrossRef]

R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001).
[CrossRef]

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

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

2000 (3)

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[CrossRef]

R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000).
[CrossRef] [PubMed]

1999 (2)

C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. 38(32), 6687–6691 (1999).
[CrossRef] [PubMed]

T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. 160(1-3), 103–108 (1999).
[CrossRef]

1998 (1)

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995).
[CrossRef]

1993 (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

1992 (1)

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38(11), 5960–5963 (1988).
[CrossRef] [PubMed]

1987 (1)

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

1984 (1)

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]

1983 (1)

G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

1982 (1)

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

1980 (2)

1979 (1)

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

1978 (1)

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]

1973 (1)

Agarwal, G. S.

Alavinejad, M.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Ando, T.

Andrews, D. L.

April, A.

Arinaga, S.

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]

Arlt, J.

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[CrossRef]

Bandres, M. A.

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Baykal, Y.

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
[CrossRef]

Borghi, R.

Bradshaw, D. S.

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Cada, M.

Cai, Y.

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]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

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(12), 1389–1391 (2008).
[CrossRef] [PubMed]

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]

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

Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
[CrossRef]

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]

Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006).
[CrossRef]

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]

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]

Carter, W. H.

Chen, C.

Chen, X.

Chen, Y.

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

Chen, Z.

Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
[CrossRef]

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Cheng, C.

Chu, X.

Ciattoni, A.

Cincotti, G.

Collett, E.

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]

Davidson, F. M.

Davidson, N.

De. Santis, P.

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Dente, G.

G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

Dholakia, K.

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[CrossRef]

Eyyuboglu, H. T.

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]

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
[CrossRef]

Friberg, A. T.

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

Friesem, A. A.

Fukuchi, N.

Fukumitsu, O.

Gbur, G.

Ghafary, B.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[CrossRef]

Gori, F.

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[CrossRef]

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Gu, J.

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

Gu, Y.

Guattari, G.

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Gui, W.

Guo, H.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Hara, T.

Hasegawa, T.

T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. 160(1-3), 103–108 (1999).
[CrossRef]

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995).
[CrossRef]

He, S.

Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[CrossRef]

Ho, Y.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Hu, L.

Huang, W.

Inoue, T.

Ishaaya, A. A.

Jarutis, V.

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

Ji, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008).
[CrossRef]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

Kato, Y.

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]

Kitagawa, Y.

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]

Kong, Q.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Korotkova, O.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Kuhn, R.

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

Lan, Y.

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

Lin, Q.

Liu, J.

Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
[CrossRef]

Liu, W.

Lu, B.

Lu, X.

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Luis, R. E.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[CrossRef]

Matsumoto, N.

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

McIver, J. K.

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
[CrossRef] [PubMed]

Mei, Z.

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

Mima, K.

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]

Miyanaga, N.

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]

Nakatsuka, M.

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]

Ohtake, Y.

Orlov, S.

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
[CrossRef]

Osgood, J. S.

G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

Palma, C.

G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002).
[CrossRef]

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Paskauskas, R.

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Peschel, U.

Ponamorenko, S. A.

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

Ponomarenko, S. A.

Porras, M. A.

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[CrossRef]

Qiu, Y.

Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Ricklin, J. C.

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995).
[CrossRef]

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

Sheppard, C. J. R.

C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express 17(5), 3690–3697 (2009).
[CrossRef] [PubMed]

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Shimizu, T.

T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. 160(1-3), 103–108 (1999).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Sidoro, K.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

Siegman, A. E.

Simon, R.

Stabinis, A.

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
[CrossRef]

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

Sudol, R. J.

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

Takenaka, T.

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

C. Tamm and C. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1034–1038 (1990).
[CrossRef]

C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38(11), 5960–5963 (1988).
[CrossRef] [PubMed]

Tervonen, E.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Turunen, J.

van Dijk, T.

Visser, T. D.

Wang, F.

Wang, P.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Wang, S.

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[CrossRef]

Wang, W.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Wang, Y.

Wang, Z.

Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009).
[CrossRef]

Weiss, C.

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

Wolf, E.

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]

Xie, Y.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Yamanaka, C.

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]

Yokota, M.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[CrossRef]

Zauderer, E.

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

Zhang, X.

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009).
[CrossRef]

Zhao, C.

Zhao, D.

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef]

Zhou, G.

G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 (2009).
[CrossRef] [PubMed]

G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008).
[CrossRef]

Zhu, S. Y.

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]

Zubairy, M. S.

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. B (3)

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[CrossRef]

Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006).
[CrossRef]

Eur. Phys. J. D (1)

Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

J. Mod. Opt. (3)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995).
[CrossRef]

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973).
[CrossRef]

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

J. Opt. Soc. Am. A (13)

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985).
[CrossRef]

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986).
[CrossRef]

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]

M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001).
[CrossRef]

R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001).
[CrossRef]

G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
[CrossRef]

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008).
[CrossRef]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef]

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]

Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
[CrossRef]

X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (12)

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
[CrossRef]

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]

P. De. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[CrossRef]

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

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 (2009).
[CrossRef]

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
[CrossRef]

X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008).
[CrossRef]

T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. 160(1-3), 103–108 (1999).
[CrossRef]

Opt. Eng. (3)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[CrossRef]

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[CrossRef]

G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

Opt. Express (7)

Opt. Laser Technol. (2)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008).
[CrossRef]

Opt. Lasers Eng. (1)

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Opt. Lett. (11)

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

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(12), 1389–1391 (2008).
[CrossRef] [PubMed]

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
[CrossRef] [PubMed]

D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. 30(22), 3039–3041 (2005).
[CrossRef] [PubMed]

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]

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 (2007).
[CrossRef] [PubMed]

S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002).
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004).
[CrossRef] [PubMed]

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]

R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000).
[CrossRef] [PubMed]

Phys. Rev. A (3)

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
[CrossRef] [PubMed]

C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38(11), 5960–5963 (1988).
[CrossRef] [PubMed]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

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]

Phys. Rev. Lett. (3)

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

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]

Other (4)

A. E. Siegman, Lasers (Mill Valley, CA: University Science Books, 1986)

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

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

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

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Figures (4)

Fig. 1
Fig. 1

Normalized intensity distribution (cross line v = 0) of a partially coherent standard LG beam for different values of the initial coherence width σ g with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

Fig. 4
Fig. 4

Effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l and coherence width σ g

Fig. 2
Fig. 2

Normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σ g with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

Fig. 3
Fig. 3

Effective beam sizes of partially coherent standard LG beam and elegant LG beam versus the propagation distance z in free space for different values of the initial coherence width

Equations (19)

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E p l ( r , φ ; 0 ) = ( q r ω 0 ) l L p l ( q 2 r 2 ω 0 2 ) exp ( r 2 ω 0 2 ) exp ( i l φ ) ,
e i l φ ρ l L p l ( ρ 2 ) = ( 1 ) p 2 2 p + l p ! m = 0 p n = 0 l i n ( p m ) ( l n ) H 2 m + l n ( x ) H 2 p 2 m + n ( y ) ,
E p l ( x , y ; 0 ) = ( 1 ) p 2 2 p + l p ! m = 0 p s = 0 l i s ( p m ) ( l s ) H 2 m + l s ( q x ω 0 ) H 2 p 2 m + s ( q y ω 0 ) exp ( x 2 + y 2 ω 0 2 ) .
W ( x 1 , y 1 , x 2 , y 2 ; 0 ) = I ( x 1 , y 1 ; 0 ) I ( x 2 , y 2 ; 0 ) g ( x 1 x 2 ; y 1 y 2 ; 0 ) ,
g ( x 1 x 2 ; y 1 y 2 ; 0 ) = exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ] ,
I ( x , y ; 0 ) = | ( 1 ) p 2 2 p + l p ! m = 0 p s = 0 l i s ( p m ) ( l s ) H 2 m + l s ( q x ω 0 ) H 2 p 2 m + s ( q y ω 0 ) exp ( x 2 + y 2 ω 0 2 ) | 2 .
W ( x 1 , y 1 , x 2 , y 2 ; 0 ) = 1 2 4 p + 2 l ( p ! ) 2 m = 0 p n = 0 l h = 0 p s = 0 l ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )                                 × H 2 m + l n ( q x 1 ω 0 ) H 2 h + l s ( q x 2 ω 0 ) H 2 p 2 m + n ( q y 1 ω 0 ) H 2 p 2 h + s ( q y 2 ω 0 )                                 × exp ( x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ) exp ( ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ) .
W ( u 1 , v 1 , u 2 , v 2 , z ) = ( 1 λ | B | ) 2 W ( x 1 , y 1 , x 2 , y 2 ; 0 )            × exp [ i k 2 B * ( A * x 1 2 2 x 1 u 1 + D * u 1 2 ) i k 2 B * ( A * y 1 2 2 y 1 v 1 + D * v 1 2 ) ]            × exp [ i k 2 B ( A x 2 2 2 x 2 u 2 + D u 2 2 ) + i k 2 B ( A y 2 2 2 y 2 v 2 + D y 2 2 ) ] d x 1 d x 2 d y 1 d y 2 ,
W ( u 1 , v 1 , u 2 , v 2 , z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 m = 0 p n = 0 l h = 0 p s = 0 l ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )                             H 2 m + l n ( q x 1 ω 0 ) H 2 h + l s ( q x 2 ω 0 ) exp ( x 1 2 + x 2 2 ω 0 2 ) exp ( ( x 1 x 2 ) 2 2 σ g 2 )                               × exp [ i k 2 B * ( A * x 1 2 2 x 1 u 1 + D * u 1 2 ) + i k 2 B ( A x 2 2 2 x 2 u 2 + D u 2 2 ) ] d x 1 d x 2                           H 2 p 2 m + n ( q y 1 ω 0 ) H 2 p 2 h + s ( q y 2 ω 0 ) exp ( y 1 2 + y 2 2 ω 0 2 ) exp ( ( y 1 y 2 ) 2 2 σ g 2 )                             × exp [ i k 2 B * ( A * y 1 2 2 y 1 v 1 + D * v 1 2 ) + i k 2 B ( A y 2 2 2 y 2 v 2 + D y 2 2 ) ] d y 1 d y 2 .
W ( u 1 , u 2 , v 1 , v 2 , z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 2 M 1 M 2 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l                          × exp [ i k D * 2 B * u 1 2 + i k D 2 B u 2 2 ) ] exp ( k 2 u 2 2 4 M 1 B 2 ) exp [ k 2 4 M 2 ( u 1 B * u 2 2 M 1 σ g 2 B ) 2 ]                          × exp [ i k D * 2 B * v 1 2 + i k D 2 B v 2 2 ) ] exp ( k 2 v 2 2 4 M 1 B 2 ) exp [ k 2 4 M 2 ( v 1 B * v 2 2 M 1 σ g 2 B ) 2 ]                          × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] ( i n ) * i s ( p m ) ( l n )                          × ( p h ) ( l s ) ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) !                          × d ! c 2 ! ( d 2 c 2 ) ! ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 i ) 2 c 1 + 2 c 2 + 2 e 1 + 2 e 2 d d 1 2 p l                          × ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1 ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2                          × H 2 h + l s d ( i q k u 2 2 B M 1 2 ω 0 2 q 2 M 1 ) H 2 m + l n + d 2 c 1 2 c 2 ( k u 2 4 M 1 M 2 σ g 2 B k u 1 2 M 2 B * )                          × H 2 p 2 h + s d 1 ( i q k v 2 2 B M 1 2 ω 0 2 q 2 M 1 ) H 2 p 2 m + n + d 1 2 e 1 2 e 2 ( k v 2 4 M 1 M 2 σ g 2 B k v 1 2 M 2 B * ) ,
exp [ ( x y ) 2 ] H n ( a x ) d x = π ( 1 a 2 ) n 2 H n ( a y ( 1 a 2 ) 1 / 2 )
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) ,
H n ( x + y ) = 1 2 n / 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
H n ( x 1 ) = m = 0 [ n / 2 ] ( 1 ) m n ! m ! ( n 2 m ) ! ( 2 x 1 ) n 2 m .
W s z ( z ) = 2 s 2 I ( x , y , z ) d x d y I ( x , y , z ) d x d y       ( s = x , y ) .
W x z = W y z = A 1 ( z ) A 2 ( z )
A 1 ( z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 3 M 1 M 2 M 3 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l           × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] f 1 = 0 [ ( 2 h + l s d ) / 2 ]           × f 2 = 0 [ ( 2 m + l n + d 2 c 1 2 c 2 ) / 2 ] g 1 = 0 [ ( 2 p 2 h + s d 1 ) / 2 ] g 2 = 0 [ ( 2 p 2 m + n + d 1 2 e 1 2 c 2 ) / 2 ] ( i n ) * i s ( p m ) ( l n ) ( p h )           × ( l s ) ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 + f 1 + f 2 + g 1 + g 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) ! d ! c 2 ! ( d 2 c 2 ) !           × ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 m + l n + d 2 c 1 2 c 2 ) ! f 2 ! ( 2 m + l n + d 2 c 1 2 c 2 2 f 2 ) !           × ( 2 h + l s d ) ! f 1 ! ( 2 h + l s d 2 f 1 ) ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 ) ! g 2 ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ) ! ( 2 p 2 h + s d 1 ) ! g 1 ! ( 2 p 2 h + s d 1 2 g 1 ) !           × ( 2 i ) 4 c 1 + 4 c 2 + 4 e 1 + 4 e 2 2 f 1 2 f 2 2 g 1 2 g 2 d d 1 6 p 3 l 2 ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1           × ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 h + l s d 2 f 1           × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 m + l n + d 2 c 1 2 c 2 2 f 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 p 2 h + s d 1 2 g 1           × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ( 1 M 3 ) 2 p + l c 1 c 2 f 1 f 2 e 1 e 2 g 1 g 2 + 1           × H 2 h + 2 m + 2 l s n 2 c 1 2 c 2 2 f 1 2 f 2 + 2 ( 0 ) H 4 p 2 h 2 m + s + n 2 e 1 2 e 2 2 g 1 2 g 2 ( 0 ) ,                                        
A 2 ( z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 3 M 1 M 2 M 3 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l                  × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] f 1 = 0 [ ( 2 h + l s d ) / 2 ]              × f 2 = 0 [ ( 2 m + l n + d 2 c 1 2 c 2 ) / 2 ] g 1 = 0 [ ( 2 p 2 h + s d 1 ) / 2 ] g 2 = 0 [ ( 2 p 2 m + n + d 1 2 e 1 2 c 2 ) / 2 ] ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )             × ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 + f 1 + f 2 + g 1 + g 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) ! d ! c 2 ! ( d 2 c 2 ) !             × ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 m + l n + d 2 c 1 2 c 2 ) ! f 2 ! ( 2 m + l n + d 2 c 1 2 c 2 2 f 2 ) !             × ( 2 h + l s d ) ! f 1 ! ( 2 h + l s d 2 f 1 ) ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 ) ! g 2 ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ) ! ( 2 p 2 h + s d 1 ) ! g 1 ! ( 2 p 2 h + s d 1 2 g 1 ) !             × ( 2 i ) 4 c 1 + 4 c 2 + 4 e 1 + 4 e 2 + 2 g 1 + 2 g 2 + 2 f 1 + 2 f 2 d d 1 6 p 3 l ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1             × ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 h + l s d 2 f 1            × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 m + l n + d 2 c 1 2 c 2 2 f 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 p 2 h + s d 1 2 g 1            × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ( 1 M 3 ) 2 p + l e 1 e 2 g 1 g 2 c 1 c 2 f 1 f 2            × H 2 h + 2 m + 2 l s n 2 c 1 2 c 2 2 f 1 2 f 2 ( 0 ) H 4 p 2 h 2 m + s + n 2 e 1 2 e 2 2 g 1 2 g 2 ( 0 ) ,                                              
( A B C D ) = ( 1 z 0 1 ) .

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