Abstract

Laguerre-Gaussian correlated Schell-model (LGCSM) vortex beam is introduced as an extension of LGCSM beam which was proposed [Opt. Lett. 38, 91 (2013) Opt. Lett. 38, 1814 (2013)] just recently. Explicit formula for a LGCSM vortex beam propagating through a stigmatic ABCD optical system is derived, and the propagation properties of such beam in free space and the focusing properties of such beam are studied numerically. Furthermore, we carry out experimental generation of a LGCSM vortex beam, and studied its focusing properties. It is found that the propagation and focusing properties of a LGCSM vortex beam are different from that of a LGCSM beam, and we can shape the beam profile of a LGCSM vortex at the focal plane (or in the far field) by varying its initial spatial coherence. Our experimental results are consistent with the theoretical predictions, and our results will be useful for particle trapping.

© 2014 Optical Society of America

1. Introduction

In the past decades, partially coherent beams with conventional Schell-model correlation functions (i.e., degrees of coherence have position-independent profiles) have been studied extensively both in theory and in experiment and has been applied widely [14]. Since Gori and collaborators discussed the sufficient condition for devising the genuine correlation function of a scalar or electromagnetic partially coherent beam [5, 6], more and more attention is being paid to partially coherent beams with nonconventional correlation functions. A variety of partially coherent beams with special correlation functions, such as beams with locally varying spatial coherence [7], special correlated partially coherent vector beam [8], nonuniformly correlated Gaussian Schell-model beam [912], multi-Gaussian Schell-model beam [1316], cosine-Gaussian Schell-model beam [1719], and Laguerre-Gaussian correlated Schell-model (also named Laguerre-Gaussian Schell-model beam) beam [2023], have been introduced. Those partially coherent beams with special correlation functions have been found to exhibit some extraordinary propagation characteristics, such as far-field flat-topped and ring intensity profile formation, self-focusing effect, and a lateral shift of the intensity maximum. Laguerre-Gaussian correlated Schell-model (LGCSM) beam was first introduced in [20], and it was found that the far field intensity of a LGCSM beam has a ring (i.e., dark hollow) intensity profile although it has the same intensity distribution with that of a Gaussian Schell-model beam in the source plane. In [21], we reported experimental generation of a LGCSM beam for the first time. The propagation properties of a LGCSM beam in turbulent atmosphere were studied in [22, 23], and it was revealed that a LGCSM beam has advantage over a Gaussian Schell-model (GSM) beam for reducing the turbulence-induced degradation, thus it has potential application in free-space optical communications.

On the other hand, it is well known that light beams with a vortex phase named vortex beams have been applied in optical trapping, optical tweezers, quantum information processing and so on [2428]. It is found that each photon of the vortex beam with a phase term exp(ilθ)carries an orbital angular momentum of l with l being the topological charge [24]. In the past several years, more and more attention is being paid to partially coherent vortex beams both in theory and in experiment [2934]. In [33], we carried out experimental study of the focusing properties of GSM vortex beam. The conventional method for determining the topological charge of a vortex beam is invalid for a partially coherent vortex beam, and we proposed a new method for determining the topological charge of a partially coherent vortex beam in [34]. More recently, we carried out experimental measurement the scintillation index of a GSM vortex beam propagating through thermally induced turbulence [35], and we have found that a GSM vortex beam has appreciably smaller scintillation than a GSM beam, which will be useful in free-space optical communication. All above mentioned partially coherent vortex beams have the conventional Schell-model correlation functions. In this paper, our aim is to introduce a partially coherent vortex beam with nonconventional correlation function, named LGCSM vortex beam, as a natural extension of the recently introduced LGCSM beam. We derive the explicit formula for the LGCSM vortex beam propagating through a stigmatic ABCD optical system, and study its propagation properties both numerically and experimentally. Some useful results are found.

2. Laguerre-Gaussian correlated Schell-model vortex beam: Theory

In the space-time domain, the statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function. For a LGCSM beam, its mutual coherence function is defined as [20]

Γ(r1,r2)=G0exp[r12+r224σ02(r1r2)22δ02]Ln0[(r1r2)22δ02],
whereG0 is a constant which has dimension of an optical intensity, r1(x1,y1) and r2(x2,y2) are two arbitrary transverse position vectors at z = 0, σ0 and δ0 are the transverse beam width and the transverse coherence width of the LGCSM beam, respectively, Ln0 denotes the Laguerre polynomial of mode order n and 0. The degree of coherence of the LGCSM beam at z = 0 is given as
μ(r1,r2)=Γ(r1,r2)Γ(r1,r1)Γ(r2,r2)=exp[(r1r2)22δ02]Ln0[(r1r2)22δ02].
From Eq. (2), one finds that the degree of coherence of the LGCSM beam doesn’t satisfy Gaussian distribution. Under the condition of n = 0, Eq. (1) reduces to the expression for the mutual coherence function of a GSM beam [1,2].

If a LGCSM beam passes through a spiral phase plate with transmission function T(φ)=exp(imφ) where m denotes the topological charge and φ denotes the azimuthal coordinate (i.e., T(x,y)=exp[imarctan(y/x)] in the Cartesian coordinates), the transmitted beam will carry a vortex phase and its mutual coherence function can be expressed as

Γ(r1,r2)=G0exp[r12+r224σ02(r1r2)22δ02]Ln0[(r1r2)22δ02]exp(imφ1+imφ2).
We call the transmitted beam as LGCSM vortex beam. Due to the vortex phase, the LGCSM vortex beam exhibits unique propagation properties as shown later. Under the condition of n = 0, Eq. (3) reduces to the expression for the mutual coherence function of a GSM vortex beam [33].

Within the validity of the paraxial approximation, the propagation of the mutual coherence function of the LGSM vortex beam through a stigmatic ABCD optical system can be studied with the help of the following generalized Collins formula [36, 37]

Γ(ρ1,ρ2)=1(λB)2exp[ikD2B(ρ12ρ22)]×Γ(r1,r2)exp[ikA2B(r12r22)+ikB(r1ρ1r2ρ2)]d2r1d2r2,
where A, B, C and D are the elements of a transfer matrix for the optical system, k=2π/λ is the wavenumber with λ being the wavelength.

For the convenience of integration, we introduce the following “sum” and “difference” coordinates:

rs=r1+r22,Δr=r1r2,
ρs=ρ1+ρ22,Δρ=ρ1ρ2.
Substituting Eqs. (3), (5) and (6) into Eq. (4), we obtain
Γ(ρs,Δρ)=G0(λB)2exp[ikDBρsΔρ]×P+*(rs+Δr2)P(rsΔr2)γ(Δr)exp[ikB(Δrρs+rsΔρ)]d2rsd2Δr,
where
P+*(rs+Δr2)=exp[(14σ02ikA2B)(rs+Δr2)2]exp(imφ+),
P(rsΔr2)=exp[(14σ02+ikA2B)(rsΔr2)2]exp(imφ),
γ(Δr)=exp[Δr22δ02]Ln0[Δr22δ02].
with φ±=arctan[(ys±Δy/2)/(xs±Δx/2)].

P+*(rs+Δr/2) andP(rsΔr/2) can be expressed in terms of their Fourier transforms P+*˜(u1/λB), P˜(u2/λB) as follows

P+*(rs+Δr2)=1(λB)2P+*˜(u1λB)exp(ikB(rs+Δr2)u1)d2u1,
P(rsΔr2)=1(λB)2P˜(u2λB)exp(ikB(rsΔr2)u2)d2u2.

Substituting Eqs. (11) and (12) into Eq. (7), after some integration, the mutual coherence function of the LGCSM vortex beam in the output plane is obtained as

Γ(ρs,Δρ)=G0(λB)4exp[ikBρsΔρ]×P˜+*(u1λB)P˜(u1+ΔρλB)γ˜(u1+ρs+Δρ/2λB)d2u1,
where γ˜(·)represents the Fourier transform of γ(·), i.e.,

γ˜(u)=γ(r)exp(2πiur)d2r.

The average intensity of the LGCSM vortex beam in the output plane is obtained as

I(ρ)=Γ(ρs,Δρ)ρ1=ρ2=G0(λB)4P˜+*(u1λB)P˜(u1λB)γ˜(u1+ρλB)d2u1,
with
γ˜(u1+ρλB)=4πδ0222n1n![8π2δ02(u1+ρλB)2]nexp[2π2δ02(u1+ρλB)2],
P˜+*(u1λB)P˜(u1λB)=π5|σ(B)|6u124(λB)2×|exp[[σ(B)π]22(λB)2u12][I12m12([σ(B)π]22(λB)2u12)I12m+12([σ(B)π]22(λB)2u12)]|2.
Here σ(B)=(1/4σ02ikA/2B)1/2, and Iα is the modified Bessel function of order α,u1 is the radial coordinate.

Under the condition of n=0, Eq. (15) reduces to the expression for the average intensity of the GSM vortex beam in the output plane [33].

Under the condition of m = 0, Eq. (15) reduces to the following expression for the average intensity of a LGCSM beam in the output plane

I(ρ,z)=2n1G0(kB)2σ*2(B)σ2(B)δ02n+2[(σ*2(B)+σ2(B)+2δ02)]n1×exp[(k2B)22δ02(σ*2(B)+σ2(B))(σ*2(B)+σ2(B)+2δ02)ρ2]Ln[(k2B)2(σ*2(B)+σ2(B))2(σ*2(B)+σ2(B)+2δ02)ρ2].

First, we study the propagation properties of a LGCSM vortex beam and a LGCSM beam in free space, comparatively. The transfer matrix for free space of distance z reads as

(ABCD)=(1z01).

Applying Eqs. (15) and (19), we calculate in Fig. 1 the normalized intensity distribution (cross line ρy=0) of a LGCSM vortex beam at several propagation distances in free space for different values of the initial coherence width δ0 with n=1, m=3, σ0=1mm and λ=632.8nm. For the convenience of comparison, applying Eqs. (18) and (19), we calculate in Fig. 2 the normalized intensity distribution (cross line ρy=0) of a LGCSM beam at several propagation distances in free space for different values of the initial coherence width δ0 with n=1, σ0=1mm and λ=632.8nm. One finds from Figs. 1 and 2 that both the LGCSM beam and the LGCSM vortex beam exhibit interesting propagation properties. When the initial coherence width is small, the intensity distribution of a LGCSM beam in the far field has a dark hollow beam profile [see Figs. 2(a-1)–2(d-1)], which is consistent with the result reported in [20]. The evolution properties of the intensity distribution of a LGCSM vortex beam with low coherence on propagation in free space is similar to that of a LGCSM beam [see Figs. 1(a-1)–1(d-1)], and it also has a dark hollow beam profile in the far field. With the increase of the initial coherence width, the far-field intensity distribution of a LGCSM beam or a LGCSM vortex beam varies. For a LGCSM beam, the far-field dark hollow beam profile disappears gradually and finally the far-field Gaussian beam profile is formed as the initial coherence width increases. For a LGCSM vortex beam, the far-field dark hollow beam profile also disappears gradually as the initial coherence width increases, while the far-field dark hollow beam profile appears again when the initial coherence width is large [see Fig. 1(d-5)], which is quite different from that of a LGCSM beam. The above interesting phenomenon can be explained in the following way. The effect of spatial correlation function on the evolution properties of a partially coherent beam plays a dominate role only when the initial coherence is low, and its effect can be neglected when the initial coherence is high. For a LGCSM beam with low coherence, its evolution properties are mainly determined by its Laguerre-Gaussian correlation function, and its far-field intensity has a dark hollow beam profile due to the effect of the Laguerre-Gaussian correlation function. For a LGCSM beam with high coherence, the effect of Laguerre-Gaussian correlation function can be neglected, and the evolution properties of a LGCSM beam is similar to that of a GSM beam. For a LGCSM vortex beam, its evolution properties are determined by the Laguerre-Gaussian correlation function and the vortex phase together. When the initial coherence is low, the Laguerre-Gaussian function plays a dominant role and the effect of vortex phase can be neglected, the far-field intensity of the LGCSM vortex beam has a dark hollow beam due to the effect of the Laguerre-Gaussian correlation function. When the initial coherence is high, the effect of the vortex phase plays a dominant role and the effect of Laguerre-Gaussian correlation function can be neglected, the far-field intensity of the LGCSM vortex beam has a dark hollow beam due to the effect of the vortex phase, which induces a phase singularity in the beam center.

 

Fig. 1 Normalized intensity distribution (cross lineρy=0) of a LGCSM vortex beam with n = 1 and m = 3 at several propagation distances in free space for different values of the initial coherence widthδ0.

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Fig. 2 Normalized intensity (cross lineρy=0) of a LGCSM beam with n = 1 at several propagation distances in free space for different values of the initial coherence widthδ0.

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Now we study the focusing properties of a LGCSM vortex beam and a LGCSM beam in free space, comparatively. Assume that a thin lens with focal length f is located at the source plane (z = 0), and the output plane is located at the geometrical focal plane, then the transfer matrix between the source plane and output plane reads as

(ABCD)=(1f01)(101/f1)=(0f1/f1).

Applying Eqs. (15), (18) and (20), we calculate in Fig. 3 the normalized intensity distribution (cross line ρy=0) of a LGCSM vortex beam at the geometrical focal plane for different values of the initial coherence width δ0 with n=1 and m=3, and in Fig. 4 the normalized intensity distribution (cross line ρy=0) of a LGCSM beam at the geometrical focal plane for different values of the initial coherence width δ0 with n=1. The other parameters are chosen as σ0=1mm, λ=632.8nm and f = 40cm. One finds from Figs. 3 and 4 that the dependence of the focused intensity of a LGCSM vortex beam or LGCSM beam on the initial coherence width is similar to the dependence of the far-field intensity of such beam on initial coherence width. In fact the intensity profile of a beam in the in the focal plane of a converging lens is necessarily the same (with suitable scaling factors) as the intensity profile of the beam in the far field. When the initial coherence width is small, the intensity of a LGCSM beam or LGCSM vortex beam at the geometrical focal plane has a dark hollow beam profile. With the increase of the initial coherence width, the dark hollow beam profile of a LGCSM beam or LGCSM vortex beam at the geometrical focal plane disappears gradually. When the initial coherence is large, at the geometrical focal plane, the intensity of a LGCSM beam has a Gaussian beam profile, while the intensity of a LGCSM vortex beam has a dark hollow beam profile. For suitable values of the initial coherence width, the intensity of a LGCSM beam or a LGCSM vortex exhibits flat-topped beam profile. Thus, modulating the spatial coherence of a LGCSM vortex beam or a LGCSM beam provides one way for shaping its focused beam profile, which will be useful for particle trapping, where a focused Gaussian or flat-topped beam spot is used to trap a Rayleigh particle whose refractive index is larger than that of the ambient and a dark hollow beam spot is used to trap a Rayleigh particle whose refractive index is smaller than that of the ambient [3840].

 

Fig. 3 Normalized intensity (cross lineρy=0) of a focused LGCSM vortex beam with n = 1 and m = 3 at the focal plane for different values of the initial coherence widthδ0.

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Fig. 4 Normalized intensity (cross lineρy=0) of a focused LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence widthδ0.

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3. Laguerre-Gaussian correlated Schell-model vortex beam: experiment

In this section, we report experimental generation of a LGCSM vortex beam with controllable spatial coherence, and carried out experimental measurement of its focusing properties.

In this paper, we generate LGCSM vortex beam through conversion of a LGCSM beam with the help of a spiral phase plate. In Ref [21], it is shown that a LGCSM beam of mode order n can be formed when an incoherent beam whose intensity distribution has a dark hollow beam profile and is expressed as I(v)=(v2/ω02)nexp(2v2/ω02)passes through free space with length f, a thin lens with focal length f and a Gaussian amplitude filter (GAF), and the spatial coherence width of the generated LGCSM beam can be approximated as δ0=λf/πw0. Part 1 of Fig. 5 shows our experimental setup for generating a LGCSM vortex beam. A beam emitted from the He-Ne laser (λ=632.8nm) passes through a beam expander, then it goes towards a spatial light modulator (SLM, Holoeye LC2002), which acts as phase grating designed by the method of computer-generated holograms. Here the pattern of the phase grating for generating a dark hollow beam with n = 1 is shown as inset in Fig. 5. The first order of the beam from the SLM is a dark hollow beam with n = 1 and is selected out by a circular aperture. After passing through a thin lens L1, the generated dark hollow beam illuminates a rotating ground-glass disk (RGGD), producing an incoherent beam with dark hollow beam profile. The beam from the RGGD can be regarded as a spatially incoherent beam if the diameter of the beam spot on the RGGD is larger than the inhomogeneity scale of the ground glass [41], and this condition is satisfied in our case. After passing through free space with length f2, the thin lens L2, and the GAF, the generated incoherent dark hollow beam becomes a LGCSM beam with n = 1 [21]. After passing through a spiral phase plate (SPP) with topological charge m = 3, the generated LGCSM beam becomes a LGCSM vortex beam. The SPP just adds a vortex phase to the LGCSM beam, and it doesn’t alter its spatial coherence and its intensity distribution in the source plane, thus the spatial coherence width and the intensity distribution of generated LGCSM vortex beam are the same with those of the generated LGCSM beam. The spatial coherence width of the generated LGCSM beam is modulated by varying the beam spot on the RGGD through varying the distance between the thin lens L1 and the RGGD.

 

Fig. 5 Experimental setup for generating a LGCSM vortex beam and measuring its focused intensity. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BPA, beam profile analyzer; PC1, PC2, personal computers.

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The degree of coherence (i.e., correlation function) and the spatial coherence width of the generated LGCSM beam can be measured by using the method proposed in Ref [42]. As illustrated in [21] and [42], the generated partially coherent beam is split into two distinct imaging optical paths by a 50:50 beam splitter, and the transmitted beam and reflected beam go to two single-photon detectors, respectively. By measuring the fourth-order correlation function between the detectors, we can obtain the distribution of the square of the degree of coherence of the generated beam with the help of the Gaussian moment theorem (i.e., internal relation between second-order and fourth-order correlation function) [1].

Part 2 of Fig. 5 shows our experimental setup of measuring the focused intensity distribution of the generated LGCSM beam. The generated beam first passes through a thin lens L3 with focal lengthf3=40cm, and then arrives at the beam profile analyzer (BPA), which measures the focused intensity. The elements of the transfer matrix between the source plane and the BPA read as

A=0,B=f3,C=1/f3,D=1.

Figure 6 shows our experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated LGCSM beam in the source plane. The solid curve is a result of the theoretical fit. It is clear from Fig. 6 that the intensity distribution of the generated beam in the source plane has a Gaussian profile as expected. Through theoretical fit (solid curve) of the experimental results, we obtain that σ0 is about 1mm. In our experiment, we generate several LGCSM beams and LGCSM vortex beams with different initial coherence widths in order to study the influence of coherence width on the focusing properties, and Fig. 7 shows our experimental results of the square of the modulus of the generated LGCSM beam for different values of the initial coherence width in the source plane. Through theoretical fit of the experimental results, we obtain δ0=0.1mm,0.2mm,0.38mm,0.52mm,0.82mm,1.35mm,2.0mmfor Figs. 7(a)7(g), respectively. With the measured beam parameters and formulae derived in section 2, we can simulate the focusing properties of the generated beam, and compare with the corresponding results.

 

Fig. 6 Experimental results of (a) the intensity distribution and (b) the corresponding cross line (dotted curve) of the generated LGCSM beam with n = 1 in the source plane. The solid curve is a result of the theoretical fit.

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Fig. 7 Experimental results of the square of the modulus of the generated LGCSM beam for different values of the initial coherence width in the source plane. The solid curve is a result of the theoretical fit.

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Figure 8 shows our experimental results of the intensity distribution and the corresponding cross line (ρy=0) of the generated LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence widthδ0. Figure 9 shows our experimental results of the intensity distribution and the corresponding cross line (ρy=0) of the generated LGCSM vortex beam with n = 1 and m = 3 at the geometrical focal plane for different values of the initial coherence widthδ0. For the case ofδ0=Infinity, the rotating round-glass disk completely removed from our experimental setup. For the convenience of comparison, the corresponding numerical results calculated by the formulae derived in section 2 are also shown in Figs. 8 and 9. One finds from Figs. 8 and 9 that the focused intensities of the generated LGCSM beams and LGCSM vortex beams indeed are modulated through varying the initial coherence width as expected by Figs. 3 and 4, and our experimental results agree well with the theoretical predictions.

 

Fig. 8 Experimental results of the intensity distribution and the corresponding cross line (ρy=0) of the generated LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence widthδ0. The solid curve denotes the theoretical results calculated by Eq. (18).

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Fig. 9 Experimental results of the intensity distribution and the corresponding cross line (ρy=0) of the generated LGCSM vortex beam with n = 1 and m = 3 at the geometrical focal plane for different values of the initial coherence widthδ0. The solid curve denotes the theoretical results calculated by Eq. (15).

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4. Summary

We have introduced the theoretical model for a new partially coherent vortex beam with special correlation function named LGCSM vortex beam as an extension of recently introduced LGCSM beam, and we have derived the explicit propagation formulae for such beam propagating through a stigmatic ABCD optical system. The propagation properties of a LGCSM beam and a LGCSM vortex beam have been studied comparatively, and it is found that they exhibit different propagation properties. Furthermore, we have carried out experimental generation of a LGCSM vortex beam through converting a LGCSM beam to such beam by a spiral phase plate, and studied the focusing properties of a LGCSM beam and a LGCSM vortex beam comparatively both in theory and in experiment. We have found that we can shape the intensity distribution of a LGCSM beam and a LGCSM vortex beam through varying its initial coherence width, and our experimental results are consistent with the theoretical results. Our results will be useful for particle trapping, where focused beam spot with special beam profile is required.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant Nos. 11274005, 11104195 and 11374222, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Province under Grant No. 11KJB140007, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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30. W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006). [CrossRef]   [PubMed]  

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

32. 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]   [PubMed]  

33. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011). [CrossRef]   [PubMed]  

34. C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012). [CrossRef]  

35. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef]   [PubMed]  

36. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]  

37. 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]  

38. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef]   [PubMed]  

39. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]  

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

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

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

References

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  4. Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
    [CrossRef]
  5. F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
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  6. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [CrossRef]
  7. L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
    [CrossRef]
  8. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
    [CrossRef]
  9. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [CrossRef] [PubMed]
  10. Z. Tong, O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
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  11. Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
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  12. Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
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  13. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
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  14. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
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  15. S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
    [CrossRef]
  16. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
    [CrossRef]
  17. Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [CrossRef] [PubMed]
  18. Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
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  19. C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [CrossRef] [PubMed]
  20. Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [CrossRef] [PubMed]
  21. F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [CrossRef] [PubMed]
  22. J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
    [CrossRef]
  23. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
    [CrossRef] [PubMed]
  24. J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
    [CrossRef] [PubMed]
  25. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [CrossRef] [PubMed]
  26. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
    [CrossRef] [PubMed]
  27. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
  28. Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
    [CrossRef]
  29. G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
    [CrossRef]
  30. W. Wang, M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
    [CrossRef] [PubMed]
  31. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
    [CrossRef] [PubMed]
  32. T. van Dijk, T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
    [CrossRef] [PubMed]
  33. F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
    [CrossRef] [PubMed]
  34. C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
    [CrossRef]
  35. X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
    [CrossRef] [PubMed]
  36. S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [CrossRef]
  37. Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [CrossRef] [PubMed]
  38. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [CrossRef] [PubMed]
  39. Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
    [CrossRef]
  40. C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
    [CrossRef] [PubMed]
  41. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
    [CrossRef]
  42. F. Wang, Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
    [CrossRef] [PubMed]

2014 (4)

2013 (9)

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

2012 (6)

2011 (4)

2010 (3)

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

2009 (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

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

2007 (2)

2006 (1)

W. Wang, M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (4)

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

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

2002 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1979 (1)

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

1970 (1)

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Bogatyryova, G. V.

Cai, Y.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[CrossRef]

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

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

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

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

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

Cang, J.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Chan, C. T.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Chen, R.

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Collins, S. A.

De Santis, P.

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

Ding, B.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Dong, Y.

Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[CrossRef]

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Du, S.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Fel’de, C. V.

Fleischer, J. W.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Gbur, G.

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[CrossRef]

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

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

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Gu, Y.

Guattari, G.

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

Han, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Jennewein, T.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Korotkova, O.

Lajunen, H.

Liang, C.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Lin, Q.

Lin, Z.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Liu, L.

Liu, X.

Liu, Y.

Mei, Z.

Ng, J.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Palma, C.

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

Pan, J. W.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Polyanskii, P. V.

Ponomarenko, S. A.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Shchepakina, E.

Shen, Y.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Situ, G.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Soskin, M. S.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Suyama, T.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

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[CrossRef] [PubMed]

Tong, Z.

van Dijk, T.

Vaziri, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
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T. van Dijk, T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
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G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
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Waller, L.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

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

Wang, W.

W. Wang, M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[CrossRef] [PubMed]

Weihs, G.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

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Wu, G.

Xiu, P.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

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Yuan, Y.

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Zeilinger, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Zhan, Q.

Zhang, Y.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

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Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

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

Zhu, S.

Appl. Phys. Lett. (1)

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Open Opt. J. (1)

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

Opt. Commun. (3)

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[CrossRef]

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

Opt. Express (4)

Opt. Laser Technol. (2)

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Opt. Lett. (14)

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

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

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

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Phys. Rev. A (3)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Phys. Rev. Lett. (3)

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

W. Wang, M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[CrossRef] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[CrossRef]

Other (2)

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007)

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Figures (9)

Fig. 1
Fig. 1

Normalized intensity distribution (cross line ρ y = 0 ) of a LGCSM vortex beam with n = 1 and m = 3 at several propagation distances in free space for different values of the initial coherence width δ 0 .

Fig. 2
Fig. 2

Normalized intensity (cross line ρ y = 0 ) of a LGCSM beam with n = 1 at several propagation distances in free space for different values of the initial coherence width δ 0 .

Fig. 3
Fig. 3

Normalized intensity (cross line ρ y = 0 ) of a focused LGCSM vortex beam with n = 1 and m = 3 at the focal plane for different values of the initial coherence width δ 0 .

Fig. 4
Fig. 4

Normalized intensity (cross line ρ y = 0 ) of a focused LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence width δ 0 .

Fig. 5
Fig. 5

Experimental setup for generating a LGCSM vortex beam and measuring its focused intensity. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BPA, beam profile analyzer; PC1, PC2, personal computers.

Fig. 6
Fig. 6

Experimental results of (a) the intensity distribution and (b) the corresponding cross line (dotted curve) of the generated LGCSM beam with n = 1 in the source plane. The solid curve is a result of the theoretical fit.

Fig. 7
Fig. 7

Experimental results of the square of the modulus of the generated LGCSM beam for different values of the initial coherence width in the source plane. The solid curve is a result of the theoretical fit.

Fig. 8
Fig. 8

Experimental results of the intensity distribution and the corresponding cross line ( ρ y = 0 ) of the generated LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence width δ 0 . The solid curve denotes the theoretical results calculated by Eq. (18).

Fig. 9
Fig. 9

Experimental results of the intensity distribution and the corresponding cross line ( ρ y = 0 ) of the generated LGCSM vortex beam with n = 1 and m = 3 at the geometrical focal plane for different values of the initial coherence width δ 0 . The solid curve denotes the theoretical results calculated by Eq. (15).

Equations (21)

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Γ ( r 1 , r 2 ) = G 0 exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] ,
μ ( r 1 , r 2 ) = Γ ( r 1 , r 2 ) Γ ( r 1 , r 1 ) Γ ( r 2 , r 2 ) = exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] .
Γ ( r 1 , r 2 ) = G 0 exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] exp ( i m φ 1 + i m φ 2 ) .
Γ ( ρ 1 , ρ 2 ) = 1 ( λ B ) 2 exp [ i k D 2 B ( ρ 1 2 ρ 2 2 ) ] × Γ ( r 1 , r 2 ) exp [ i k A 2 B ( r 1 2 r 2 2 ) + i k B ( r 1 ρ 1 r 2 ρ 2 ) ] d 2 r 1 d 2 r 2 ,
r s = r 1 + r 2 2 , Δ r = r 1 r 2 ,
ρ s = ρ 1 + ρ 2 2 , Δ ρ = ρ 1 ρ 2 .
Γ ( ρ s , Δ ρ ) = G 0 ( λ B ) 2 exp [ i k D B ρ s Δ ρ ] × P + * ( r s + Δ r 2 ) P ( r s Δ r 2 ) γ ( Δ r ) exp [ i k B ( Δ r ρ s + r s Δ ρ ) ] d 2 r s d 2 Δ r ,
P + * ( r s + Δ r 2 ) = exp [ ( 1 4 σ 0 2 i k A 2 B ) ( r s + Δ r 2 ) 2 ] exp ( i m φ + ) ,
P ( r s Δ r 2 ) = exp [ ( 1 4 σ 0 2 + i k A 2 B ) ( r s Δ r 2 ) 2 ] exp ( i m φ ) ,
γ ( Δ r ) = exp [ Δ r 2 2 δ 0 2 ] L n 0 [ Δ r 2 2 δ 0 2 ] .
P + * ( r s + Δ r 2 ) = 1 ( λ B ) 2 P + * ˜ ( u 1 λ B ) exp ( i k B ( r s + Δ r 2 ) u 1 ) d 2 u 1 ,
P ( r s Δ r 2 ) = 1 ( λ B ) 2 P ˜ ( u 2 λ B ) exp ( i k B ( r s Δ r 2 ) u 2 ) d 2 u 2 .
Γ ( ρ s , Δ ρ ) = G 0 ( λ B ) 4 exp [ i k B ρ s Δ ρ ] × P ˜ + * ( u 1 λ B ) P ˜ ( u 1 + Δ ρ λ B ) γ ˜ ( u 1 + ρ s + Δ ρ / 2 λ B ) d 2 u 1 ,
γ ˜ ( u ) = γ ( r ) exp ( 2 π i u r ) d 2 r .
I ( ρ ) = Γ ( ρ s , Δ ρ ) ρ 1 = ρ 2 = G 0 ( λ B ) 4 P ˜ + * ( u 1 λ B ) P ˜ ( u 1 λ B ) γ ˜ ( u 1 + ρ λ B ) d 2 u 1 ,
γ ˜ ( u 1 + ρ λ B ) = 4 π δ 0 2 2 2 n 1 n ! [ 8 π 2 δ 0 2 ( u 1 + ρ λ B ) 2 ] n exp [ 2 π 2 δ 0 2 ( u 1 + ρ λ B ) 2 ] ,
P ˜ + * ( u 1 λ B ) P ˜ ( u 1 λ B ) = π 5 | σ ( B ) | 6 u 1 2 4 ( λ B ) 2 × | exp [ [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ] [ I 1 2 m 1 2 ( [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ) I 1 2 m + 1 2 ( [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ) ] | 2 .
I ( ρ , z ) = 2 n 1 G 0 ( k B ) 2 σ * 2 ( B ) σ 2 ( B ) δ 0 2 n + 2 [ ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ] n 1 × exp [ ( k 2 B ) 2 2 δ 0 2 ( σ * 2 ( B ) + σ 2 ( B ) ) ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ρ 2 ] L n [ ( k 2 B ) 2 ( σ * 2 ( B ) + σ 2 ( B ) ) 2 ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ρ 2 ] .
( A B C D ) = ( 1 z 0 1 ) .
( A B C D ) = ( 1 f 0 1 ) ( 1 0 1 / f 1 ) = ( 0 f 1 / f 1 ) .
A = 0 , B = f 3 , C = 1 / f 3 , D = 1.

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