Abstract

In this paper, we introduce a new kind of vector multi-Gaussian Schell-model (MGSM) beam named radially polarized MGSM beam as an extension of recently introduced scalar MGSM beam [Opt. Lett. 37, 2970 (2012)], and we obtain the realizability conditions of such beam. The tight focusing properties of a radially polarized MGSM beam passing through a high numerical aperture objective lens are investigated numerically based on the vectorial diffraction theory. It is interesting to find that the intensity distributions of both transverse and longitudinal fields near the focus display flat-top beam profiles due to the special correlation functions of the incident beam. Our results clearly show that engineering the structures of the correlation functions of a radially polarized partially coherent beam provides a novel way for shaping both transverse and longitudinal fields distributions, which have potential applications in particle acceleration and laser machining. Finally, we report experimental generation of a radially polarized MGSM beam with the help of a spatial light modulator and a radial polarization converter.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Study of tight focusing properties of a vector beam passing through a high numerical aperture objective lens is an important research topic in several branches of optics, and the polarization properties of the incident beam has a strong influence on the distributions of the three-dimensional electric and magnetic fields near the focal region [1–4]. Radially polarized beam and azimuthal polarized beam are typical kinds of vector beams with spatially non-uniform states of polarization. When a radially polarized beam is tightly focused by a high numerical aperture (NA) objective lens, a strong longitudinal electric field will be produced near the focus, and the focused beam spot size is much smaller than that of a focused linearly polarized beam. These unique features are useful for imaging of molecules, mapping the orientation of a single molecule, trapping metallic Rayleigh particles, particle acceleration, microscopy, material processing, high-resolution metrology, lithography and free-space optical communications [4–13]. When an azimuthally polarized beam is tightly focused, a strong magnetic field on the optical axis will be generated but the electric field exhibits a doughnut profile with pure azimuthal polarization near the focal region, which is useful for atom guiding [1, 4] and enhancement of Fano resonance in a gold spilt-ring resonator nanoarc structure [14]. By engineering polarization proportion properties of the incident beam, one can produce electric/magnetic field distributions with special patterns near the focal region, which are useful in selectively switching on and off electric/magnetic multipole resonances in single dielectric nanoparticles [15]. In addition, other interesting tightly focused field patterns such as multiple focal spots, needle-like focus and optical chain also can be achieved [16–18]. Besides the polarization, the phase also affects the electric and magnetic field distributions near the focus. Through manipulating the polarization and phase of the incident beam simultaneously, various special focused field patterns can be achieved [19–23]. For example, Wang et. al derived a purely longitudinal electric field by focusing a radially polarized beam with phase modulation [19]. Yuan et. al reported generation of a nondiffracting transversally polarized light by focusing an azimuthally polarized beam through a multibelt spiral phase hologram [20]. As one of the intrinsic attributes, spatial coherence plays an important role in determining the beam’s focusing properties and the performance in imaging system [24, 25]. The effects of spatial coherence on the intensity distribution and the polarization properties of the tightly focused partially coherent beams were investigated in [26–30], and it was found that the transverse field distribution can be shaped by varying the initial coherence width. However, the correlation functions (i.e., degrees of coherence) of the partially coherent beams in previous literatures on tight focusing are confined to Gaussian functions.

In recent years, partially coherent beams with prescribed correlation functions have been a subject of intense investigation since Gori and collaborators discussed the sufficient conditions for devising physically realizable correlation functions [31–56]. By manipulating the correlation functions of the partially coherent beams in the source plane, one can obtain any pre-established beam profile in the far field, which are useful in material processing, optical trapping, free-space optical communications, optical imaging, and information transfer [31–48]. One intriguing feature induced by correlation function is that partially coherent beam displays strong self-healing ability when scattered by opaque obstacles, e.g., both the beam profile and the state of polarization can be recovered [49]. Furthermore, the information of the orbital angular momentum of a partially coherent vortex beam is hidden in the correlation function in the far field [50], which also displays self-healing ability [51]. Multi-Gaussian Schell-model (MGSM) beam is a typical kind of partially coherent beam with prescribed correlation function introduced by Sahin and Korotkova [38, 39], whose degree of coherence is expressed as a sum of suitably weighted Gaussian functions with different widths and sign-alternating coefficients. Owing to its special correlation function, the far-field beam profile generated by a MGSM beam displays flat-topped shape, which is useful for material thermal processing. The MGSM beam has lower scintillation compared to the conventional partially coherent beam [Gaussian Schell-model (GSM) beam] upon propagation in turbulent atmosphere [52–55], which makes it attractive for free-space optical communications. Vector MGSM beam with spatially uniform state of polarization named electromagnetic MGSM beam was introduced in [56], and the changes of polarization and intensity pattern upon propagation in free space and in turbulent atmosphere were analyzed in detail.

In this paper, we introduce a new kind of vector MGSM beam with spatially non-uniform state of polarization named radially polarized MGSM beam, which combines the properties of non-uniform polarization and special correlation functions together, and we discuss the realizability conditions for such beam. We also explore tight focusing properties of a radially polarized MGSM beam passing through a high NA objective lens numerically, and we find that both transverse and longitudinal field patterns can be shaped by engineering the structures of the correlation functions of the incident beam. At last, we report experimental generation of a radially polarized MGSM beam.

2. Radially polarized multi-Gaussian Schell-model beam and its realizability conditions

According to the unified theory of optical coherence and polarization [57], the statistical properties of a partially coherent vector beam-like field, propagating along z-axis, can be characterized by the cross-spectral density (CSD) matrix W(r1,r2,ω,z) in the space-frequency domain, where ω denotes the angular frequency, r1 and r2 are two arbitrary position vectors in the plane of z, perpendicular to the propagation axis. In the source plane (z = 0), the elements of the CSD matrix are expressed as

Wαβ(r1,r2,ω,0)=Eα*(r1,ω,0)Eβ(r2,ω,0),(α,β=x,y),
Ex(r) and Ey(r) represent two orthogonally stochastic electric field components along x and y axis, respectively. The asterisks stand for the complex conjugate and the angular brackets denote an ensemble average. For simplicity, the dependence of the CSD matrix and other derived quantities on ω will be omitted throughout the paper.

Assume that the vector source is a Schell-model source whose degree of coherence (DOC) only depends on the difference of two spatial position vectors. Under this circumstance, the elements of the CSD matrix can be written as

Wαβ(r1,r2,0)=Sαα(r1,0)Sββ(r2,0)μαβ(r1r2),(α,β=x,y),
where Sαα(r1,0)=Wαα(r1,r1,0) (α = x, y) denotes x-x or y-y component of the spectral density,μαβis the correlation function between α and β components of the electric field.

Furthermore, we suppose that the beam source displays radial polarization and its correlation functions are expressed in terms of multi-Gaussian functions. In this situation, the elements of the CSD matrix take the following forms

Wαβ(r1,r2,0)=α1β2exp(r12+r224w02)μαβ(r1r2),(α,β=x,y),
with
μαβ(r1r2)=Bαβ1C0m=1M(Mm)(1)m1mexp[(r1r2)22mδαβ2],
where C0=m=1M(Mm)(1)m1m is the normalized factor, M is the number of Gaussian functions, (Mm) stands for binomial coefficients, w0 is the beam width of the source, δxx, δyyδxyare the coherence widths of the correlation functions of x-x, y-y and x-y components, respectively. Bαβ=|Bαβ|exp(iϕαβ) is the complex correlation coefficient between α and β component of the electric field with ϕαβ being the phase difference. The beam source whose CSD matrix elements are given by Eq. (3) is termed as radially polarized MGSM source. When the parameter M equals 1, the beam source reduces to a radially polarized Gaussian Schell-model (GSM) source.

Now, let us discuss the realizability conditions of a radially polarized MGSM source. First, the CSD matrix must be Hermitian conjugate from its definition shown in Eq. (1) [25]. Hence, the following conditions must be met

Bxx=Byy=1,Bxy=Byx*,δxy=δyx.

Second, to be a physically realizable CSD matrix [32], it is fulfilled to have a representation of the following integral form

Wαβ(r1,r2,0)=pαβ(v)Hα(r1,v)Hβ(r2,v)d2v,(α,β=x,y),
where Hx(r,v) and Hy(r,v) are two arbitrary functions,pαβ(v) is the element of the following weighting matrix
p(v)=(pxx(v)pxy(v)pyx(v)pyy(v)).
The elements of the weighting matrix should satisfy the conditions [32]

pxx(v)0,pyy(v)0,pxx(v)pyy(v)|pxy(v)|20.

The functions Hx(r,v), Hy(r,v) and pαβ(v) for a radially polarized MGSM source are given as follows

Hα(r,v)=αexp(r24w02)exp(ivr),(α=x,y),
pαβ(v)=Bαβδαβ2C0×m=1M(Mm)(1)m1exp(mδαβ2v22).
By applying the binomial theorem, Eq. (10) can be expressed alternatively as
pαβ(v)=Bαβδαβ2C0×{1[1exp(δαβ2v2/2)]M}.
It follows from Eq. (11) that the elements pxx(v) andpyy(v) are non-negative functions because exp(δαα2v2/2) is smaller than one for any value v. Thus, the first and the second inequalities in Eq. (8) are always true. Substituting Eq. (11) into the third inequality in Eq. (8), we obtain

δxx2δyy2×{1[1exp(δxx2v2/2)]M}×{1[1exp(δyy2v2/2)]M}.|Bxy|2δxy4×{1[1exp(δxy2v2/2)]M}2

Due to fact that the elements of the weighting matrix are all monotonically decreasing functions via v2, by setting v = 0, we obtain the following inequality easily

δxyδxxδyy|Bxy|,
then Eq. (12) can be simplified as
Min{1[1exp(δxx2v2/2)]M,1[1exp(δyy2v2/2)]M},1[1exp(δxy2v2/2)]M
which implies
δxyMax{δxx,δyy}.
Combining inequalities Eqs. (13) and (15), we obtain the double inequality
Max{δxx,δyy}δxyδxxδyy|Bxy|.
We find that Eq. (16) is independent on the parameter M. Let us recall such condition for a GSM beam (M = 1) derived by Gori et al. [58], given by
δxx2+δyy22δxyδxxδyy|Bxy|.
Compared to Eq. (16), one finds that Eq. (17) is more tight than Eq. (16). Hence, the realizablity condition of a radially polarized MGSM beam also can be given by Eq. (17).

Besides the conditions shown in Eqs. (5) and (17), a radially polarized beam source should obey two additional restrictions on its parameters [59]

(a) Any point in the source plane is linearly polarized, which means that the minor semi-axes of the polarization ellipse equals to zero.

(b) The orientation angle of the polarization at any point in the source plane should satisfies θ(x,y)=arctan(y/x).

According to [60], the orientation angle θ and ellipticity ε of the polarization ellipse in source plane can be expressed in terms of the elements of the CSD matrix, i.e.,

θ(r,0)=12arctan[2Re[Wxy(r,r,0)]Wxx(r,r,0)Wyy(r,r,0)],
ε(r,0)=A(r,0)/A+(r,0),(0ε1),
with
A±(r)=12[[Wxx(r,r,0)Wyy(r,r,0)]2+4|Wxy(r,r,0)|2±[Wxx(r,r,0)Wyy(r,r,0)]2+4(Re[Wxy(r,r,0)])2].
HereA+ and A in Eqs. (19)-(20) denote the major and minor semi-axes of the polarization ellipse, respectively. Substituting Eq. (3) into Eqs. (18)-(20), we obtain the following expressions for θ(r,0) and A(r,0)
tan2θ(r,0)=Re[Bxy]2y/x1y2/x2,
A(r,0)=12[(x2y2)2+4|Bxy|2x2y2(x2y2)2+4Re[Bxy]2x2y2],
where Re denotes taking the real part of Bxy. By applying the conditions (a) and (b), one can obtain
Re[Bxy]=1,|Bxy|=1.
It follows from Eq. (23) that Bxy=Byx=1. Under this condition and combining Eq. (17), one can easily obtain δxy=δxx=δyy=δ0. Finally, the realibality conditions of a radially polarized MGSM beam can be simplified as

Bxy(yx)=Bxx=Byy=1,δxy(yx)=δxx=δyy=δ0.

The degree of coherence of a vector partially coherent beam is defined as [61]

μ2(r1r2)=α,β|Wαβ(r1,r2,0)|2α,βWαα(r1,r1,0)Wββ(r2,r2,0).
Substituting from Eq. (3) into Eq. (25) and applying Eq. (24), we obtain
μ2(r1r2)=μαβ2(r1r2),(α,β=x,y).
One sees that the degree of coherence and the correlation functions of a radially polarized MGSM beam are the same.

3. Tight focusing properties of a radially polarized multi-Gaussian Schell-model beam passing through a high NA objective lens

The most interesting property of a scalar MGSM beam is that its initial Gaussian beam profile will evolve into a flat-topped beam profile in the far field in free space or in the focal plane when it was focused by a thin lens [38, 39]. In [38, 39], only transverse field distribution is discussed because the longitudinal field distribution can be ignored. What will happen when a radially polarized MGSM beam is focused by a high NA object lens? In this section, we will study the tight focusing properties of a radially polarized MGSM beam, and discuss the influence of correlation functions of the incident beam on the transverse and longitudinal fields distributions.

Let us refer to the scheme of tight focusing of a light beam focused by a high NA objective lens as shown in Fig. 1. According to the Richards–Wolf vectorial diffraction integral, the vectorial electric field in the focal region of a light beam passing through a high NA objective lens can be expressed as follows [1–4]

Ef(r,φ,z)=(EfxEfyEfz)=ik1f2π02πdϕ0αsinθcos12θexp(ik1ζ){lx(θ,ϕ)[cosθ+sinϕ(1cosθ)]+ly(θ,ϕ)cosϕsinϕ(cosθ1)lx(θ,ϕ)cosϕsinϕ(cosθ1)+ly(θ,ϕ)[cosθ+cos2ϕ(1cosθ)]lx(θ,ϕ)cosϕsinθly(θ,ϕ)sinϕsinθ}dθ,
where ζ=zcosθ+rsinθcos(ϕφ), r, φ, z are the cylindrical coordinates of an observation point near the focal region, ϕ is the azimuthal angle of an incident beam, θ is the NA angle, and α is the maximal NA angle given by the formula α=arcsinNNA/n1 with NNA and n1 being the NA number and the refractive index of the surrounding medium, k1=2πn1/λ is the wave number with λ being the wavelength of the incident beam, f is the focal length of the lens, lx and lyare the pupil apodization functions in the plane of aperture surface and are derived by setting x=fsinθcosϕand y=fsinθsinϕ in Ex(x,y) and Ey(x,y) of the incident beam, respectively.

 

Fig. 1 Scheme of tight focusing of a light beam focused by a high NA objective lens.

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Suppose that the incident beam is a radially polarized MGSM beam proposed in section 2, after passing through a high NA objective lens, the statistical properties of the field near the focal region can be characterized by the 3×3 CSD matrix, and the elements is expressed

Wfαβ(r1,φ1,r2,φ2,z)=Efα*(r1,φ1,z)Efβ(r2,φ2,z),(α,β=x,y,z).

Applying Eqs. (3), (4), (27) and (28), we can obtain the integral formulae for the elements of the CSD matrix of a tightly focused radially polarized MGSM beam in the focal region. In this paper, we focus on the average intensity distribution of the focused radial polarized MGSM beam in the focal region, therefore, we only give the expressions for the diagonal elements of the CSD matrix in the focal plane, i.e.,

Wfxx(r1,φ1,r2,φ2,z)=f2n1λ20θmax0θmax02π02πdθ1dθ2dϕ1dϕ2W0(θ1,ϕ1,θ2,ϕ2),×exp[ik1(ζ2ζ1)]cos3/2θ1cos3/2θ2sin2θ1sin2θ2cosϕ1cosϕ2
Wfyy(r1,φ1,r2,φ2,z)=f2n1λ20θmax0θmax02π02πdθ1dθ2dϕ1dϕ2W0(θ1,ϕ1,θ2,ϕ2),×exp[ik1(ζ2ζ1)]cos3/2θ1cos3/2θ2sin2θ1sin2θ2sinϕ1sinϕ2
Wfzz(r1,φ1,r2,φ2,z)=f2n1λ20θmaxdθ10θmaxdθ202πdϕ102πdϕ2W0(θ1,ϕ1,θ2,ϕ2),×exp[ik1(ζ2ζ1)]cos1/2θ1cos1/2θ2sin3θ1sin3θ2
with
ζi=zcosθi+risinθicos(ϕiφi),(i=1,2),
W0(θ1,ϕ1,θ2,ϕ2)=f2C0exp[f2sin2θ1+sin2θ24ω02]m=1M(Mm)(1)m1m.×exp[f2sin2θ1+sin2θ22mδ02]exp(f2sinθ1sinθ2cos(ϕ1ϕ2)mδ02)
In above derivations, the realizabilitiy conditions derived in section 2 have been applied. The transverse, longitudinal and total intensity distributions in the focal region can be obtained by setting r1=r2=r and φ1=φ2=φ in Eqs. (29)-(31), i.e.,

Itra(r,φ,z)=Wfxx(r,φ,r,φ,z)+Wfyy(r,φ,r,φ,z),
Iz(r,φ,z)=Wfzz(r,φ,r,φ,z),
Itotal(r,φ,z)=Itra(r,φ,z)+Iz(r,φ,z)

To explore the tight focusing properties of a radially polarized MGSM beam passing through a high NA objective lens, we carry out numerical calculations using above derived formulae. In the following numerical examples, we set n1 = 1, NNA = 0.95, f = 3.0mm, ω0 = 10mm and λ = 632.8nm. We calculate in Figs. 2 and 3 the contour graphs of the total intensity Itotal, transverse intensity Itra and longitudinal intensity Iz of the tightly focused radially polarized MGSM beam in the focal plane for different values of beam index M and coherence width δ0. The intensity distributions in each row shown in Figs. 2 and 3 are normalized by the maximum value of the total intensity distribution, and the white solid line in Figs. 2 and 3 denote the cross line of the intensity distribution at ρy = 0. One finds from Fig. 2 that the intensity distributions Itotal, Itra and Iz in the transverse plane keep rotational symmetries irrespectively of the beam index M. The reason is that the intensity distribution, the correlation functions and the polarization distribution of the incident beam all are of circular symmetries. When M = 1, Itotal, Itra and Iz all display quasi-Gaussian profiles due to the fact that the correlation functions of the incident beam are of Gaussian distributions [see Figs. 2(a)-2(c)]. It is interesting to find that with the increase of the beam index M, Itotal, Itra and Iz all gradually turn to flat-topped beam profiles [see in Figs. 2(h) and 2(i)], which is due to the effects of the special correlation functions of the incident beam. One can expect to produce other particular longitudinal intensity distributions such as doughnut and lattice profiles through manipulating the correlation functions of the incident beam. The ratio of the energy carried by the longitudinal field to the total energy of the beam in our case is about 50%. From Fig. 3, one sees that the focused intensity distributions are also controlled by the coherence width δ0 of the incident beam, e.g., with the increase of δ0, the longitudinal and transverse intensity distributions gradually lose their flat-topped profiles. For the case of δ0 = 1.5mm, the longitudinal intensity distribution still remains a flat-topped profile, while the transverse intensity distribution degenerates to a single-peak profile [see Figs. 3(b) and 3(c)], which implies that the longitudinal field is more resistant to variance of δ0. When δ0 = 10mm, the distributions of Itotal, Itra and Iz are almost the same as those of a tightly focused coherent radially polarized beam [1–4]. Combining Fig. 2 and Fig. 3, we may come to the conclusion that to achieve a longitudinal field with flat-top profile, large beam order M and small coherence width δ0 are required. Anyway, our numerical results clearly show that both transverse and longitudinal field distributions of a tightly focused radially polarized partially coherent beam can be shaped through manipulating the structures of its correlation functions, which have potential applications in particle acceleration and laser machining. On the other hand, it is known that the phase distribution of a tightly focused vector beam in the focal plane plays an important role from the point view of the light-matter interaction [15], and our numerical results (not presented here to save space) have also revealed that the phase distribution of a tightly focused radially polarized MGSM beam was almost not affected by its correlation functions. More information about the phase distribution of a tightly focused vector beam can be found in [15].

 

Fig. 2 Contour graphs of the total intensity Itotal, transverse intensity Itra and longitudinal intensity Iz of a tightly focused radially polarized MGSM beam in the focal plane for different values of beam index M. The white solid line denotes the cross line of the intensity distribution at ρy = 0.

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Fig. 3 Contour graphs of the total intensity Itotal, transverse intensity Itra and longitudinal intensity Iz of a tightly focused radially polarized MGSM beam in the focal plane for different values of coherence width δ0. The white solid line denotes the cross line of the intensity distribution at ρy = 0.

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Now, we study the influences of the beam parameters M and δ0 on the behavior of the longitudinal field in the ρx-z plane near the focal region. We calculate in Fig. 4 the contour graph of the longitudinal intensity Iz of the focused radial polarized MGSM beam in x-z plane for different values of M and δ0 and the cross lines of the longitudinal intensity distribution at different distances. As illustrated in Fig. 4, the beam parameters M and δ0 seriously affect the spot size of the longitudinal field in the ρx -z plane. When δ0 = 0.5mm, the depth of focus (DOF) defined as the FHWM of its intensity profile in the longitudinal direction is calculated to be 2.40λ, 3.04λ and 3.74λ for M = 1, 3 and 10, respectively, which means the DOF increases with the increase of the beam index M. The FHWM of the intensity distribution in ρx direction at z = 0 is calculated to be 2.56λ, 3.44λ and 4.40λ for M = 1, 3 and 10, respectively, implying that the ability for focusing in the transverse plane is weakened as M increases. The cross line of the longitudinal intensity distribution for M = 10 and δ0 = 0.5mm shows that in the plane z = 0.5λ, the beam profile in the transverse plane can still keep flat-topped shape, but in the plane z = λ, the beam profile degenerates to a Gaussian-like shape [see Fig. 4(f)]. As δ0 increases [see Figs. 4(g)-4(i)], both the longitudinal spot size in ρx direction (transverse direction) and that in z direction decrease, and the intensity distribution in the transverse plane at z = 0 no longer exhibits flat-topped profile as we discussed in Fig. 3. Note that the spot size in ρx direction becomes much smaller than that in z direction for larger δ0. Hence, the focal spot in the ρx-z plane turns to an elliptical shape [see Fig. 4(i)].

 

Fig. 4 Contour graph of the longitudinal intensity Iz of the focused radial polarized MGSM beam in x-z plane for different values of M and δ0 [see (a)-(c) and (g)-(i)]. The cross lines of the longitudinal intensity at y = 0 at z = 0 (black solid curve), z = 0.5λ (red dashed curve) and z = λ (blue dash-doted curve) are shown in (d)-(f) and (j)-(l).

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What’s more, our numerical results (not present here) also show that when the NA of the lens is low, the longitudinal field near the focus can be neglected, while the transverse field near the focus still can be shaped through engineering the structures of the correlation functions of the radial polarized MGSM beam and it also displays flat-topped beam profile by choosing suitable beam parameters M and δ0.

4. Experimental generation of a radially polarized multi-Gaussian Schell-model beam

In this section, we report experimental generation of a radially polarized MGSM beam. In general, there are several outer-cavity methods for generating a partially coherent beam with prescribed correlations function. The first one is based on the Fourier transform of the intensity distribution involving the dynamic diffuser, i.e., rotating ground grass disk [43]. The second one resorts to random phase screens with designed correlation function with the help of spatial light modulator (SLM) [47]. The third one relies on a deformable mirror to synthesize a partially coherent beam with prescribed correlation function [48]. Here, we adopt the second method to generate a radially polarized MGSM beam with the help of a SLM and a radial polarization converter (RPC) [62], and the experiment setup is shown in Fig. 5. A horizontally polarized beam generated by a ND:YAG laser (λ = 532nm) first goes through a beam expander (BE), then reflected by a beam splitter (BS). The reflected beam from the BS goes towards a SLM, which act as random phase screens controlled by the personal computer. The approach for synthesizing random phase screens with designed correlation function is described in [47]. When we load the random phase screens whose correlation function is of multi-Gaussian functions, the first-order diffraction pattern reflected from the SLM can be regarded as a linearly polarized MGSM beam. After passing through the BS, the diffraction light from the SLM goes through a 4f-optical system comprised of two thin lenses (L1 and L2) and a circular aperture (CA). Here the CA which is located in the rear focal plane L1 (front focal plane of L2) is used to block the unwanted light and other order diffraction patterns except the first-order diffraction pattern. In order to convert the linearly polarized beam to a radially polarized beam, a half-wave plate (HWP) and a RPC are placed in the rear focal plane of L2 which is used to rotate the linear polarization to the vertical direction and convert linearly polarized beam to a radially polarized beam, respectively. The output beam from the RPC can be considered as a radially polarized MGSM beam. A charge-coupled device (CCD) is used to measure the intensity distribution or the squared modulus of the degree of coherence of the generated radially polarized MGSM beam. The principle and the procedure for measuring the degree of coherence by a CCD can be found in [34, 35].

 

Fig. 5 Experimental setup for generating a radially polarized MGSM beam and measuring its intensity distribution and squared modulus of the degree of coherence. BE: beam expander; SLM: spatial light modulator; BS: intensity beam splitter; L1 and L2, thin lenses; CA: circular aperture; HWP: half-wave plate; RPC: radial polarization converter; CCD: charge-coupled device.

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Figure 6(a) shows our experimental result of the intensity distribution of the generated radially polarized MGSM beam with M = 10 captured by the CCD just behind the RPC. As expected, the intensity distribution displays a doughnut beam shape due to that the state of the polarization in the central point of the beam is undefined, i.e., polarization singularity. The beam width w0 obtained from the experiment data is about 0.12mm. To verify that the generated beam displays radial polarization, we insert a linear polarizer between the RPC and the CCD, and Figs. 6(b)-6(d) show the experiment result of the intensity distribution captured by the CCD for different transmission axes of the polarizer. One finds from Fig. 6 that the generated beam indeed displays radial polarization. We want to point out that the beam profile of the generated radially polarized MGSM beam just after the RPC keeps almost unchanged if we change the beam parameter M. Note that the intensity distribution of the generated beam shown in Fig. 6(a) is somewhat inhomogeneous, which was caused by two factors. One is the protecting glass window located in front of the photo-sensitive area of the CCD, which leads to the interference fringes shown in Fig. 6(a) due to the interference effects between the front and rear surface of the protecting glass window, while this factor doesn’t affect the quality of the focal spot when the generated beam is tightly focused by a high NA lens. Another is the variable phase shifter embedded in the RPC. Such phase shifter has a π-phase jump line that coincides with the x-axis. Thus the intensity near the x-axis region is slightly weaker than the top or the low region of the beam. Figure 7 shows our experimental result of the squared modulus of the degree of coherence |μ(r1r2)|2of the generated radially polarized MGSM beam with M = 1 and M = 10 just after the RPC. The coherence width δ0 obtained from the experiment data for M = 1 or M = 10 is about 0.17mm. To show the influence of the correlation functions on the focusing properties, we also measure the focused intensity distribution of the generated radially polarized MGSM beam focused by a thin lens with focal length f = 15cm in the focal plane with M = 1 and M = 10, and the results are shown Fig. 8. In this case, the longitudinal field can be neglected, and Fig. 8 only shows the transverse field intensity distribution in the focal plane. As expected, the generated radially polarized MGSM beam with M = 1 displays quasi-Gaussian beam profile in the focal plane, while the generated radially polarized MGSM beam with M = 10 displays flat-topped beam profile in the focal plane.

 

Fig. 6 Experimental results of (a) the intensity distribution of the generated radially polarized MGSM beam with M = 10 just after the RPC, and (b)-(d) the corresponding intensity distribution of the generated beam after passing through a linear polarizer. The white arrows denote the transmission axis of the polarizer.

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Fig. 7 Experimental result of the squared modulus of the degree of coherence |μ(r1-r2)|2of the generated radially polarized MGSM beam with (a) M = 1 and (b) M = 10 just after the RPC.

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Fig. 8 Experimental result of the intensity distribution of the generated radially polarized MGSM beam focused by a thin lens with focal length f = 15cm in the focal plane with (a) M = 1 and (b) M = 10.

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5. Summary

We have introduced a new kind of vector MGSM beam named radially polarized MGSM beam as a natural extension of scalar MGSM beam and discussed its realizability conditions. Through exploring the tight focusing properties of the radially polarized MGSM beam passing through a high NA objective lens, we have found that one can shape both the transverse and longitudinal field distributions near the focus through varying the structures of the correlation functions. It is known that there are several methods in literatures for shaping the transverse filed distribution of a tightly focused beam, while our results have clearly show that engineering the structures of the correlation functions of a radially polarized partially coherent beam provides a novel way for shaping both transverse and longitudinal field distributions, which have potential applications in particle acceleration and laser machining. Furthermore, we have reported experimental generation of a radially polarized MGSM beam.

Funding

National Natural Science Foundation of China (NSFC) (91750201&11474213); National Natural Science Fund for Distinguished Young Scholar (11525418); Project of the Priority Academic Program Development of Jiangsu Higher Education Institutions; Qing Lan Project of Jiangsu Province; Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_2024).

References and links

1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]   [PubMed]  

2. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [CrossRef]   [PubMed]  

3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

4. Q. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]  

5. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef]   [PubMed]  

6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef]   [PubMed]  

7. M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004). [CrossRef]  

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

9. J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012). [CrossRef]  

10. Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010). [CrossRef]  

11. S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012). [CrossRef]  

12. H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010). [CrossRef]  

13. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef]   [PubMed]  

14. W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017). [CrossRef]   [PubMed]  

15. P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015). [CrossRef]  

16. M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013). [CrossRef]   [PubMed]  

17. B. Gu, J. L. Wu, Y. Pan, and Y. Cui, “Achievement of needle-like focus by engineering radial-variant vector fields,” Opt. Express 21(25), 30444–30452 (2013). [CrossRef]   [PubMed]  

18. Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015). [CrossRef]  

19. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]  

20. G. H. Yuan, S. B. Wei, and X. C. Yuan, “Non-diffracting transversally polarized beam,” Opt. Lett. 36(17), 3479–3481 (2011). [CrossRef]   [PubMed]  

21. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010). [CrossRef]   [PubMed]  

22. Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, “Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes,” Opt. Express 24(2), 874–882 (2016). [CrossRef]   [PubMed]  

23. P. Suresh, C. Mariyal, K. B. Rajesh, T. V. Pillai, and Z. Jaroszewicz, “Generation of a strong uniform transversely polarized nondiffracting beam using a high-numerical-aperture lens axicon with a binary phase mask,” Appl. Opt. 52(4), 849–853 (2013). [CrossRef]   [PubMed]  

24. J. W. Goodman, Statistical optics (Wiley, 1995).

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

26. L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011). [CrossRef]  

27. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011). [CrossRef]  

28. C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014). [CrossRef]   [PubMed]  

29. Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012). [CrossRef]  

30. Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015). [CrossRef]  

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

32. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

33. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009). [CrossRef]   [PubMed]  

34. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017). [CrossRef]  

35. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef]   [PubMed]  

36. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008). [CrossRef]   [PubMed]  

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

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

39. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]   [PubMed]  

40. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014). [CrossRef]   [PubMed]  

41. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015). [CrossRef]   [PubMed]  

42. Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016). [CrossRef]  

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

44. S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013). [CrossRef]   [PubMed]  

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

46. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015). [CrossRef]  

47. M. W. Hyde IV, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015). [CrossRef]  

48. M. W. Hyde IV, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017). [CrossRef]  

49. F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016). [CrossRef]   [PubMed]  

50. 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]  

51. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017). [CrossRef]  

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

53. O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

54. S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016). [CrossRef]   [PubMed]  

55. C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016). [CrossRef]  

56. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]  

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

58. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]   [PubMed]  

59. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012). [CrossRef]   [PubMed]  

60. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005). [CrossRef]  

61. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef]   [PubMed]  

62. http://www.arcoptix.com/radial_polarization_converter.htm

References

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  1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [Crossref] [PubMed]
  2. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref] [PubMed]
  3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  4. Q. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  5. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
    [Crossref] [PubMed]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
    [Crossref] [PubMed]
  7. M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004).
    [Crossref]
  8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [Crossref] [PubMed]
  9. J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
    [Crossref]
  10. Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
    [Crossref]
  11. S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
    [Crossref]
  12. H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
    [Crossref]
  13. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
    [Crossref] [PubMed]
  14. W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
    [Crossref] [PubMed]
  15. P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
    [Crossref]
  16. M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013).
    [Crossref] [PubMed]
  17. B. Gu, J. L. Wu, Y. Pan, and Y. Cui, “Achievement of needle-like focus by engineering radial-variant vector fields,” Opt. Express 21(25), 30444–30452 (2013).
    [Crossref] [PubMed]
  18. Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
    [Crossref]
  19. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
    [Crossref]
  20. G. H. Yuan, S. B. Wei, and X. C. Yuan, “Non-diffracting transversally polarized beam,” Opt. Lett. 36(17), 3479–3481 (2011).
    [Crossref] [PubMed]
  21. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
    [Crossref] [PubMed]
  22. Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, “Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes,” Opt. Express 24(2), 874–882 (2016).
    [Crossref] [PubMed]
  23. P. Suresh, C. Mariyal, K. B. Rajesh, T. V. Pillai, and Z. Jaroszewicz, “Generation of a strong uniform transversely polarized nondiffracting beam using a high-numerical-aperture lens axicon with a binary phase mask,” Appl. Opt. 52(4), 849–853 (2013).
    [Crossref] [PubMed]
  24. J. W. Goodman, Statistical optics (Wiley, 1995).
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  26. L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
    [Crossref]
  27. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
    [Crossref]
  28. C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014).
    [Crossref] [PubMed]
  29. Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
    [Crossref]
  30. Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
    [Crossref]
  31. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  32. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).
  33. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
    [Crossref] [PubMed]
  34. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
    [Crossref]
  35. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref] [PubMed]
  36. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008).
    [Crossref] [PubMed]
  37. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  38. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  39. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  40. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [Crossref] [PubMed]
  41. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
    [Crossref] [PubMed]
  42. Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
    [Crossref]
  43. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [Crossref] [PubMed]
  44. S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
    [Crossref] [PubMed]
  45. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref] [PubMed]
  46. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
    [Crossref]
  47. M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
    [Crossref]
  48. M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
    [Crossref]
  49. F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
    [Crossref] [PubMed]
  50. 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]
  51. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
    [Crossref]
  52. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
    [Crossref]
  53. O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
  54. S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
    [Crossref] [PubMed]
  55. C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
    [Crossref]
  56. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
    [Crossref]
  57. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  58. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
    [Crossref] [PubMed]
  59. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
    [Crossref] [PubMed]
  60. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
    [Crossref]
  61. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
    [Crossref] [PubMed]
  62. http://www.arcoptix.com/radial_polarization_converter.htm

2017 (4)

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

2016 (5)

2015 (6)

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

2014 (4)

2013 (8)

P. Suresh, C. Mariyal, K. B. Rajesh, T. V. Pillai, and Z. Jaroszewicz, “Generation of a strong uniform transversely polarized nondiffracting beam using a high-numerical-aperture lens axicon with a binary phase mask,” Appl. Opt. 52(4), 849–853 (2013).
[Crossref] [PubMed]

M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013).
[Crossref] [PubMed]

B. Gu, J. L. Wu, Y. Pan, and Y. Cui, “Achievement of needle-like focus by engineering radial-variant vector fields,” Opt. Express 21(25), 30444–30452 (2013).
[Crossref] [PubMed]

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

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
[Crossref] [PubMed]

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

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

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

2012 (7)

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

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]

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

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

2011 (4)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

G. H. Yuan, S. B. Wei, and X. C. Yuan, “Non-diffracting transversally polarized beam,” Opt. Lett. 36(17), 3479–3481 (2011).
[Crossref] [PubMed]

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

2010 (3)

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[Crossref] [PubMed]

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

2009 (3)

Q. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

2008 (4)

2007 (1)

2005 (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

2004 (2)

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

2000 (2)

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
[Crossref] [PubMed]

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

Banzer, P.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Baykal, Y.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Borghi, R.

Bose-Pillai, S. R.

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
[Crossref]

Brown, D. P.

Brown, T.

Brown, T. G.

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Cai, M.

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
[Crossref] [PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014).
[Crossref] [PubMed]

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

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

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

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]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
[Crossref] [PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

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

Chong, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Cui, Y.

Dong, Y.

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

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]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Du, L.

Eyyuboglu, H. T.

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

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Fan, H.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Fourmaux, S.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Friberg, A.

Galow, B. J.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Gbur, G.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

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

Gori, F.

Gu, B.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Gu, M.

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Gu, Y.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

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

Guo, L.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Guth, S.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
[Crossref] [PubMed]

Han, Y.

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]

He, H.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Huang, K.

Huang, W.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Hyde, M. W.

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Jaroszewicz, Z.

Jia, B.

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Kang, H.

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Kang, X. L.

Keitel, C.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Kieffer, J. C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Korotkova, O.

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
[Crossref] [PubMed]

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

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

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Kumar, A.

Lajunen, H.

Légaré, F.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Leger, J.

Leuchs, G.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Li, J.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Li, Y.

Li, Y. P.

Liang, C.

C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014).
[Crossref] [PubMed]

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Lieb, M. A.

Lin, Q.

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Liu, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
[Crossref] [PubMed]

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

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

Lou, K.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Ma, L.

MacLean, J. P.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Malek-Madani, R.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

Man, Z.

Mariyal, C.

Martínez-Herrero, R.

Mei, T.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Mei, Z.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

Mejías, P. M.

Min, C.

Morrish, D.

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

Nelson, C.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
[Crossref] [PubMed]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).

Novotny, L.

M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21(6), 1210–1215 (2004).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Pan, Y.

Payeur, S.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Peng, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Piché, M.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Pillai, T. V.

Ponomarenko, S. A.

Prabhakar, S.

Premaratne, M.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Qian, S.

Qu, J.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

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

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Rajesh, K. B.

Ramírez-Sánchez, V.

Reddy, S. G.

Saastamoinen, T.

Sahin, S.

Salamin, Y.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Salamin, Y. I.

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

Santarsiero, M.

Schmidt, B. E.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Setälä, T.

Shang, W.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Shchepakina, E.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, P.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref] [PubMed]

Singh, R. P.

Suresh, P.

Tan, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Tang, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Tchervenkov, C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

Tervo, J.

Tu, C.

Voelz, D. G.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Wang, F.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
[Crossref] [PubMed]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

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

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and 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, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Wang, H. T.

Wang, K.

Wei, S. B.

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Wood, R. A.

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
[Crossref]

Wozniak, P.

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Wu, J. L.

Xiao, F.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Xiao, X.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Xu, H.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Youngworth, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Yu, Y.

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

Yuan, G. H.

Yuan, X.

Yuan, X. C.

Yuan, Y.

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

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

Zavislan, J. M.

Zhan, Q.

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

Q. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

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

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

Zhang, H.

Zhang, X.

Zhang, Y.

Zhang, Z.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Zhao, C.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014).
[Crossref] [PubMed]

C. Liang, C. Zhao, C. Zhao, K. Wang, and Y. Cai, “Degree of polarization of a tightly focused, partially coherent anomalous hollow beam,” J. Opt. Soc. Am. A 31(12), 2753–2758 (2014).
[Crossref] [PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

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]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Zhao, J.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Zhu, S.

Zhu, W.

W. Shang, F. Xiao, W. Zhu, H. He, M. Premaratne, T. Mei, and J. Zhao, “Fano resonance with high local field enhancement under azimuthally polarized excitation,” Sci. Rep. 7(1), 1049 (2017).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Appl. Phys. Lett. (6)

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012).
[Crossref]

H. Kang, B. Jia, J. Li, D. Morrish, and M. Gu, “Enhanced photothermal therapy assisted with gold nanorods using a radially polarized beam,” Appl. Phys. Lett. 96(6), 063702 (2010).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett. 111(10), 101106 (2017).
[Crossref]

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]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

J. Appl. Phys. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

J. Opt. (4)

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

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

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

Laser Photonics Rev. (1)

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015).
[Crossref]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Opt. Commun. (3)

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

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Opt. Express (12)

J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013).
[Crossref] [PubMed]

B. Gu, J. L. Wu, Y. Pan, and Y. Cui, “Achievement of needle-like focus by engineering radial-variant vector fields,” Opt. Express 21(25), 30444–30452 (2013).
[Crossref] [PubMed]

Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, “Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes,” Opt. Express 24(2), 874–882 (2016).
[Crossref] [PubMed]

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008).
[Crossref] [PubMed]

Opt. Laser Technol. (1)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Opt. Lett. (10)

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

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

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

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

G. H. Yuan, S. B. Wei, and X. C. Yuan, “Non-diffracting transversally polarized beam,” Opt. Lett. 36(17), 3479–3481 (2011).
[Crossref] [PubMed]

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref] [PubMed]

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

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
[Crossref] [PubMed]

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

Phys. Rev. A (4)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

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[Crossref]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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Proc. SPIE (1)

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[Crossref]

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http://www.arcoptix.com/radial_polarization_converter.htm

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Figures (8)

Fig. 1
Fig. 1 Scheme of tight focusing of a light beam focused by a high NA objective lens.
Fig. 2
Fig. 2 Contour graphs of the total intensity Itotal, transverse intensity Itra and longitudinal intensity Iz of a tightly focused radially polarized MGSM beam in the focal plane for different values of beam index M. The white solid line denotes the cross line of the intensity distribution at ρy = 0.
Fig. 3
Fig. 3 Contour graphs of the total intensity Itotal, transverse intensity Itra and longitudinal intensity Iz of a tightly focused radially polarized MGSM beam in the focal plane for different values of coherence width δ0. The white solid line denotes the cross line of the intensity distribution at ρy = 0.
Fig. 4
Fig. 4 Contour graph of the longitudinal intensity Iz of the focused radial polarized MGSM beam in x-z plane for different values of M and δ0 [see (a)-(c) and (g)-(i)]. The cross lines of the longitudinal intensity at y = 0 at z = 0 (black solid curve), z = 0.5λ (red dashed curve) and z = λ (blue dash-doted curve) are shown in (d)-(f) and (j)-(l).
Fig. 5
Fig. 5 Experimental setup for generating a radially polarized MGSM beam and measuring its intensity distribution and squared modulus of the degree of coherence. BE: beam expander; SLM: spatial light modulator; BS: intensity beam splitter; L1 and L2, thin lenses; CA: circular aperture; HWP: half-wave plate; RPC: radial polarization converter; CCD: charge-coupled device.
Fig. 6
Fig. 6 Experimental results of (a) the intensity distribution of the generated radially polarized MGSM beam with M = 10 just after the RPC, and (b)-(d) the corresponding intensity distribution of the generated beam after passing through a linear polarizer. The white arrows denote the transmission axis of the polarizer.
Fig. 7
Fig. 7 Experimental result of the squared modulus of the degree of coherence | μ ( r 1 - r 2 ) | 2 of the generated radially polarized MGSM beam with (a) M = 1 and (b) M = 10 just after the RPC.
Fig. 8
Fig. 8 Experimental result of the intensity distribution of the generated radially polarized MGSM beam focused by a thin lens with focal length f = 15cm in the focal plane with (a) M = 1 and (b) M = 10.

Equations (36)

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W α β ( r 1 , r 2 , ω , 0 ) = E α * ( r 1 , ω , 0 ) E β ( r 2 , ω , 0 ) , ( α , β = x , y ) ,
W α β ( r 1 , r 2 , 0 ) = S α α ( r 1 , 0 ) S β β ( r 2 , 0 ) μ α β ( r 1 r 2 ) , ( α , β = x , y ) ,
W α β ( r 1 , r 2 , 0 ) = α 1 β 2 exp ( r 1 2 + r 2 2 4 w 0 2 ) μ α β ( r 1 r 2 ) , ( α , β = x , y ) ,
μ α β ( r 1 r 2 ) =B α β 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m exp [ ( r 1 r 2 ) 2 2 m δ α β 2 ] ,
B x x = B y y = 1 , B x y = B y x * , δ x y = δ y x .
W α β ( r 1 , r 2 , 0 ) = p α β ( v ) H α ( r 1 , v ) H β ( r 2 , v ) d 2 v , ( α , β = x , y ) ,
p ( v ) = ( p x x ( v ) p x y ( v ) p y x ( v ) p y y ( v ) ) .
p x x ( v ) 0 , p y y ( v ) 0 , p x x ( v ) p y y ( v ) | p x y ( v ) | 2 0.
H α ( r , v ) = α exp ( r 2 4 w 0 2 ) exp ( i v r ) , ( α = x , y ) ,
p α β ( v ) = B α β δ α β 2 C 0 × m = 1 M ( M m ) ( 1 ) m 1 exp ( m δ α β 2 v 2 2 ) .
p α β ( v ) = B α β δ α β 2 C 0 × { 1 [ 1 exp ( δ α β 2 v 2 / 2 ) ] M } .
δ x x 2 δ y y 2 × { 1 [ 1 exp ( δ x x 2 v 2 / 2 ) ] M } × { 1 [ 1 exp ( δ y y 2 v 2 / 2 ) ] M } . | B x y | 2 δ x y 4 × { 1 [ 1 exp ( δ x y 2 v 2 / 2 ) ] M } 2
δ x y δ x x δ y y | B x y | ,
M i n { 1 [ 1 exp ( δ x x 2 v 2 / 2 ) ] M , 1 [ 1 exp ( δ y y 2 v 2 / 2 ) ] M } , 1 [ 1 exp ( δ x y 2 v 2 / 2 ) ] M
δ x y M a x { δ x x , δ y y } .
M a x { δ x x , δ y y } δ x y δ x x δ y y | B x y | .
δ x x 2 + δ y y 2 2 δ x y δ xx δ yy | B x y | .
θ ( r , 0 ) = 1 2 arc tan [ 2 Re [ W x y ( r , r , 0 ) ] W x x ( r , r , 0 ) W y y ( r , r , 0 ) ] ,
ε ( r , 0 ) = A ( r , 0 ) / A + ( r , 0 ) , ( 0 ε 1 ) ,
A ± ( r ) = 1 2 [ [ W x x ( r , r , 0 ) W y y ( r , r , 0 ) ] 2 + 4 | W x y ( r , r , 0 ) | 2 ± [ W x x ( r , r , 0 ) W y y ( r , r , 0 ) ] 2 + 4 ( Re [ W x y ( r , r , 0 ) ] ) 2 ] .
tan 2 θ ( r , 0 ) = Re [ B x y ] 2 y / x 1 y 2 / x 2 ,
A ( r , 0 ) = 1 2 [ ( x 2 y 2 ) 2 + 4 | B x y | 2 x 2 y 2 ( x 2 y 2 ) 2 + 4 Re [ B x y ] 2 x 2 y 2 ] ,
Re[ B x y ] = 1 , | B x y | = 1.
B x y ( y x ) = B x x = B y y = 1 , δ x y ( y x ) = δ x x = δ y y = δ 0 .
μ 2 ( r 1 r 2 ) = α , β | W α β ( r 1 , r 2 , 0 ) | 2 α , β W α α ( r 1 , r 1 , 0 ) W β β ( r 2 , r 2 , 0 ) .
μ 2 ( r 1 r 2 ) = μ α β 2 ( r 1 r 2 ) , ( α , β = x , y ) .
E f ( r , φ , z ) = ( E f x E f y E f z ) = i k 1 f 2 π 0 2 π d ϕ 0 α sin θ cos 1 2 θ exp ( i k 1 ζ ) { l x ( θ , ϕ ) [ cos θ + sin ϕ ( 1 cos θ ) ] + l y ( θ , ϕ ) cos ϕ sin ϕ ( cos θ 1 ) l x ( θ , ϕ ) cos ϕ sin ϕ ( cos θ 1 ) + l y ( θ , ϕ ) [ cos θ + cos 2 ϕ ( 1 cos θ ) ] l x ( θ , ϕ ) cos ϕ sin θ l y ( θ , ϕ ) sin ϕ sin θ } d θ ,
W f α β ( r 1 , φ 1 , r 2 , φ 2 , z ) = E f α * ( r 1 , φ 1 , z ) E f β ( r 2 , φ 2 , z ) , ( α , β = x , y , z ) .
W f x x ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ max 0 θ max 0 2 π 0 2 π d θ 1 d θ 2 d ϕ 1 d ϕ 2 W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) , × exp [ i k 1 ( ζ 2 ζ 1 ) ] cos 3 / 2 θ 1 cos 3 / 2 θ 2 sin 2 θ 1 sin 2 θ 2 cos ϕ 1 cos ϕ 2
W f y y ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ max 0 θ max 0 2 π 0 2 π d θ 1 d θ 2 d ϕ 1 d ϕ 2 W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) , × exp [ i k 1 ( ζ 2 ζ 1 ) ] cos 3 / 2 θ 1 cos 3 / 2 θ 2 sin 2 θ 1 sin 2 θ 2 sin ϕ 1 sin ϕ 2
W f z z ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ max d θ 1 0 θ max d θ 2 0 2 π d ϕ 1 0 2 π d ϕ 2 W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) , × exp [ i k 1 ( ζ 2 ζ 1 ) ] cos 1 / 2 θ 1 cos 1 / 2 θ 2 sin 3 θ 1 sin 3 θ 2
ζ i = z cos θ i + r i sin θ i cos ( ϕ i φ i ) , ( i = 1 , 2 ) ,
W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = f 2 C 0 exp [ f 2 sin 2 θ 1 + sin 2 θ 2 4 ω 0 2 ] m = 1 M ( M m ) ( 1 ) m 1 m . × exp [ f 2 sin 2 θ 1 + sin 2 θ 2 2 m δ 0 2 ] exp ( f 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 2 ) m δ 0 2 )
I t r a ( r , φ , z ) = W f x x ( r , φ , r , φ , z ) + W f y y ( r , φ , r , φ , z ) ,
I z ( r , φ , z ) = W f z z ( r , φ , r , φ , z ) ,
I t o t a l ( r , φ , z ) = I t r a ( r , φ , z ) + I z ( r , φ , z )

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