Abstract

In this paper, we have introduced a new class of partially coherent vector vortex beams, named radially polarized multi-Gaussian Schell-model (MGSM) vortex beam, carrying the vortex phase with tunable topological charges (i.e., both integral and fractional values) as a natural extension of the radially polarized MGSM beam. The tight focusing properties of the radially polarized MGSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structure of the correlation functions and the topological charge of vortex phase. In contrast with the integral vortex beam, the most intriguing property of the fractional vortex beam is that the focal intensity distribution at the focal plane can be nonuniformity and asymmetry, while such unique characteristics will vanish when the spatial coherence length is sufficiently small. Furthermore, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, can be formed by choosing suitable fractional values of topological charge and spatial coherence length. Our results will be useful for optical trapping, especially for trapping of irregular particles or manipulation of absorbing particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The shaping and construction of the desired focal field to meet the needs and improve the performance of optical manipulation is currently an important research topic [116]. For a long time, numerous efforts have been paid to find the effective methods for shaping the focal field. In recent years, the tight focusing of radially polarized beam has attracted considerable attention due to its peculiar properties of a strong longitudinal electric field in the focal region [28]. Compared with the transverse component, the longitudinal electrical field is dominant and makes no contribution to the energy flow along the optical axis, which is of great importance for the trapping of metallic nanoparticles [57]. Furthermore, a three-dimensional orientated focal field can be obtained by configuring the relative weighting factor between the longitudinal and transverse field components [8]. In particular, the integration of conventional tight focusing with other beam engineering techniques including binary optics, diffractive optical element (DOE), 4π focusing system and planar plasmonic metalens offers additional flexibility for focus shaping [916]. Correspondingly, a series of novel focal field with different characteristic spatial distributions, such as flat-top focus, optical needle, optical cage, optical chain and multiple focal spots has been theoretically proposed and experimentally demonstrated [916].

On the other hand, as one of the most important features of light beams, the spatial coherence of partially coherent beams plays an important role in determining the focusing properties, the transverse field distribution are greatly affected by its spatial coherence width [1719]. More recently, it also has been revealed that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which provides a novel way for shaping the focal field [20,21]. In addition to the coherence, the phase also has significant effects on the tight focusing properties, and the focal field patterns can be modulated by engineering the vortex phase of the incident beam [2224]. For instance, an equilateral-polygon-like flat-top focus can be generated by engineering the arrangement and positions of off-axis vortices of a radially polarized beam superposed with off-axis vortex arrays [24]. Nevertheless, over the past decades, previous researches were restricted to optical vortex beams with integer-order topological charges. In fact, the value of the topological charge can be non-integral [25]. A Fermat’s spiral slit can be used to modulate the topological charge of vortex beams to be an arbitrary value, both integral and fractional, within a continuous range [26]. In recent years, vortex beams with fractional topological charges called as fractional vortex beams or non-integer vortex beams have been aroused great interest due to their unique properties, such as a radial opening in the annular intensity ring, alternating charge vortices, birth and annihilation of vortices [2630].

To our knowledge, most of previous published works are restricted to only produce circular-symmetric focus patterns, which is capable of confining spherical particles. In reality, in the field of optical trapping, particles are usually irregular in shape or diversity of property. Therefore, it is essential to generate asymmetric fields or even arbitrary shaped fields to meet diverse demands from different aspects in real environment. In this paper, we have introduced a new class of partially coherent vector vortex beam named as radially polarized multi-Gaussian Schell-model (MGSM) vortex beam with tunable topological charges (i.e., both integral and fractional values) as a nature extension of MGSM beam [3134]. The tight focusing properties of such vortex beams passing through a high numerical aperture (NA) objective lens are investigated numerically. It is particularly interesting to find that optical vortex beams with fractional topological charge can break the circular symmetry of focal intensity distribution of vector beams allowing versatile means for focus shaping. Moreover, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, are obtained by tailoring the fractional values of topological charge and spatial coherence length. Our results may find potential applications in optical trapping, especially for the trapping of irregular particles or manipulation of absorbing particles.

2. Radially polarized multi-Gaussian Schell-model vortex beam

In the Cartesian coordinate system, the vectorial electric field of a coherent radially polarized TEM beam at the source plane can be characterized by [35],

$$\begin{aligned} {\textbf E}({{\textbf r};\omega } )&= {E_x}({{\textbf r};\omega } ){{\textbf e}_x} + {E_y}({{\textbf r};\omega } ){{\textbf e}_y} \\ & = \frac{x}{{{w_0}}}\exp \left( { - \frac{{{{\textbf r}^2}}}{{w_0^2}}} \right){{\textbf e}_x} + \frac{y}{{{w_0}}}\exp \left( { - \frac{{{{\textbf r}^2}}}{{w_0^2}}} \right){{\textbf e}_y}, \end{aligned}$$
where $\omega$ denotes the angular frequency and ${w_0}$ is the transverse beam size; ${\textbf r} \equiv ({x,\;y} )$ denotes an arbitrary transverse position vector at the source plane; ${{\textbf e}_x}$ and ${{\textbf e}_y}$ represent the unit vectors in the x and y directions, respectively.

On the basis of the unified theory of coherence and polarization [35,36], the second-order spatial coherence properties of a vector partially coherent beam in the spatial frequency domain can be characterized by the 2×2 the cross-spectral density (CSD) matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\textbf r}_1},{{\textbf r}_2},\omega } )$ with the matrix elements

$$W_{\alpha \beta }^{(0 )}({{{\textbf r}_1},{{\textbf r}_2},\omega } )= \langle{E_\alpha^\ast ({{{\textbf r}_1};\omega } ){E_\beta }({{{\textbf r}_2};\omega } )} \rangle ,\quad({\alpha ,\beta = x,\;y} )$$
where the asterisk denotes the complex conjugate and the angular brackets represent an ensemble average. For brevity, the angular $\omega$ dependence of all the derived quantities of interest will be omitted but implied.

To be a physically genuine CSD function, the elements of the CSD matrix should satisfy the condition of nonnegative definiteness in the form of [37]

$$W_{\alpha \beta }^{(0 )}({{{\textbf r}_1},{{\textbf r}_2}} )= \int {{p_{\alpha \beta }}} ({\textbf v} )H_\alpha ^{\ast }({{{\textbf r}_1},{\textbf v}} ){H_\beta }({{{\textbf r}_2},{\textbf v}} ){d^2}{\textbf v}, $$
where ${p_{\alpha \beta }}({\textbf v} )$ are the elements of the 2×2 weighting matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over p} ({\textbf v} )$. For obtaining the Schell-model sources, the kernel ${H_\alpha }({{\textbf r},{\textbf v}} )$ must have a Fourier-like structure, i.e.,
$${H_\alpha }({{\textbf r},{\textbf v}} ) = {F_\alpha }({\textbf r} )\exp ({ - 2\pi i{\textbf r} \cdot {\textbf v}} ),$$
where ${F_\alpha }({\textbf r} )$ are possible complex profile functions. In most previous studies, ${F_\alpha }({\textbf r} )$ is usually simply set as a real Gaussian function with no phase factor in the constructed CSD matrices. In this paper, we assume that the complex profile function in the source plane is a Laguerre Gaussian mode with separable phase [34],
$${F_\alpha }({\textbf r} ) = \frac{\alpha }{{{w_0}}}{\left( {\frac{{\sqrt 2 r}}{{{w_0}}}} \right)^l}\exp \left( { - \frac{{{r^2}}}{{w_0^2}}} \right)\exp ({ - il\phi } ),$$
where r and $\phi$ are the radial and azimuthal (angle) coordinates, l is the topological charge.

On the other hand, the choice of ${p_{\alpha \beta }}({\textbf v} )$ defines a family of sources with different correlation functions. Let us suppose ${p_{\alpha \beta }}({\textbf v} )$ to be of the form [20,3134]

$${p_{\alpha \beta }}({\textbf v} ) = \frac{{{B_{\alpha \beta }}\delta _{\alpha \beta }^2}}{{{C_0}}} \times \left\{{1 - {{\left[{1 - \textrm{exp}({ - \delta_{\alpha \beta }^2{{\textbf v}^2}/2} )} \right]}^M}} \right\}.$$
Here the variable M denotes the beam index, ${B_{\alpha \beta }} = |{{B_{\alpha \beta }}} |\textrm{exp}({i{\varphi_{\alpha \beta }}} )$ is the single-point correlation coefficient and ${\delta _{\alpha \beta }}$ is the characteristic source correlations. Meanwhile, the realizability conditions of a radially polarized source are obtained as [20]
$${B_{xy({yx} )}} = {B_{xx({yy} )}} = 1,\; {\delta _{xy({yx} )}} = {\delta _{xx({yy} )}} = {\delta _0}.$$

Substituting Eqs. (4)–(6) into Eq. (3), the elements of the CSD matrix take the following forms

$${W_{\alpha \beta }}({{{\textbf r}_1},{{\textbf r}_2},0} )= \frac{{{\alpha _1}{\beta _2}}}{{w_0^2}}{\left( {\frac{{2{r_1}{r_2}}}{{w_0^2}}} \right)^l}\exp \left( { - \frac{{r_1^2 + r_2^2}}{{w_0^2}}} \right)\exp [{il({{\phi_1} - {\phi_2}} )} ]{\mu _{\alpha \beta }}({{{\textbf r}_1} - {{\textbf r}_2}} ),$$
and the SDOC is given by
$${\mu _{\alpha \beta }}({{{\textbf r}_1} - {{\textbf r}_2}} ) = {\mu _0}({{{\textbf r}_1} - {{\textbf r}_2}} )= \frac{1}{{{C_0}}}\sum\limits_{m = 1}^M {\left( {\begin{array}{{c}} M\\ m \end{array}} \right)} \frac{{{{({ - 1} )}^{m - 1}}}}{m}\exp \left[ { - \frac{{{{({{{\textbf r}_1} - {{\textbf r}_2}} )}^2}}}{{2m\delta_0^2}}} \right],$$
where ${\delta _0}$ represents the spatial correlation length that describes the coherence properties of the effective beam source. ${C_0} = \sum\limits_{m = 1}^M {\left( {\begin{array}{{c}} M\\ m \end{array}} \right)} \frac{{{{({ - 1} )}^{m - 1}}}}{m}$ is the normalized factor with $\left( {\begin{array}{{c}} M\\ m \end{array}} \right)$ being the binomial coefficients. Here, Eq. (8) denotes the elements of CSD matrix of a radially polarized MGSM vortex beam, which carry a vortex phase with l being the topological charge, it can be an arbitrary value, both integral and fractional. One can call it as the radially polarized MGSM integral vortex (IV) beam when its topological charge is integral and radially polarized MGSM fractional vortex (FV) beam when its topological charge is fractional (or non-integer). The beam source reduces to a radially polarized MGSM source under the condition of $l = 0$ [20]. When $M = 1$, the beam source reduces to the radially polarized Gaussian Schell-model (GSM) vortex source.

3. Tight focusing properties of a radially polarized MGSM vortex beam focused by a high NA objective lens

In this section, we consider the focusing of a radially polarized MGSM vortex beam by a high NA objective lens. The geometry scheme of problem is shown as in Fig. 1.

 

Fig. 1. Schematic diagram of tight focusing of a light beam focused by a high NA objective lens. $Q({r,\;\varphi,\;z} )$ is an observation point in the focal plane.

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After being focused by a high NA objective lens, according to the Richards-Wolf vectorial diffraction, the vectorial electric field of a tightly focused cylindrical vector beam at the focal plane can be expressed as follows [1921],

$$\begin{array}{c} {E_f}({r,\;\varphi,\;z} )= \left[ {\begin{array}{{c}} {{E_{fx}}}\\ {{E_{fy}}}\\ {{E_{fz}}} \end{array}} \right] = - \frac{{i{k_1}f}}{{2\pi }}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {\left[ {\begin{array}{{c}} {{l_x}({\theta ,\phi } )\Im ({\theta ,\phi } )+ {l_y}({\theta ,\phi } )\Omega ({\theta ,\phi } )}\\ {{l_x}({\theta ,\phi } )\Omega ({\theta ,\phi } )+ {l_y}({\theta ,\phi } )\Im ({\theta ,\phi } )}\\ { - {l_x}({\theta ,\phi } )\sin \theta \cos \phi - {l_y}({\theta ,\phi } )\sin \theta \sin \phi } \end{array}} \right]} } \\ \times \sqrt {\cos \theta } \sin \theta \exp [{i{k_1}({z\;{\cos}\theta + r\;{\sin}\theta {\cos}({\phi - \varphi } )} )} ]d\phi d\theta \end{array}$$
with
$$\Im ({\theta ,\phi } ) = {\cos}\theta + \textrm{si}{\textrm{n}^2}\phi ({1 - {\cos}\theta } ),\Omega ({\theta ,\phi } ) = {\cos}\phi {\sin}\phi ({{\cos}\theta - 1} ), $$
where $r,\;\varphi,\;z$ are the cylindrical coordinates of an observation point, $\phi$ is the azimuthal angle of an incident beam, f is the focal length of the lens, ${k_1} = k{n_1} = 2\pi {n_1}/\lambda$ is the wave number in the surrounding medium with ${n_1}$ being the refractive index of the surrounding medium, $\theta$ is the NA angle, and ${\theta _{max }}$ is the maximal NA angle given by the formula ${\theta _{max }} = \textrm{arcsin}({{N_{NA}}/{n_1}} )$ with ${N_{NA}}$ being the NA number, ${l_x}({\theta ,\phi } )$ and ${l_y}({\theta ,\phi } )$ are the pupil apodization functions at aperture surface, which is derived by setting $x = f\;{\sin}\theta\;{\cos}\phi$ and $y = f\;{\sin}\theta\;{\sin}\phi$ in ${E_x}({x,\;y} )$ and ${E_y}({x,\;y} )$, respectively.

Considering a partially coherent radially polarized beam focused by a high NA objective lens, the statistical properties of the beam in the vicinity of the focal region can be characterized by the 3×3 CSD matrix with the elements [1921]

$${W_{f\alpha \beta }}({{r_1},\;{\varphi_1},\;{r_2},\;{\varphi_2},\;z} )= \langle{E_{f\alpha }^\ast ({{r_1},\;{\varphi_1},\;z} ){E_{f\beta }}({{r_2},\;{\varphi_2},\;z} )} \rangle ,\quad ({\alpha ,\beta = x,\;y,\;z} )$$
In this paper, we only consider the average intensity distribution of the tightly focused radially polarized MGSM vortex beam near the focal region, thus only the expressions for the diagonal elements of the CSD matrix in the focal plane are shown here, i.e.,
$$\begin{array}{c} {W_{fxx}}({{r_1},{\varphi_1},{r_2},{\varphi_2},{\boldsymbol z}} )= \frac{{{f^2}{n_1}}}{{{\lambda ^2}}}\int_0^{{\theta _{max }}} {\int_0^{{\theta _{max }}} {\int_0^{2\pi } {\int_0^{2\pi } {{W_0}({{\theta_1},{\phi_1},{\theta_2},{\phi_2}} )} } } } \times \textrm{exp}[{i{k_1}({{\Re_2} - {\Re_1}} )} ]\\ \times {({{\cos}{\theta_1}{\cos}{\theta_2}} )^{3/2}}{({{\sin}{\theta_1}{\sin}{\theta_2}} )^2}\;{\cos}{\phi _1}{\cos}{\phi _2}d{\theta _1}d{\theta _2}d{\phi _1}d{\phi _2}, \end{array}$$
$$\begin{array}{c} {W_{fyy}}({{r_1},{\varphi_1},{r_2},{\varphi_2},{\boldsymbol z}} )= \frac{{{f^2}{n_1}}}{{{\lambda ^2}}}\int_0^{{\theta _{max }}} {\int_0^{{\theta _{max }}} {\int_0^{2\pi } {\int_0^{2\pi } {{W_0}({{\theta_1},{\phi_1},{\theta_2},{\phi_2}} )} } } } \times \textrm{exp}[{i{k_1}({{\Re_2} - {\Re_1}} )} ]\\ \times {({{\cos}{\theta_1}{\cos}{\theta_2}} )^{3/2}}{({{\sin}{\theta_1}{\sin}{\theta_2}} )^2}{\sin}{\phi _1}{\sin}{\phi _2}d{\theta _1}d{\theta _2}d{\phi _1}d{\phi _2}, \end{array}$$
$$\begin{array}{c} {W_{fzz}}({{r_1},\;{\varphi_1},\;{r_2},\;{\varphi_2},\;z} )= \frac{{{f^2}{n_1}}}{{{\lambda ^2}}}\int_0^{{\theta _{max }}} {\int_0^{{\theta _{max }}} {\int_0^{2\pi } {\int_0^{2\pi } {{W_0}({{\theta_1},{\phi_1},{\theta_2},{\phi_2}} )} } } } \times \textrm{exp}[{i{k_1}({{\Re_2} - {\Re_1}} )} ]\\ \times {({{\cos}{\theta_1}{\cos}{\theta_2}} )^{1/2}}{({{\sin}{\theta_1}{\sin}{\theta_2}} )^3}d{\theta _1}d{\theta _2}d{\phi _1}d{\phi _2} \end{array}$$
with
$${\Re _i} = z\;{\cos}{\theta _i} + {r_i}\;{\sin}{\theta _i}\;{\cos}({{\phi_i} - {\varphi_i}} ),\; ({i = 1,2} ),$$
$$\begin{aligned} &{W_0}({{\theta_1},{\phi_1},{\theta_2},{\phi_2}} )\\ & = \frac{1}{{{C_0}}}\frac{{{f^2}}}{{w_0^2}}{\left( {\frac{{2{f^2}{\sin}{\theta_1}{\sin}{\theta_2}{\cos}({{\phi_1} - {\phi_2}} )}}{{w_0^2}}} \right)^l}\textrm{exp}\left[ { - \frac{{{f^2}}}{{w_0^2}}({\textrm{si}{\textrm{n}^2}{\theta_1} + \textrm{si}{\textrm{n}^2}{\theta_2}} ) + il({{\phi_1} - {\phi_2}} )} \right] \\ & \times \sum\limits_{m = 1}^M {\left( {\begin{array}{{c}} M\\ m \end{array}} \right)} \frac{{{{({ - 1} )}^{m - 1}}}}{m}\textrm{exp}\left[ { - \frac{{{f^2}}}{{2m\delta_0^2}}({\textrm{si}{\textrm{n}^2}{\theta_1} + \textrm{si}{\textrm{n}^2}{\theta_2} - 2{\sin}{\theta_1}{\sin}{\theta_2}{\cos}({{\phi_1} - {\phi_2}} )} )} \right]. \end{aligned}$$

By setting ${r_1} = {r_2} = r,\;{\varphi _1} = {\varphi _2} = \varphi$, one can obtain the transverse, longitudinal and total component of the focal intensity in the focal region,

$${I_{tra}}({r,\;\varphi,\;z} )= {W_{fxx}}({r,\;\varphi,\;r,\;\varphi,\;z} )+ {W_{fyy}}({r,\;\varphi,\;r,\;\varphi,\;z} ),$$
$${I_z}({r,\;\varphi,\;z} )= {W_{fzz}}({r,\;\varphi,\;r,\;\varphi,\;z} ),$$
$${I_{total}}({r,\;\varphi,\;z} )= {I_{tra}}({r,\;\varphi,\;z} )+ {I_z}({r,\;\varphi,\;z} ).$$

4. Numerical results

In this section, we will explore the focal intensity distribution of a radially polarized MGSM vortex beam passing through a high NA objective lens. In the following numerical examples, the parameters are set as ${n_1} = 1,\;{N_{NA}} = 0.95,\;f = 3.0\textrm{mm},\;{w_0} = 10\textrm{mm}$, and $\lambda = 632.8\textrm{nm}$, other beam parameters are given in each captions. Note that the range of each picture is normalized in units of eight wavelengths.

Figure 2 shows the normalized focal intensity distributions of the total component ${I_{total}}$, transverse component ${I_{tra}}$, and longitudinal component ${I_{\boldsymbol z}}$ of the radially polarized MGSM beam at the focal plane for different values of beam index M, respectively. All the intensity distributions are normalized by the maximum value of the total focal intensity component, and the green solid line denotes the corresponding cross line of the intensity distribution at ${\rho _x}({{\rho_y}} ) = 0$. It is clearly shown that both transverse and longitudinal field distributions of a tightly focused radially polarized MGSM beam can be shaped into a flat-topped profile, thus leading to a flat-topped focal intensity distribution for small spatial coherence width ${\delta _0}$ (i.e., ${\delta _0} = 0.5\textrm{mm}$), while the focal intensity distributions of the radially polarized GSM beam (i.e., $M = 1$) all are of Gaussian profiles. Furthermore, the area of the flattop region increases as the beam index M increases. Our results are consistent with those reported in [20].

 

Fig. 2. Normalized focal intensity distribution of the total component ${I_{total}}$, transverse component ${I_{tra}}$, longitudinal component ${I_z}$ and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM beam with ${\delta _0} = 0.5\textrm{mm}$ and $l = 0$ at the focal plane for different values of beam index M.

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In this paper, we mainly consider the influences of the spatial coherence width ${\delta _0}$, the beam index M and the topological charge l (both integral and fractional value) of the radially polarized MGSM vortex beam on the focal intensity distribution. We show in Figs. 3 and 4 the normalized focal intensity distribution of a tightly focused radially polarized MGSM IV beam with integral topological charge $l = 3$ and FV beam with fractional topological charge $l = 3.5$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$, respectively. One can deduce from Fig. 3 that the normalized focal intensity distribution of the radially polarized MGSM IV beam displays an anomalous hollow shaped pattern (a full-doughnut-shaped pattern for larger topological charge $l$) when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$), while it will gradually degenerates to a flat-topped profile and finally evolves into a Gaussian profile with the decrease of ${\delta _0}$. Moreover, the focal intensity distribution of the radially polarized MGSM IV beam always display circular symmetry for arbitrary spatial coherence width ${\delta _0}$. However, as shown in Fig. 4, it is particularly interesting to find that the focal intensity distribution of the radially polarized MGSM FV beam is a nonuniform asymmetric focal spot when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$), which is quite different from that of the radially polarized MGSM IV beam. In particular, with the decreases of ${\delta _0}$, the focal spot gradually loses its nonuniform asymmetric characteristics, and finally becomes a Gaussian profile with circular symmetry. Thus, it is hard to distinguish the radially polarized MGSM IV beam and FV beam from their focal intensity distributions at the focal plane when ${\delta _0}$ is sufficiently small (e.g., ${\delta _0} = 0.5\textrm{mm}$). Moreover, the beam index M has relatively small influence on the focal intensity distributions for both of the radially polarized MGSM IV beam and FV beam. This is because that the topological charge l plays a decisive role in determining the focal intensity distribution, while this dominated effect will be weakened with the decreases of spatial coherence width ${\delta _0}$.

 

Fig. 3. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with integral topological charge $l = 3$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.

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Fig. 4. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with fractional topological charge $l = 3.5$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.

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To learn more about the influences of topological charge on the focal intensity distribution, we calculate in Figs. 5 and 6 the normalized focal intensity distribution of the radially polarized MGSM IV beam and FV beam with $M = 5$ at the focal plane for different values of topological charge l and spatial coherence length ${\delta _0}$, respectively. In Fig. 5, the case of the radially polarized MGSM beam (i.e., $l = 0$) is also shown as a comparison. One can see from Fig. 5 that the focal intensity distribution of the radially polarized MGSM IV beam can form a focal spot or a doughnut-shaped focal field for different values of integral topological charge l when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$). However, it will gradually degenerates to a flat-topped shape focal field and finally becomes a quasi-Gaussian profile as spatial coherence length ${\delta _0}$ decreases, which is different from that of the radially polarized MGSM beam. As shown in Fig. 6, the focal intensity distribution can exhibit a nonuniform asymmetric focal spot or an anomalous asymmetric hollow focal field at the focal plane for the radially polarized MGSM FV beam with different values of fractional topological charge l. Of particular note is that such a nonuniform asymmetric focal spot or anomalous asymmetric hollow focal field will be useful for trapping irregular particles or manipulating absorbing particles. A comparison of Figs. 5 and 6 shows that, the degree of asymmetry of the focal intensity distribution is associated with the fractional part of the topological charge, while the extent of hollow is closely related to its integral part of the topological charge. In particular, the pattern of the focal intensity distribution will be reversed for the radially polarized MGSM FV beam with opposite topological charge (see $l = 5.5$ and $l = - \textrm{5}.5$). However, with the decrease of ${\delta _0}$, the focal intensity distributions of the radially polarized MGSM FV beam gradually lose their nonuniform asymmetric characteristics, and finally all focal intensity profiles become quasi-Gaussian profiles with circular symmetry when ${\delta _0}$ is too small (i.e., ${\delta _0} = 0.5\textrm{mm}$). In a word, we can shape the focal intensity distribution of the radially polarized MGSM vortex beam by optimizing its beam parameters, such as the topological charge l, spatial coherence length ${\delta _0}$ and beam index M.

 

Fig. 5. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with $M = 5$ at the focal plane for different values of integral topological charge l and spatial coherence length ${\delta _0}$.

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Fig. 6. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with $M = 5$ at the focal plane for different values of fractional topological charge l and spatial coherence length ${\delta _0}$.

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5. Summary

In conclusion, a new kind of partially coherent vector vortex beam named radially polarized MGSM vortex beam carrying vortex phase with tunable topological charge (both integral and fractional values) was introduced as a natural extension of radially polarized MGSM beam. The focal intensity distributions of the radially polarized MGSM vortex beam passing through a high NA objective lens have been explored numerically. The combined effect of both spatial coherence width and the topological charge on the tight focusing properties has been investigated in detail. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structures of the correlation functions and the vortex phase. Some peculiar focal field with novel structure, e.g., a flat-topped focal field, a circular focal spot or a nonuniform asymmetric focal spot, a doughnut profiles or an anomalous asymmetric hollow focal field, are obtained by choosing suitable values of beam index M, spatial coherence length ${\delta _0}$ and topological charge l of the radially polarized MGSM vortex beam. In contrast with the integral vortex beam, the nonuniformity and asymmetry of a formed focal spot or anomalous hollow focal field is the most attractive property for the radially polarized MGSM FV beam. However, such nonuniform and asymmetric characteristics will be gradually vanished, and finally all focal intensity profiles become quasi-Gaussian profiles with circular symmetry as the spatial coherence length decreases. Our results will be rewarding in various applications, such as optical data storage and optical trapping, especially for trapping irregular particles or manipulating absorbing particles, where there some unique or desired focal fields are required.

Funding

National Natural Science Foundation of China (11747065); Natural Science Foundation of Anhui Province (1808085QA10).

References

1. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008). [CrossRef]  

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

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

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

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

6. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010). [CrossRef]  

7. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012). [CrossRef]  

8. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006). [CrossRef]  

9. 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]  

10. 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]  

11. 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]  

12. S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011). [CrossRef]  

13. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015). [CrossRef]  

14. Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015). [CrossRef]  

15. Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016). [CrossRef]  

16. X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017). [CrossRef]  

17. L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011). [CrossRef]  

18. 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]  

19. 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]  

20. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]  

21. H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018). [CrossRef]  

22. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009). [CrossRef]  

23. L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016). [CrossRef]  

24. X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017). [CrossRef]  

25. I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995). [CrossRef]  

26. Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018). [CrossRef]  

27. S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005). [CrossRef]  

28. Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017). [CrossRef]  

29. J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018). [CrossRef]  

30. J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019). [CrossRef]  

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

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

33. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). [CrossRef]  

34. Z. Mei, Y. Mao, and Y. Wang, “Electromagnetic multi-Gaussian Schell-model vortex light sources and their radiation field properties,” Opt. Express 26(17), 21992–22000 (2018). [CrossRef]  

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

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

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

References

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  • |

  1. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
    [Crossref]
  2. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  3. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref]
  4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref]
  5. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [Crossref]
  6. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
    [Crossref]
  7. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012).
    [Crossref]
  8. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
    [Crossref]
  9. 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]
  10. 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]
  11. 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]
  12. S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011).
    [Crossref]
  13. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
    [Crossref]
  14. Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
    [Crossref]
  15. Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
    [Crossref]
  16. X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017).
    [Crossref]
  17. L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
    [Crossref]
  18. 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]
  19. 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]
  20. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
    [Crossref]
  21. H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
    [Crossref]
  22. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
    [Crossref]
  23. L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
    [Crossref]
  24. X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
    [Crossref]
  25. I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
    [Crossref]
  26. Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
    [Crossref]
  27. S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
    [Crossref]
  28. Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
    [Crossref]
  29. J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
    [Crossref]
  30. J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
    [Crossref]
  31. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref]
  32. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref]
  33. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
    [Crossref]
  34. Z. Mei, Y. Mao, and Y. Wang, “Electromagnetic multi-Gaussian Schell-model vortex light sources and their radiation field properties,” Opt. Express 26(17), 21992–22000 (2018).
    [Crossref]
  35. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [Crossref]
  36. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University).
  37. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]

2019 (1)

2018 (4)

2017 (4)

2016 (2)

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
[Crossref]

2015 (3)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (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]

2014 (1)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

2012 (4)

2011 (3)

S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011).
[Crossref]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

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]

2010 (2)

2009 (3)

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

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

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

2008 (2)

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
[Crossref]

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]

2006 (1)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[Crossref]

2005 (1)

2004 (1)

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]

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

2002 (1)

1995 (1)

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Basistiy, I.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Bo, F.

Cai, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[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]

Chen, B.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Chen, H.

Chen, L.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[Crossref]

Chen, Z.

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[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]

Ding, J.

Dong, Y.

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[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]

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]

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]

Fang, Y.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Feng, B.

Gong, L.

Gori, F.

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

Gu, B.

Guo, H.

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]

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Hua, L.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Huang, K.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

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]

Huang, L.

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]

Jabbour, T. G.

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Kang, X. L.

Korotkova, O.

Kozawa, Y.

Kuebler, S. M.

Leger, J.

Leuchs, G.

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

Li, J.

Li, X.

Li, Y. P.

Li, Z.

Liang, C.

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

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]

Lin, J.

Ling, L.

Liu, L.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Liu, X.

Lu, Q.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Lu, X.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

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]

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Mao, Y.

Mei, Z.

Niu, H.

Peng, X.

Ping, C.

Pu, J.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Qian, B.

Qin, F.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Qu, J.

H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
[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]

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]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Rupp, R.

Sahin, S.

Sanchez, V. R.

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

Santarsiero, M.

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

Sato, S.

Shchepakina, E.

Sheng, Z.

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. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Soskin, M.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

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]

Tao, S.

Urbach, H.

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Vasnetsov, M.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Wang, F.

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, L.

Wang, S.

Wang, X.

Wang, Y.

Wei, L.

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Wen, J.

Wolf, E.

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

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

Wu, H.

Wu, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Xu, H.

H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
[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]

Yan, S.

Yang, X.

Yang, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Yao, B.

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Yuan, X. C.

Zeng, J.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
[Crossref]

Zeng, T.

Zhan, Q.

Zhang, J.

Zhang, W.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Zhang, X.

Zhang, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

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.

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[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]

Zhou, Y.

Zhu, B.

Zhu, S.

Zhu, X.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Zhu, Z.

Adv. Opt. Photonics (1)

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

J. Opt. (3)

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]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

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

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

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

Nanophotonics (1)

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[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. (2)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[Crossref]

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Opt. Express (14)

X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
[Crossref]

J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

Z. Mei, Y. Mao, and Y. Wang, “Electromagnetic multi-Gaussian Schell-model vortex light sources and their radiation field properties,” Opt. Express 26(17), 21992–22000 (2018).
[Crossref]

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
[Crossref]

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

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

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
[Crossref]

S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

Opt. Laser Technol. (2)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

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. (4)

Phys. Lett. A (2)

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

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Phys. Rev. A (2)

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[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]

Phys. Rev. Lett. (1)

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

Sci. Rep. (1)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Other (1)

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

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of tight focusing of a light beam focused by a high NA objective lens. $Q({r,\;\varphi,\;z} )$ is an observation point in the focal plane.
Fig. 2.
Fig. 2. Normalized focal intensity distribution of the total component ${I_{total}}$, transverse component ${I_{tra}}$, longitudinal component ${I_z}$ and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM beam with ${\delta _0} = 0.5\textrm{mm}$ and $l = 0$ at the focal plane for different values of beam index M.
Fig. 3.
Fig. 3. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with integral topological charge $l = 3$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.
Fig. 4.
Fig. 4. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with fractional topological charge $l = 3.5$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.
Fig. 5.
Fig. 5. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with $M = 5$ at the focal plane for different values of integral topological charge l and spatial coherence length ${\delta _0}$.
Fig. 6.
Fig. 6. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with $M = 5$ at the focal plane for different values of fractional topological charge l and spatial coherence length ${\delta _0}$.

Equations (20)

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E ( r ; ω ) = E x ( r ; ω ) e x + E y ( r ; ω ) e y = x w 0 exp ( r 2 w 0 2 ) e x + y w 0 exp ( r 2 w 0 2 ) e y ,
W α β ( 0 ) ( r 1 , r 2 , ω ) = E α ( r 1 ; ω ) E β ( r 2 ; ω ) , ( α , β = x , y )
W α β ( 0 ) ( r 1 , r 2 ) = p α β ( v ) H α ( r 1 , v ) H β ( r 2 , v ) d 2 v ,
H α ( r , v ) = F α ( r ) exp ( 2 π i r v ) ,
F α ( r ) = α w 0 ( 2 r w 0 ) l exp ( r 2 w 0 2 ) exp ( i l ϕ ) ,
p α β ( v ) = B α β δ α β 2 C 0 × { 1 [ 1 exp ( δ α β 2 v 2 / 2 ) ] M } .
B x y ( y x ) = B x x ( y y ) = 1 , δ x y ( y x ) = δ x x ( y y ) = δ 0 .
W α β ( r 1 , r 2 , 0 ) = α 1 β 2 w 0 2 ( 2 r 1 r 2 w 0 2 ) l exp ( r 1 2 + r 2 2 w 0 2 ) exp [ i l ( ϕ 1 ϕ 2 ) ] μ α β ( r 1 r 2 ) ,
μ α β ( r 1 r 2 ) = μ 0 ( r 1 r 2 ) = 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m exp [ ( r 1 r 2 ) 2 2 m δ 0 2 ] ,
E f ( r , φ , z ) = [ E f x E f y E f z ] = i k 1 f 2 π 0 θ max 0 2 π [ l x ( θ , ϕ ) ( θ , ϕ ) + l y ( θ , ϕ ) Ω ( θ , ϕ ) l x ( θ , ϕ ) Ω ( θ , ϕ ) + l y ( θ , ϕ ) ( θ , ϕ ) l x ( θ , ϕ ) sin θ cos ϕ l y ( θ , ϕ ) sin θ sin ϕ ] × cos θ sin θ exp [ i k 1 ( z cos θ + r sin θ cos ( ϕ φ ) ) ] d ϕ d θ
( θ , ϕ ) = cos θ + si n 2 ϕ ( 1 cos θ ) , Ω ( θ , ϕ ) = cos ϕ sin ϕ ( cos θ 1 ) ,
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 m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 3 / 2 ( sin θ 1 sin θ 2 ) 2 cos ϕ 1 cos ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W f y y ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 3 / 2 ( sin θ 1 sin θ 2 ) 2 sin ϕ 1 sin ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W f z z ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 1 / 2 ( sin θ 1 sin θ 2 ) 3 d θ 1 d θ 2 d ϕ 1 d ϕ 2
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 ) = 1 C 0 f 2 w 0 2 ( 2 f 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 2 ) w 0 2 ) l exp [ f 2 w 0 2 ( si n 2 θ 1 + si n 2 θ 2 ) + i l ( ϕ 1 ϕ 2 ) ] × m = 1 M ( M m ) ( 1 ) m 1 m exp [ f 2 2 m δ 0 2 ( si n 2 θ 1 + si n 2 θ 2 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 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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