Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams

Hua-Feng Xu; Yuan Zhou; Hong-Wei Wu; Hua-Jun Chen; Zong-Qiang Sheng; Jun Qu

doi:10.1364/OE.26.020076

1. Introduction

Polarization is one of the most important features of light [1]. According to the state of polarization (SOP), vector beams can be classified as beams with spatially uniform SOP and beams with spatially non-uniform SOP [2]. As the typical kinds of cylindrical vector beams with non-uniform SOP, the radially polarized beams and azimuthally polarized beams have been explored extensively due to their wide applications in free-space optical communications, optical trapping, optical data storage and so on [2–11]. A strong longitudinally polarized field component will be produced when a radially polarized beam is tightly focused by a high numerical aperture (NA) objective lens, thus a tighter spot beyond the diffraction limit can be created compared with a focused linearly polarized one [6]. For a tightly focused azimuthally polarized beam, a strong magnetic field will be generated on the axis while the electric field is purely transverse [6]. Due to their quite different focusing properties, Zhan et. al used a polarization rotator setup to shape the focal field by adjusting the rotating angle of the polarization rotator [6]. Besides the polarization, the phase also plays an important role in shaping the focal field. Some novel and peculiar focal field patterns, such as optical chain, optical needle, optical dark channel and optical cage, can be realized by engineering the polarization and phase of the incident beam [12–16]. For instance, a needle of longitudinally polarized light can be created by focusing a radially polarized beam with the help of phase modulation [12,13]. Moreover, shaping a subwavelength optical needle with ultra-long length was designed and experimentally demonstrated by use of planar metalens [14]. In our previous work, we have proposed a 3D focus shaping technique by focusing partially coherent circularly polarized vortex beams with a binary diffractive optical element (DOE) [15]. Focus shaping has become a subject of considerable importance in optical tweezers and optical high-resolution microscopy [6,12–18].

Coherence is another one of the most important features of light [1]. It has been revealed that spatial coherence of partially coherent beams has significant effects on the tight focusing properties, and the transverse field distribution can be shaped by changing the initial coherence width [19–23]. However, previous studies on the tight focusing properties of partially coherent beams were confined to conventional partially coherent beams whose correlation function satisfies Gaussian distribution [19–23]. More recently, partially coherent beams with nonconventional correlation functions have attracted considerable attention since the sufficient conditions for devising physically realizable correlation functions were established by Gori et al [24,25]. A slew of novel models of partially coherent beams with different correlation characteristics were devised and generated, and some intriguing properties were extensively explored [26–36]. It has been pointed out in [26–31] that customizing the spatial correlation properties of the source field is an efficient way to generate desired far-field beam patterns. However, only few papers have been concerned with the tight focusing properties of partially coherent beams with special correlation functions [34,35].

In this paper, we introduce a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized LGCSM vortex beam as a natural extension of scalar LGCSM vortex beam [36], and discuss its realizability conditions. The tight focusing properties of a radially polarized LGCSM vortex beam passing through a high NA objective lens are investigated numerically. It is particularly interesting to find 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 is quite different from those of conventional radially polarized partially coherent beams [21,22]. Apart from the spatial coherence, the influences of the vortex phase of such a beam on the focal field intensity distributions are also considered. At last, the focus shaping can be achieved by choosing suitable values of beam parameters of the incident beam. Our results indicate that modulating the structures of the correlation functions of the partially coherent beams provide a novel approach for shaping the focal field, which may find valuable applications in optical trapping, optical data storage and optical high-resolution microscopy.

2. Theoretical model and its realizability conditions for a radially polarized LGCSM vortex beam

Based on the unified theory of coherence and polarization, in the space-frequency domain, the second-order statistical properties of a partially coherent vector beam can be characterized by the cross-spectral density (CSD) matrix $\overset{\leftrightarrow}{W} (r_{1}, r_{2}, ω, z)$ [1]. In the source plane, the elements of the CSD matrix are expressed as

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)

where

r_{1} \equiv (x_{1}, y_{1})

and

r_{2} \equiv (x_{2}, y_{2})

are two arbitrary position vectors in the source plane. The asterisk denotes the complex conjugate and the angular brackets represent ensemble average. For brevity, the dependence of the derived quantities on the angular frequency

ω

will be omitted.

In order to be physically realizable, the elements of the CSD matrix must have an integral representation in the form of [25]

W_{α β}^{(0)} (r_{1}, r_{2}) = \int p_{α β} (v) H_{α}^{*} (r_{1}, v) H_{β} (r_{2}, v) d^{2} v, (α, β = x, y)

where

H_{x}

and

H_{y}

are two arbitrary kernel functions;

p_{α β} (v)

are the elements of the 2 × 2 weighting matrix

\overset{\leftrightarrow}{p} (v) = (\begin{matrix} p_{x x} (v) & p_{x y} (v) \\ p_{y x} (v) & p_{y y} (v) \end{matrix})

The elements of the weighting matrix should satisfy the conditions for any $v$ [25]

p_{x x} (v) \geq 0, p_{y y} (v) \geq 0, p_{x x} (v) p_{y y} (v) - {| p_{x y} (v) |}^{2} \geq 0.

As a natural extension of a scalar LGCSM beam shown in [27], if we set the functions $H_{x} (r, v)$ , $H_{y} (r, v)$ and $p_{α β} (v)$ as follows

H_{α} (r, v) = \frac{α}{w_{0}} e x p (- \frac{r^{2}}{4 w_{0}^{2}}) e x p (- i r \cdot v), (α = x, y),

p_{α β} (v) = B_{α β} (π^{2 n + 1} δ_{0 α β}^{2 n + 2} 2^{n + 1} / n!) v^{2 n} e x p (- 2 π^{2} δ_{0 α β}^{2} v^{2}) .

Substituting Eqs. (5) and (6) into Eq. (2), the elements of the CSD matrix take the following forms

W_{α β} (r_{1}, r_{2}) = \frac{α_{1} β_{2}}{w_{0}^{2}} e x p [- \frac{r_{1}^{2} + r_{2}^{2}}{4 w_{0}^{2}}] μ_{α β} (r_{1} - r_{2}), (α, β = x, y)

with

μ_{α β} (r_{1} - r_{2}) = B_{α β} e x p [- \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0 α β}^{2}}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0 α β}^{2}}]

where

L_{n}^{0} (\cdot)

denotes the Laguerre polynomial of mode order n and 0,

w_{0}

is the transverse beam width of the source,

δ_{0 x x}, δ_{0 y y}, δ_{0 x y}

are the coherence widths of the correlation functions of x-x, y-y and x-y components, respectively.

B_{α β} = | B_{α β} | e x p (i ϕ_{α β})

is the complex correlation coefficient between

E_{α}

and

E_{β}

components with

ϕ_{α β}

being the phase difference. The beam source whose CSD matrix elements are given by Eq. (7) is called as a radially polarized LGCSM source.

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

B_{x x} = B_{y y} = 1, B_{x y} = B_{y x}^{*}, δ_{0 x y} = δ_{0 y x}

Second, the CSD matrix of the radially polarized LGCSM source should satisfy the nonnegative conditions as shown in Eq. (4). It is evident that the first two inequalities are always true for any value $v$ , while the third inequality reduces to,

δ_{0 x x}^{2 n + 2} δ_{0 y y}^{2 n + 2} e x p [- 2 π^{2} (δ_{0 x x}^{2} + δ_{0 y y}^{2}) v^{2}] \geq {| B_{x y} |}^{2} {(δ_{0 x y}^{2 n + 2})}^{2} e x p (- 4 π^{2} δ_{0 x y}^{2} v^{2})

Since both sides of the above inequality equation are monotonic function with $v^{2}$ , we can obtain the following inequality easily by setting $v = 0$ ,

\sqrt{(δ_{0 x x}^{2} + δ_{0 y y}^{2}) / 2} \leq δ_{0 x y} \leq \sqrt{δ_{0 x y} δ_{0 x y} / {| B_{x y} |}^{1 / (n + 1)}}

Equation (11) is the realizability (i.e., necessary and sufficient) conditions for a radially polarized LGCSM source.

Moreover, for a radially polarized LGCSM source, besides the general condition discussed above, two additional restrictions should be satisfied [37], i.e.,

(a) Any point in the source plane is linearly polarized, which means that the minor semiaxes of the polarization ellipse equals to zero.
(b) The orientation angle of the polarization at any point in the source plane should satisfies $θ (x, y) = a r c t a n (y / x)$ .

In general, a partially coherent vector beam can be decomposed into a superimposition of completely polarized portion and completely unpolarized portion. According to [38], the orientation angle $θ$ and the degree of ellipticity $ε$ in the source plane can be expressed in terms of the elements of the CSD matrix, i.e.,

θ (r, 0) = \frac{1}{2} a r c \tan [\frac{2 R e [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 \leq ε \leq 1) .

where

A_{+}

and

A_{-}

stand for the major and minor semi-axes of the polarization ellipse and take the form as

\begin{matrix} A_{\pm} (r, 0) = \frac{1}{\sqrt{2}} {\sqrt{{[W_{x x} (r, r, 0) - W_{y y} (r, r, 0)]}^{2} +4 {| W_{x y} (r, r, 0) |}^{2}} \\ \pm \sqrt{{[W_{x x} (r, r, 0) - W_{y y} (r, r, 0)]}^{2} +4 {(R e [W_{x y} (r, r, 0)])}^{2}}} . \end{matrix}

Under the conditions (a) and (b), one can obtain

R e [B_{x y}] = 1, | B_{x y} | = 1

It follows from Eq. (15) that $B_{x y} = B_{y x} = 1$ . Finally, the realizability conditions of a radially polarized LGCSM beam can be determined as

B_{x x} = B_{y y} = B_{x y} = B_{y x} = 1

δ_{0 x x} = δ_{0 y y} = δ_{0 x y} = δ_{0 y x} = δ_{0}

Suppose that a radially polarized LGCSM beam passes through a spiral phase plate with transmission function $T (φ) = e x p (i l φ)$ where $l$ denotes the topological charge and $φ$ denotes the azimuthal coordinate (i.e., $T (x, y) = e x p [i l a r c t a n (y / x)]$ in the Cartesian coordinates). After passing through the spiral phase plate, the radially polarized LGCSM beam is converted into a radially polarized LGCSM vortex beam [36]. The elements of the CSD matrix for a radially polarized LGCSM vortex beam in the source plane ( $z = 0$ ) are described as

\begin{matrix} W_{α β} (r_{1}, r_{2}) = \frac{α_{1} β_{2}}{w_{0}^{2}} e x p [- \frac{r_{1}^{2} + r_{2}^{2}}{4 w_{0}^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0}^{2}}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0}^{2}}] \\ \times e x p [- i l (φ_{1} - φ_{2})], (α, β = x, y) \end{matrix}

The radially polarized LGCSM vortex beam carry a vortex phase with $l$ being the topological charge. Under the condition of $l = 0$ , the beam source reduces to a radially polarized LGCSM source.

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

In this section, we study the tight focusing properties of a radially polarized LGCSM vortex beam passing through a high NA objective lens. First, 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.

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

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According to the Richards-Wolf vectorial diffraction integral, in the cylindrical coordinate system, the vectorial electric field of a tightly focused cylindrical vector beam in the focal region can be expressed as follows [21,34]

\begin{matrix} E_{f} (r, φ, z) = [\begin{matrix} E_{f x} \\ E_{f y} \\ E_{f z} \end{matrix}] = - \frac{i k_{1} f}{2 π} \int_{0}^{θ_{\max}} \int_{0}^{2 π} [\begin{matrix} l_{x} (θ, ϕ) [\cos θ + \sin^{2} 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 + \sin^{2} ϕ (1 - \cos θ)] \\ - l_{x} (θ, ϕ) \cos ϕ \sin θ - l_{y} (θ, ϕ) \sin ϕ \sin θ \end{matrix}] \\ \times \sqrt{\cos θ} \sin θ \exp [i k_{1} (z \cos θ + r \sin θ \cos (ϕ - φ))] d ϕ d θ \end{matrix}

where

r, φ, z

are the cylindrical coordinates of an observation point,

ϕ

is the azimuthal angle of an incident beam,

f

is the focal length of the lens,

k_{1} = k n_{1} = 2 π n_{1} / λ

is the wave number in the surrounding medium with

n_{1}

being the refractive index of the surrounding medium,

θ

is the NA angle, and

θ_{\max}

is the maximal NA angle given by the formula

θ_{\max} = a r c s i n (N_{N A} / n_{1})

with

N_{N A}

being the NA number,

l_{x} (θ, ϕ)

and

l_{y} (θ, ϕ)

are the pupil apodization functions at aperture surface and are derived by setting

x = f s i n θ c o s ϕ

and

y = f s i n θ s i n ϕ

in

E_{x} (x, y)

and

E_{y} (x, y)

, respectively.

After passing through a high NA objective lens, the statistical properties of the electric field near the focal region can be characterized by the 3 × 3 CSD matrix, and the elements is expressed [21,34]

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)

In this paper, we only consider the average intensity distribution of the tightly focused radially polarized LGCSM 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{matrix} W_{f x x} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = \frac{f^{2} n_{1}}{λ^{2}} \int_{0}^{θ_{\max}} \int_{0}^{θ_{\max}} \int_{0}^{2 π} \int_{0}^{2 π} W_{0} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \times e x p [i k_{1} (ζ_{2} - ζ_{1})] \\ \times {(\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}, \end{matrix}

\begin{matrix} W_{f y y} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = \frac{f^{2} n_{1}}{λ^{2}} \int_{0}^{θ_{\max}} \int_{0}^{θ_{\max}} \int_{0}^{2 π} \int_{0}^{2 π} W_{0} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \times e x p [i k_{1} (ζ_{2} - ζ_{1})] \\ \times {(\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}, \end{matrix}

\begin{matrix} W_{f z z} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = \frac{f^{2} n_{1}}{λ^{2}} \int_{0}^{θ_{\max}} \int_{0}^{θ_{\max}} \int_{0}^{2 π} \int_{0}^{2 π} W_{0} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \times e x p [i k_{1} (ζ_{2} - ζ_{1})] \\ \times {(\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} \end{matrix}

with

ζ_{i} = z \cos θ_{i} + r_{i} \sin θ_{i} \cos (ϕ_{i} - φ_{i}), (i = 1, 2)

\begin{matrix} W_{0} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = \frac{f^{2}}{w_{0}^{2}} e x p [- (\frac{f^{2}}{4 w_{0}^{2}} + \frac{f^{2}}{2 δ_{0}^{2}}) (\sin^{2} θ_{1} + \sin^{2} θ_{2}) - i l (ϕ_{1} - ϕ_{2})] \\ \times L_{n}^{0} [\frac{f^{2}}{2 δ_{0}^{2}} (\sin^{2} θ_{1} + \sin^{2} θ_{2} - 2 \sin θ_{1} \sin θ_{2} \cos (ϕ_{1} - ϕ_{2}))] \end{matrix}

By setting $r_{1} = r_{2} = r, φ_{1} = φ_{2} = φ$ , one can obtain the transverse, longitudinal and total intensity distributions in the focal region,

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),

4. Numerical results

In this section, we will explore the average intensity distribution of a radially polarized LGCSM vortex beam passing through a high NA objective lens with the help of the above derived formulae. In the following numerical examples, the parameters are set as $n_{1} = 1, N_{N A} = 0.95, f = 3.0 m m, w_{0} = 10 m m$ , and $λ = 632.8 n m$ , other beam parameters are given in each captions. All the lengths are normalized in units of wavelength.

Figures 2-4 show the intensity distributions of the total intensity $I_{t o t a l}$ , transverse intensity $I_{t r a}$ , and longitudinal intensity $I_{z}$ of the radially polarized LGCSM vortex beam in the focal plane for different values of mode order $n$ , coherence width $δ_{0}$ and topological charge $l$ , respectively. All the intensity distributions are normalized by the maximum value of the total intensity distribution, and the green solid line denotes the corresponding cross line of the intensity distribution at $ρ_{x} (ρ_{y}) = 0$ . In Fig. 2, the case of the radially polarized GSM (i.e., $n = 0$ ) vortex beam is also shown as a comparison under the same tight focusing conditions. One clearly sees that the intensity distributions $I_{t o t a l}$ , $I_{t r a}$ and $I_{z}$ of the radially polarized GSM vortex beam all display quasi-Gaussian profiles [see in Figs. 2(a-1)-2(a-3)] due to the fact that the correlation functions of the incident beam are of Gaussian distributions. It is particularly interesting to find that the transverse intensity $I_{t r a}$ and the longitudinal intensity $I_{z}$ gradually evolve into a flat-topped or doughnut beam profile by increasing the mode order $n$ , thus the total focused intensity distribution $I_{t o t a l}$ can form a flat-topped [see in Fig. 2(b-1)] or doughnut beam profile [see in Fig. 2(c-1) and 2(d-1)] for the radially polarized LGCSM vortex beam. This indicates that not only the transverse component but also the longitudinal component of the focal field distributions can be modulated by regulating the structures of correlation functions, which is due to the effects of the special correlation functions (i.e., the Laguerre-Gaussian correlation function). However, as shown in Fig. 3, one can see that all the intensity distributions of $I_{t o t a l}$ , $I_{t r a}$ and $I_{z}$ gradually lose their doughnut beam profiles with the increase of coherent length $δ_{0}$ . This is because that the effects of the correlation functions on the focused intensity distributions will be weakened with the increase of coherent length $δ_{0}$ . Apart from the spatial coherence, as shown in Fig. 4, the vortex phase also has significant influence on the focused intensity distribution, and this effect will gradually plays a dominate role in the formation of focal field as the topological charge $l$ increases. In a word, we can shape the focused intensity distribution by regulating the beam parameters, such as the mode order $n$ , the topological charge $l$ and coherence width $δ_{0}$ of the radially polarized LGCSM vortex beam.

Fig. 2 Intensity distributions of the total intensity $I_{t o t a l}$ , transverse intensity $I_{t r a}$ , and longitudinal intensity $I_{z}$ of a tightly focused radially polarized LGCSM vortex beam with coherence width $δ_{0} = 1 mm$ and topological charge $l = 1$ in the focal plane for different values of mode order $n$ . The green solid line denotes the corresponding cross line of the intensity distribution at $ρ_{x} (ρ_{y}) = 0$ .

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Fig. 3 Intensity distributions of the total intensity $I_{t o t a l}$ , transverse intensity $I_{t r a}$ , and longitudinal intensity $I_{z}$ of a tightly focused radially polarized LGCSM vortex beam with mode order $n = 3$ and topological charge $l = 1$ in the focal plane for different values of coherence width $δ_{0}$ . The green solid line denotes the corresponding cross line of the intensity distribution at $ρ_{x} (ρ_{y}) = 0$ .

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Fig. 4 Intensity distributions of the total intensity $I_{t o t a l}$ , transverse intensity $I_{t r a}$ , and longitudinal intensity $I_{z}$ of a tightly focused radially polarized LGCSM vortex beam with coherence width $δ_{0} = 1 mm$ and mode order $n = 3$ in the focal plane for different values of topological charge $l$ . The green solid line denotes the corresponding cross line of the intensity distribution at $ρ_{x} (ρ_{y}) = 0$ .

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To view the evolution behavior of a radially polarized LGCSM vortex beam under the tight focusing conditions in the vicinity of the focal plane, we calculate in Figs. 5-7 the total intensity distributions of the radially polarized LGCSM vortex beam for different values of the mode order $n$ , the coherence width $δ_{0}$ and the topological charge $l$ in the $ρ - z$ plane, respectively. As shown in Fig. 5(a), the focal intensity profile for radially polarized GSM vortex beam exhibits a peak-centered shape due to its Gaussian correlation functions. Particularly interesting, for the radially polarized LGCSM vortex beam, the needle-like focal field [see Fig. 5(b) and Fig. 6(c)] or a controllable 3D optical cage [see Fig. 5(c) and 5(d)] can be generated by regulating the mode order $n$ or the coherence width $δ_{0}$ due to its special structure of the correlation functions. The 3D optical cages will be useful for trapping low-index particles [9]. However, as illustrated in Figs. 6 and 7, the 3D optical cage gradually disappears and degenerates to a peak-centered shape with the increase of coherence width $δ_{0}$ . One can explain by the fact that the effect of the special correlation functions will be weakened with the decrease of the mode order n or with the increase of the coherent length $δ_{0}$ . On the other hand, apart from the source spectral degree of coherence, vortex phase will also has significant effect on the focal field intensity distribution. This effect will gradually play a dominate role in the formation of focal field, and the 3D cage structure will be lost as the topological charge $l$ increases. These results show that the focal intensity distributions can be shaped through modulating the structures of the correlation functions and choosing appropriate beam parameters for a radially polarized LGCSM vortex beam.

Fig. 5 Intensity distributions of the total intensity $I_{t o t a l}$ of a tightly focused radially polarized LGCSM vortex beam with coherence width $δ_{0} = 1 mm$ and topological charge $l = 1$ in the $ρ - z$ plane for different values of mode order $n$ .

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Fig. 6 Intensity distributions of the total intensity $I_{t o t a l}$ of a tightly focused radially polarized LGCSM vortex beam with mode order $n = 3$ and topological charge $l = 3$ in the $ρ - z$ plane for different values of coherence width $δ_{0}$ .

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Fig. 7 Intensity distributions of the total intensity $I_{t o t a l}$ of a tightly focused radially polarized LGCSM vortex beam with mode order $n = 3$ and coherence width $δ_{0} = 1 mm$ in the $ρ - z$ plane for different values of topological charge $l$ .

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5. Summary

In this paper, a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized LGCSM vortex beam was introduced as a natural extension of scalar LGCSM vortex beam, and its realizability conditions were discussed. The tight focusing properties of the radially polarized LGCSM vortex beam passing through a high NA objective lens were explored numerically. It is particularly interest to find 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 is quite different from those of conventional radially polarized partially coherent beams. Furthermore, we have achieved the purpose of the focus shaping and obtained some widely used focal field with novel structure, such as focal spot, flat-topped or doughnut beam profiles, needle-like focal field and controllable 3D optical cage, by choosing some suitable values of mode order $n$ , topological charge $l$ and coherence width $δ_{0}$ . These results may have some potential applications in optical data storage, laser processing and optical manipulation.

Funding

National Natural Science Foundation of China (NSFC) (11747065, 11374015); Natural Science Foundation of Anhui Province (1808085QA10, 1708085QA11).

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Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams

Abstract

1. Introduction

2. Theoretical model and its realizability conditions for a radially polarized LGCSM vortex beam

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

4. Numerical results

5. Summary

Funding

References and links

Cited By

Figures (7)

Equations (28)

Optics Express