Tunable optical cage array generated by Dammann vector beam

Xiaoyu Weng; Luping Du; Peng Shi; Xiaocong Yuan

doi:10.1364/OE.25.009039

1. Introduction

Optical cages with an intensity cavity have attracted great interest in recent years. Owing to the peculiar light intensity distribution, optical cages give rise to many applications, such as optical trapping [1–4], optical cloaking [5,6], optical imaging [7,8], etc. Known as optical tweezers, optical cages can be as an effective tool for the manipulation of particles with low refractive index [9,10]. For example, the dark spots generated by an azimuthally-polarized beam can trap low refractive index particles in two-dimensions [9], and a more stable trap can be achieved by providing an additional gradient force along the z-axis. For the optical imaging in a stimulated emission depletion (STED) microscopy, hollow dark spot formed by a circularly-polarized vortex beam or an azimuthally-polarized beam is usually utilized as an erase beam [11,12] to promote the lateral resolution. However, it was not until 2008 that the system of STED has the comparable axial and lateral resolution with optical cage [7]. In term of optical cloaking, particles inside an optical cage could be invisible, which makes cloaking of macro-particles possible in the visible spectral range [5].

So far, many technologies were proposed to generate optical cages, such as special incident vector beam [13–16], diffractive optical element (DOE) [5,8,17,18], optical fiber [19], etc. However, the above methods can only generate a single uniform optical cage, which cannot meet the technical requirement. For example, a regular STED has only a quarter of imaging speed compared to the parallelized STED nanoscopy that with four dark spots [20], because the imaging speed is mainly restricted by the number of optical cages. Thus, it is desired to create multiple optical cages simultaneously, with the number and their positions adjustable. In recent years, uniform optical cages along the optical axis have been realized already based on the reversal radiation methods [21,22]. This type of optical cage array along optical axis, in principle, can be used for acquiring more information at different depth of a sample. However, it cannot provide better imaging speed in the process of imaging. In most cases, high imaging speed is a key aspect to realize the real-time and in-vivo imaging of a sample. This requires multiple optical cages been generated simultaneously in the transverse plane, which, to the best of our knowledge, is not achieved up to now.

In this work, we demonstrate for the first time in theory that optical cage arrays with high tunability can be created in the transverse plane with a full polarization-controlled method. In contrast to the formal techniques with the amplitude or phase modulation, optical cage arrays in this work can be generated by a so-called “Dammann vector beam (DVB)”, both under the conditions of high- and low-NA focusing system. The DVB is formed by combining two orthogonally-polarized beams modulated respectively with a Dammann grating (DG). They are radially- and azimuthally-polarized beams for the high-NA focusing condition, and the x- and y-polarized beams for the low-NA condition. By changing the polarization state of DVB with the parameters of DGs, the optical cage array with adjustable number and positions can be obtained in the focal region by superposing the bright spot arrays and the hollow dark spot array in the transverse plane. This optical cage array in the transverse plane is vital to enhance the efficiency of optical systems, such as parallelized optical imaging, dynamic optical trapping, etc.

2. Theoretical model

To form a uniform optical cage, two issues need to be considered. One is to generate dark and bright spots with adjustable number and desired position in the focal region. For example, a uniform optical cage can be realized by combining a center dark spot together with two side bright spots. The distance between the side bright spots must be adjusted carefully to fit the depth of focus of the dark spot. Another is to create the dark and bright spots with orthogonal electric field, to avoid the interference between the spots that may affect the uniformity and eventually the effectiveness of the generated optical cage.

The optical system used for generating the optical cage array in this work is shown in Fig. 1. An incident beam is firstly split into two branches with a beam splitter, going respectively through a polarization converter (PC) and a Dammann grating (DG), and combined together again with another beam splitter to form a DVB at point C. The polarization converters at the two branches (PC₁ and PC₂) guarantee the orthogonal polarization state of the two basic vector beams in Points A and B. There are two different types of DVBs respectively for the high- and low-NA focusing systems. In the case of high-NA focusing system, the output vector beams after PC₁ and PC₂ are radially- and azimuthally-polarized beams, respectively; while in the situation of low-NA system, the basic vector beams are x- and y-polarized beams. Such combination of vector beams in Points A and B can make sure that the bright and dark spots in the focal region are of orthogonal electric fields, namely the dominate longitudinal and the pure transversal components for the first pair [23], and the x- and y-polarized components for the latter pair [24]. By adjusting the polarization state of the DVB with the phases in the Dammann gratings (DG₁ and DG₂), optical cage array with adjustable number and positions can be achieved.

Fig. 1 The schematic of optical system used for generating the optical cage arrays in the transverse plane with pure polarization modulation. (a) DG for 4 × 4 focus array; (b) DG for 1 × 2 focuses with topological charge of 1.

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2.1 The polarization state of DVB

The criterion of choosing the DVB at different NA is dependent on the vector properties of light focused by an objective lens. When NA is high, the vector properties are prominent and both the x- and y-polarized beams will yield a large longitudinal field that will cause the interference. A radial-azimuthal polarization combination however can work well in which the radial polarization component yield a tight bright spot with strong longitudinal field while the azimuthal one yield a dark spot with purely transversal field. On the contrary, in the low NA case, the vector properties can be ignored and the focused x- and y-polarized beams will maintain their polarization and hence the orthogonally. In the following, we will introduce the DVBs at different NAs in detail.

Firstly, for the case of high NA focusing system, the DVB is the combination of radially-polarized (RP) and azimuthally-polarized (AP) beams in points A and B, which can be described as:

E_{D V B H} = η_{H} D G_{1} e_{r} + D G_{2} e_{φ}

where e_r and e_φ are the unit vector in radial and azimuthal direction, respectively. η_H controls the weight of the input RP beam provided the weight for the AP is normalized. DG_n is the transmittance of the Dammann grating, which can be expressed as

D G_{j} = D G_{x j} (N_{x j}, d_{x j}, t_{p j}) \times D G_{y j} (N_{y j}, d_{y j}) \times D G_{z j} (N_{z j}, d_{z j}) j = 1, 2

where DG₍_x_~_z₎_j represent the standard Dammann gratings that control the focus in the x, y and z axis, respectively. N₍_x~z₎_j and d₍_x~z₎_j indicate the number of focal spots in the coordinate axes and the distance between each focus, respectively, both of which can be adjusted by the transition points and the period of the Dammann grating [25,26]. t_pj controls the topological charge of each focus. For instance, two standard DGs are shown in Fig. 1 where (a) is for 4 × 4 focus array and (b) for 1 × 2 focuses with topological charge of 1 along x axis. It is unnecessary to adjust t_pj to generate dark and bright spot in this case, because the AP and RP components of DVB naturally provides perfect dark and bright spot in the focal region under a tight-focusing condition. Thus, t_pj is set to be zero.

Secondly, for the case of low-NA focusing system, the focal spots generated by x- and y-polarized beams are bright spots dominated by the x- and y-polarized components of electric field. Optical cage array cannot be achieved simply by overlapping the bright spots. In this case, one of the input linearly-polarized beams should be added with a spiral phase in order to obtain a dark spot in the focal region. For example, double dark spots with topological charge of ± 1 can be generated along y-axis with the modulation of DG₂ when N₂ = 2, t_p₂ = 1. Hence, the number of hollow dark spots can be controlled with

D G_{2} = D G_{x 2} (N_{x 2}, d_{x 2}) \times D G_{y 2} (2, d_{y 2}, 1) \times D G_{y 2} (N_{y 2}, d_{y 2}^{'}) \times D G_{z 2} (N_{z 2}, d_{z 2})

where

D G_{y 2} (2, d_{y 2}, 1)

denotes the Dammann grating for generating a couple of hollow dark spots with topological charge of ± 1 along the y-axis. By multiplying

D G_{y 2} (N_{y 2}, d_{y 2}^{'})

, the number of focuses in the y-axis is equal to 2N_y. The parameter

d_{y 2}^{'}

determines the separation of the whole spots created by

D G_{y 2} (2, d_{y 2}, 1)

. Here,

d_{y 2}^{'} =4 d_{y 2}

for odd N_y and

d_{y 2}^{'} =2 d_{y 2}

for even N_y, so that the distance between adjacent hollow dark spots is equal, i.e.,

2 d_{y 2}

.

In term of bright spots for the x-polarized component in point A, DG₁ can be described as

D G_{1} = D G_{x 1} (N_{x 1}, d_{x 1}, t_{p 1}) \times D G_{y 1} (N_{y 1}, d_{y 1}) \times D G_{z 1} (N_{z 1}, d_{z 1})

where t_p₁ = 0. Finally, the DVB in point C composing of the x- and y-polarized beams can be expressed as

E_{D V B L} = η_{L} D G_{1} e_{x} + D G_{2} e_{y}

where e_x and e_y are the unit vector in x and y direction, respectively. Similarly, η_L controls the relative weight of the x-polarized beam with respect to the y-polarized beam.

To sum up, the first type DVB is consisted of the radially- and azimuthally-polarized beams, while the combination of x- and y-polarized beams forms the second type DVB. Both are modulated by the DGs with transmittance of ± 1 to form a specific spatial-variant polarized beam. They can be obtained by the interferometric method as shown in Fig. 1, and also by using the liquid crystal modulation technique, such as Q-plate [27–30], vortex retarder [31], etc. Once the spatial-variant polarizing element is designed, the DVB for the particular optical cage array can be generated directly.

2.2 Focal light intensity of DVB

By focusing the aforementioned two types of DVBs, the electric field in the vicinity of focus are derived as [23]

\begin{array}{l} E (r, ϕ, z) = - \frac{i A}{π} \int_{0}^{2 π} \int_{0}^{α} \sin θ \cos^{1 / 2} θ l_{0} (θ) V \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & \exp [i k (r \sin θ \cos (φ - ϕ) + z \cos θ)] d θ d φ \end{matrix} \end{array}

where θ and φ represent the convergent and azimuthal angle, A the normalized constant, α = arcsin(NA/n) the maximum angle allowed by the objective lens, k = 2nπ/λ the incident wave number, n the refractive index in the imaging space, λ the incident wavelength, and l₀(θ) the electric field amplitude function of the incident DVB, which can be expressed as

l_{0} (θ) = J_{1} (2 β_{0} \frac{\sin θ}{\sin α}) e x p [- {(β_{0} \frac{\sin θ}{\sin α})}^{2}]

In Eq. (7), β₀ is the ratio of the pupil radius to the incident beam waist, J_n (•) is the Bessel function of the first kind with order n. According to the expressions for the DVBs [Eq. (1) and Eq. (5)], the focusing vector V for the high- and low-NA conditions can be written respectively as:

V_{h i g h} = η_{H} D G_{1} V_{r} + D G_{2} V_{φ}

V_{l o w} = η_{L} D G_{1} V_{x} + D G_{2} V_{y}

where

V_{x} = \cos φ V_{r} - \sin φ V_{φ}

,

V_{y} = \sin φ V_{r} + \cos φ V_{φ}

.

V_{r}

and

V_{φ}

are the focusing vector for the RP and AP beams and can be expressed as [23]

V_{r} = {[\begin{matrix} \cos θ \cos φ & \cos θ \sin φ & \sin θ \end{matrix}]}^{T}

V_{φ} = {[\begin{matrix} - \sin φ & \cos φ & 0 \end{matrix}]}^{T}

T denotes the matrix transposition operator. Hence, the focal light intensity for DVB can be obtained by

I = | E |^{2}

.

3. Results and analyses

Before calculating the intensity distribution of optical cage array generated in the focal region, the uniformity U and cleanness C are defined to characterize the quality of the optical cage, where U is the ratio of the peak intensity along the radial direction containing the smallest intensity to the peak intensity along the direction containing the largest intensity in the light shell; and the cleanness C is the ratio of the center intensity to the maximum intensity along the smallest intensity direction in the light shell [14,32]. An optical cage is considered to be perfect when U = 1 and C = 0, while it is not effective if U < 0.5 or C > 0.5. In the following simulations, n = 1, β₀ = 1.5; the length unit in all figures are the wavelength λ. The intensity distribution is normalized. We take NA = 0.95 for the high NA condition and NA = 0.3 for the low one as the examples to generate the optical cage array, respectively.

3.1 Optical cage array with NA = 0.95

Figure 2 shows the generation of an optical cage by focusing a type 1 DVB, the polarization state of which is shown in Fig. 2(c). The DVB is formed by combining two modulated orthogonal beams, namely the RP [Fig. 2(a)] and AP [Fig. 2(b)] beams. To obtain an optical cage, a hollow dark spot in the x-y plane together with a pair of bright spots along z axis are needed. As shown in Fig. 2(d), double bright spots with distance of 4λ are generated by focusing the modulated RP component in Fig. 2(a), when the parameters of DG₁ are set to N_x1 = N_y1 = 1, d_x1 = d_y1 = 0, t_p1 = 0, N_z₁ = 2, dz₁ = 2. By introducing the AP component and optimizing the parameters of DG₂ and the weight (η_H) of RP component, high quality of optical cage can be realized by overlapping the hollow dark spot [Fig. 2(e)] generated by the modulated AP component in Fig. 2(b). The corresponding parameters of DG₂ are N_x2 = N_y2 = 1, d_x2 = d_y2 = 0, t_p2 = 0, N_z₂ = 2, dz₁ = 0.54 and η_H = 1.25. The reason why N_z₂ = 2, dz₂ = 0.54 is that the depth of focus (DOF) of the focused AP component in Fig. 2(e) can be enlarged to get high quality of optical cage. The final optical cage is achieved with U = 0.92 and C = 0.228, as shown in Fig. 2(f).

Fig. 2 The generation of an optical cage (f) under a high-NA focusing condition with a DVB (c) composed of a pair of Dammann grating modulated RP (a) and AP (b) beams. (d) and (e) the field intensity distributions in the focal region from the individual RP and AP beams, respectively.

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It is vital to increase the number of optical cages in the x-y plane to promote the efficiency of their applications in optical trapping, imaging, cloaking, etc. The generation of optical cage array involves two steps: one is the control of focal bright spots in different z-planes; another is the adjustment of the hollow dark spots in the z = 0 plane. For the first process, the number and positions of the bright spots achieved by focusing the RP component can be easily manipulated by DG_x₁ and DG_y₁ in Eq. (2). Such bright spot array can further be divided into double arrays with controllable positions along z axis by DG_z₁. For the second process, the hollow dark spot array with tunable number and positions are attained with the modulation of DG_x₂ and DG_y₂. By superposing the double bright spot arrays with the hollow dark spot array, an optical cage array with prescribed number and tunable positions can be generated.

Figure 3 shows the electric field distributions of two generated optical cage arrays at different z-planes. For the 2 × 2 optical cage array, the 2 × 2 bright spot array is obtained with DG_x₁(N_x₁,d_x₁,t_p₁) × DG_y₁(N_y₁,d_y₁) in Eq. (2), where N_x₁ = N_y₁ = 2, d_x₁ = d_y₁ = 6, t_p₁ = 0. By multiplying the additional Dammann Grating component DG_z₁ with N_z₁ = 2 and dz₁ = 2, the 2 × 2 bright spot array is separated into two arrays in the plane of z = −2λ in Fig. 3(a) and z = 2λ in Fig. 3(c). The hollow dark spots from azimuthal component is obtained by the modulation of DG₂ with the same phase of DG₁, namely N_x₂ = N_y₂ = 2, d_x₂ = d_y₂ = 6, t_p₂ = 0. Again, the DOF of the hollow dark spots can be enhanced by DG_z₂ with N_z₂ = 2, dz₂ = 0.54. Finally, the 2 × 2 optical cage array is obtained by overlapping the double 2 × 2 bright spot arrays and the 2 × 2 hollow dark spot array. Likewise, the 3 × 3 optical cage array is achieved with N_x₁ = N_y₁ = 3, d_x₁ = d_y₁ = 6, t_p₁ = 0, N_z₁ = 2, dz₁ = 2 for DG₁ and N_x₂ = N_y₂ = 3, d_x₂ = d_y₂ = 6, t_p₂ = 0, N_z₂ = 2, dz₂ = 0.54 for DG₂. The corresponding field intensity distributions in the y-z plane for the optical cages within the dashed line [Fig. 3(b, e)] are shown in Fig. 4(b, e), together with the DVBs for generating the above optical cage arrays [Fig. 4(a, d)]. Based on the above analysis, the number and positions of optical cages are determined by DG_x_n and DG_y_n while the quality of each cage mainly depends on DG_z_n. Owing to the same DG_z_n as that in Fig. 1, the uniformity U and cleanness C of each optical cage almost remain the same. In addition, we define FWHM as the full width at (I_m + I_c)/2, where I_m and I_c are the maximum and center light intensity of optical cage, respectively. The transverse FWHM is determined by NA, while the axial FWHM can be adjusted by the distance of double bright spot arrays, as shown in Fig. 3(a, c). Closer distance leads to smaller axial FWHM, but higher C. By taking the quality of whole optical cage array into consideration, as shown in Fig. 4(c, f), FWHMs along y and z axis are about 0.455λ and 2.434λ, respectively.

Fig. 3 The generation of optical cage arrays with DVBs under a high-NA focusing condition. Top panels: 2 × 2 and bottom ones: 3 × 3 optical cage array, respectively. The figures show the field intensity distributions in the focal region at different z-planes; (a,d) z = −2λ; (b,e) z = 0; (c,f) z = 2λ.

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Fig. 4 The field intensity distributions in the y-z plane for the optical cages as shown in Fig. 3, where (b) x = 2λ and (e) x = 0 plane. (a) and (d) are the polarization state of DVBs for generating the 2 × 2 and 3 × 3 optical cage arrays, respectively. (c, f) are the light intensity along y (red dash line) and z axis (blue line), respectively. The FWHMs of optical cage along y and z axis are about 0.455λ and 2.434λ for all optical cages.

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3.2 Optical cage array with NA = 0.3

In the case of low NA focusing system, the focused fields of x- and y-polarized beams are dominated with the x- and y-polarized field components, respectively, with negligible longitudinal field. In order to obtain a dark intensity in the focal point for the y-polarized component, the topological charge of DG₂ is set to 1 [DG_y₂(2, d_y₂,1)] in Eq. (3), thus all of the focal spots for the y-component are the hollow spots with topological charge of ± 1. Based on the principle of DG, the hollow spots come in pairs and the optical cages are achieved in even number.

Figure 5 illustrates the generation of an optical cage under the low NA focusing condition with a type 2 DVB [Fig. 5(c)], which is the combination of an x-polarized [Fig. 5(a)] and a y-polarized [Fig. 5(b)] beam with the modulation of DG₁ and DG₂, where N_x₁ = N_x₂ = N_y₂ = 1, N_y₁ = N_z₁ = N_z₂ = 2, dx₁ = dx₂ = 0, t_p₁ = 0, d_y₁ = d_y₂ = 15, dz₁ = 34, dz₂ = 17. As seen in Fig. 5(d, e), the x- and y-polarized beams give rise to a 2 × 2 bright spots and a 1 × 2 hollow spots respectively in the y-z plane. The distance between the bright spots along z-axis can be adjusted with dz₁. The uniformity and cleanness of the optical cages are increased with the decreasing of dz₁, and vice versa. The key to keep the high uniformity while maintaining a small cleanness lays on the DOF of the hollow spots. By optimizing dz₂, a 1 × 2 optical cage array with U = 0.76 and C = 0.289 is achieved in the focal region, as shown in Fig. 5(f).

Fig. 5 The generation of a 1 × 2 optical cage (f) under a low-NA focusing condition generated by a DVB (c) composed of a pair of Dammann grating modulation x- (a) and y-polarized (b) beams. (d) and (e) the field intensity distributions in the focal region from the individual x- and y-polarized beams, respectively.

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To further generate optical cage array with tunable number and positions, double bright spot arrays along z-axis and dark spot array in x-y plane are needed. As discussed above, the dark spots with topological charge of ± 1 always come in pairs, thus the optical cage array is with even number. Figure 6 shows the field intensity distributions of a 2 × 2 and a 4 × 4 optical cage array at different z planes. For the 2 × 2 array, bright spot array with adjustable number and position are obtained with the modulation of DG₁, where N_x1 = N_y1 = 2, d_x₁ = d_y₁ = 15, t_p₁ = 0. By multiplying DG_z₁ with N_z1 = 2, d_z₁ = 34, the 2 × 2 bright spot array is divided into two arrays at z = −34λ [Fig. 6(a)] and z = 34λ [Fig. 6(c)]. Similarly, double dark spots with topological charge of ± 1 are obtained with the modulation of DG_y₂(2, d_y₂,1), the 2 × 2 dark spot array can further be achieved by DG_x₂, where N_x2 = 2, d_x₂ = d_y₂ = 15. In addition, the DOF of the dark spots can be enhanced to increase the quality of optical cages by DG_z₂ with N_z₂ = 2, dz₂ = 17. By superposing the pair of 2 × 2 bright spot arrays (at z = ± 34λ planes) with the dark spot array at z = 0 plane, a 2 × 2 optical cage array with U = 0.76 and C = 0.289 is obtained. The corresponding field intensity distribution in the y-z plane indicated by the dashed lines [Fig. 6(b)] is shown in Fig. 7(b), together with the polarization state of the DVB [Fig. 7(a)].

Fig. 6 The generation of optical cage arrays with DVBs under a low-NA focusing condition. Top panels: 2 × 2 and bottom ones: 4 × 4 optical cage array, respectively. The figures show the field intensity distributions in the focal region at different z-planes; (a,d) z = −34λ; (b,e) z = 0; (c,f) z = 34λ.

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Fig. 7 The field intensity distributions in the y-z plane for the optical cages as shown in Fig. 6, where (b) x = 15λ and (e) x = 12λ plane. (a) and (d) are the polarization state of DVBs for generating the 2 × 2 and 4 × 4 optical cage arrays, respectively. (c, f) are the light intensity along y (red dash line) and z axis (blue line), respectively. The FWHMs of optical cage along y and z axis are about 1.90λ and 40.14λ for all optical cages.

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For the case of 4 × 4 optical cage array in the second row of Fig. 6, a pair of 4 × 4 bright spot arrays at z = ± 34λ planes are generated with N_x₁ = N_y₁ = 4, d_x₁ = d_y₁ = 12, t_p₁ = 0, N_z₁ = 2, d_z₁ = 34 [Fig. 6(d, f)]. However, the number of dark spots cannot be adjusted directly by changing only the DG, which cannot guarantee the topological charge of the dark spots. Thereby the way to adjust the number of hollow dark spots relies on an additional Dammann Grating, namely DG_y₂(2, d_y₂,1) × DG_y₂(N_y2, 2d_y₂). In this case, we take N_y2 = 2, d_y₂ = 12. Then, the number of optical cages along y axis is four and the distance between each focus is 24λ. The 4 × 4 dark spot array can be generated by multiplying DG_x₂ with N_x₂ = 4, dx₂ = 12. The z-direction DG_z₂ with parameters of N_z₂ = 2, dz₂ = 17 is utilized to enhance the DOF of the dark spot array, which can further increase the quality of the entire optical cage array. Figure 7(e) shows the field intensity of the 4 × 4 optical cage array in the y-z plane. The uniformity of each optical cage is larger than 0.55 while the cleanness is smaller than 0.385. The polarization state of the DVB for generating the 4 × 4 optical cage array is shown in Fig. 7(d) and the FWHMs of optical cage along y and z axis [Fig. 7(c, f)] are about 1.90λ and 40.14λ, respectively.

4. Conclusion

In conclusion, the first generation of optical cage array in the transverse plane is demonstrated in this paper, which is obtained by focusing a Dammann vector beam. The DVB consists of a pair of orthogonally-polarized beams modulated by the Dammann gratings, i.e., the radially- and azimuthally-polarized beams for the high-NA focusing condition, and the x- and y-polarized beams for the low-NA case. By modulating merely the polarization state of the DVB with the parameters of DGs, the number and position of optical cages can be adjusted flexibly. This work uncovers the link between the input polarization state of incident vector beam and the output optical cage array in the focal region, which is of great potential to enhance the efficiency of the parallel optical imaging system, the dynamic optical trapping, etc.

Funding

National Natural Science Foundation of China (NSFC) (61427819, 61490712, 61622504,11504244); National Key Basic Research Program of China (973) (2015CB352004); Science and Technology Innovation Commission of Shenzhen (KQTD2015071016560101, KQCS2015032416183980, KQCS201532416183981); the Leading talents of Guangdong province program (00201505); and the Natural Science Foundation of Guangdong Province under No.2016A030312010.

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Tunable optical cage array generated by Dammann vector beam

Abstract

1. Introduction

2. Theoretical model

2.1 The polarization state of DVB

2.2 Focal light intensity of DVB

3. Results and analyses

3.1 Optical cage array with NA = 0.95

3.2 Optical cage array with NA = 0.3

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (11)

Optics Express