Abstract

We propose a method for producing multifocal spot arrays (MSAs) capable of controlling the position and polarization orientation of each focal spot with radially polarized Bessel-Gaussian beam. Based on a simple analytical equation that can be used to manipulate the position of the focal spot, we design a type of multi-zone plate (MZP) composed of many fan-shaped subareas which accordingly generate lateral position-controllable multifocal spots. By adding a π-phase difference between a division line passing through the center of the back aperture with different orientations to corresponding subareas of the MZP, we realize MSAs in which orientations of the linear polarization in each focal spot can be arbitrarily manipulated. Such position and polarization controllable MSAs may potentially have applications in many fields.

© 2015 Optical Society of America

1. Introduction

The polarization manipulation of tightly focused light have become a subject of intense research in recent years due to advances in modern nano-optics [1–7]. The polarization control of focused light is important in a great variety of optical phenomena. For example, light can selectively interact with anisotropic materials through its polarization orientations which can be imaged with a high resolution by probing the focal region using gold nanorods [8–11]. The different polarization orientations of the light lead to different excitation rates of a dipolar molecule when they interact with each other. It can be used for high resolution fluorescence molecular imaging [12]. It has also been shown that polarized light can determine the three dimensional orientation of individual molecules [13–15]. In addition, polarization control in the focal region of a high numerical aperture (NA) focusing system also provides a new technology for polarization microscopy [16] which can be applied to multi-dimensional optical data storage [17,18], polarization information encryption [19,20], nonlinear optics imaging [21], and surface plasmon-based photonic devices [22].

However, research on the polarization manipulation has so far concentrated predominantly on a single foal spot of a high NA objective [23–27]. There were few reports about the research on the polarization manipulation of multifocal spot arrays (MSAs). On the one hand, for generation of MSAs with identical polarization in each focal spot of a high-NA objective, phase-only modulation via a spatial light modulator (SLM) is the preferable choice owing to the SLM’s programmable ability to dynamically update the intensity distributions in the focal plane by varying the incident phase patterns [28–38]. But, the phase patterns used in the method are mainly produced by iterative algorithms [39–45] which are inconvenient and lack flexibility. It is not possible to produce MSAs with controllable polarization in each focal spot using the iterative algorithms. On the other hand, although superposing two orthogonal field components through an interference [46] or a double-path [47–49] approaches can provide the possibility to realize polarization manipulations in multiple diffraction orders, these methods exhibit a complexity of the experimental configurations and are intrinsically restricted to low NA lens. Therefore, it is still a challenge to achieve a MSA with controllable polarization in each focal spot over the entire focal region of a high NA objective. Recently, M. Gu et al. reported a polarization-multiplexed multifocal array capable of individually manipulating the focal polarization state in each focal spot using an azimuthally polarized beam [50]. However, their method was limited to four spots, and the positions of each focal spot in the array was not flexible. Creating tightly focused multifocal array with a different number of focal spots, and capable of controlling position and polarization in each focal spot has never been demonstrated.

Here we introduce a method for generating MSAs with arbitrarily controllable orientations of the linear polarization in each focal spot over the tightly focused optical field of a high NA objective with radially polarized Bessel-Gaussian beam illumination. In addition, this method enables the MSAs to arbitrarily change the position and number of the focal spots. In section 2, the Fourier transform theory of tight focusing filed is discussed. Based on the theory, a simple phase-only analytical equation is derived, which can be used for dynamic focus control in the focal region of a high-NA objective. In section 3, based on the phase-only analytical equation, we design a new type of multi-zone plate (MZP) composed of a large number of fan-shaped subareas, which contains phase-only distributions resulting in different positions of each focal spot. In section 4, by adding a π-phase difference at two sides of a division line passing through the center of the back aperture with different orientations to corresponding subareas of the MZP, MSAs with controllable polarization and position in each focal spot are realized. The dynamic properties of MSAs in the process of controlling the position and polarization are also discussed. The MSAs can keep a high good uniformity and stability during the dynamic control of the position and polarization.

2. Phase-only analytical equation for manipulating the position of the focal spot

According to the Richards–Wolf vectorial integral [51, 52], the electric field distribution at any point in the focal volume of an aberration-free high-NA objective can be expressed as

E(x,y,z)=A0α02πP(θ)Et(θ,ϕ)×exp{ikx2+y2sinθcos[tan1(y/x)ϕ]}exp(ikzcosθ)sinθdϕdθ,
where A is a constant that is related to the focal length and the wavelength; k is the wave number; α is the maximum aperture angle of the objective; θ is the converge angle given by sinθ = rNA/(Rnt), assuming that the focusing system obeys Abbe’s sine condition, where R is the aperture stop radius, NA is the numerical aperture of the objective, and nt is the refractive index of the immersion medium; and P(θ) is the so-called apodization function. The coordinates r and ϕ are polar coordinates at the back aperture plane, and x, y and z are Cartesian coordinates in the focal region as shown in Fig. 1.

 

Fig. 1 Schematic diagram showing the geometry for the calculation of the focused field distribution and the variables involved.

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In the Debye approximation, the transmitted field Et is the vector angular spectrum of the focal field E [53]. Hence, the electric field E at a point (x, y, z) in the focal region can be obtained using

E(x,y,z)=0α02πP(θ)Et(θ,ϕ)exp[i(kzzkxxkyy)]sinθdϕdθ,
where the constant coefficients have been omitted for clarity. The wave vector kt = kxex + kyey + kzez (here ex, ey and ez are Cartesian unit vectors) can be expressed with spherical coordinates θ and ϕ as kx = -kcosϕsinθ, ky = -ksinϕsinθ, and kz = kcosθ. Neglecting the constant coefficients, Eq. (2) can be rewritten as
E(x,y,z)=P(θ)Et(θ,ϕ)/cosθ×exp(ikzz)exp[i2π(ξx+ηy)]dξdη,
where ξ and η denote the spatial frequency in x, y directions, which can be given by ξ = -cosϕsinθ/λ and η = -sinϕsinθ/λ. From Eq. (3), We can see that the Debye diffraction integral has been rewritten as an Fourier transform (FT) of the weighted field Et; that is, the electric field distribution can be expressed as
Ez(x,y)=F{G(ξ,η)},
where F{·} denotes the FT, and where the function G(ξ, η) is
G(ξ,η)=P(θ)Et(θ,ϕ)/cosθ×exp(ikzz).
The Debye integral of the highly focused field is expressed as an FT of the field distribution at the back aperture of the objective. Then, based on the shift theorem of the FT, the additional phase shift introduces a linear displacement in the spatial domain; i.e.,
Ez(xΔx,yΔy)=F{exp[i2π(ξΔx+ηΔy)]G(ξ,η)},
which shows that the focused spot of the high-NA objective has a displacement of Δx and Δy in the focal region of the objective. Therefore, by modulating the additional phase distribution at the back plane of the objective, we can control the position of the focal spot in the focal region of the high-NA objective.

Based on the relationship sinθ = rNA/(Rnt), the phase-only analytical expression for controlling the 2D lateral displacement of the highly focused spot can be expressed as

ψ(x0,y0)=2πλNARnt(x0Δx+y0Δy),
where Δx and Δy are the relative displacement components compared with the original focal spot of the objective without phase modulation in the x and y directions in the focal plane, and x0 and y0 are orthogonal coordinates at back aperture plane of the objective (see Fig. 1). Thus, the position of the tightly focused spots can be controlled in the 2D focal plane of a high-NA objective using the phase-only distribution in Eq. (7).

3. Generation of MSA with controllable position in each focal spot

In this section, we describe a new type of MZP for generating a tunable array with controllable position of each spot at the focal region of the objective by modulating only the phase distributions using Eq. (7).

To generate multiple focal spots, we divide the back aperture equally into N fan-shaped areas with an equal central angle Δφ = 2π/N, where N an even number, as shown in Fig. 2(a). Each fan-shaped area is then further divided into smaller fan-shaped subareas, in which the central angle of each subareas is also equal, given by δφ = Δφ/M, where M is a positive integer, as shown in Fig. 2(b). Each of these subareas is filled with different phase-only distributions, as described by Eq. (7). That is, if one fan-shaped area with angle Δφ = 2π/N is divided into M subareas with δφ = Δφ/M, the one fan-shaped area will be filled with M phase-only distributions of different displacements. The other N-1 fan-shaped areas are filled with similar phase-only distributions ψ1, ψ2ψM. MSAs with M spots can be generated using this MZP to modulate the back aperture of the objective.

 

Fig. 2 Schematic diagram showing the multi-zone phase distributions used to generate the MSA. (a) The back aperture plane of the objective. (b) A single fan-shaped area with multiple subareas, where the blue line denotes one fan-shaped area as shown in (a), and red lines denote smaller fan-shaped subarea.

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As an example, we assume a radially polarized Bessel-Gaussian (RPBG) beam with a wavelength of 532 nm impinges onto the back aperture of a 1.32 NA objective and an index of refraction (nt) 1.518. The amplitude distribution of the Bessel-Gaussian beam is given by [54]

P(θ)=exp[β2(sinθsinα)2]J1(2βsinθsinα),
where β = 1.5 is the ratio of the pupil radius and the beam waist. J1(θ) denotes the Bessel function.

Figure 3(a) shows the phase-only pattern of a MZP, in which four types (M = 4) of phase-only distributions with different positions be filled. The phase-only distributions are calculated using Eq. (7) when the four position parameters are (Δx, Δy) = [(1, 1); (1, −1); (−1, 1); (−1, −1)] μm. The corresponding intensity distribution in the focal plane is shown in Fig. 3(d). We can see that a square array was created with four spots in the exact position. Thus, by directly changing the parameters Δx and Δy in Eq. (7), we can exactly control the focal spot at any position of the 2D lateral focal plane. Figure 3(b) shows the phase-only pattern of a MZP with M = 5. The phase-only distributions filled in the MZP are calculated using Eq. (7) when (Δx, Δy) = [(0, 0); (0, 1); (0, −1); (−1, 0); (1, 0)] μm. Compared with Fig. 3(a), the MZP shown in Fig. 3(b) just adds a subarea with position parameter of (Δx, Δy) = (0, 0). Although the difference between Fig. 3(a) and Fig. 3(b) is not easy to distinguish, the phase distributions are different. We can see different periodic arrays of the phase patterns, especially in the center area. Figure 3(e) shows the corresponding intensity distribution of Fig. 3(b). We can see that a centered-square array with five spots was created. For other MSAs with different numbers of spots and different periodic lattices, we need only to change M, and the displacements calculated using Eq. (7). Figure 3(c) shows an MZP pattern which is composed of seven subareas (i.e., M = 7), in which each subarea is filled with different 2D lateral phase-only distributions. The filled phases are calculated using Eq. (7) with seven different displacements, so that Δxm = 1.5cos(mπ/3) μm, Δym = 1.5sin(mπ/3) μm, m = 1, 2…6, and Δx = Δy = 0, which can be used for generating a hexagonal multifocal lattice. The corresponding intensity distribution is shown in Fig. 3(f). We can see that a hexagonal array with seven spots was created using the MZP shown in Fig. 3(c). Thus, by directly changing the parameters Δx and Δy in Eq. (7), we can exactly control the position in each focal spot of the MSA. And by changing the parameter M of subareas, we can easily control the number of the focal spots of the MSA.

 

Fig. 3 MZP patterns (N = 50) filled with phase-only distribution as described by Eq. (7) when (a) M = 4, (b) M = 5 and (c) M = 7. (d), (e) and (f) are the corresponding intensity distribution in the focal plane when the high NA objective are modulated by the phase shown in (a), (b) and (c), respectively. Dynamic MSAs with individually controllable position can be created easily using that MZPs (see Visualization 1).

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It should be noted that the more the number of N used in the MZP, the better quality and uniformity for the focal spots array. Figure 4(a) shows the intensity distribution in the focal plane when the high NA objective is modulated by the MZP with the same parameters in Fig. 3(d), except the number of N = 4. Figure 4(c) shows the corresponding 3D iso-intensity surface of one focal spot in Fig. 4(a). It is clear that the focal spots were significantly distorted when N was a small number. It is because the beams from each circular sector interfere with each other. When N is large, the distortion of focal spots resulted from interference can be minimized. Figures 4(b) shows the intensity distribution in the focal plane of the high NA objective when N = 40. The corresponding 3D iso-intensity surface in the focal region is shown in Fig. 4(d). Figure 4(e) shows the 3D iso-intensity surface of one focal spot when N = 50. It is shown that the quality of the focal spots is better with a large N than that with a small one. However, the intensities of the four focal spots are different in the focal plane, which can be seen from Fig. 4(b). The corresponding intensity cross-section profiles of Fig. 4(b) are shown in Figs. 4(f) and 4(g). It is clearly shown that, the number of N used in the MZP not only can influence the quality of focal spots but also can change the intensity distribution of the MSA in the focal region of a high NA objective. To quantify this, a uniformity factor is defined as η = 1 - (Imax-Imin) / (Imax-Imin), where Imax and Imin are the maximum and minimum intensities in foci. Figure 4(h) shows the uniformity of the four focal spots as a function of the number N. It is clear to see that the high uniformity (> 99%) was achieved when the number N ≥ 50.

 

Fig. 4 The intensity distributions of four spots array in the focal plane when the high NA objective are modulated by the MZPs with the same parameters used in the Fig. 3(a), except the numbers of N: (a) N = 4 and (b) N = 40. (c) and (d) are the corresponding 3D iso-intensity surfaces of single spot shown in Figs. 4(a) and 4(b), respectively. (e) shows the 3D iso-intensity surface of single spot in the square array when N = 50 (as shown in Fig. 3(d)). (f) and (g) show the intensity cross-section profiles of Fig. 4(b). (h) The uniformity changes as a function of the number N.

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4. Controlling the polarization in each focal spot

From the above simulated results, we see that position-controllable MSAs have been created successfully using MZPs, and the number of spots in the MSAs solely depends on the number of subareas M, which are divided from one fan-shaped area, whereas the positions of each spot can be controlled by varying the displacement parameters (Δx, Δy) using the phase-only analytical expressions. The MZPs have the most flexibility to control the position of each focal spot, because each focal spot of the MSA is generated by independent parts. So, we can realize other optical modulations in each focal spot, such as amplitude, phase, and polarization. In this article, we mainly discuss the controllable polarization in each focal spot of the MSA. Through adding a π-phase shift to each subarea of the MZPs, MSAs with controllable polarization in each focal spot can be achieved.

When the high NA objective is illuminated by a radially polarized Bessel-Gaussian beam (see Fig. 5(a)), the polarization of each focal spot in the MSA is radial distribution, as shown in Fig. 5(f). It is well know that π-phase filters have been used for optimized confinement or enhancing the longitudinal field component, because the interference effect induced by the π-phase. Based on the principle of constructive or destructive interference, we can manipulate the polarization of the focal spot. If we add a π-phase shift filter with a radially polarized or azimuthally polarized beam, the rotational phase symmetry in the transverse plane will be broken, and a linearly polarized focal spot will be produced. Figures 5(b)-5(e) show π-phase filters with different symmetrical directions. The corresponding intensity and polarization distributions of the focal spot are shown in Fig. 5(g)-5(j), respectively. We can see that, by adding a π-phase shift filter to a radially polarized Bessel-Gaussian beam, the filled component orientated along with the phase-step line can interfere destructively. Consequently, a linearly polarized focal fields with different polarization direction can be achieved. Therefore, the orientation of the linearly polarization of the focal spot can be controlled by rotating the π-phase shift filter.

 

Fig. 5 (a) Amplitude and polarization distribution of a radially polarized Bessel-Gaussian beam. The corresponding intensity and polarization distributions in the focal plane is shown in (f). (b) - (e) show the π-phase shift modulations with different diametric directions. (g) - (j) are the corresponding intensity and polarization distributions of the focal spot when the high NA objective modulated by the adding π-phase shift as show in (b) - (e), respectively.

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To achieve a controllable polarization MSA in the focal region of a high NA objective, the phase modulation of each individual focal spot in MSA consists of two independent parts: a position-controllable phase for manipulating the structure (number and position of the spots) of the MSA and a π-phase shift filter for manipulating the focal polarization. Thus, using a phase-only pattern which is combination of the MZP and the π-phase shift filters with different rotational symmetric directions to each focal spot, a MSA with individually controllable polarization and position of each focal spot can be created. Figure 6(a) shows a MZP pattern which is composed of seven subareas filled with the same phase parameters (as shown in Fig. 3(c)). In addition, each subarea of the MZP adds different π-phase shifts for controlling the polarization in each focal spot. The corresponding intensity and polarization distributions of the MSA in the focal plane is shown in Fig. 6(b). It is clear that a MSA with different polarization in each focal spot was generated.

 

Fig. 6 (a) Phase distribution of a MZP with different π-phase shift in each subarea. (b) The corresponding intensity and polarization distributions of the MSA in the focal plane. Dynamic MSAs with controllable polarization in each focal spot can be realized. (see Visualization 2)

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From the above results, we see that polarization and position controllable MSAs have been created successfully using MZPs. In addition to the dynamically controllable polarization and position (see Visualization 1 and Visualization 2), the MSAs with diffraction-limited focal spots, high uniformity and high stability can also be achieved. Figures 7(a) and 7(b) show the intensity plots of the central focal spot in the array along the x and y directions, respectively, when the high NA objective was illuminated by the RPBG beam. It is clear to see that the full width at half-maximum (FWHM) of the focal spot is smaller with the π-phase MZP modulation than the focal spot without the π-phase MZP modulation in both directions. The FWHM of each polarization-controlled focal spot is about 0.47λ along the linearly polarized direction, while the FWHM perpendicular to the linearly polarized direction is about 0.37λ. It can be seen from Figs. 7(a) and 7(b) that the feature of smaller FWHM in the polarization-controlled MSAs is mainly attribution to the π-phase shift filter used in the MZP. Figure 7(c) shows the uniformity of seven spots and the maximum intensity of one spot in hexagonal MSAs as a function of the radial position displacements. It is shown that the intensity distribution of each spot in the MSAs is highly uniform and stable, even with continuous changes in distance. Such tunable, uniform, and highly stable MSAs with dynamically controllable position and polarization is not easily created using other methods.

 

Fig. 7 Comparison of intensity plots of single focal spot illuminated by the RPBG with or without π-MZP modulation along the (a) x and (b) y directions. (c) The uniformity of seven spots and the maximum intensity of the central spot in hexagonal MSAs as a function of the radial shifting distances.

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5. Conclusion

We have demonstrated the generation of MSAs with controllable polarization and position in each focal spot using a new type of MZP. Based on an analytical equation which was derived using FT theory of the Debye vectorial integral, the position of focal spots in the MSAs can be arbitrarily controlled. The number of subareas in MZP was used to control the number of focal spots in the MSAs. By adding a π-phase difference between a division line passing through the center of the back aperture with different orientations to corresponding subareas of the MZP, the MSAs with individually controllable polarization in each focal spot have been realized. In addition, due to the flexibility of the MZP in each subarea, dynamic MSAs with controllable position and polarization orientations in each focal spot have been realized, as well as high uniformity and high stability during the dynamic control process. To the best of our knowledge, this work represents the first demonstration of dynamic control over such tunable MSAs with controllable number, controllable position, and controllable polarization in each focal spot. This controllable position and polarization multifocal array with high uniformity and stability has potential applications in the parallel fabrication of polarization-sensitive nano-photonic devices, optical recording or multi-dimensional optical data storage, parallel polarization optical imaging and dynamic optical manipulation.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (61205014, 61378060, and 61308077), the Project of Shandong Province Higher Educational Science and Technology Program (J12LJ02), and the Doctoral Foundational of Shandong Province (BS2012DX006). It is also partially supported by Dawn Program of Shanghai Education Commission (11SG44).

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41. H. Duadi and Z. Zalevsky, “Optimized design for realizing a large and uniform 2-D spot array,” J. Opt. Soc. Am. A 27(9), 2027–2032 (2010). [CrossRef]   [PubMed]  

42. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15(4), 1913–1922 (2007). [CrossRef]   [PubMed]  

43. D. Engström, A. Frank, J. Backsten, M. Goksör, and J. Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express 17(12), 9989–10000 (2009). [CrossRef]   [PubMed]  

44. D. R. Burnham, T. Schneider, and D. T. Chiu, “Effects of aliasing on the fidelity of a two dimensional array of foci generated with a kinoform,” Opt. Express 19(18), 17121–17126 (2011). [CrossRef]   [PubMed]  

45. J. Yu, C. Zhou, W. Jia, W. Cao, S. Wang, J. Ma, and H. Cao, “Three-dimensional Dammann array,” Appl. Opt. 51(10), 1619–1630 (2012). [CrossRef]   [PubMed]  

46. K. Lou, S.-X. Qian, Z. C. Ren, C. Tu, Y. Li, and H. T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013). [CrossRef]   [PubMed]  

47. D. Preece, S. Keen, E. Botvinick, R. Bowman, M. Padgett, and J. Leach, “Independent polarisation control of multiple optical traps,” Opt. Express 16(20), 15897–15902 (2008). [CrossRef]   [PubMed]  

48. S. Hasegawa and Y. Hayasaki, “Polarization distribution control of parallel femtosecond pulses with spatial light modulators,” Opt. Express 21(11), 12987–12995 (2013). [CrossRef]   [PubMed]  

49. J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. 38(7), 1043–1045 (2013). [CrossRef]   [PubMed]  

50. H. Ren, X. Li, and M. Gu, “Polarization-multiplexed multifocal arrays by a π-phase-step-modulated azimuthally polarized beam,” Opt. Lett. 39(24), 6771–6774 (2014). [CrossRef]   [PubMed]  

51. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959). [CrossRef]  

52. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1247), 358–379 (1959). [CrossRef]  

53. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef]   [PubMed]  

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

References

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  30. L. Zhu, M. Sun, M. Zhu, J. Chen, X. Gao, W. Ma, and D. Zhang, “Three-dimensional shape-controllable focal spot array created by focusing vortex beams modulated by multi-value pure-phase grating,” Opt. Express 22(18), 21354–21367 (2014).
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  31. H. Lin and M. Gu, “Creation of diffraction-limited non-Airy multifocal arrays using a spatially shifted vortex beam,” Appl. Phys. Lett. 102(8), 084103 (2013).
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  32. M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013).
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  33. Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
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  40. E. H. Waller and G. von Freymann, “Multi foci with diffraction limited resolution,” Opt. Express 21(18), 21708–21713 (2013).
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  41. H. Duadi and Z. Zalevsky, “Optimized design for realizing a large and uniform 2-D spot array,” J. Opt. Soc. Am. A 27(9), 2027–2032 (2010).
    [Crossref] [PubMed]
  42. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15(4), 1913–1922 (2007).
    [Crossref] [PubMed]
  43. D. Engström, A. Frank, J. Backsten, M. Goksör, and J. Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express 17(12), 9989–10000 (2009).
    [Crossref] [PubMed]
  44. D. R. Burnham, T. Schneider, and D. T. Chiu, “Effects of aliasing on the fidelity of a two dimensional array of foci generated with a kinoform,” Opt. Express 19(18), 17121–17126 (2011).
    [Crossref] [PubMed]
  45. J. Yu, C. Zhou, W. Jia, W. Cao, S. Wang, J. Ma, and H. Cao, “Three-dimensional Dammann array,” Appl. Opt. 51(10), 1619–1630 (2012).
    [Crossref] [PubMed]
  46. K. Lou, S.-X. Qian, Z. C. Ren, C. Tu, Y. Li, and H. T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
    [Crossref] [PubMed]
  47. D. Preece, S. Keen, E. Botvinick, R. Bowman, M. Padgett, and J. Leach, “Independent polarisation control of multiple optical traps,” Opt. Express 16(20), 15897–15902 (2008).
    [Crossref] [PubMed]
  48. S. Hasegawa and Y. Hayasaki, “Polarization distribution control of parallel femtosecond pulses with spatial light modulators,” Opt. Express 21(11), 12987–12995 (2013).
    [Crossref] [PubMed]
  49. J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. 38(7), 1043–1045 (2013).
    [Crossref] [PubMed]
  50. H. Ren, X. Li, and M. Gu, “Polarization-multiplexed multifocal arrays by a π-phase-step-modulated azimuthally polarized beam,” Opt. Lett. 39(24), 6771–6774 (2014).
    [Crossref] [PubMed]
  51. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
    [Crossref]
  52. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1247), 358–379 (1959).
    [Crossref]
  53. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
    [Crossref] [PubMed]
  54. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
    [Crossref]

2015 (1)

L. Lermusiaux, V. Maillard, and S. Bidault, “Widefield spectral monitoring of nanometer distance changes in DNA-templated plasmon rulers,” ACS Nano 9(1), 978–990 (2015).
[Crossref] [PubMed]

2014 (9)

M. Gu, X. Li, and Y. Cao, “Optical storage arrays: a perspective for future big data storage,” Light Sci. Appl. 3(5), e177 (2014).
[Crossref]

M. P. Busson and S. Bidault, “Selective excitation of single molecules coupled to the bright mode of a plasmonic cavity,” Nano Lett. 14(1), 284–288 (2014).
[Crossref] [PubMed]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
[Crossref]

P. R. Dolan, X. Li, J. Storteboom, and M. Gu, “Complete determination of the orientation of NV centers with radially polarized beams,” Opt. Express 22(4), 4379–4387 (2014).
[Crossref] [PubMed]

C. Macias-Romero, P. R. T. Munro, and P. Török, “Polarization-multiplexed encoding at nanometer scales,” Opt. Express 22(21), 26240–26245 (2014).
[Crossref] [PubMed]

H. Ren, H. Lin, X. Li, and M. Gu, “Three-dimensional parallel recording with a Debye diffraction-limited and aberration-free volumetric multifocal array,” Opt. Lett. 39(6), 1621–1624 (2014).
[Crossref] [PubMed]

L. Zhu, J. Yu, D. Zhang, M. Sun, and J. Chen, “Multifocal spot array generated by fractional Talbot effect phase-only modulation,” Opt. Express 22(8), 9798–9808 (2014).
[Crossref] [PubMed]

L. Zhu, M. Sun, M. Zhu, J. Chen, X. Gao, W. Ma, and D. Zhang, “Three-dimensional shape-controllable focal spot array created by focusing vortex beams modulated by multi-value pure-phase grating,” Opt. Express 22(18), 21354–21367 (2014).
[Crossref] [PubMed]

H. Ren, X. Li, and M. Gu, “Polarization-multiplexed multifocal arrays by a π-phase-step-modulated azimuthally polarized beam,” Opt. Lett. 39(24), 6771–6774 (2014).
[Crossref] [PubMed]

2013 (9)

S. Hasegawa and Y. Hayasaki, “Polarization distribution control of parallel femtosecond pulses with spatial light modulators,” Opt. Express 21(11), 12987–12995 (2013).
[Crossref] [PubMed]

J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. 38(7), 1043–1045 (2013).
[Crossref] [PubMed]

E. H. Waller and G. von Freymann, “Multi foci with diffraction limited resolution,” Opt. Express 21(18), 21708–21713 (2013).
[Crossref] [PubMed]

K. Lou, S.-X. Qian, Z. C. Ren, C. Tu, Y. Li, and H. T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
[Crossref] [PubMed]

H. Lin and M. Gu, “Creation of diffraction-limited non-Airy multifocal arrays using a spatially shifted vortex beam,” Appl. Phys. Lett. 102(8), 084103 (2013).
[Crossref]

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

M. Gu, H. Lin, and X. Li, “Parallel multiphoton microscopy with cylindrically polarized multifocal arrays,” Opt. Lett. 38(18), 3627–3630 (2013).
[Crossref] [PubMed]

F. Shafiei, C. Wu, Y. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7(5), 367–372 (2013).
[Crossref]

B. van den Broek, B. Ashcroft, T. H. Oosterkamp, and J. van Noort, “Parallel nanometric 3D tracking of intracellular gold nanorods using multifocal two-photon microscopy,” Nano Lett. 13(3), 980–986 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (5)

2010 (4)

2009 (3)

D. Engström, A. Frank, J. Backsten, M. Goksör, and J. Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express 17(12), 9989–10000 (2009).
[Crossref] [PubMed]

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

P. Zijlstra, J. W. M. Chon, and M. Gu, “Five-dimensional optical recording mediated by surface plasmons in gold nanorods,” Nature 459(7245), 410–413 (2009).
[Crossref] [PubMed]

2008 (3)

2007 (2)

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15(4), 1913–1922 (2007).
[Crossref] [PubMed]

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
[Crossref]

2006 (2)

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
[Crossref] [PubMed]

2005 (3)

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13(10), 3777–3786 (2005).
[Crossref] [PubMed]

J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2005).
[Crossref] [PubMed]

C. Sönnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5(2), 301–304 (2005).
[Crossref] [PubMed]

2003 (2)

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

J. N. Forkey, M. E. Quinlan, M. A. Shaw, J. E. Corrie, and Y. E. Goldman, “Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization,” Nature 422(6930), 399–404 (2003).
[Crossref] [PubMed]

2002 (1)

J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
[Crossref]

2001 (2)

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

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202(1), 72–76 (2001).
[Crossref] [PubMed]

2000 (1)

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

1999 (1)

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399(6732), 126–130 (1999).
[Crossref]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1247), 358–379 (1959).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

Alivisatos, A. P.

C. Sönnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5(2), 301–304 (2005).
[Crossref] [PubMed]

Ashcroft, B.

B. van den Broek, B. Ashcroft, T. H. Oosterkamp, and J. van Noort, “Parallel nanometric 3D tracking of intracellular gold nanorods using multifocal two-photon microscopy,” Nano Lett. 13(3), 980–986 (2013).
[Crossref] [PubMed]

Backsten, J.

Banzer, P.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
[Crossref]

Bauer, T.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
[Crossref]

Bawendi, M. G.

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399(6732), 126–130 (1999).
[Crossref]

Bengtsson, J.

Beversluis, M. R.

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

Bidault, S.

L. Lermusiaux, V. Maillard, and S. Bidault, “Widefield spectral monitoring of nanometer distance changes in DNA-templated plasmon rulers,” ACS Nano 9(1), 978–990 (2015).
[Crossref] [PubMed]

M. P. Busson and S. Bidault, “Selective excitation of single molecules coupled to the bright mode of a plasmonic cavity,” Nano Lett. 14(1), 284–288 (2014).
[Crossref] [PubMed]

Botvinick, E.

Bowman, R.

Brown, T. G.

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M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332(6030), 702–704 (2011).
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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
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B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
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B. van den Broek, B. Ashcroft, T. H. Oosterkamp, and J. van Noort, “Parallel nanometric 3D tracking of intracellular gold nanorods using multifocal two-photon microscopy,” Nano Lett. 13(3), 980–986 (2013).
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T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
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J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
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F. Shafiei, C. Wu, Y. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7(5), 367–372 (2013).
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J. N. Forkey, M. E. Quinlan, M. A. Shaw, J. E. Corrie, and Y. E. Goldman, “Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization,” Nature 422(6930), 399–404 (2003).
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T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202(1), 72–76 (2001).
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T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202(1), 72–76 (2001).
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J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
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J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
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Schonbrun, E.

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K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
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F. Shafiei, C. Wu, Y. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7(5), 367–372 (2013).
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J. N. Forkey, M. E. Quinlan, M. A. Shaw, J. E. Corrie, and Y. E. Goldman, “Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization,” Nature 422(6930), 399–404 (2003).
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H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
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K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
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M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332(6030), 702–704 (2011).
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Sun, M.

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J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
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Yang, P.

J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
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Zalevsky, Z.

Zhan, Q.

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284(1), 52–56 (2011).
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W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
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Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
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Zhang, H.

Zhou, C.

Zhu, L.

Zhu, M.

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ACS Nano (1)

L. Lermusiaux, V. Maillard, and S. Bidault, “Widefield spectral monitoring of nanometer distance changes in DNA-templated plasmon rulers,” ACS Nano 9(1), 978–990 (2015).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Lin and M. Gu, “Creation of diffraction-limited non-Airy multifocal arrays using a spatially shifted vortex beam,” Appl. Phys. Lett. 102(8), 084103 (2013).
[Crossref]

J. Microsc. (1)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202(1), 72–76 (2001).
[Crossref] [PubMed]

J. Opt. (1)

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
[Crossref]

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

Light Sci. Appl. (1)

M. Gu, X. Li, and Y. Cao, “Optical storage arrays: a perspective for future big data storage,” Light Sci. Appl. 3(5), e177 (2014).
[Crossref]

Nano Lett. (4)

B. van den Broek, B. Ashcroft, T. H. Oosterkamp, and J. van Noort, “Parallel nanometric 3D tracking of intracellular gold nanorods using multifocal two-photon microscopy,” Nano Lett. 13(3), 980–986 (2013).
[Crossref] [PubMed]

C. Sönnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5(2), 301–304 (2005).
[Crossref] [PubMed]

M. P. Busson and S. Bidault, “Selective excitation of single molecules coupled to the bright mode of a plasmonic cavity,” Nano Lett. 14(1), 284–288 (2014).
[Crossref] [PubMed]

J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2(4), 279–283 (2002).
[Crossref]

Nat. Commun. (1)

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Nat. Photonics (4)

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Supplementary Material (2)

NameDescription
» Visualization 1: MP4 (723 KB)      Video 1
» Visualization 2: MP4 (609 KB)      Video 2

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram showing the geometry for the calculation of the focused field distribution and the variables involved.
Fig. 2
Fig. 2 Schematic diagram showing the multi-zone phase distributions used to generate the MSA. (a) The back aperture plane of the objective. (b) A single fan-shaped area with multiple subareas, where the blue line denotes one fan-shaped area as shown in (a), and red lines denote smaller fan-shaped subarea.
Fig. 3
Fig. 3 MZP patterns (N = 50) filled with phase-only distribution as described by Eq. (7) when (a) M = 4, (b) M = 5 and (c) M = 7. (d), (e) and (f) are the corresponding intensity distribution in the focal plane when the high NA objective are modulated by the phase shown in (a), (b) and (c), respectively. Dynamic MSAs with individually controllable position can be created easily using that MZPs (see Visualization 1).
Fig. 4
Fig. 4 The intensity distributions of four spots array in the focal plane when the high NA objective are modulated by the MZPs with the same parameters used in the Fig. 3(a), except the numbers of N: (a) N = 4 and (b) N = 40. (c) and (d) are the corresponding 3D iso-intensity surfaces of single spot shown in Figs. 4(a) and 4(b), respectively. (e) shows the 3D iso-intensity surface of single spot in the square array when N = 50 (as shown in Fig. 3(d)). (f) and (g) show the intensity cross-section profiles of Fig. 4(b). (h) The uniformity changes as a function of the number N.
Fig. 5
Fig. 5 (a) Amplitude and polarization distribution of a radially polarized Bessel-Gaussian beam. The corresponding intensity and polarization distributions in the focal plane is shown in (f). (b) - (e) show the π-phase shift modulations with different diametric directions. (g) - (j) are the corresponding intensity and polarization distributions of the focal spot when the high NA objective modulated by the adding π-phase shift as show in (b) - (e), respectively.
Fig. 6
Fig. 6 (a) Phase distribution of a MZP with different π-phase shift in each subarea. (b) The corresponding intensity and polarization distributions of the MSA in the focal plane. Dynamic MSAs with controllable polarization in each focal spot can be realized. (see Visualization 2)
Fig. 7
Fig. 7 Comparison of intensity plots of single focal spot illuminated by the RPBG with or without π-MZP modulation along the (a) x and (b) y directions. (c) The uniformity of seven spots and the maximum intensity of the central spot in hexagonal MSAs as a function of the radial shifting distances.

Equations (8)

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E ( x , y , z ) = A 0 α 0 2 π P ( θ ) E t ( θ , ϕ ) × exp { i k x 2 + y 2 sin θ cos [ tan 1 ( y / x ) ϕ ] } e x p ( i k z cos θ ) sin θ d ϕ d θ ,
E ( x , y , z ) = 0 α 0 2 π P ( θ ) E t ( θ , ϕ ) e x p [ i ( k z z k x x k y y ) ] sin θ d ϕ d θ ,
E ( x , y , z ) = P ( θ ) E t ( θ , ϕ ) / cos θ × e x p ( i k z z ) e x p [ i 2 π ( ξ x + η y ) ] d ξ d η ,
E z ( x , y ) = F { G ( ξ , η ) } ,
G ( ξ , η ) = P ( θ ) E t ( θ , ϕ ) / cos θ × e x p ( i k z z ) .
E z ( x Δ x , y Δ y ) = F { exp [ i 2 π ( ξ Δ x + η Δ y ) ] G ( ξ , η ) } ,
ψ ( x 0 , y 0 ) = 2 π λ N A R n t ( x 0 Δ x + y 0 Δ y ) ,
P ( θ ) = exp [ β 2 ( sin θ sin α ) 2 ] J 1 ( 2 β sin θ sin α ) ,

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