- Email: [email protected]
Focusing properties of spirally polarized sinh Gaussian beam
Optics and Laser Technology 111 (2019) 623–628
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Focusing properties of spirally polarized sinh Gaussian beam a
M. Senthilkumar , K.B. Rajesh
b,⁎
b
c,d
e
, M. Udhayakumar , Z. Jaroszewicz , G. Mahadevan
T
a
Department of Physics, R & D Centre, Bharathiar University, Coimbatore, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India Institute of Applied Optics, Department of Physical Optics, Warsaw, Poland d National Institute of Telecommunications, Warsaw, Poland e The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu 624 302, India b c
HIGHLIGHTS
tight focusing properties of spirally polarized sinh-Gaussian beam is studied. • The properties are investigated theoretically by vector diffraction theory. • Focusing intensity in focal region can be altered considerably by the spiral parameter. • Optical • Focal patterns including flattop profile, focal hole, axially separated focal spots are generated. ARTICLE INFO
ABSTRACT
Keywords: Focusing property spirally polarized sinh Gaussian beam Vector diffraction theory
The tight focusing properties of spirally polarized sinh-Gaussian beam is investigated numerically by vector diffraction theory. Results show that the optical intensity in focal region of spirally polarized sinh Gaussian beam can be altered considerably by the beam order, relative waist width and the spiral parameter that indicates the polarization spiral degree of the spirally polarized sinh Gaussian beam. Many novel focal patterns including flattop profile, focal hole axially separated focal spots and focal spot with long focal depth are evolved considerably for the suitable beam parameters. We expect such a tunable focal patterns are useful for optical manipulation of micro particles.
1. Introduction Creating suitable focal patterns with precise structure and polarization in a focal region of high NA lens system plays an important role in applications including super-resolution imaging [1–3], particle trapping and manipulating [4–6], laser micromachining [7], optical lithography [8,9] and so on. A variety of peculiar optical field distributions in the focal region such as optical needle [10–14], optical chain [15–20], optical bubble or cage [21–23], spherical spot [24–26], and super long dark channel [27,28], etc. have been generated based on amplitude, phase and polarization Engineering. Beam shaping using spatially variant polarization has been a topic of much interest in the recent years owing to its advantages of achieving the desired focal structures and polarization in the focal region without using amplitude or phase filters. Recently, there is an increasing interest in laser beams with cylindrical symmetry in polarization. These so-called cylindrical vector beams can be generated by active or passive methods [29–32] and have been the topic of numerous recent theoretical and ⁎
experimental investigations [33–36]. Radial and azimuthal polarizations are two basic components of cylindrical polarization. Achieving desired focal patterns using amplitude, phase and polarization within the pupil to generate many novel focal pattern have been studied [10–28]. However, the presence of phase filters or Amplitude filters makes some applications more difficult or even impossible. Bing Hao and James Leger have recently investigated the numerical aperture (NA) invariant focus shaping using spirally polarized beams. They proposed that spirally polarization is another kind of spatially variant polarization bearing radial symmetry [37]. The spirally polarized beam possess axially symmetric polarization patterns, with linear polarization at any point of the transverse profile and with the electric field lines being logarithmic spiral. Recent investigation of the tight focusing properties of spirally polarized beam suggested several possible applications such as as optical tweezers, particle trapping, laser cutting, material processing, microscopy, etc. [38–41]. Recently, Sun et al. introduce a new class of sinh-Gaussian beam called hollow sinh-Gaussian beams which is characterized by single ringed transverse intensity
Corresponding author. E-mail address: [email protected] (K.B. Rajesh).
https://doi.org/10.1016/j.optlastec.2018.10.048 Received 27 July 2018; Received in revised form 3 October 2018; Accepted 22 October 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 1. Diagram of the focusing configuration.
Fig. 2. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8.
distributions [42]. Jie Lin et al. analyzed the focusing performance of radial polarized sinh-Gaussian beam by lens with large NA based on the Richards-Wolf’s formulas. They reported that under suitable beam parameter high beam quality and sub wavelength focusing can be achieved simultaneously [43]. Followed by this work, several work on tight focusing properties of sinh Gaussian beam have been reported [44–46]. In this paper, we numerically demonstrate the focusing performance of spirally polarized hollow sinh-Gaussian beam by high NA objective lens. We expect such a study may pave the way for the deeper understanding of hallow sinh Gaussian beam extended to possible applications.
2. Principle of the optical focusing spirally polarized sinh Gaussian beam A schematic configuration of the proposed method is shown in Fig. 1. In this paper we employed the Richards and Wolf vector diffraction theory to analyze the tight focusing properties of the spirally polarized beam by taking into account the spatial inhomogeneity vector property [47]. This method is extensively used in applications such as optical trapping and recording where high NA imaging and focusing are considered. Furthermore, in dealing with radial and azimuthal polarization or the generalized cylindrical vector beam, this method has been routinely employed and the integral formula can be shown to take a
624
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 3. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0.4. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8.
relatively simple form. The electric field in focal region of spirally polarized can be given by [40]
0 ≤ θ ≤ arcsin(NA/n). Where, n = 1.0 is the index of refraction of free space. Obviously, as defined in Eq. (5), the amplitude distribution of hollow sinh-Gaussian beam is determined by w0 and m. Therefore, one can control the amplitude distribution of hollow sinh-Gaussian beam by choosing w0 or m reasonably The term ϕ(θ) in Eqs. (2)–(4) characterizes the polarization of focusing spirally polarized beam and can be in different forms [37–41].we assume ϕ(θ) in the form as in Ref. [40]
(1)
E (r , , z ) = Er er + Ez ez + E e
where er, eϕ and ez are the unit vectors in the radial, azimuthal, and propagating directions, respectively. Er, Ez, and Eϕ are amplitudes of the three orthogonal components and can be expressed as
Er (r ,z ) = A
cos · cos[ ( )]·E0 ( )sin(2 )·J1 (kr sin( ))·
0
(2)
exp(ikzcos( ))d Ez (r , z ) = 2iA
0
0
(3)
In the investigation of spirally polarized sinh Gaussian beam, without loss of validity and generality, it is proposed that the numerical aperture of the focusing optical system NA = 0.95 and n = 1. Fig. 2 shows the focal structure generated for m = 4 and 8 for the relative waist size w0 = 0.500 and the spirality C = 0. The 2D intensity along rz plane shown in Fig. 2(a) and their corresponding radial and axial intensity shown in Fig. 2(b&c) shows that the generated focal segment is highly confined focal spot having FWHM 0.51λ and focal depth of 1.74λ . It is noted from Fig. 2(b), the focal structure is dominated by Ez component. Fig. 2(d,e & f) shows same as Fig. 2(a–c) but for m = 8. It is
(4)
For a high NA objective lens, the electric field of hollow sinhGaussian incident beam at a pupil can be defined as follows [43]
Emw0 ( ) = sinhm
sin( ) ·exp w0
sin2 ( ) w02
(6)
3. Result and discussion
cos · sin[ ( )]·E0 ( ) sin( )·J1 (kr sin( ))·
exp(ikzcos( )) d
tan( ) × tan( )
where C is spiral parameter indicating polarization spiral degree, = arcsin(NA) is convergence angle corresponding to the radius of incident optical aperture.
cos · cos[ ( )]·E0 ( )sin2 ( )·J0 (kr sin( ))·
exp(ikzcos( )) d
E (r , z ) = 2A
( )= C×
(5)
where m (m = 0, 1, 2,…) is the order of the hollow sinh-Gaussian beam. However, a new kind of sinh-Gaussian beam is obtained when m is greater than 1.0. θ is determined by NA of objective lens and
625
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 4. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0.9. For w0 = 0.500. Figure (d-f) Same as (a–c) but for m = 8.
noted from figure increasing the beam order m to 8 slightly reduced the FWHM the spot size as (0.41λ) and improved the focal depth as 2.54λ . This reduction in the spot size is due to the contribution of the longitudinal field component near the focal plane which is mainly derived from the outer most ray of the beam [43]. As order to the beam is increased the central dark area is increased and hence the bend angle of the outermost ray is increased and so are the longitudinal component and the total field leads to a sharper focal spot. In view of the transfer function, incident hollow sinh Gaussian beam with smaller w0 and m include much of component with low frequency and hence exhibit larger lateral width at the focal plane. Whereas the large m and w0 are attributed to much of component with high frequency and hence a sharper focal spot is observed. Fig. 3(a) shows the flattop structure generated for different beam order of sinh Gaussian beam. It is noted from Fig. 3(a–c) for the beam order m = 4 increasing the spirality C to 0.4 generated the flattop profile with FWHM of 0.955λ with focal depth of 1.85λ. Fig. 3(d–f) shows increasing m to 8 generated a flattop profile when C = 0.42. The FWHM of the generated flattop profile is measured as 0.855λ whereas its focal depth is found to be 2.48λ. This type of beam profile is useful in improved printing filling factor, improved uniformity and quality in materials processing, particle acceleration and microlithography. Fig. 4(a–c) shows that by setting C = 0.9, generated a 3D optical bubble for m = 4. We also noted that the FWHM the generated optical bubble is 1.14 λ where is focal depth is 2.1λ. However a 3D optical bubble with FWHM 1.05λ and focal depth of 4.24λ is obtained for
m = 8 and C = 0.9 shown in Fig. 4(d–f). Such an optical bubble with uniform intensity find the application in stable optical traps for those particles whose refractive indices are smaller than the ambient [48]. These micromanipulations have been reported to be implemented both in fluids and in air [49,50] and hence the optical bottle beams can trap atoms and molecules [51–53]. Apart from this, the 3D hollow focus has been drawing great interest in the recent year due to its wide applications in many research domains: such as nonlinear optics [54], biooptics [55] and excitation of surface wave [56]. Fig. 5(a–c) shows that by setting C = 2.4 and m = 4, the generated focal structure has an axially splitted focal spot search having flattop profile with FWHM of 0.892λ and focal depth of 1.7λ, separated by an axial distance of 1.7λ. Fig. 5(d–f), we also noted that for m = 8, the splitted focal spot appeared when C = 3.4. It is noted that the FWHM of generated focal spot is 0.5λ whereas its focal depth is 2.12λ. Such an axially splitted focal spot is useful to trap and manipulate small particles, such as atoms, molecules, cells [15,57–61]. It has wide applications in many areas, such as DNA sequencing, genes transplant, microassembly and micromachining. Hence from the above discussion we shows that by using single focusing unit and by properly tuning the beam order and spiral parameter of spirally polarized Hollow sinh Gaussian beam with suitable m and w0, one can achieve many novel focal patterns including sub wavelength focal spot with long focal depth, flat top profile, optical bubble, splitted focal spots with flat top profile etc. The authors expect that such a study exhibiting the possibility of manipulating the focal patterns will be of useful for
626
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 5. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 2.4. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8, C = 3.4.
applications such as optical data storage, lithography, and optical manipulation.
[5] O.M. Maragò, P.H. Jones, P.G. Gucciardi, G. Volpe, A.C. Ferrari, Optical trapping and manipulation of nanostructures, Nat. Nanotechnol. 8 (11) (2013) 807–819. [6] S. Zhu, J. Wang, X. Liu, Y. Cai, Z. Li, Generation of arbitrary radially polarized array beams by manipulating correlation structure, Appl. Phys. Lett. 109 (16) (2016) 161904. [7] Ting-Hsuan Chen, Romain Fardel and Craig B Arnold Ultrafast z-scanning for highefficiency laser micro-machining, Light Sci. Appl. 7 (2018) 17181. [8] Z.S. Gan, Y.Y. Cao, R.A. Evans, et al., Three-dimensional deep sub-diffraction optical beam lithography with 9 nm feature size, Nat. Commun. 4 (2013) 2061. [9] D. Fan, L. Wangand, Y. Ekinci, Nanolithography using Bessel beams of extreme ultraviolet wavelength, Sci. Rep. 6 (2016) 31301. [10] Z. Nie, Z. Li, G. Shi, X. Zhang, Y. Wang, Y. Song, Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ringshaped azimuthally polarized beam, Opt. Lasers Eng. 59 (2014) 93–97. [11] F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, M. Hong, Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light, Sci. Rep. 5 (2015) 9977. [12] S.H.I. Chang-kun, et al., Super-resolution longitudinally polarized light needle achieved by tightly focusing radially polarized beams, Optoelectron. Lett. 14 (2018) 1–5. [13] Z. Man, et al., Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes, Opt. Express 24 (2016) 874–882. [14] K. Hu, Z. Chen, J. Pu, Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface, Opt. Lett. 37 (2012) 3303–3305. [15] J. Wang, W. Chen, Q. Zhan, Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams, J. Opt. 14 (2012) 055004. [16] J. Lin, R. Chen, P. Jin, M. Cada, Y. Ma, Generation of longitudinally polarized optical chain by 4π focusing system, Opt. Commun. 340 (2015) 69–73. [17] J.M. Wang, Q.L. Liu, C.J. He, Y.W. Liu, Reversal construction of polarization controlled focusing field with multiple focal spots, Opt. Eng. 52 (2013) 048002. [18] B. Gu, Y. Pan, J. Wu, Y. Cui, Manipulation of radial-variant polarization for creating tunable bifocusing spots, J. Opt. Soc. Am. A: 31 (2014) 253–257. [19] Q.F. Xu, J.N. Chen, The creation of double tight focus by a concentric multi-belt
4. Conclusion The tight focusing properties of spirally polarized sinh Gaussian beam by High NA Lens is investigated theoretically by vector diffraction theory. Results show that the optical intensity in focal region of spirally polarized sinh Gaussian beam can be altered considerably by the beam order and spiral parameter. Spiral parameter can induce focal pattern change in axial direction remarkably and many novel focal patterns including flattop profile, focal hole and focal spots of long focal depth and axially separated focal spots are evolved. We expect such a tunable focal patterns are useful for optical manipulation of micro particles super-resolution imaging, particle trapping and manipulating, laser micromachining, optical lithography and so on. References [1] E.T.F. Rogers, N.I. Zheludev, et al., Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging, J. Opt. 15 (9) (2013) 094008. [2] Wan-Chinkim, et al., Investigation on achieving super-resolution by solid immersion lens based STED microscopy, Opt. Express 25 (14) (2017) 16629. [3] H. Gao, M. Pu, X. Li, X. Ma, Z. Zhao, Y. Guo, X. Luo, Super-resolution imaging with a Bessel lens realized by a geometric meta surface, Opt. Express 25 (12) (2017) 13933–13943. [4] A.P. Porfirev, R.V. Skidanov, Dark-hollow optical beams with a controllable shape for optical trapping in air, Opt. Express 23 (7) (2015) 8373–8382.
627
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al. pure phase filter, Opt. Commun. 285 (2002) 1642–1645. [20] T. Liu, J.B. Tan, J. Liu, J. Lin, Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel, J. Mod. Opt. 60 (2013) 378–381. [21] N. Bokor, N. Davidson, A three-dimensional dark focal spot uniformly surrounded by light, Opt. Commun. 279 (2007) 229–233. [22] X.L. Wang, et al., Configurable three-dimensional optical cage generated from cylindrical vector beams, Opt. Commun. 282 (2009) 3421–3425. [23] Zhongsheng Man, Zhidong Bai, Jinjian Li, Shuoshuo Zhang, Xiaoyu Li, Yuquan Zhang, Xiaolu Ge, Shenggui Fu, Optical cage generated by azimuthal- and radial-variant vector beams, Appl. Opt. 57 (13) (2018) 3592–3597. [24] W.B. Chen, Q.W. Zhan, Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy, Opt. Lett. 34 (2009) 2444–2446. [25] S.H. Yan, B.L. Yao, W. Zhao, M. Lei, Generation of multiple spherical spots with a radially polarized beam in a 4Pi focusing system, J. Opt. Soc. Am. A 27 (2010) 2033–2037. [26] S.N. Khonina, I. Golub, engineering the smallest 3D symmetrical bright and dark focal spots, J. Opt. Soc. Am. A: 30 (2013) 2029–2033. [27] K. Lalithambigai, P. Suresh, V. Ravi, K. Prabakaran, Z. Jaroszewicz, K.B. Rajesh, P.M. Anbarasan, T.V.S. Pillai, Generation of sub wavelength super-long dark channel using high NA lens axicon, Opt. Lett. 37 (2012) 999–1001. [28] J. William Charles, K. Prabakaran, K.B. Rajesh, Haresh M. Pandya, Generation of sub wavelength super long dark channel using azimuthally polarized annular multiGaussian beam, Opt. Quant. Electron. 46 (2014) 1079–1086. [29] Tingting Wang, Cuifang Kuang, Xiang Hao, Xu Liu, Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik 124 (2013) 4762–4765. [30] X. Yi, et al., Generation of cylindrical vector vortex beams by two cascaded meta surfaces, Opt. Exp. 22 (14) (2014) 17207–17215. [31] Alexey P. Porfirev, Andrey V. Ustinov, Svetlana N. Khonina, Polarization conversion when focusing cylindrically polarized vortex beams, Scientific Reports 6 (2016) 6. [32] Huan Chen, Xiaohui Ling, Qianguang Li, Hao Lv, Yu Huaqing, Xunong Yi, Generation of double-ring-shaped cylindrical vector beams by modulating Pancharatnam-Berry phase, Optik 134 (2017) 227–232. [33] Q. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [34] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon. 1 (2009) 1–57. [35] W. Chen, Q. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2006) 411–417. [36] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindrical vector beams, Opt. Express 7 (2000) 77–87. [37] B. Hao, J. Leger, Numerical aperture invariant focus shaping using spirally polarized beams, Opt. Commun. 281 (8) (2008) 1924–1928. [38] V. Ramírez-Sánchez, G. Piquero, M. Santarsiero, Generation and characterization of spirally polarized fields, J. Opt. A Pure Appl. Opt. 11 (2009) 085708–1 085708-6. [39] J. Pu, Z. Zhang, Tight focusing of spirally polarized vortex beams, Opt. Laser Technol. 42 (2010) 186–191. [40] X. Gao, Q. Zhan, J. Li, J. Wang, S. Zhuang, Spirally polarized axisymmetric Besselmodulated Gaussian beam, Optik 122 (2011) 524–528.
[41] P. Suresh, C. Mariyal, R.C. Sarswathi, K.B. Rajesh, T.V.S. Pillai, Z. Jaroszewicz, Tightly focusing of spirally polarized Quadratic Bessel Gaussian beam through a dielectric interface, Optik 125 (2014) 1264–1266. [42] Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, S.T. Liu, Hollow sinh-Gaussian beams and their paraxial properties, Opt. Express 20 (2012) 9682–9691. [43] J. Lin, Y. Ma, P. Jin, G. Davies, J.B. Tan, Longitudinal polarized focusing of radially polarized sinh-Gauss beam, Opt. Exp. 21 (2013) 13193–13198. [44] Yong Liu, Cuifang Kuang, Xu Liu, The use of azimuthally polarized sinh-Gauss beam in STED microscopy, J. Opt. 17 (2015) 045609. [45] C.M. Sundaram, K. Prabakaran, P.M. Anbarasan, K.B. Rajesh, A.M. Musthafa, V. Aroulmoji, Tight focusing properties of phase modulated transversely polarized sinh Gaussian beam, Opt. Quant. Electron. 49 (2017) 11. [46] R. Murugesan, N. Pasupathy, M. Udhayakumar, K.B. Rajesh, Z. Jaroszewicz, Properties of surface plasmon polaritons excited by radially polarized sinh Gaussian beams, Plasmonics (2017) 1–7. [47] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proc. Roy. Soc. A 253 (1959) 358–379. [48] V.C. Daria, P.J. Rodrigo, J. Glückstad, Dynamic array of dark optical traps, Appl. Phys. Lett. 84 (2004) 323–325. [49] P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D.N. Christodoulides, Z. Chen, Trapping and transporting aerosols with a single optical bottle beam generated by Moire techniques, Opt. Lett. 36 (2011) 1491–1493. [50] V.G. Shvedov, A.S. Desyatnikov, A.V. Rode, Y.V. Izdebskaya, W.Z. Kroli kowski, Y.S. Kivshar, Optical vortex beams for trapping and transport of particles in air, Appl. Phys. A 100 (2010) 327–331. [51] A. Ashkin, J.M. Dziedzic, J.M. Bjorkholm, S. Chuk, Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett. 11 (1986) 288–290. [52] L. Isenhower, W. Williams, A. Dally, M. Saffman, Atom trapping in an inter ferometrically generated bottle beam trap, Opt. Lett. 34 (2009) 1159–1161. [53] R. Ozeri, L. Khaykovich, N. Davidson, Long spin relaxation times in a single-beam blue-detuned optical trap, Phys. Rev. A 59 (R1) (1999) 750–R1753. [54] K.I. Willig, R.R. Kellner, R. Medda, B. Hein, S. Jakobs, S.W. Hell, Nanoscale resolution in GFP-based microscopy, Nat. Methods 3 (9) (2006) 721–723. [55] D.W. Zhang, X.C. Yuan, Optical doughnut for optical tweezers, Opt. Lett. 28 (9) (2003) 740–742. [56] L.P. Du, G.H. Yuan, D.Y. Tang, X.C. Yuan, Tightly focused radially polarized beam for propagating surface plasmon-assisted gap-mode raman spectroscopy, Plasmonics 6 (4) (2011) 651–657. [57] Y. Konawa, S. Sato, Focusing property of a double-ring-shaped radially polarized beam, Opt. Lett. 31 (2006) 820–822. [58] Y. Zhao, Q. Zhan, Y. Zhang, Y.P. Li, Creation of a three-dimensional optical chain for controllable particle delivery, Opt. Lett. 30 (2005) 848–850. [59] Y. Yu, Q. Zhan, Creation of identical multiple focal spots with prescribed axial distribution, Sci. Rep. 5 (2015) 14673. [60] Z. Zhou, Q. Tan, C. Yang, L. Zhu, Near-field multiple optical trapping using high order axially symmetric polarized beams, Opt. Appl. 2 (2013) 287. [61] R.L. Eriksen, V.R. Daria, J. Gluckstad, Fully dynamic multiple beam optical tweezers, Opt. Express 14 (2002) 597.
628
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Focusing properties of spirally polarized sinh Gaussian beam a
M. Senthilkumar , K.B. Rajesh
b,⁎
b
c,d
e
, M. Udhayakumar , Z. Jaroszewicz , G. Mahadevan
T
a
Department of Physics, R & D Centre, Bharathiar University, Coimbatore, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India Institute of Applied Optics, Department of Physical Optics, Warsaw, Poland d National Institute of Telecommunications, Warsaw, Poland e The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu 624 302, India b c
HIGHLIGHTS
tight focusing properties of spirally polarized sinh-Gaussian beam is studied. • The properties are investigated theoretically by vector diffraction theory. • Focusing intensity in focal region can be altered considerably by the spiral parameter. • Optical • Focal patterns including flattop profile, focal hole, axially separated focal spots are generated. ARTICLE INFO
ABSTRACT
Keywords: Focusing property spirally polarized sinh Gaussian beam Vector diffraction theory
The tight focusing properties of spirally polarized sinh-Gaussian beam is investigated numerically by vector diffraction theory. Results show that the optical intensity in focal region of spirally polarized sinh Gaussian beam can be altered considerably by the beam order, relative waist width and the spiral parameter that indicates the polarization spiral degree of the spirally polarized sinh Gaussian beam. Many novel focal patterns including flattop profile, focal hole axially separated focal spots and focal spot with long focal depth are evolved considerably for the suitable beam parameters. We expect such a tunable focal patterns are useful for optical manipulation of micro particles.
1. Introduction Creating suitable focal patterns with precise structure and polarization in a focal region of high NA lens system plays an important role in applications including super-resolution imaging [1–3], particle trapping and manipulating [4–6], laser micromachining [7], optical lithography [8,9] and so on. A variety of peculiar optical field distributions in the focal region such as optical needle [10–14], optical chain [15–20], optical bubble or cage [21–23], spherical spot [24–26], and super long dark channel [27,28], etc. have been generated based on amplitude, phase and polarization Engineering. Beam shaping using spatially variant polarization has been a topic of much interest in the recent years owing to its advantages of achieving the desired focal structures and polarization in the focal region without using amplitude or phase filters. Recently, there is an increasing interest in laser beams with cylindrical symmetry in polarization. These so-called cylindrical vector beams can be generated by active or passive methods [29–32] and have been the topic of numerous recent theoretical and ⁎
experimental investigations [33–36]. Radial and azimuthal polarizations are two basic components of cylindrical polarization. Achieving desired focal patterns using amplitude, phase and polarization within the pupil to generate many novel focal pattern have been studied [10–28]. However, the presence of phase filters or Amplitude filters makes some applications more difficult or even impossible. Bing Hao and James Leger have recently investigated the numerical aperture (NA) invariant focus shaping using spirally polarized beams. They proposed that spirally polarization is another kind of spatially variant polarization bearing radial symmetry [37]. The spirally polarized beam possess axially symmetric polarization patterns, with linear polarization at any point of the transverse profile and with the electric field lines being logarithmic spiral. Recent investigation of the tight focusing properties of spirally polarized beam suggested several possible applications such as as optical tweezers, particle trapping, laser cutting, material processing, microscopy, etc. [38–41]. Recently, Sun et al. introduce a new class of sinh-Gaussian beam called hollow sinh-Gaussian beams which is characterized by single ringed transverse intensity
Corresponding author. E-mail address: [email protected] (K.B. Rajesh).
https://doi.org/10.1016/j.optlastec.2018.10.048 Received 27 July 2018; Received in revised form 3 October 2018; Accepted 22 October 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 1. Diagram of the focusing configuration.
Fig. 2. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8.
distributions [42]. Jie Lin et al. analyzed the focusing performance of radial polarized sinh-Gaussian beam by lens with large NA based on the Richards-Wolf’s formulas. They reported that under suitable beam parameter high beam quality and sub wavelength focusing can be achieved simultaneously [43]. Followed by this work, several work on tight focusing properties of sinh Gaussian beam have been reported [44–46]. In this paper, we numerically demonstrate the focusing performance of spirally polarized hollow sinh-Gaussian beam by high NA objective lens. We expect such a study may pave the way for the deeper understanding of hallow sinh Gaussian beam extended to possible applications.
2. Principle of the optical focusing spirally polarized sinh Gaussian beam A schematic configuration of the proposed method is shown in Fig. 1. In this paper we employed the Richards and Wolf vector diffraction theory to analyze the tight focusing properties of the spirally polarized beam by taking into account the spatial inhomogeneity vector property [47]. This method is extensively used in applications such as optical trapping and recording where high NA imaging and focusing are considered. Furthermore, in dealing with radial and azimuthal polarization or the generalized cylindrical vector beam, this method has been routinely employed and the integral formula can be shown to take a
624
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 3. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0.4. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8.
relatively simple form. The electric field in focal region of spirally polarized can be given by [40]
0 ≤ θ ≤ arcsin(NA/n). Where, n = 1.0 is the index of refraction of free space. Obviously, as defined in Eq. (5), the amplitude distribution of hollow sinh-Gaussian beam is determined by w0 and m. Therefore, one can control the amplitude distribution of hollow sinh-Gaussian beam by choosing w0 or m reasonably The term ϕ(θ) in Eqs. (2)–(4) characterizes the polarization of focusing spirally polarized beam and can be in different forms [37–41].we assume ϕ(θ) in the form as in Ref. [40]
(1)
E (r , , z ) = Er er + Ez ez + E e
where er, eϕ and ez are the unit vectors in the radial, azimuthal, and propagating directions, respectively. Er, Ez, and Eϕ are amplitudes of the three orthogonal components and can be expressed as
Er (r ,z ) = A
cos · cos[ ( )]·E0 ( )sin(2 )·J1 (kr sin( ))·
0
(2)
exp(ikzcos( ))d Ez (r , z ) = 2iA
0
0
(3)
In the investigation of spirally polarized sinh Gaussian beam, without loss of validity and generality, it is proposed that the numerical aperture of the focusing optical system NA = 0.95 and n = 1. Fig. 2 shows the focal structure generated for m = 4 and 8 for the relative waist size w0 = 0.500 and the spirality C = 0. The 2D intensity along rz plane shown in Fig. 2(a) and their corresponding radial and axial intensity shown in Fig. 2(b&c) shows that the generated focal segment is highly confined focal spot having FWHM 0.51λ and focal depth of 1.74λ . It is noted from Fig. 2(b), the focal structure is dominated by Ez component. Fig. 2(d,e & f) shows same as Fig. 2(a–c) but for m = 8. It is
(4)
For a high NA objective lens, the electric field of hollow sinhGaussian incident beam at a pupil can be defined as follows [43]
Emw0 ( ) = sinhm
sin( ) ·exp w0
sin2 ( ) w02
(6)
3. Result and discussion
cos · sin[ ( )]·E0 ( ) sin( )·J1 (kr sin( ))·
exp(ikzcos( )) d
tan( ) × tan( )
where C is spiral parameter indicating polarization spiral degree, = arcsin(NA) is convergence angle corresponding to the radius of incident optical aperture.
cos · cos[ ( )]·E0 ( )sin2 ( )·J0 (kr sin( ))·
exp(ikzcos( )) d
E (r , z ) = 2A
( )= C×
(5)
where m (m = 0, 1, 2,…) is the order of the hollow sinh-Gaussian beam. However, a new kind of sinh-Gaussian beam is obtained when m is greater than 1.0. θ is determined by NA of objective lens and
625
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 4. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 0.9. For w0 = 0.500. Figure (d-f) Same as (a–c) but for m = 8.
noted from figure increasing the beam order m to 8 slightly reduced the FWHM the spot size as (0.41λ) and improved the focal depth as 2.54λ . This reduction in the spot size is due to the contribution of the longitudinal field component near the focal plane which is mainly derived from the outer most ray of the beam [43]. As order to the beam is increased the central dark area is increased and hence the bend angle of the outermost ray is increased and so are the longitudinal component and the total field leads to a sharper focal spot. In view of the transfer function, incident hollow sinh Gaussian beam with smaller w0 and m include much of component with low frequency and hence exhibit larger lateral width at the focal plane. Whereas the large m and w0 are attributed to much of component with high frequency and hence a sharper focal spot is observed. Fig. 3(a) shows the flattop structure generated for different beam order of sinh Gaussian beam. It is noted from Fig. 3(a–c) for the beam order m = 4 increasing the spirality C to 0.4 generated the flattop profile with FWHM of 0.955λ with focal depth of 1.85λ. Fig. 3(d–f) shows increasing m to 8 generated a flattop profile when C = 0.42. The FWHM of the generated flattop profile is measured as 0.855λ whereas its focal depth is found to be 2.48λ. This type of beam profile is useful in improved printing filling factor, improved uniformity and quality in materials processing, particle acceleration and microlithography. Fig. 4(a–c) shows that by setting C = 0.9, generated a 3D optical bubble for m = 4. We also noted that the FWHM the generated optical bubble is 1.14 λ where is focal depth is 2.1λ. However a 3D optical bubble with FWHM 1.05λ and focal depth of 4.24λ is obtained for
m = 8 and C = 0.9 shown in Fig. 4(d–f). Such an optical bubble with uniform intensity find the application in stable optical traps for those particles whose refractive indices are smaller than the ambient [48]. These micromanipulations have been reported to be implemented both in fluids and in air [49,50] and hence the optical bottle beams can trap atoms and molecules [51–53]. Apart from this, the 3D hollow focus has been drawing great interest in the recent year due to its wide applications in many research domains: such as nonlinear optics [54], biooptics [55] and excitation of surface wave [56]. Fig. 5(a–c) shows that by setting C = 2.4 and m = 4, the generated focal structure has an axially splitted focal spot search having flattop profile with FWHM of 0.892λ and focal depth of 1.7λ, separated by an axial distance of 1.7λ. Fig. 5(d–f), we also noted that for m = 8, the splitted focal spot appeared when C = 3.4. It is noted that the FWHM of generated focal spot is 0.5λ whereas its focal depth is 2.12λ. Such an axially splitted focal spot is useful to trap and manipulate small particles, such as atoms, molecules, cells [15,57–61]. It has wide applications in many areas, such as DNA sequencing, genes transplant, microassembly and micromachining. Hence from the above discussion we shows that by using single focusing unit and by properly tuning the beam order and spiral parameter of spirally polarized Hollow sinh Gaussian beam with suitable m and w0, one can achieve many novel focal patterns including sub wavelength focal spot with long focal depth, flat top profile, optical bubble, splitted focal spots with flat top profile etc. The authors expect that such a study exhibiting the possibility of manipulating the focal patterns will be of useful for
626
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al.
Fig. 5. (a) The intensity distribution in the r-z plane. (b) The 2D intensity distribution in the corresponding radial direction, (c) shows the respective axial intensity distribution measured at r = 0. m = 4, C = 2.4. For w0 = 0.500. Figure (d–f) Same as (a–c) but for m = 8, C = 3.4.
applications such as optical data storage, lithography, and optical manipulation.
[5] O.M. Maragò, P.H. Jones, P.G. Gucciardi, G. Volpe, A.C. Ferrari, Optical trapping and manipulation of nanostructures, Nat. Nanotechnol. 8 (11) (2013) 807–819. [6] S. Zhu, J. Wang, X. Liu, Y. Cai, Z. Li, Generation of arbitrary radially polarized array beams by manipulating correlation structure, Appl. Phys. Lett. 109 (16) (2016) 161904. [7] Ting-Hsuan Chen, Romain Fardel and Craig B Arnold Ultrafast z-scanning for highefficiency laser micro-machining, Light Sci. Appl. 7 (2018) 17181. [8] Z.S. Gan, Y.Y. Cao, R.A. Evans, et al., Three-dimensional deep sub-diffraction optical beam lithography with 9 nm feature size, Nat. Commun. 4 (2013) 2061. [9] D. Fan, L. Wangand, Y. Ekinci, Nanolithography using Bessel beams of extreme ultraviolet wavelength, Sci. Rep. 6 (2016) 31301. [10] Z. Nie, Z. Li, G. Shi, X. Zhang, Y. Wang, Y. Song, Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ringshaped azimuthally polarized beam, Opt. Lasers Eng. 59 (2014) 93–97. [11] F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, M. Hong, Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light, Sci. Rep. 5 (2015) 9977. [12] S.H.I. Chang-kun, et al., Super-resolution longitudinally polarized light needle achieved by tightly focusing radially polarized beams, Optoelectron. Lett. 14 (2018) 1–5. [13] Z. Man, et al., Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes, Opt. Express 24 (2016) 874–882. [14] K. Hu, Z. Chen, J. Pu, Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface, Opt. Lett. 37 (2012) 3303–3305. [15] J. Wang, W. Chen, Q. Zhan, Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams, J. Opt. 14 (2012) 055004. [16] J. Lin, R. Chen, P. Jin, M. Cada, Y. Ma, Generation of longitudinally polarized optical chain by 4π focusing system, Opt. Commun. 340 (2015) 69–73. [17] J.M. Wang, Q.L. Liu, C.J. He, Y.W. Liu, Reversal construction of polarization controlled focusing field with multiple focal spots, Opt. Eng. 52 (2013) 048002. [18] B. Gu, Y. Pan, J. Wu, Y. Cui, Manipulation of radial-variant polarization for creating tunable bifocusing spots, J. Opt. Soc. Am. A: 31 (2014) 253–257. [19] Q.F. Xu, J.N. Chen, The creation of double tight focus by a concentric multi-belt
4. Conclusion The tight focusing properties of spirally polarized sinh Gaussian beam by High NA Lens is investigated theoretically by vector diffraction theory. Results show that the optical intensity in focal region of spirally polarized sinh Gaussian beam can be altered considerably by the beam order and spiral parameter. Spiral parameter can induce focal pattern change in axial direction remarkably and many novel focal patterns including flattop profile, focal hole and focal spots of long focal depth and axially separated focal spots are evolved. We expect such a tunable focal patterns are useful for optical manipulation of micro particles super-resolution imaging, particle trapping and manipulating, laser micromachining, optical lithography and so on. References [1] E.T.F. Rogers, N.I. Zheludev, et al., Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging, J. Opt. 15 (9) (2013) 094008. [2] Wan-Chinkim, et al., Investigation on achieving super-resolution by solid immersion lens based STED microscopy, Opt. Express 25 (14) (2017) 16629. [3] H. Gao, M. Pu, X. Li, X. Ma, Z. Zhao, Y. Guo, X. Luo, Super-resolution imaging with a Bessel lens realized by a geometric meta surface, Opt. Express 25 (12) (2017) 13933–13943. [4] A.P. Porfirev, R.V. Skidanov, Dark-hollow optical beams with a controllable shape for optical trapping in air, Opt. Express 23 (7) (2015) 8373–8382.
627
Optics and Laser Technology 111 (2019) 623–628
M. Senthilkumar et al. pure phase filter, Opt. Commun. 285 (2002) 1642–1645. [20] T. Liu, J.B. Tan, J. Liu, J. Lin, Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel, J. Mod. Opt. 60 (2013) 378–381. [21] N. Bokor, N. Davidson, A three-dimensional dark focal spot uniformly surrounded by light, Opt. Commun. 279 (2007) 229–233. [22] X.L. Wang, et al., Configurable three-dimensional optical cage generated from cylindrical vector beams, Opt. Commun. 282 (2009) 3421–3425. [23] Zhongsheng Man, Zhidong Bai, Jinjian Li, Shuoshuo Zhang, Xiaoyu Li, Yuquan Zhang, Xiaolu Ge, Shenggui Fu, Optical cage generated by azimuthal- and radial-variant vector beams, Appl. Opt. 57 (13) (2018) 3592–3597. [24] W.B. Chen, Q.W. Zhan, Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy, Opt. Lett. 34 (2009) 2444–2446. [25] S.H. Yan, B.L. Yao, W. Zhao, M. Lei, Generation of multiple spherical spots with a radially polarized beam in a 4Pi focusing system, J. Opt. Soc. Am. A 27 (2010) 2033–2037. [26] S.N. Khonina, I. Golub, engineering the smallest 3D symmetrical bright and dark focal spots, J. Opt. Soc. Am. A: 30 (2013) 2029–2033. [27] K. Lalithambigai, P. Suresh, V. Ravi, K. Prabakaran, Z. Jaroszewicz, K.B. Rajesh, P.M. Anbarasan, T.V.S. Pillai, Generation of sub wavelength super-long dark channel using high NA lens axicon, Opt. Lett. 37 (2012) 999–1001. [28] J. William Charles, K. Prabakaran, K.B. Rajesh, Haresh M. Pandya, Generation of sub wavelength super long dark channel using azimuthally polarized annular multiGaussian beam, Opt. Quant. Electron. 46 (2014) 1079–1086. [29] Tingting Wang, Cuifang Kuang, Xiang Hao, Xu Liu, Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik 124 (2013) 4762–4765. [30] X. Yi, et al., Generation of cylindrical vector vortex beams by two cascaded meta surfaces, Opt. Exp. 22 (14) (2014) 17207–17215. [31] Alexey P. Porfirev, Andrey V. Ustinov, Svetlana N. Khonina, Polarization conversion when focusing cylindrically polarized vortex beams, Scientific Reports 6 (2016) 6. [32] Huan Chen, Xiaohui Ling, Qianguang Li, Hao Lv, Yu Huaqing, Xunong Yi, Generation of double-ring-shaped cylindrical vector beams by modulating Pancharatnam-Berry phase, Optik 134 (2017) 227–232. [33] Q. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [34] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon. 1 (2009) 1–57. [35] W. Chen, Q. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2006) 411–417. [36] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindrical vector beams, Opt. Express 7 (2000) 77–87. [37] B. Hao, J. Leger, Numerical aperture invariant focus shaping using spirally polarized beams, Opt. Commun. 281 (8) (2008) 1924–1928. [38] V. Ramírez-Sánchez, G. Piquero, M. Santarsiero, Generation and characterization of spirally polarized fields, J. Opt. A Pure Appl. Opt. 11 (2009) 085708–1 085708-6. [39] J. Pu, Z. Zhang, Tight focusing of spirally polarized vortex beams, Opt. Laser Technol. 42 (2010) 186–191. [40] X. Gao, Q. Zhan, J. Li, J. Wang, S. Zhuang, Spirally polarized axisymmetric Besselmodulated Gaussian beam, Optik 122 (2011) 524–528.
[41] P. Suresh, C. Mariyal, R.C. Sarswathi, K.B. Rajesh, T.V.S. Pillai, Z. Jaroszewicz, Tightly focusing of spirally polarized Quadratic Bessel Gaussian beam through a dielectric interface, Optik 125 (2014) 1264–1266. [42] Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, S.T. Liu, Hollow sinh-Gaussian beams and their paraxial properties, Opt. Express 20 (2012) 9682–9691. [43] J. Lin, Y. Ma, P. Jin, G. Davies, J.B. Tan, Longitudinal polarized focusing of radially polarized sinh-Gauss beam, Opt. Exp. 21 (2013) 13193–13198. [44] Yong Liu, Cuifang Kuang, Xu Liu, The use of azimuthally polarized sinh-Gauss beam in STED microscopy, J. Opt. 17 (2015) 045609. [45] C.M. Sundaram, K. Prabakaran, P.M. Anbarasan, K.B. Rajesh, A.M. Musthafa, V. Aroulmoji, Tight focusing properties of phase modulated transversely polarized sinh Gaussian beam, Opt. Quant. Electron. 49 (2017) 11. [46] R. Murugesan, N. Pasupathy, M. Udhayakumar, K.B. Rajesh, Z. Jaroszewicz, Properties of surface plasmon polaritons excited by radially polarized sinh Gaussian beams, Plasmonics (2017) 1–7. [47] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proc. Roy. Soc. A 253 (1959) 358–379. [48] V.C. Daria, P.J. Rodrigo, J. Glückstad, Dynamic array of dark optical traps, Appl. Phys. Lett. 84 (2004) 323–325. [49] P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D.N. Christodoulides, Z. Chen, Trapping and transporting aerosols with a single optical bottle beam generated by Moire techniques, Opt. Lett. 36 (2011) 1491–1493. [50] V.G. Shvedov, A.S. Desyatnikov, A.V. Rode, Y.V. Izdebskaya, W.Z. Kroli kowski, Y.S. Kivshar, Optical vortex beams for trapping and transport of particles in air, Appl. Phys. A 100 (2010) 327–331. [51] A. Ashkin, J.M. Dziedzic, J.M. Bjorkholm, S. Chuk, Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett. 11 (1986) 288–290. [52] L. Isenhower, W. Williams, A. Dally, M. Saffman, Atom trapping in an inter ferometrically generated bottle beam trap, Opt. Lett. 34 (2009) 1159–1161. [53] R. Ozeri, L. Khaykovich, N. Davidson, Long spin relaxation times in a single-beam blue-detuned optical trap, Phys. Rev. A 59 (R1) (1999) 750–R1753. [54] K.I. Willig, R.R. Kellner, R. Medda, B. Hein, S. Jakobs, S.W. Hell, Nanoscale resolution in GFP-based microscopy, Nat. Methods 3 (9) (2006) 721–723. [55] D.W. Zhang, X.C. Yuan, Optical doughnut for optical tweezers, Opt. Lett. 28 (9) (2003) 740–742. [56] L.P. Du, G.H. Yuan, D.Y. Tang, X.C. Yuan, Tightly focused radially polarized beam for propagating surface plasmon-assisted gap-mode raman spectroscopy, Plasmonics 6 (4) (2011) 651–657. [57] Y. Konawa, S. Sato, Focusing property of a double-ring-shaped radially polarized beam, Opt. Lett. 31 (2006) 820–822. [58] Y. Zhao, Q. Zhan, Y. Zhang, Y.P. Li, Creation of a three-dimensional optical chain for controllable particle delivery, Opt. Lett. 30 (2005) 848–850. [59] Y. Yu, Q. Zhan, Creation of identical multiple focal spots with prescribed axial distribution, Sci. Rep. 5 (2015) 14673. [60] Z. Zhou, Q. Tan, C. Yang, L. Zhu, Near-field multiple optical trapping using high order axially symmetric polarized beams, Opt. Appl. 2 (2013) 287. [61] R.L. Eriksen, V.R. Daria, J. Gluckstad, Fully dynamic multiple beam optical tweezers, Opt. Express 14 (2002) 597.
628