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Tight focusing properties of anomalous vortex beams
Optik 154 (2018) 133–138
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Tight focusing properties of anomalous vortex beams Mingyan Zhang, Yuanjie Yang ∗ School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu, 611731, China
a r t i c l e
i n f o
Article history: Received 24 June 2017 Accepted 3 October 2017 Keywords: Singular optics Diffraction theory Laser beam characterization
a b s t r a c t Tight focusing properties of an anomalous vortex beam (AVB) passing through a high numerical aperture (NA) lens system are investigated, based on vector Debye integral. The numerical examples show that intensity distribution for the AVBs in focal region depend on the beam parameters including the topological charge m, the order of the AVBs n. Furthermore, the phase distribution and the intensity distribution at the different propagation distance z are discussed in detail. Our results will be useful to find the potential applications of AVBs. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Vortex is a general phenomenon in wave fields, in which the flow rotates around the dark center [1]. In an optical vortex, its central light intensity is zero and the phase is undetermined [2]. In 1992, Allen et al. identified that vortex beams with a phase term of exp (ilϕ) carry an orbital angular momentum of l per photon, where l is the topological charge, ϕ is the azimuthal angle and is Planck’s constant [3]. The properties of vortex beams have attracted the strong interest of many researchers [4–6], and so far, vortex beams have found important applications in free-space information transfer and communications [7] and optical manipulation [8], etc. In 2013, a new type of vortex beam called anomalous vortex beam(AVB) has been proposed both theoretically and experimentally, the AVB will eventually become an elegant Laguerre-Gaussian beam in the far field (or in the focal plane) in free space [9]. Recently, the characteristic of AVBs through paraxial optical systems [10,11] and in strongly nonlocal nonlinear media [12] has been discussed. On the other hand, the focusing properties of the beams focused by a high NA lens system have attracted much attention due to their potential applications in high density optical data storage [13], material processing [14] and particle trapping, etc. [15,16]. Therefore, in recent years, the tight focusing properties of vortex beams have been studied extensively [17–23], such as, the degree of polarization has been discussed by considering the distribution of the spectral densities of the three electric field components in the focusing region [17], and the electric fields have been calculated in the study of azimuthally and radially polarized beams focused by a high NA lens system [19], besides, the tight properties of an asymmetric Bessel beam and Gaussian beam are studied in detail as well [21,22]. However, up to now, to our knowledge, the tight focusing properties of AVBs passing through a high NA lens system have not been studied yet. The purpose of this letter is to investigate the tight focusing properties of AVBs, and which will be important to the applications of AVBs in some fields. In Section 2, the theoretical electric fields for AVBs passing through a high NA lens system are derived. In Section 3, some numerical simulation results are presented and discussed. The Section 4 is a summary of the paper.
∗ Corresponding author. E-mail address: [email protected] (Y. Yang). https://doi.org/10.1016/j.ijleo.2017.10.013 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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M. Zhang, Y. Yang / Optik 154 (2018) 133–138
Fig. 1. Schematic diagram of the tight focusing system.
2. Theory The electric field of the AVB at z = 0 is defined as follows En,m (r, ϕ) = E0
r 2n+|m| w0
exp
r2 − w0 2
exp(−imϕ).
(1)
where E0 is a constant, n is the beam order of the AVB, m is the topological charge, and w0 is beam waist size of the fundamental Gaussian beam (m = n = 0), r and ϕ are radial and azimuthal coordinates, respectively. According to the vector Debye integral, for a linearly polarized incident beam along x axis, the electric field in the vicinity of the focal region of a high NA lens system can be expressed as [24–26].
⎡
E(, , z) =
i
→
[cos + sin2 ϕ(1 − cos )]ex
⎢ ⎢ → ⎣ [cos ϕ sin ϕ(cos − 1)]ey
2 ˛
E(, ϕ) × T () × ⎢ 0
0
⎤ ⎥ ⎥ ⎥ ⎦
(2)
→ )ez
(cos ϕ sin × exp[−ik sin cos(ϕ − )] exp(−ikz cos ) sin d dϕ 1
where T() is the apodization function (for an aplanatic lens T () = cos 2 [18]), E(, ϕ) is the pupil apodization function at the entrance pupil, is the angle of convergence. The schematic diagram of the tight focusing system is shown in Fig. 1. As shown in Fig. 1, is wavelength of the incident beam;k(k = 2 ) is the wavenumber; f is the focal length of the high numerical → → →
aperture objective; The parameters , , z are the cylindrical coordinates of an observation; ex , ey , ez are the unit vectors along the x, y, z directions, respectively. ␣ is given by numerical aperture value sin␣. Under the sine condition, we get r = fsin, so that the pupil apodization function of the AVB can be written as
f sin w0
En,m (, ϕ) = E0
2n+|m|
exp
−
f 2 sin2 w0 2
exp(−imϕ),
(3)
According to Eqs. (1)–(3), after some simplification, the x, y, and z components of the electrical field in the vicinity of the focus can be simplified as
˛
A {(−1)m
Ex (, , z) = ik
1 + cos m i Jm (k sin ) exp(−im) 2
0
− (−1) m+2
1 − cos m+2 i Jm+2 (k sin ) exp[−i(m + 2)] 4
− (−1) m−2
1 − cos m−2 i Jm−2 k sin exp[−i(m − 2)]}d, 4
k Ey (, , z) = − 4
˛
(4a)
A (cos − 1){(−1) m+2 im+2 Jm+2 k sin exp[−i(m + 2)] 0
− (−1) m−2 im−2 Jm−2 k sin
exp[−i(m − 2)]}d,
(4b)
M. Zhang, Y. Yang / Optik 154 (2018) 133–138
135
Fig. 2. Calculated three-dimensional intensity distribution (a-l) and the corresponding two-dimensional distribution (a -l ) of |E|2 for a highly focused AVB with n = 3, NA = 0.8, = 632.8 nm, w0 = 2 mm, f = 2 mm. (a, e, i) are total field, (b, f, j), (c, g, k) and (d, h, l) are component|Ex |2 , |Ey |2 , |Ez |2 , respectively. a, b, c, d m = 1, e, f, g, h m = 2, i, j, k, l m = 3.
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M. Zhang, Y. Yang / Optik 154 (2018) 133–138
Fig. 3. Calculated two-dimensional distribution of |E|2 for a highly focused AVB with m = 3, NA = 0.8, = 632.8 nm, w0 = 2 mm, f = 2 mm. (a, e, i) are total field, (b, f, g), (c, g, k) and (d, h, l) are component |Ex |2 , |Ey |2 , |Ez |2 , respectively. a, b, c, d n = 1, e, f, g, h n = 5, i, j, k, l n = 10.
1 Ez (, , z) = ik 2
˛
A sin {(−1) m+1 im+1 Jm+1 k sin exp[−i(m + 1)] 0
(4c)
+ (−1) m−1 im−1 Jm−1 k sin exp[−i(m − 1)]}d, where A() = E0
f sin w0
2n+|m|
exp
−
f 2 sin2 w0 2
1
exp −ikz cos sin cos 2 .
(4d)
and Jm (x) is a Bessel function of the first kind of order m. From the above derived equations, we can see that a linearly polarized incident beam along x passing through a high NA lens system, there will produce an orthogonal and a longitudinal component along the incident polarization direction in the vicinity of the focus. 3. Numerical examples and results discussion In this section, we performed some numerical calculations on the focusing properties for the AVBs through a high NA lens system. Fig. 2 shows the total intensity distribution and its x, y, and z components in the focal plane for different topological charges with m = 1, 2, and 3, respectively. The other parameters for the calculations are n = 3, NA = 0.8, = 632.8nm, w0 = 2mm, f = 2mm. Fig. 2(a, a’ ), (e, e’ ), and (i, i’ ) show that the total intensity distribution in the focal plane has two peaks along the y direction, the dark region in the center of the total intensity pattern appear and become larger gradually with increasing topological charge number m. It is noted that the intensity distribution of the x-component in the focal plane is similar with the total field, which indicates that the x-component dominates over the total intensity distribution. While, the intensity patterns of y-component and z-component are quite different from that of x-component. For the y-component, there is the maximum light intensity at the intensity pattern for the topological charge m = 2. We found that when m > 2, a central dark appear in the intensity pattern of y-component and which increases with the increasing m. Besides, we can see that the central bright ring is not a circle but resembling closely a square. For the z-component, there is the maximum light intensity at the center of AVB for the topological charge m = 1, the central light intensity will become zero when the topological charge m = 2(m = 3), and the dark area increases with the increase of the topological charge m. Fig. 3 shows the total intensity distribution and its x, y, and z components in the focal plane with n = 1, 5, and 10, respectively. It is shown that with the increasing n, the outer rings around the central ring of the total field are gradually becoming brighter,which because that the outer rings of the x-component brighten gradually [see Fig. 3(b),(f) and (g)]. However, as a
M. Zhang, Y. Yang / Optik 154 (2018) 133–138
137
Fig. 4. Diagrams of phase distribution (a, c, e) and the corresponding two-dimensional distribution (b, d, f) of the x-component of the electric field as an AVB focused by a high NA lens system, m = 2, n = 2, NA = 0.9, = 632.8 nm, w0 = 2 mm, f = 2 mm a, b z = − ; c, d z = 0; e, f z = .
whole, with the parameter n increases, the two-dimensional distributions of the total field and each component in the focal plane keep a similar pattern. Therefore, it can be concluded that the influence of parameter n on the shape of the intensity patterns is insignificant. Owing to that in the three components of electric field in the focal plane, the magnitude of x-component is the largest [27], here we give the diagrams of phase distribution and the corresponding two-dimensional distribution in the cross sections of the x-component along z-axis in Fig. 4. The locations of Fig. 4(a, b), (c, d), (e, f) are z = − , 0, , respectively, where z = 0 refers to the focal plane. As shown in Fig. 4(a), (c), (e), we can see that the helical phase distribution occurs. As the value of the parameter z is changed from negative to positive, it is found that the rotation direction of the phase is changed from anticlockwise rotated to clockwise rotated. Accordingly, the light intensity patterns in Fig. 4(b) and (f) are mirror symmetric. This because that when a light beam passing through the focal plane, the propagation mode of the light will change from the converging wave to the divergent wave, which will lead to the phase reversal and the rotation of the intensity pattern. 4. Conclusions As a summary, we have studied the tight focusing properties of an AVB passing through a high NA lens system. The analytical formulas for the AVBs through the high NA lens system are derived by using the Vector diffraction theory. The intensity distributions in the focal region are investigated, the influences of the topological charge number m and the beam order n on the intensity distributions are discussed in detail. We found that the total intensity pattern is not a rotationally symmetric any longer for the tight focusing cases. Furthermore, it is shown that the influence of parameter n on the shape of the intensity patterns is not as significant as that of parameter m. The theoretical results obtained in this paper might be useful to find the potential application of AVBs in optical trapping. Acknowledgments We acknowledge the support of the National Natural Science Foundation of China under Grant nos. 61205122 and 11474048. References [1] W.H. Lee, Computer-generated holograms: techniques and applications, Prog. Opt. 16 (3) (1978) 119–232. [2] P. Coullet, L. Gill, F. Rocca, Optical vortices, Opt. Commun. 73 (5) (1989) 403–408. [3] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phy. Rev. A 45 (11) (1992) 8185–8189. [4] A. Ya. Bekshaev, Manifestation of mechanical properties of light waves in vortex beam optical systems, Opt. Spectrosc. 88 (6) (2000) 904–910. [5] Y. Yang, D. Jiang, Y. Liu, G. Thirunavukkarasu, Topological charges shift of vectorial nonparaxial vortex beam, Optik 127 (24) (2016) 11644–11648. [6] J. Li, W. Wang, M. Duan, J. Well, Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams, Opt. Express 24 (18) (2016) 20413–20423.
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[7] J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. Willner, Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photonics 6 (2012) 488–496. [8] K.T. Dholakia Cizmar, Shaping the future of manipulation, Nat. Photonics 5 (6) (2011) 335–342. [9] Y. Yang, Y. Dong, C. Zhao, Y. Cai, Generation and propagation of an anomalous vortex beam, Opt. Lett. 38 (24) (2013) 5418–5421. [10] Y. Xu, S. Wang, Characteristic study of anomalous vortex beam through a paraxial optical system, Opt. Commun. 331 (2014) 32–38. [11] Y. Yuan, Y. Yang, Propagation of anomalous vortex beams through an annular apertured paraxial ABCD optical system, Opt. Quant. Electron. 47 (7) (2015) 2289–2297. [12] Z. Dai, Z. Yang, S. Zhang, Propagation of anomalous vortex beams in strongly nonlocal nonlinear media, Opt. Commun. 350 (2015) 19–27. [13] M. Gu, X.P. Li, Y.Y. Cao, Optical storage arrays: a perspective for future big data storage, Light: Sci. Appl. 3 (2014) e177. [14] K. Lou, S.X. Qian, Z.C. Ren, C.H. Tu, Y.N. Li, H.T. Wang, Femtosecond laser processing by using patterned vector optical fields, Sci. Rep. 3 (2281) (2013). [15] Q. Zhan, Trapping metallic rayleigh particles with radial polarization, Opt. Express 12 (15) (2004) 3377–3382. [16] Y. Kozawa, S. Sato, Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams, Opt. Express 18 (10) (2010) 10828–10833. [17] K. Lindfors, T. Setala, M. Kaivola, A.T. Friberg, Degree of polarization in tightly focused optical fields, J. Opt. Soc. Am. A 22 (3) (2005) 561–568. [18] B. Chen, Z. Zhang, J. Pu, Tight focusing of partially coherent and circularly polarized vortex beams, J. Opt. Soc. Am. A 26 (4) (2009) 862–869. [19] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindrical-vector beams, Opt. Express 7 (2) (2000) 77–87. [20] Q. Zhan, Properties of circularly polarized vortex beams, Opt. Lett. 31 (7) (2006) 867–869. [21] V.V. Kotlyar, S.S. Stafeev, A.P. Porfirev, Tight focusing of an asymmetric Bessel beam, Opt. Commun. 357 (2015) 45–51. [22] Z. Chen, J. Pu, D. Zhao, Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices, Phys. Lett. A 375 (32) (2011) 2958–2963. [23] J. Pu, Z. Zhang, Tight focusing of spirally polarized vortex beams, Opt. Laser. Technol. 42 (1) (2010) 186–191. [24] E. Wolf, Electromagnetic diffraction in optical systems I. An integral representation of the image field, Proc. R. Soc. A 253 (1959) 349–357. [25] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proc. R. Soc. A 253 (1959) 358–379. [26] M. Gu, Advanced Optical Imaging Theory, Springer-Verlag, Berlin Heidelberg, 2000. [27] J.W.M. Chon, X.S. Gan, M. Gu, Splitting of the focal spot of a high numerical-aperture objective in free space, Appl. Phys. Lett. 81 (9) (2002) 1576–1578.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Tight focusing properties of anomalous vortex beams Mingyan Zhang, Yuanjie Yang ∗ School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu, 611731, China
a r t i c l e
i n f o
Article history: Received 24 June 2017 Accepted 3 October 2017 Keywords: Singular optics Diffraction theory Laser beam characterization
a b s t r a c t Tight focusing properties of an anomalous vortex beam (AVB) passing through a high numerical aperture (NA) lens system are investigated, based on vector Debye integral. The numerical examples show that intensity distribution for the AVBs in focal region depend on the beam parameters including the topological charge m, the order of the AVBs n. Furthermore, the phase distribution and the intensity distribution at the different propagation distance z are discussed in detail. Our results will be useful to find the potential applications of AVBs. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Vortex is a general phenomenon in wave fields, in which the flow rotates around the dark center [1]. In an optical vortex, its central light intensity is zero and the phase is undetermined [2]. In 1992, Allen et al. identified that vortex beams with a phase term of exp (ilϕ) carry an orbital angular momentum of l per photon, where l is the topological charge, ϕ is the azimuthal angle and is Planck’s constant [3]. The properties of vortex beams have attracted the strong interest of many researchers [4–6], and so far, vortex beams have found important applications in free-space information transfer and communications [7] and optical manipulation [8], etc. In 2013, a new type of vortex beam called anomalous vortex beam(AVB) has been proposed both theoretically and experimentally, the AVB will eventually become an elegant Laguerre-Gaussian beam in the far field (or in the focal plane) in free space [9]. Recently, the characteristic of AVBs through paraxial optical systems [10,11] and in strongly nonlocal nonlinear media [12] has been discussed. On the other hand, the focusing properties of the beams focused by a high NA lens system have attracted much attention due to their potential applications in high density optical data storage [13], material processing [14] and particle trapping, etc. [15,16]. Therefore, in recent years, the tight focusing properties of vortex beams have been studied extensively [17–23], such as, the degree of polarization has been discussed by considering the distribution of the spectral densities of the three electric field components in the focusing region [17], and the electric fields have been calculated in the study of azimuthally and radially polarized beams focused by a high NA lens system [19], besides, the tight properties of an asymmetric Bessel beam and Gaussian beam are studied in detail as well [21,22]. However, up to now, to our knowledge, the tight focusing properties of AVBs passing through a high NA lens system have not been studied yet. The purpose of this letter is to investigate the tight focusing properties of AVBs, and which will be important to the applications of AVBs in some fields. In Section 2, the theoretical electric fields for AVBs passing through a high NA lens system are derived. In Section 3, some numerical simulation results are presented and discussed. The Section 4 is a summary of the paper.
∗ Corresponding author. E-mail address: [email protected] (Y. Yang). https://doi.org/10.1016/j.ijleo.2017.10.013 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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M. Zhang, Y. Yang / Optik 154 (2018) 133–138
Fig. 1. Schematic diagram of the tight focusing system.
2. Theory The electric field of the AVB at z = 0 is defined as follows En,m (r, ϕ) = E0
r 2n+|m| w0
exp
r2 − w0 2
exp(−imϕ).
(1)
where E0 is a constant, n is the beam order of the AVB, m is the topological charge, and w0 is beam waist size of the fundamental Gaussian beam (m = n = 0), r and ϕ are radial and azimuthal coordinates, respectively. According to the vector Debye integral, for a linearly polarized incident beam along x axis, the electric field in the vicinity of the focal region of a high NA lens system can be expressed as [24–26].
⎡
E(, , z) =
i
→
[cos + sin2 ϕ(1 − cos )]ex
⎢ ⎢ → ⎣ [cos ϕ sin ϕ(cos − 1)]ey
2 ˛
E(, ϕ) × T () × ⎢ 0
0
⎤ ⎥ ⎥ ⎥ ⎦
(2)
→ )ez
(cos ϕ sin × exp[−ik sin cos(ϕ − )] exp(−ikz cos ) sin d dϕ 1
where T() is the apodization function (for an aplanatic lens T () = cos 2 [18]), E(, ϕ) is the pupil apodization function at the entrance pupil, is the angle of convergence. The schematic diagram of the tight focusing system is shown in Fig. 1. As shown in Fig. 1, is wavelength of the incident beam;k(k = 2 ) is the wavenumber; f is the focal length of the high numerical → → →
aperture objective; The parameters , , z are the cylindrical coordinates of an observation; ex , ey , ez are the unit vectors along the x, y, z directions, respectively. ␣ is given by numerical aperture value sin␣. Under the sine condition, we get r = fsin, so that the pupil apodization function of the AVB can be written as
f sin w0
En,m (, ϕ) = E0
2n+|m|
exp
−
f 2 sin2 w0 2
exp(−imϕ),
(3)
According to Eqs. (1)–(3), after some simplification, the x, y, and z components of the electrical field in the vicinity of the focus can be simplified as
˛
A {(−1)m
Ex (, , z) = ik
1 + cos m i Jm (k sin ) exp(−im) 2
0
− (−1) m+2
1 − cos m+2 i Jm+2 (k sin ) exp[−i(m + 2)] 4
− (−1) m−2
1 − cos m−2 i Jm−2 k sin exp[−i(m − 2)]}d, 4
k Ey (, , z) = − 4
˛
(4a)
A (cos − 1){(−1) m+2 im+2 Jm+2 k sin exp[−i(m + 2)] 0
− (−1) m−2 im−2 Jm−2 k sin
exp[−i(m − 2)]}d,
(4b)
M. Zhang, Y. Yang / Optik 154 (2018) 133–138
135
Fig. 2. Calculated three-dimensional intensity distribution (a-l) and the corresponding two-dimensional distribution (a -l ) of |E|2 for a highly focused AVB with n = 3, NA = 0.8, = 632.8 nm, w0 = 2 mm, f = 2 mm. (a, e, i) are total field, (b, f, j), (c, g, k) and (d, h, l) are component|Ex |2 , |Ey |2 , |Ez |2 , respectively. a, b, c, d m = 1, e, f, g, h m = 2, i, j, k, l m = 3.
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M. Zhang, Y. Yang / Optik 154 (2018) 133–138
Fig. 3. Calculated two-dimensional distribution of |E|2 for a highly focused AVB with m = 3, NA = 0.8, = 632.8 nm, w0 = 2 mm, f = 2 mm. (a, e, i) are total field, (b, f, g), (c, g, k) and (d, h, l) are component |Ex |2 , |Ey |2 , |Ez |2 , respectively. a, b, c, d n = 1, e, f, g, h n = 5, i, j, k, l n = 10.
1 Ez (, , z) = ik 2
˛
A sin {(−1) m+1 im+1 Jm+1 k sin exp[−i(m + 1)] 0
(4c)
+ (−1) m−1 im−1 Jm−1 k sin exp[−i(m − 1)]}d, where A() = E0
f sin w0
2n+|m|
exp
−
f 2 sin2 w0 2
1
exp −ikz cos sin cos 2 .
(4d)
and Jm (x) is a Bessel function of the first kind of order m. From the above derived equations, we can see that a linearly polarized incident beam along x passing through a high NA lens system, there will produce an orthogonal and a longitudinal component along the incident polarization direction in the vicinity of the focus. 3. Numerical examples and results discussion In this section, we performed some numerical calculations on the focusing properties for the AVBs through a high NA lens system. Fig. 2 shows the total intensity distribution and its x, y, and z components in the focal plane for different topological charges with m = 1, 2, and 3, respectively. The other parameters for the calculations are n = 3, NA = 0.8, = 632.8nm, w0 = 2mm, f = 2mm. Fig. 2(a, a’ ), (e, e’ ), and (i, i’ ) show that the total intensity distribution in the focal plane has two peaks along the y direction, the dark region in the center of the total intensity pattern appear and become larger gradually with increasing topological charge number m. It is noted that the intensity distribution of the x-component in the focal plane is similar with the total field, which indicates that the x-component dominates over the total intensity distribution. While, the intensity patterns of y-component and z-component are quite different from that of x-component. For the y-component, there is the maximum light intensity at the intensity pattern for the topological charge m = 2. We found that when m > 2, a central dark appear in the intensity pattern of y-component and which increases with the increasing m. Besides, we can see that the central bright ring is not a circle but resembling closely a square. For the z-component, there is the maximum light intensity at the center of AVB for the topological charge m = 1, the central light intensity will become zero when the topological charge m = 2(m = 3), and the dark area increases with the increase of the topological charge m. Fig. 3 shows the total intensity distribution and its x, y, and z components in the focal plane with n = 1, 5, and 10, respectively. It is shown that with the increasing n, the outer rings around the central ring of the total field are gradually becoming brighter,which because that the outer rings of the x-component brighten gradually [see Fig. 3(b),(f) and (g)]. However, as a
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Fig. 4. Diagrams of phase distribution (a, c, e) and the corresponding two-dimensional distribution (b, d, f) of the x-component of the electric field as an AVB focused by a high NA lens system, m = 2, n = 2, NA = 0.9, = 632.8 nm, w0 = 2 mm, f = 2 mm a, b z = − ; c, d z = 0; e, f z = .
whole, with the parameter n increases, the two-dimensional distributions of the total field and each component in the focal plane keep a similar pattern. Therefore, it can be concluded that the influence of parameter n on the shape of the intensity patterns is insignificant. Owing to that in the three components of electric field in the focal plane, the magnitude of x-component is the largest [27], here we give the diagrams of phase distribution and the corresponding two-dimensional distribution in the cross sections of the x-component along z-axis in Fig. 4. The locations of Fig. 4(a, b), (c, d), (e, f) are z = − , 0, , respectively, where z = 0 refers to the focal plane. As shown in Fig. 4(a), (c), (e), we can see that the helical phase distribution occurs. As the value of the parameter z is changed from negative to positive, it is found that the rotation direction of the phase is changed from anticlockwise rotated to clockwise rotated. Accordingly, the light intensity patterns in Fig. 4(b) and (f) are mirror symmetric. This because that when a light beam passing through the focal plane, the propagation mode of the light will change from the converging wave to the divergent wave, which will lead to the phase reversal and the rotation of the intensity pattern. 4. Conclusions As a summary, we have studied the tight focusing properties of an AVB passing through a high NA lens system. The analytical formulas for the AVBs through the high NA lens system are derived by using the Vector diffraction theory. The intensity distributions in the focal region are investigated, the influences of the topological charge number m and the beam order n on the intensity distributions are discussed in detail. We found that the total intensity pattern is not a rotationally symmetric any longer for the tight focusing cases. Furthermore, it is shown that the influence of parameter n on the shape of the intensity patterns is not as significant as that of parameter m. The theoretical results obtained in this paper might be useful to find the potential application of AVBs in optical trapping. Acknowledgments We acknowledge the support of the National Natural Science Foundation of China under Grant nos. 61205122 and 11474048. References [1] W.H. Lee, Computer-generated holograms: techniques and applications, Prog. Opt. 16 (3) (1978) 119–232. [2] P. Coullet, L. Gill, F. Rocca, Optical vortices, Opt. Commun. 73 (5) (1989) 403–408. [3] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phy. Rev. A 45 (11) (1992) 8185–8189. [4] A. Ya. Bekshaev, Manifestation of mechanical properties of light waves in vortex beam optical systems, Opt. Spectrosc. 88 (6) (2000) 904–910. [5] Y. Yang, D. Jiang, Y. Liu, G. Thirunavukkarasu, Topological charges shift of vectorial nonparaxial vortex beam, Optik 127 (24) (2016) 11644–11648. [6] J. Li, W. Wang, M. Duan, J. Well, Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams, Opt. Express 24 (18) (2016) 20413–20423.
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