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Properties of a Tightly Focused Circularly Polarized Anomalous Vortex Beam and Its Optical Forces on Trapped Nanoparticles
Nanoscale Research Letters volume 14, Article number: 252 (2019)
The characteristics of a circularly polarized anomalous vortex beam (CPAVB), focused by an objective lens with a high numerical aperture (NA), are studied analytically and theoretically. It shows that the topological charge can affect the beam profile significantly and a flat-topped (FT) beam can be obtained by modulating the NA and topological charge. It is interesting to find that spin-to-orbital angular momentum conversion can occur in the longitudinal component after tight focusing. Furthermore, optical forces of the tightly focused CPAVB on nanoparticles are analyzed in detail. It can be expected to trap two kinds of nanoparticles using such beam near the focus.
Vortex beams with a spiral phase factor exp(imθ) have attracted extensive attention over the past two decades, where m is a topological charge and can be any integer value and θ is the azimuthal angle on a plane transverse to optical axis [1, 2]. Vortex beams have been widely used in numerous applications owing to their “doughnut” intensity profile and orbital angular momentum (OAM), such as optical tweezers [3,4,5,6,7], free-space optical communication , and quantum information . Recently, researchers have paid more attention to the study of circularly polarized vortex beam because of its unique characteristics [10,11,12,13,14,15], for instance, it carries both spin angular momentum (SAM) and OAM at the same time. These unique characteristics can significantly expand and enhance the applications of vortex beams.
The tightly focusing characteristics of various beams under a lens system with high NA is another hot topic [16,17,18,19,20] for their important applications in particles trapping , microscopy , optical data storage , etc. Thus far, different beams have been studied, ranging from scalar vortex beams to vector vortex beams [10, 24,25,26,27,28,29,30,31]. For instance, Hao et al.  and Pu et al.  studied the properties of spirally polarized vortex beam under a high NA lens. It was shown that a flat-topped (FT) profile can be achieved and the OAM can be adjusted by choosing a proper polarized state in the focal plane. Zhan et al. studied the properties of tightly focused vortex beams with circularly polarization , showing that a strong longitudinal component can be produced.
Anomalous vortex beam (AVB), a novel beam which can evolve into elegant Laguerre-Gaussian beam in the far field, was proposed recently . Such beam has attracted much attention and been widely investigated [33,34,35,36,37,38], owing to its extraordinary propagation properties. To the best of our knowledge, there is no report on the CPAVBs focused by a high NA lens. In this paper, the mathematical expressions of the CPAVBs after tight focusing are derived. Then we analyze the effect of beam order, topological charge, and NA value on the beam profile and phase distribution. At the last part, optical forces of tightly focused CPAVBs are studied.
A circularly polarized beam can be written as follow, which indicates the linear superposition of radially and azimuthally polarized beams :
where P(r) is the amplitude distribution. The sign “+” and “−” are left-hand and right-hand circular polarization, respectively. eρ and eφ are the radial and azimuthal vectors in the cylindrical coordinates, respectively. And expressions of the radially and azimuthally polarized beam can be obtained in [39,40,41].
where f is the focal length, θ varies from 0 to α, α is the maximal angle of NA, and E0 and w0 are a constant and waist radius, respectively. n, φ, and m are the beam order, azimuthal coordinates, and the topological charge, respectively.
According to the vector Debye theory, the expressions of the electrical field, of the tightly focused CPAVB in cylindrical coordinates, can be derived as Eq. (3):
where Jn(α) is a n-order Bessel function of the first kind and k = 2π/λ. We define E+ and E− as the expression of the electrical field of right-hand and left-hand CPAVB, respectively.
In the above equations, the following formulas are used :
Then, we can calculate the total intensity of the tightly focused CPAVB as follow:
where Eρ, Eφ, and Ez are the amplitudes of corresponding components.
Results and Discussion
Tight-Focusing Characteristics of the CPAVB
In this section, using the above equations, we study the properties of the tightly focused CPAVB. In the simulation, we set NA = 0.85, λ = 632.8 nm, w0 = 2 mm, and f = 2 mm. In Fig. 1, the total intensity profile and corresponding longitudinal and radial components of the left-hand CPAVBs with n = 1 for different topological charges in the focal plane are depicted, respectively. We can find that the total intensity is nonzero at the center when m ≤ 2, while there exists a dark spot in the center when m > 2. In addition, the radial component of focused fields is not zero on the axis when m = 0, 2, and the same as the longitudinal component when m = 1. These results can be explained from Eq. (3) and Eq. (5), owing to the fact that Jm always equals to zero at the origin except for m = 0. The Bessel function of the first kind in all three components is zero at the center when m > 2, and thus the total intensity is zero. Otherwise, there exists at least one component containing J0, which means the central intensity can be nonzero and maximum. What is more, for total and radial components, focal spot size increases as the topological charge increases. Therefore, we can conclude that the total intensity and focal spot size in the focal field are affected by topological charge.
In Fig. 2, the total intensity profile and corresponding longitudinal and radial components of the left-hand CPAVBs with m = 1 for different beam orders in the focal plane are depicted, respectively. One can see that as n increases, the outer rings of each component and total intensity are gradually becoming brighter, while the pattern of the intensity does not change. Thus the beam order n does not affect the shape of the intensity patterns greatly.
Then we study how the NA value influences the focusing properties of CPAVBs with n = 2 for m = 1 and m = 4, respectively. As shown in Fig. 3, it is noticeable that the central intensity remains nonzero for the case of topological charge m = 1, while central intensity is dark in the focal plane for m = 4. Comparing Fig. 3 d-1 with d-2, we can find that the intensity increases and gathers to the center with increasing NA. Especially, for the case of m = 1, a FT beam can be obtained when NA increases to 0.8.
Based on Eq. (3c), we calculated the phase distributions of longitudinal component CPAVBs in the vicinity of focus, as shown in Fig. 4. The first and second rows of Fig. 4 are the left-hand and right-hand CPAVBs, respectively. The location for Fig. 4 a–c are z = − 0.005zr, 0, 0.005zr, respectively, where zr = kw02/2 is the Rayleigh range. Other parameters are set as n = 1 and NA = 0.85. As shown in Fig. 4, the contour of phase patterns changes from clockwise to anticlockwise after passing through the focal plane. Comparing Fig. 4 a-1 to c-1 with Fig. 4 a-2 to c-2, it is interesting to find that the topological charge near the focus changes from 3 to 5 when the left-hand CPAVB is replaced by a right-hand one. This phenomenon may be explained as a left-hand CPAVB with m = 4 carries SAM ls = −ħ and OAM m = 4ħ. Owing to the compensation of the opposite OAM converted from SAM, the topological charges decrease to three after tight focusing. By analogy, we can expect the similar behavior of the right-hand CPAVB with m = 4, which carries SAM ls = ħ and OAM m = 4ħ. Owing to OAM converted from SAM, the topological charges increase to five. Therefore, we can conclude that there is a conversion from SAM into OAM in the longitudinal component after tight focusing.
Trapping Nanoparticles Using the Tightly Focused CPAVB
Based on the Rayleigh scattering theory , the scattering force and gradient force should be considered when discussing the optical trapping. The scattering force, written as Fscat = eznmαIout/c, tends to destabilize the optical trap, where c is light velocity, ez is a unit vector along the z direction, Iout is the intensity of focused beam, α = (8/3)π(ka)4a2[(η2 − 1)2/(η2 + 2)2], ɑ is the nanoparticle’s radius, η = np/nm, and nm and np are refractive index of surrounding media and nanoparticle, respectively. And the gradient force (Fgrad) trends to pull a nanoparticle back to the focus, which can be expressed asFgrad = 2πnmβ ∇ Iout/c, where β = a3(η2 − 1)/(η2 + 2).
In the simulation experiment, we set np = 1.59 and np = 1 for glass and air bubble, respectively, nm = 1.332, NA = 0.85, and ɑ = 50 nm. Figure 5 represents the radial, longitudinal gradient forces and scattering forces of a left-hand CPAVB on a nanoparticle with np = 1 for different m and n. The previous work shows that the total intensity is dark at the center when m ≥ 3. Therefore, as expected, for low refraction index nanoparticle, the radial and longitudinal gradient force will always pull the nanoparticle back to the focus, as shown in Fig. 5 a–d. Comparing with the gradient force, the scattering force is very small. Therefore, the low refraction index nanoparticle can be trapped stably.
Figure 6 represents the radial, longitudinal gradient forces, and scattering forces of a left-hand CPAVB on a nanoparticle with np = 1.59 for different topological charges m and the beam orders n. From Fig. 6, we can see that there are several equilibrium points near the focus and the scattering force can be neglected compared with the gradient force. Therefore, the high refraction index nanoparticle can be captured near the focus.
In this paper, the characteristics of tightly focused CPAVBs and their optical forces on nanoparticle have been discussed. We find that SAM of CPAVB can convert into OAM when such beam is tightly focused. Furthermore, tightly focused CPAVB can be used to trap two different kinds of nanoparticles, with low and high refraction index, near the focal plane. Our research will be of help for finding potential applications of CPAVB.
Availability of Data and Materials
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
Anomalous vortex beam
Circularly polarized anomalous vortex beam
Orbital angular momentum
Spin angular momentum
Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP (1992) Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys Rev A 45:8185–8189
Alison MY, Miles JP (2011) Orbital angular momentum: origins, behavior and applications. Adv Opt Photonics 3:161–204
He H, Friese MEJ, Heckenberg NR, Rubinsztein-Dunlop H (1995) Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. Phys Rev Lett 75:826–829
Chen MZ, Mazilu M, Arita Y, Wright EM, Dholakia K (2013) Dynamics of microparticles trapped in a perfect vortex beam. Opt Lett 38:4919–4922
Li XZ, Ma HX, Zhang H, Tang MM, Li HH, Tang J, Wang YS (2019) Is it possible to enlarge the trapping range of optical tweezers via a single beam? Appl Phys Lett 114:081903
Tkachenko G, Chen MZ, Dholakia K, Mazilu M (2017) Is it possible to create a perfect fractional vortex beam? Optica 4:330–333
Arita Y, Chen MZ, Wright EM, Dholakia K (2017) Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential. J Opt Soc Am B 34:C14–C19
Wang J, Yang JY, Fazal IM, Ahmed N, Yan Y, Huang H, Ren YX, Yue Y et al (2012) Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat Photonics 6:488–496
Mafu M, Dudley A, Goyal S, Giovannini D, McLaren M, Padgett MJ, Konrad T, Petruccione F et al (2013) Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases. Phys Rev A 88:032305
Zhan QW (2007) Properties of circularly polarized vortex beams. Opt Lett 31(7):867–869
Rao LZ, Wang ZC, Zheng XX (2008) Tightly focusing of circularly polarized vortex beams through a uniaxial birefringent crystal. Chin Phys Lett 25(9):3223–3226
Chen BS, Zhang ZM, Pu JX (2009) Tight focusing of partially coherent and circularly polarized vortex beams. J Opt Soc Am A 26(4):862–869
Pang XY (2015) Gouy phase and phase singularities of tightly focused, circularly polarized vortex beams. Opt Commun 338:534–539
Wang S, Deng ZL, Cao YY, Hu DJ, Xu Y, Cai BY, Jin L, Bao Y, Wang XL, Li XP (2018) Angular momentum-dependent transmission of circularly polarized vortex beams through a plasmonic coaxial nanoring. IEEE Photonics J 10(1):5100109
Ciattoni A, Cincotti G, Palma G (2003) Circularly polarized beams and vortex generation in uniaxial media. J Opt Soc Am A 20(1):163–171
Zhao YQ, Edgar SJ, Jeffries GDM, McGloin D, Chiu DT (2007) Spin-to-orbital angular momentum conversion in a strongly focused optical beam. Phys Rev Lett 99:073901
Ganic D, Gan XS, Gu M (2003) Focusing of doughnut laser beams by a high numerical-aperture objective in free space. Opt Express 11(21):2747–2752
Quabis S, Dorn R, Eberler M, Glöckl O, Leuchs G (2001) The focus of light-theoretical calculation and experimental tomographic reconstruction. Appl Phys B 72:109–113
Quabis S, Dorn R, Eberler M, Glöckl O, Leuchs G (2000) Focusing light into a tighter spot. Opt Commun 179:1–7
Diehl DW, Visser TD (2004) Phase singularities of the longitudinal field components in the focal region of a high-aperture optical system. J Opt Soc Am A 21(11):2103–2108
Jeffries GDM, Edgar JS, Zhao YQ, Shelby JP, Fong C, Chiu DT (2007) Using polarization-shaped optical vortex traps for single-cell nanosurgery. Nano Lett 7(2):415–420
Youngworth KS, Biss DP, Brown TG (2001) Point spread functions for particle imaging using inhomogeneous polarization in scanning optical microscopy. Proc of SPIE 4261:14–23
Kim W-C, Park N-C, Yoon Y-J, Chol Y, Park Y-P (2007) Investigation of near-field imaging characteristics of radial polarization for application to optical data storage. Opt Rev 14(4):236–242
Zhang ZM, Pu JX, Wang XQ (2008) Tightly focusing of linearly polarized vortex beams through a dielectric interface. Opt Commun 281:3421–3426
Pang XY, Visser TD, Wolf E (2011) Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems. Opt Commun 284:5517–5522
Hao B, Leger J (2008) Numerical aperture invariant focus shaping using spirally polarized beams. Opt Commun 281:1924–1928
Pu JX, Zhang ZM (2010) Tight focusing of spirally polarized vortex beams. Opt Laser Technol 42:186–191
Singh RK, Senthilkumaran P, Singh K (2009) Tight focusing of vortex beams in presence of primary astigmatism. J Opt Soc Am A 26(3):576–588
Singh RK, Senthilkumaran P, Singh K (2008) Effect of primary spherical aberration on high numerical-aperture focusing of a Laguerre-Gaussian beam. J Opt Soc Am A 25(6):1307–1317
Chen BS, Pu JX (2009) Tight focusing of elliptically polarized vortex beams. Appl Opt 48(7):1288–1294
Rao LZ, Pu JX, Chen ZY, Yei P (2009) Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens. Opt Laser Technol 41:241–246
Yang YJ, Dong Y, Zhao CL, Cai YJ (2013) Generation and propagation of an anomalous vortex beam. Opt Lett 38(24):5418–5421
Xu YG, Wang SJ (2014) Characteristic study of anomalous vortex beam through a paraxial optical system. Opt Commun 331:32–38
Zhang DJ, Yang YJ (2015) Radiation forces on Rayleigh particles using a focused anomalous vortex beam under paraxial approximation. Opt Commun 336:202–206
Zhang X, Wang HY, Tang L (2018) Propagation of partially coherent vector anomalous vortex beam in turbulent atmosphere. Proc of SPIE 10617
Dai ZP, Yang ZJ, Zhang SM, Pang ZG (2015) Propagation of anomalous vortex beams in strongly nonlocal nonlinear media. Opt Commun 350:19–27
Yuan YP, Yang YJ (2015) Propagation of anomalous vortex beams through an annular apertured paraxial ABCD optical system. Opt Quant Electron 47:2289–2297
Zhang MY, Yang YJ (2018) Tight focusing properties of anomalous vortex beams. OPTIK 154:133–138
Wolf E (1959) Electromagnetic diffraction in optical systems. I. An integral representation of the image field. Proc R Soc London Ser A 253:349–357
Gu M Advanced optical imaging theory. Springer, Berlin, p 2000
Youngworth KS, Brown TG (2000) Focusing of high numerical aperture cylindrical vector beams. Opt Express 7(2):77–87
Richards B, Wolf E (1959) Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proc R Soc London Ser A 253:358–379
Erdelyi A (1954) Tables of integral transforms. Mc Graw-Hill Book Company, New York
Harada Y, Asakura T (1996) Radiation forces on a dielectric sphere in the Rayleigh scattering regime. Opt Commun 124:529–541
This work is supported by the National Natural Science Foundation of China (11874102, 11474048).
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Bai, Y., Dong, M., Zhang, M. et al. Properties of a Tightly Focused Circularly Polarized Anomalous Vortex Beam and Its Optical Forces on Trapped Nanoparticles. Nanoscale Res Lett 14, 252 (2019) doi:10.1186/s11671-019-3089-5
- Optical trapping
- Circularly polarized anomalous vortex beam