November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
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Dynamics of microparticles trapped in a perfect vortex beam Mingzhou Chen,1,* Michael Mazilu,1 Yoshihiko Arita,1 Ewan M. Wright,1,2 and Kishan Dholakia1,2 1
2
SUPA, School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK College of Optical Sciences, The University of Arizona, 1630 East University Boulevard, Tuscon, Arizona 85721-0094, USA *Corresponding author: mc225@st‑andrews.ac.uk Received August 12, 2013; accepted September 24, 2013; posted October 18, 2013 (Doc. ID 195621); published November 15, 2013 We analyze microparticle dynamics within a “perfect” vortex beam. In contrast to other vortex fields, for any given integer value of the topological charge, a “perfect” vortex beam has the same annular intensity profile with fixed radius of peak intensity. For a given topological charge, the field possesses a well-defined orbital angular momentum density at each point in space, invariant with respect to azimuthal position. We experimentally create a perfect vortex and correct the field in situ, to trap and set in motion trapped microscopic particles. For a given topological charge, a single trapped particle exhibits the same local angular velocity moving in such a field independent of its azimuthal position. We also investigate particle dynamics in “perfect” vortex beams of fractional topological charge. This light field may be applied for novel studies in optical trapping of particles, atoms, and quantum gases. © 2013 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (020.7010) Laser trapping; (050.4865) Optical vortices; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.38.004919
Vortices are ubiquitous in numerous areas of physics and are at the heart of many phenomena in optics, fluid dynamics, superconductivity, and quantum gases. In the optical domain, vortices have seen widespread application in the last 20 years since the identification that such fields, described particularly by Laguerre–Gaussian (LG) transverse modes, may possess a well-defined orbital angular momentum (OAM) [1]. This property is directly related to the topological charge (azimuthal index) l of the beam, which denotes the multiple of 2π the field phase accumulates upon circling the beam center. This field has gained prominence due to a diverse range of applications in optical manipulation of microparticles, studies in cold atoms, quantum gases, and quantum information processing. For the family of LG transverse modes, the mode profile and radius of peak intensity vary with the topological charge of the beam [2,3]. This interdependence introduces a degree of complexity to studies and hampers direct experimental comparisons of the intricate nature of OAM and the local OAM density including studies at the single atom level [4]. For the case of trapped microparticles, these can rotate due to the inclined nature of the Poynting vector of such fields. Previous studies have shown the quantitative dependency of trapped microparticle motion upon the topological charge of the vortex beam, the trapping power and particle position with respect to the optical axis of the trapping beam (a high-order Bessel beam or a LG beam) [5–8]. In the context of all of these studies, the creation of a “perfect” vortex beam whose radial intensity profile and radius are both independent of topological charge would be of significant interest. The creation of such a beam and its application to microparticle trapping is the subject of this Letter. The concept of a “perfect optical vortex” was first introduced by Ostrovsky et al. [9]. The corresponding annular field profile at a fixed propagation distance for topological charge l may be written as 0146-9592/13/224919-04$15.00/0
ρ − ρ0 2 ilθ V l ρ; θ exp − e ; Δρ2
(1)
where ρ; θ are polar coordinates, ρ0 and Δρ define the radius of the annulus and its width, and l ∈ N defines the azimuthal index. It is key that the annular field profile be independent of l. Reference [9] considered a deltafunction profile as the ideal case, but in general the annular field profile will have an annular width Δρ that should be less than the annular radius ρ0 , i.e., Δρ ≪ ρ0 . To proceed, we first remark that an annular laser beam with a topological charge of zero may be created by Gaussian beam illumination of an axicon in combination with a lens [10–13]. The radius ρ0 of the generated annular beam is determined by the axicon parameters along with the propagation distance D past the axicon, whereas the annulus width Δρ is inversely proportional to the incident Gaussian beam waist. The lens is applied after the propagation distance D to provide an annular field on its focal plane. By imprinting the vortex phase lθ upon the resulting annular beam, one can realize a perfect vortex beam that may be represented by Eq. (1). There are two essential differences in our optical setup compared with other methods to generate a vortex field with a spatial light modulator (SLM) [7,9]. The first one relates to the ring illumination of the SLM. The second one corresponds to the “perfect” vortex beam being created by directly imaging the SLM plane onto the trapping (sample) plane in contrast to taking its Fourier transform. Our approach fully utilizes the input laser power while maintaining full control of the complex field created by the SLM. As shown in Fig. 1, a collimated and linearly polarized CW fiber laser (λ 1070 nm, 5 W) is transmitted through an axicon (apex angle γ 178°). The emergent beam is collected by a lens (L1 ) such that the resultant annulus (2ρ0 9.25 mm) is incident on the SLM (Hamamatsu LCOS-SLM X10468-03). The topological charge of the beam is controlled by the applied vortex phase, which is displayed together with a wavefront © 2013 Optical Society of America
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Fig. 1. Schematic of the experimental setup used for trapping with a perfect vortex beam. A1 is an axicon, and L1 ∼ L5 are lenses. MO1 is a microscope objective. Inset figures show the created perfect vortex beams with (a) l 15, (b) l 25, (c) l 35, and (d) l 25 with the scattered light of 13 trapped microparticles (Media 1), and (e) a vortex mask with l 20.
correction mask [14] and a grating phase mask on the SLM. The modulated first diffraction order is then selected by a pinhole (P 1 ). Figure 2 shows a paraxial simulation of the propagation of a perfect vortex starting from the axicon-generated annular beam at the SLM. In the experimental setup (Fig. 1) this beam is demagnified by lens L4 and the microscope objective (MO1 ) (40×, NA 1.3, oil immersion). With this approach, the SLM plane is imaged onto the trapping plane (TP) where the created perfect vortex beam is used in order to trap microparticles (polymer particles, 5 0.3 μm) dispersed in water within a glass chamber (100 μm thickness). The particles are trapped in two dimensions and pushed against the microscope coverslip surface of the chamber. The laser power used for trapping is measured as 0.425 W at the back aperture of MO1 . The same MO1 also images the TP onto a high-speed CMOS camera. The achievable topological charge is mainly constrained by the back aperture of the microscope objective. Depending on l, this may truncate the spectrum of the beam due to the ring illumination used. In our system, we can realize topological charges up to l 40. As shown in Figs. 1(a)–1(c), perfect vortex beams that are created in the TP have very similar intensity profiles
regardless of their different topological charges. The beam is slightly truncated when l 35 in Fig. 1(c) due to the finite back aperture of MO1 . The motion of the trapped particles is dictated by the gradient and scattering forces resulting from the beam. The particles move around the annulus due to scattering from the inclined wavefront of the field with the sense of rotation dictated by the sign of the topological charge. Since the refractive index of the particles used is greater than that of the surrounding water, the particles tend to be trapped in beam “hot spots” at local intensity maxima introduced by aberrations in the optical system. A necklace arrangement of several touching particles can overcome the hot spots due to the interparticle interactions, and are easier to set into continuous rotation [15]. For the first studies, the created perfect vortex beam is filled with 13 microparticles to make a necklace as shown by the scattered light from these particles in Fig. 1(d) and Media 1. To analyze the rotation of the particles, videos are recorded at a rate of 150 fps. The topological charge l was varied from l −35 to 35 in steps of 5. The orbital rotation rate of the particle necklace is calculated from positions of a trapped particle in each video frame. The relationship between the rotation rate and the topological charge of the perfect vortex beams is shown in Fig. 3. From the fitted lines, a linear relationship can be clearly seen for both negative and positive topological charges due to the well-defined OAM density in each beam. However, these two lines do not meet at the origin as one might expect. This discrepancy can be explained by the friction force between the particles and the glass coverslip. As a result, due to the frictional forces present and the laser power used for our particular geometry, the particles do not start to rotate unless l ≥ 2. Interestingly, using the same approach it is possible to probe the OAM of beams having a noninteger topological charge. The fractional part of the topological charge leads to a phase jump that is associated with azimuthal intensity modulations; see Figs. 2(d) and 2(e). With respect to the definition of the perfect vortex, this intensity 1.2
Rotation rate (Hz)
0.8
0.4
0
−0.4 Positive vortices y=0.032x−0.044 Negative vortices y=−0.032x+0.046
−0.8
−1.2
Fig. 2. Paraxial simulation of the generation of perfect vortex beams with both integer and noninteger charges. (a), (d) show the beams in the x–z plane for the cases l 3 and l 3.3, respectively. Both beams propagate from left to right. (b), (c) and (e), (f) show the transverse intensity and phase profiles on the final plane, respectively, for these two topological charges.
−30
−20
−10
0
10
20
30
Fig. 3. Relationship between the particle rotation rate and the integer topological charge for a perfect vortex beam. Positive rotation rate is defined as counterclockwise rotation. The solid lines represent a least-square linear regression of the original data.
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
Rotation rate (Hz)
−0.25
−0.3
−0.35
−0.4 −12
Experimental data y=0.032[x−sin(2 x)/(2 )]+0.024 −11.5
−11
−10.5
−10
−9.5
−9
Fig. 4. Relationship between the particle rotation rate and the fractional topological charge for a perfect vortex beam. The solid line, including our specific prefactor and offset, shows the trend equivalent to the theoretical prediction of OAM per photon of a fractional vortex as l − sin2πl∕2π [16].
variation changes the amplitude part of Eq. (1) to include an azimuthal dependence. We remark that this intensity is invariant for integer steps of the fractional topological charge l. A single particle would typically be trapped by these intensity modulations; however, for multiple particles a collective “averaged” rotation of the particle necklace is seen. In particular, we explore the rotation rate of the 13 polymer microparticles necklace when varying the topological charge l ranging from −12 to −9 in steps of Δl 0.2. As shown in Fig. 4, the rotation rate does not vary linearly for fractional variations of the topological charge. Theoretical models [16–18] considering the specific phase jump distribution used in our experiment are in good agreement. Readdressing the case of integer topological charge, the experimental realization of such vortex beams often has azimuthal intensity variations due to optical aberrations. It is thus difficult for a single particle to maintain a constant velocity around the trap due to local intensity hot spots. Figures 5(a) and 5(b) show two perfect vortex beams that can successfully rotate particle necklaces while they can only rotate a single particle over a small part of the ring. For the case shown in Fig. 5(b), the trapped particle stops at the position marked by the red arrow (Media 2). This is a problem in any singleparticle vortex trapping experiment due to the presence of unavoidable hot spots [7,17]. The experimental implementation of a perfect vortex beam can be improved experimentally to achieve a θ
Fig. 5. (a) shows a perfect vortex beam with l 25 before the amplitude correction. (b) shows the beam with the scattered light from a single trapped particle. The red arrow indicates the position where the particle stops (Media 2). (c) shows the beam after the amplitude correction. The trapped particle can sustain the rotation along the annulus as shown by (d) and in (Media 3).
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independent intensity profile. Since the SLM can only control the phase of the beam, the procedure of our amplitude correction consists of reducing the local efficiency of the SLM diffraction grating to decrease the intensity at the locations of the hot spots. In order to achieve this, we first need to register the camera plane and the SLM plane with a precise image registration method [19]. This allows us to determine the positions on the SLM that need to be corrected in amplitude. Obviously, this amplitude correction reduces the total laser power used for the trapping beam. As shown in Figs. 5(c) and 5(d), the uniformity of the beam is improved significantly while the power in the beam is reduced by 40% after amplitude correction. A single trapped particle is seen to have a continuous orbital rotation for integer l as shown in the video (Media 3). Although a single trapped particle can sustain full cycles of rotation after the amplitude correction, the recorded instantaneous rotation velocity at any annular position is not very uniform. Based on the knowledge that the particle rotation rate is linear to the OAM density, the rotation velocity can be adjusted locally by changing the local OAM density. This local OAM density correction procedure is described in Fig. 6. A recorded video with a single trapped particle is registered to the SLM plane frame by frame. The angular velocity of the particle can be calculated based on the position of the particle seen in each frame. A correction phase mask that is created based on the angular velocity is displayed on the SLM together with the original vortex mask and previous correction masks. In this way, the local OAM density can be adjusted by the SLM if the local angular velocity is lower or higher than the mean velocity. In order to correct the beam smoothly, and in an adiabatic manner, a small amount of OAM change is applied at each iteration to avoid introducing additional hot spots. Using this local OAM density correction procedure, we can improve the uniformity of instantaneous particle rotation velocity over the ring. For the example shown in Figs. 7(a) and 7(b) and Media 4 and 5, the standard deviation of rotation speed over the ring is reduced from 0.37 (rad/s) before the correction to 0.20 (rad/s) after the
Fig. 6. Flow chart of local OAM density correction.
Fig. 7. Bright field images of trapped particles. (a) and (b) show a single particle trapped by a perfect vortex beam before (Media 4) and after (Media 5) local OAM density correction. (c) and (d) show two (Media 6) and three (Media 7) well-separated particles trapped by a perfect vortex beam after local OAM density correction.
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correction. This improvement makes it possible to trap several particles with a well-defined interparticle distance within a perfect vortex beam and enable a quasiconstant rotation. Examples for two and three trapped particles are shown in Figs. 7(c) and 7(d) and Media 6 and 7, where we observe the particles maintaining their mutual distances. In summary, we have introduced a novel approach for generating perfect vortex beams using an axicon to create an annular beam of fixed profile in conjunction with an SLM to impose a chosen topological charge. The utility of the approach was demonstrated by experimentally measuring the rotation of a particle necklace for integer and fractional topological charge. In situ correction of the OAM density of the perfect vortex beam enabled rotation of a single particle at a constant velocity, thereby overcoming problems of trapping at intensity hot spots, and further demonstrating the flexibility of the approach. In the broader context of cold atomic gases, the annular intensity profile of the perfect vortex should be of utility in creating ring traps for atoms. The constant velocity for particles around the perfect vortex lends itself to creating persistent currents on a ring, akin to those realized using LG beams [20]. Where the perfect vortex comes into its own right is for ring geometries that employ two overlapping and coherent laser fields of differing topological charge and polarization, such as generation of superflows in Bose–Einstein condensates [21], wave function engineering, and spin current generation [22], and creation of artificial gauge fields and magnetism on a ring [23,24]. Using perfect vortices for such studies would remove issues related to the different spatial profiles of the LG beams involved. The authors thank the UK Engineering and Physical Sciences Research Council (EPSRC) for funding. K. D. acknowledges support from a Royal Society WolfsonMerit Award. References 1. L. Allen, M. W. Beijersbergen, R. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 2. R. L. Phillips and L. C. Andrews, Appl. Opt. 22, 643 (1983).
3. N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996). 4. S. H. Simpson and S. Hanna, J. Opt. Soc. Am. A 27, 2061 (2010). 5. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, J. Opt. B 4, S82 (2002). 6. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, Phys. Rev. Lett. 91, 093602 (2003). 7. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003). 8. J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Čižmár, P. Fischer, and K. Dholakia, Opt. Express 16, 10117 (2008). 9. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, Opt. Lett. 38, 534 (2013). 10. I. Manek, Y. B. Ovchinnikov, and R. Grimm, Opt. Commun. 147, 67 (1998). 11. J. Arlt, R. Kuhn, and K. Dholakia, J. Mod. Opt. 48, 783 (2001). 12. T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, Opt. Express 16, 14024 (2008). 13. C. Alpmann, R. Bowman, M. Woerdeman, M. Padgett, and C. Denz, Opt. Express 18, 26084 (2010). 14. T. Čižmár, M. Mazilu, and K. Dholakia, Nat. Photonics 4, 388 (2010). 15. S. H. Tao, X.-C. Yuan, J. Lin, and Y. Y. Sun, J. Appl. Phys. 100, 043105 (2006). 16. J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004). 17. S. H. Tao, X.-C. Yuan, and J. Lin, Opt. Express 13, 7726 (2005). 18. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, J. Mod. Opt. 54, 1723 (2007). 19. P. Thévenaz, U. Ruttimann, and M. Unser, IEEE Trans. Image Process. 7, 27 (1998). 20. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006). 21. C. Ryu, M. F. Andersen, P. Cladé, V. Natarajan, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 99, 260401 (2007). 22. J.-J. Song, B. A. Foreman, X.-J. Liu, and C. H. Oh, Europhys. Lett. 84, 20012 (2008). 23. P. Öhberg, J. Opt. 13, 064024 (2011). 24. J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011).
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Dynamics of microparticles trapped in a perfect vortex beam Mingzhou Chen,1,* Michael Mazilu,1 Yoshihiko Arita,1 Ewan M. Wright,1,2 and Kishan Dholakia1,2 1
2
SUPA, School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK College of Optical Sciences, The University of Arizona, 1630 East University Boulevard, Tuscon, Arizona 85721-0094, USA *Corresponding author: mc225@st‑andrews.ac.uk Received August 12, 2013; accepted September 24, 2013; posted October 18, 2013 (Doc. ID 195621); published November 15, 2013 We analyze microparticle dynamics within a “perfect” vortex beam. In contrast to other vortex fields, for any given integer value of the topological charge, a “perfect” vortex beam has the same annular intensity profile with fixed radius of peak intensity. For a given topological charge, the field possesses a well-defined orbital angular momentum density at each point in space, invariant with respect to azimuthal position. We experimentally create a perfect vortex and correct the field in situ, to trap and set in motion trapped microscopic particles. For a given topological charge, a single trapped particle exhibits the same local angular velocity moving in such a field independent of its azimuthal position. We also investigate particle dynamics in “perfect” vortex beams of fractional topological charge. This light field may be applied for novel studies in optical trapping of particles, atoms, and quantum gases. © 2013 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (020.7010) Laser trapping; (050.4865) Optical vortices; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.38.004919
Vortices are ubiquitous in numerous areas of physics and are at the heart of many phenomena in optics, fluid dynamics, superconductivity, and quantum gases. In the optical domain, vortices have seen widespread application in the last 20 years since the identification that such fields, described particularly by Laguerre–Gaussian (LG) transverse modes, may possess a well-defined orbital angular momentum (OAM) [1]. This property is directly related to the topological charge (azimuthal index) l of the beam, which denotes the multiple of 2π the field phase accumulates upon circling the beam center. This field has gained prominence due to a diverse range of applications in optical manipulation of microparticles, studies in cold atoms, quantum gases, and quantum information processing. For the family of LG transverse modes, the mode profile and radius of peak intensity vary with the topological charge of the beam [2,3]. This interdependence introduces a degree of complexity to studies and hampers direct experimental comparisons of the intricate nature of OAM and the local OAM density including studies at the single atom level [4]. For the case of trapped microparticles, these can rotate due to the inclined nature of the Poynting vector of such fields. Previous studies have shown the quantitative dependency of trapped microparticle motion upon the topological charge of the vortex beam, the trapping power and particle position with respect to the optical axis of the trapping beam (a high-order Bessel beam or a LG beam) [5–8]. In the context of all of these studies, the creation of a “perfect” vortex beam whose radial intensity profile and radius are both independent of topological charge would be of significant interest. The creation of such a beam and its application to microparticle trapping is the subject of this Letter. The concept of a “perfect optical vortex” was first introduced by Ostrovsky et al. [9]. The corresponding annular field profile at a fixed propagation distance for topological charge l may be written as 0146-9592/13/224919-04$15.00/0
ρ − ρ0 2 ilθ V l ρ; θ exp − e ; Δρ2
(1)
where ρ; θ are polar coordinates, ρ0 and Δρ define the radius of the annulus and its width, and l ∈ N defines the azimuthal index. It is key that the annular field profile be independent of l. Reference [9] considered a deltafunction profile as the ideal case, but in general the annular field profile will have an annular width Δρ that should be less than the annular radius ρ0 , i.e., Δρ ≪ ρ0 . To proceed, we first remark that an annular laser beam with a topological charge of zero may be created by Gaussian beam illumination of an axicon in combination with a lens [10–13]. The radius ρ0 of the generated annular beam is determined by the axicon parameters along with the propagation distance D past the axicon, whereas the annulus width Δρ is inversely proportional to the incident Gaussian beam waist. The lens is applied after the propagation distance D to provide an annular field on its focal plane. By imprinting the vortex phase lθ upon the resulting annular beam, one can realize a perfect vortex beam that may be represented by Eq. (1). There are two essential differences in our optical setup compared with other methods to generate a vortex field with a spatial light modulator (SLM) [7,9]. The first one relates to the ring illumination of the SLM. The second one corresponds to the “perfect” vortex beam being created by directly imaging the SLM plane onto the trapping (sample) plane in contrast to taking its Fourier transform. Our approach fully utilizes the input laser power while maintaining full control of the complex field created by the SLM. As shown in Fig. 1, a collimated and linearly polarized CW fiber laser (λ 1070 nm, 5 W) is transmitted through an axicon (apex angle γ 178°). The emergent beam is collected by a lens (L1 ) such that the resultant annulus (2ρ0 9.25 mm) is incident on the SLM (Hamamatsu LCOS-SLM X10468-03). The topological charge of the beam is controlled by the applied vortex phase, which is displayed together with a wavefront © 2013 Optical Society of America
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Fig. 1. Schematic of the experimental setup used for trapping with a perfect vortex beam. A1 is an axicon, and L1 ∼ L5 are lenses. MO1 is a microscope objective. Inset figures show the created perfect vortex beams with (a) l 15, (b) l 25, (c) l 35, and (d) l 25 with the scattered light of 13 trapped microparticles (Media 1), and (e) a vortex mask with l 20.
correction mask [14] and a grating phase mask on the SLM. The modulated first diffraction order is then selected by a pinhole (P 1 ). Figure 2 shows a paraxial simulation of the propagation of a perfect vortex starting from the axicon-generated annular beam at the SLM. In the experimental setup (Fig. 1) this beam is demagnified by lens L4 and the microscope objective (MO1 ) (40×, NA 1.3, oil immersion). With this approach, the SLM plane is imaged onto the trapping plane (TP) where the created perfect vortex beam is used in order to trap microparticles (polymer particles, 5 0.3 μm) dispersed in water within a glass chamber (100 μm thickness). The particles are trapped in two dimensions and pushed against the microscope coverslip surface of the chamber. The laser power used for trapping is measured as 0.425 W at the back aperture of MO1 . The same MO1 also images the TP onto a high-speed CMOS camera. The achievable topological charge is mainly constrained by the back aperture of the microscope objective. Depending on l, this may truncate the spectrum of the beam due to the ring illumination used. In our system, we can realize topological charges up to l 40. As shown in Figs. 1(a)–1(c), perfect vortex beams that are created in the TP have very similar intensity profiles
regardless of their different topological charges. The beam is slightly truncated when l 35 in Fig. 1(c) due to the finite back aperture of MO1 . The motion of the trapped particles is dictated by the gradient and scattering forces resulting from the beam. The particles move around the annulus due to scattering from the inclined wavefront of the field with the sense of rotation dictated by the sign of the topological charge. Since the refractive index of the particles used is greater than that of the surrounding water, the particles tend to be trapped in beam “hot spots” at local intensity maxima introduced by aberrations in the optical system. A necklace arrangement of several touching particles can overcome the hot spots due to the interparticle interactions, and are easier to set into continuous rotation [15]. For the first studies, the created perfect vortex beam is filled with 13 microparticles to make a necklace as shown by the scattered light from these particles in Fig. 1(d) and Media 1. To analyze the rotation of the particles, videos are recorded at a rate of 150 fps. The topological charge l was varied from l −35 to 35 in steps of 5. The orbital rotation rate of the particle necklace is calculated from positions of a trapped particle in each video frame. The relationship between the rotation rate and the topological charge of the perfect vortex beams is shown in Fig. 3. From the fitted lines, a linear relationship can be clearly seen for both negative and positive topological charges due to the well-defined OAM density in each beam. However, these two lines do not meet at the origin as one might expect. This discrepancy can be explained by the friction force between the particles and the glass coverslip. As a result, due to the frictional forces present and the laser power used for our particular geometry, the particles do not start to rotate unless l ≥ 2. Interestingly, using the same approach it is possible to probe the OAM of beams having a noninteger topological charge. The fractional part of the topological charge leads to a phase jump that is associated with azimuthal intensity modulations; see Figs. 2(d) and 2(e). With respect to the definition of the perfect vortex, this intensity 1.2
Rotation rate (Hz)
0.8
0.4
0
−0.4 Positive vortices y=0.032x−0.044 Negative vortices y=−0.032x+0.046
−0.8
−1.2
Fig. 2. Paraxial simulation of the generation of perfect vortex beams with both integer and noninteger charges. (a), (d) show the beams in the x–z plane for the cases l 3 and l 3.3, respectively. Both beams propagate from left to right. (b), (c) and (e), (f) show the transverse intensity and phase profiles on the final plane, respectively, for these two topological charges.
−30
−20
−10
0
10
20
30
Fig. 3. Relationship between the particle rotation rate and the integer topological charge for a perfect vortex beam. Positive rotation rate is defined as counterclockwise rotation. The solid lines represent a least-square linear regression of the original data.
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
Rotation rate (Hz)
−0.25
−0.3
−0.35
−0.4 −12
Experimental data y=0.032[x−sin(2 x)/(2 )]+0.024 −11.5
−11
−10.5
−10
−9.5
−9
Fig. 4. Relationship between the particle rotation rate and the fractional topological charge for a perfect vortex beam. The solid line, including our specific prefactor and offset, shows the trend equivalent to the theoretical prediction of OAM per photon of a fractional vortex as l − sin2πl∕2π [16].
variation changes the amplitude part of Eq. (1) to include an azimuthal dependence. We remark that this intensity is invariant for integer steps of the fractional topological charge l. A single particle would typically be trapped by these intensity modulations; however, for multiple particles a collective “averaged” rotation of the particle necklace is seen. In particular, we explore the rotation rate of the 13 polymer microparticles necklace when varying the topological charge l ranging from −12 to −9 in steps of Δl 0.2. As shown in Fig. 4, the rotation rate does not vary linearly for fractional variations of the topological charge. Theoretical models [16–18] considering the specific phase jump distribution used in our experiment are in good agreement. Readdressing the case of integer topological charge, the experimental realization of such vortex beams often has azimuthal intensity variations due to optical aberrations. It is thus difficult for a single particle to maintain a constant velocity around the trap due to local intensity hot spots. Figures 5(a) and 5(b) show two perfect vortex beams that can successfully rotate particle necklaces while they can only rotate a single particle over a small part of the ring. For the case shown in Fig. 5(b), the trapped particle stops at the position marked by the red arrow (Media 2). This is a problem in any singleparticle vortex trapping experiment due to the presence of unavoidable hot spots [7,17]. The experimental implementation of a perfect vortex beam can be improved experimentally to achieve a θ
Fig. 5. (a) shows a perfect vortex beam with l 25 before the amplitude correction. (b) shows the beam with the scattered light from a single trapped particle. The red arrow indicates the position where the particle stops (Media 2). (c) shows the beam after the amplitude correction. The trapped particle can sustain the rotation along the annulus as shown by (d) and in (Media 3).
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independent intensity profile. Since the SLM can only control the phase of the beam, the procedure of our amplitude correction consists of reducing the local efficiency of the SLM diffraction grating to decrease the intensity at the locations of the hot spots. In order to achieve this, we first need to register the camera plane and the SLM plane with a precise image registration method [19]. This allows us to determine the positions on the SLM that need to be corrected in amplitude. Obviously, this amplitude correction reduces the total laser power used for the trapping beam. As shown in Figs. 5(c) and 5(d), the uniformity of the beam is improved significantly while the power in the beam is reduced by 40% after amplitude correction. A single trapped particle is seen to have a continuous orbital rotation for integer l as shown in the video (Media 3). Although a single trapped particle can sustain full cycles of rotation after the amplitude correction, the recorded instantaneous rotation velocity at any annular position is not very uniform. Based on the knowledge that the particle rotation rate is linear to the OAM density, the rotation velocity can be adjusted locally by changing the local OAM density. This local OAM density correction procedure is described in Fig. 6. A recorded video with a single trapped particle is registered to the SLM plane frame by frame. The angular velocity of the particle can be calculated based on the position of the particle seen in each frame. A correction phase mask that is created based on the angular velocity is displayed on the SLM together with the original vortex mask and previous correction masks. In this way, the local OAM density can be adjusted by the SLM if the local angular velocity is lower or higher than the mean velocity. In order to correct the beam smoothly, and in an adiabatic manner, a small amount of OAM change is applied at each iteration to avoid introducing additional hot spots. Using this local OAM density correction procedure, we can improve the uniformity of instantaneous particle rotation velocity over the ring. For the example shown in Figs. 7(a) and 7(b) and Media 4 and 5, the standard deviation of rotation speed over the ring is reduced from 0.37 (rad/s) before the correction to 0.20 (rad/s) after the
Fig. 6. Flow chart of local OAM density correction.
Fig. 7. Bright field images of trapped particles. (a) and (b) show a single particle trapped by a perfect vortex beam before (Media 4) and after (Media 5) local OAM density correction. (c) and (d) show two (Media 6) and three (Media 7) well-separated particles trapped by a perfect vortex beam after local OAM density correction.
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correction. This improvement makes it possible to trap several particles with a well-defined interparticle distance within a perfect vortex beam and enable a quasiconstant rotation. Examples for two and three trapped particles are shown in Figs. 7(c) and 7(d) and Media 6 and 7, where we observe the particles maintaining their mutual distances. In summary, we have introduced a novel approach for generating perfect vortex beams using an axicon to create an annular beam of fixed profile in conjunction with an SLM to impose a chosen topological charge. The utility of the approach was demonstrated by experimentally measuring the rotation of a particle necklace for integer and fractional topological charge. In situ correction of the OAM density of the perfect vortex beam enabled rotation of a single particle at a constant velocity, thereby overcoming problems of trapping at intensity hot spots, and further demonstrating the flexibility of the approach. In the broader context of cold atomic gases, the annular intensity profile of the perfect vortex should be of utility in creating ring traps for atoms. The constant velocity for particles around the perfect vortex lends itself to creating persistent currents on a ring, akin to those realized using LG beams [20]. Where the perfect vortex comes into its own right is for ring geometries that employ two overlapping and coherent laser fields of differing topological charge and polarization, such as generation of superflows in Bose–Einstein condensates [21], wave function engineering, and spin current generation [22], and creation of artificial gauge fields and magnetism on a ring [23,24]. Using perfect vortices for such studies would remove issues related to the different spatial profiles of the LG beams involved. The authors thank the UK Engineering and Physical Sciences Research Council (EPSRC) for funding. K. D. acknowledges support from a Royal Society WolfsonMerit Award. References 1. L. Allen, M. W. Beijersbergen, R. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 2. R. L. Phillips and L. C. Andrews, Appl. Opt. 22, 643 (1983).
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