It is well known that the angular momentum carried by a paraxial light beam can be characterized by the “spin” angular momentum associated with the polarization state [1] and the “orbital” angular momentum associated with the spatial distribution of the optical field [2]. Interaction of radiation with matter may produce a transfer of angular momentum from the optical field to a body which in turn suffers a mechanical torque that can be measured. Beth [1] was the first to measure the optical torque produced by the transfer of spin angular momentum of light to a transparent quartz plate. Similar experiments have been then repeated on a small half-wave suspended dipole using microwave radiation [3], on liquid crystal films [4], and on small birefringent particles trapped by optical tweezers [5, 6]. More recently, it has been pointed out that the orbital part of the angular momentum of light could be equally transferred to macroscopic bodies, using Laguerre-Gauss laser modes with a well defined orbital angular momentum [2]. Laguerre-Gauss beams have been exploited, in fact, to rotate small absorptive particles [7, 8]. The orbital angular momentum of laser beams with phase singularities has also been used to move transparent and isotropic particles along circular trajectories around the beam axis [9, 10]. Light beams with a well defined orbital angular momentum, however, cannot produce a torque on a transparent body putting it into rotation around its center of mass. This paradox was first noted by Van Enk and Nienhuis [11]. This prevents the exact replication of Beth’s experiment with the spin of light replaced by its orbital angular momentum. The question is then raised if Beth’s transfer mechanism, where photons are not absorbed but just scattered into a different angular momentum state, could be applied to the orbital angular momentum too.

The present work is aimed at demonstrating, with an experiment, that the orbital angular momentum of light can be transferred to trapped transparent bodies with a mechanism analogous to Beth’s experiment by using a suitable astigmatic light beam which is not an angular momentum eigenstate. In particular, we used an astigmatic laser beam carrying zero average angular momentum. No permanent spinning rotation of the particle can be sustained by such a beam, but a mechanical torque is nevertheless generated. Nevertheless, our technique may be actually useful in medical and biological applications, since it allows to change at will the orientation of molecules or micro-organisms without necessarily displacing their centers of mass and avoiding heating problems due to light absorption.

When a transparent body moves under the action of light (spin or orbital) angular momentum transfer, photons are not destroyed, but put into a different angular momentum state. The angular momentum lost by photons is gained by the body, thus originating the optical torque. Energy also is transferred in the scattering process. In usual experimental conditions this energy is dissipated by viscous forces during the motion of the body. Since the photon number is preserved, the energy transfer induces a red-shift in the photon frequency, that, in some experiments on spin angular momentum transfer has been also measured [3, 12]. The overall optical process has some resemblance with Stimulated Raman Scattering [3] and, in the case of the spin transfer, it was recognized as Self-Induced Stimulated Light Scattering (SISLS) [13]. It is worthwhile presenting here a brief outline on the SISLS process in the case of transfer of the orbital angular momentum of light.

Let us consider a collimated light beam with frequency ω, wavevector k, and astigmatic intensity profile with different waists w ₁ and w ₂ along the transverse directions x and y, respectively. Without loss of generality, we may assume w ₁ > w ₂. The beam passes through a small transparent body having an elongated shape at an angle α with respect to the x-axis. If the linear dimensions of the body are much larger than the optical wavelength, refraction prevails and the main effect of the elongated body may be approximated with the effect produced by a thin cylindrical lens with effective focal lengths f ₁ and f ₂ (f ₁ > f ₂), rotated at an angle α from the x-axis [2]. The optical field emerging from the body is then given by E(r, ϕ, α) = E ₀(r, ϕ) exp[iψ(r, α)], where E ₀(r,ϕ) = Aexp(-x ² cos ² ϕ/$w_1^2$ - y ² sin² ϕ/$w_2^2$), and ψ(r,α) is the phase change produced by the body. In the thin lens approximation, ψ(α) is given by

where r and ϕ are the polar coordinates in the plane transverse to the beam and we posed 1/f = 1/f ₁ + 1/f ₂ and 1/a = 1/f ₁ - 1/f ₂. If the body orientation is held fixed, the angle α is constant and the output field E(r, ϕ, α) is proportional to the input field E ₀(r, ϕ). In our case, however, the body moves under the action of the optical field itself, changing its orientation. As a consequence, the phase of the optical field emerging from the body changes in time at a rate which depends crucially on the shape of the beam intensity profile: no effect is induced, for example, when the beam profile is cylindrically symmetric. At the early stages of the body rotation we have δα = α(t) - α(0) ≃ Ωt, with Ω = -Asin2α(0) [see Eq. (7), below], and the change δψ = (∂ψ/∂α)₀ δα in the optical phase is proportional to the incident power P, as in self-focusing. The occurrence of self-induced phase change modulation depending on the light intensity profile is a clear demonstration of the nonlinear character of the underlying optical process. A deeper insight in this process is gained when it is regarded as a SISLS of photons among states with different but well defined orbital angular momentum. The treatment of the SISLS is here more complicated than in the spin case [13], as the eigenstates of the orbital angular momentum are infinite rather than two. The SISLS formulation for the photon orbital angular momentum transfer to a macroscopic body may be carried out as follows. We expand E(r, ϕ, α) in the orbital angular momentum basis

with

where $L_p^{\left|l\right|}$ (x) are the generalized Laguerre polynomials, x = √2r/w, and 1/w = 1/w ₁ + 1/w ₂. The numbers p and l are the radial and azimuthal number, respectively. The modes f _p,l have orbital angular momentum lħ per photon and have been normalized so that ∫∫ |f _p,l|²dxdy = 1. Since in our case E ₀(r, ϕ) = E ₀(r, - ϕ) and E ₀(r, ϕ) = E ₀(r, ϕ + π), the coefficients c _p,l(α) all vanish for odd l and obey the symmetry conditions c _p,-l(α) = c _p,l(-α). Let us assume that the body orientation is changing in time, so that α = α(t) and the coefficients c _p,l(α(t)) are time dependent as well. Taking the time derivative of E(r, ϕ, α(t)) and using recurrence relations of Laguerre’s polynomials, we obtain the following set of coupled differential equations for the mode coefficients

In order to obtain a set of equations involving the optical field alone, we must eliminate from Eqs. (4) the material parameter α(t). This is done adding to Eqs. (4) the equation of motion of the body. Neglecting inertial terms, the body equation of motion is

where γ is a viscosity coefficient, P is the power carried by the beam and M_z is the mechanical torque on the body. The right-hand side of this equation is the mechanical torque due to the transfer of orbital angular momentum. In writing Eq. (6) we assumed an incident beam with no net orbital angular momentum. When the “matter” equations are eliminated inserting Eq. (6) back into Eqs. (4), we are left with a closed set of differential equations for the mode coefficients c _p,l, describing the SISLS process in a pure electromagnetic way. From these equations, it is evident that the SISLS is a third-order nonlinear optical process and that the nonlinear coupling transfers power among adjacent even-l modes. In this respect, the SISLS process is more complicated here than in the case of the transfer of spin angular momentum, where only the two modes, corresponding to the spin s = ±1 are involved [13].

In the present case, however, we know E(x, y, α) explicitly, so we may obtain the mechanical torque M_z (t) in closed form using the alternative expression

where A = P($w_1^2$ - $w_2^2$)(f ₁ - f ₂)/4cf ₁ f ₂. Then Eq. (6), inserting Eq. (7), can be solved explicitly, obtaining

with time constant τ = γ/2A. The asymptotic equilibrium position is α = 0, i.e. the body is finally aligned along the x-axis, corresponding to the larger dimension of the intensity profile of the incident beam. This could have been anticipated just regarding the intensity profile of the beam as a sort of elliptically shaped potential well accommodating our elongated body. Although this picture has some content of truth, since it can be shown that the the body orientation at equilibrium maximizes the electromagnetic energy excess due to the presence of the body, Eqs. (4) and (6) above show clearly the nonlinear nature of the underlying optical process during the body rotation.

 

Fig. 1. Rotational dragging of three trapped latex spheres each 14 μm in diameter.

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In the present work, we tested Eq. (8) using an astigmatic laser beam with an elliptical intensity profile to induce a mechanical torque on small glass rods dispersed in water. The same beam was used also as optical tweezers to trap the rods in the focal region [14]. The trap used a 40× high numerical aperture (NA = 0.87 in water) microscope objective. The light source was a cw commercial frequency-doubled Nd:YVO₄ laser, working at a wavelength of λ = 532 nm. The incident laser power ranged from 100 mW to 300 mW at the sample position. The intensity profile of the trapping beam was made elliptical by using a cylindrical lens, with focal length f_x = 80 mm and cylindrical axis along the x direction. The beam radii (1/e ² intensity) at the trapping position were found to be w_x = 75 ±2 mm, and w_y = 1.0 ±0.1 mm. The cylindrical lens could be rotated around the beam axis so to change the orientation of the intensity profile and the trapped particles were observed by means of a CCD camera through the same optical port used to inject the laser beam into the microscope. With this device we trapped both small latex spheres (15 μm in diameter) and glass rods obtained by crushing a small piece of glass wool. When the microspheres stand near the optical well, they are trapped and form linear aggregates as shown in Fig. 1. While rotating the cylindrical lens, the aggregates rotate as a whole following the motion of the lens with high accuracy. When used in this way, our device exploits the optical gradient forces present in the trapping region to confine the particles and then uses the rotation of the trapping region — owing to the rotation of the cylindrical lens — to induce the particle rotation. A similar orientational dragging effect was achieved by rotating the spiral intensity pattern produced by interference in the cross-section of a Laguerre-Gauss laser beam [15]. Also trapped small bodies were dragged by rotating the pattern produced in the optical tweezers by a high-order Hermite-Gauss TEM_0n mode laser beam [16], or even by a slit [17] or by interference fringes [18, 19]. These experiments of rotational dragging have little to do with the nonlinear optical process described above, which is essentially dynamic, as shown in the movie in Fig. 2. In order to study more quantitatively the transfer of the photon orbital angular momentum to matter, we made the following experiment: a glass rod, prepared as above mentioned, was preliminarily trapped and aligned along the intensity profile by switching the laser beam on. Then, the laser was switched off (to avoid the fragment to be dragged by the tweezers) and the cylindrical lens was suddenly rotated of about 90°. Later on, the laser was switched on again and the fragment was immediately observed to start rotating spontaneously until the alignment along the new orientation of the intensity profile had been reestablished. The rotation was recorded by the CCD camera and some frames are shown in Fig. 3 in the proper sequence. The intensity profile of the laser beam at the trapping region was kept fixed during the motion of the particle. In these conditions, the rotation of the body is entirely due to the torque exerted on it by light. The mechanical torque vanishes and the motion stops when the rod is aligned along the major axis of the beam intensity profile (last frame in Fig. 3). In our tweezers, the incident beam carries no angular momentum and the angular momentum gained by the body was extracted from the beam by the body itself. In order to test Eq. (8), we reconstructed the motion of the glass rod from the frames recorded by the CCD camera and compared the experimental motion law with Eq. (8) using τ as fitting parameter. The fitting is shown in Fig. 4. From the best fit we found τ = 0.75±0.01 s. Using the experimental value of τ and Stokes’ relation γ = 8πL ³ η, where η = 1 cp is the viscosity of water and L = 13 μm is the length of the rod (see Fig. 3), we evaluated from Eq. (6) the maximum torque A acting on the body as A = 3.7×10^-17 Nm, corresponding to a trapping force F ≈ A/L ≈ 3 pN, which is a typical order of magnitude for the forces present in optical tweezers. From the experimental value of A and assuming f ₁ ≫ f ₂, as is appropriate for cylindrical rods, we estimate the effective focal length f ₂ to be 25 mm. This focal length is much longer than the diameter a ≃ 3 μm of the glass rod, showing the single thin lens approximation to be inadequate in the present case. The rod, in fact, seems to approximate better the behavior of a thick nearly confocal two-lens system. The good fit between our experimental data and Eq. (6) strongly supports that it is really the light orbital angular momentum to orient our trapped fragments. We repeated the experiment on glass fragments having different shapes and sizes yielding similar results.

 

Fig. 2. Movie showing the laser-induced spinning of a trapped small glass rod (length 13 μm, diameter 2 μm). [Media 1]

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Fig. 3. Nine frames of a trapped glass rods showing the alignment motion along the major axis of the trapping beam shape. As the optical torque acting on the particle depends on its orientation [see Eq. (8)], the rotation speed is not constant. The frames are 200 ms apart. The scale bar is 10 μm.

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Fig. 4. The angular position of the particle as a function of time during its motion. The laser power at the sample position was P = 200 mW.

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In conclusion, we presented the quantitative proof that the orbital angular momentum of light can be transferred to a transparent isotropic body. We derived the equations governing the underlying optical process, showing that it is a third-order nonlinear process, similar, in some respects, to Raman scattering and previously studied in the case of light spin transfer [13]. We realized a directional optical tweezers by which microscopic particles can be trapped and rotated at will with high angular accuracy. The optical torque connected with the transfer of orbital angular momentum from the laser beam to a small glass rods was measured by testing the motion law of the rod itself. With our apparatus we cannot measure the orbital angular momentum content of the beam emerging from the optical tweezers so we are planning to check Eqs. (4) for the optical modes in a future experiment. We think, however, that the good quantitative agreement between the observed motion and the calculated optical torque strongly supports that the orbital angular momentum of light can be used to rotate transparent bodies using suitable laser beams although not in an angular momentum eigenstate.

We thank the Istituto Nazionale per la Fisica della Materia for financial support and prof. E. Arimondo for loaning part of the experimental equipment.