Spin to orbital light momentum conversion visualized by particle trajectory

In a tightly focused beam of light having both spin and orbital angular momentum, the beam exhibits the spin-orbit interaction phenomenon. We demonstrate here that this interaction gives rise to series of subtle, but observable, effects on the dynamics of a dielectric microsphere trapped in such a beam. In our setup, we control the strength of spin-orbit interaction with the width, polarization and vorticity of the beam and record how these parameters influence radius and orbiting frequency of the same single orbiting particle pushed by the laser beam. Using Richard and Wolf model of the non-paraxial beam focusing, we found a very good agreement between the experimental results and the theoretical model based on calculation of the optical forces using the generalized Lorenz-Mie theory extended to a non-paraxial vortex beam. Especially the radius of the particle orbit seems to be a promising parameter characterizing the spin to orbital momentum conversion independently on the trapping beam power.

So the radial stiffness κ r can be obtained either from Eq. (1) or from fitting Gaussian distribution of probability to the experimental profile (see Fig. A1d). Using the power spectrum density of radial positions S( f ) = 2|F {R(t)} | 25, 6 the drag coefficient γ is related to the 'corner frequency' by ω c = 2π f c = κ r /γ (using either σ or κ r ): where k B is Boltzman constant and T is thermodynamical temperature. Independently, one can compare with the value ∆R( f = 0) which yields (using either σ or κ r ): The hydrodynamic drag coefficient is influenced by particle-surface proximity as γ = γ 0 ψ where γ 0 = 6πaη denotes the Stokes formula with a and η denoting the particle radius and medium viscosity, respectively, and ψ denotes the Faxen's correction [7][8][9] to the proximity of the surface ψ(a, h) = 1 − 9 16 (a/h) where h is the distance of the particle center from the glass interface. Since we have not measured the particle-surface distance we could not used the above described approach. Instead we determined the corner frequency from the fit of experimental data to S( f ) and using the trap stiffness κ r determined from Eq. (1) we obtained the drag coefficient γ . To obtain the correction term ψ we used the expected size of the particle and local viscosity corresponding to local temperature 25 o C as where A = 2.414E − 5K, B = 247.8K, T is in Kelvins. Obtained values are summarized in Table A1.  Figure A1. Examples of processed data a) Particle trajectory in lateral plane b) x and y positions of the particle in time c) The power spectral density of radial position of the particle with corner frequency f c = 14Hz d) Probability distribution of radial position of the particle giving radial stiffness κ r = 2.4 N/m. Appendix Figure A2. Local speed of the particle as a function of the angular position for different topological charges and for the LH polarization. Appendix Table A1. Parameters of the experiment.

Beam description
Let us start with a description of the paraxial beam entering the objective at plane A, i.e. at the front focal plane of the objective ( Fig. 1d in the main manuscript). Let us assume the monochromatic wave of electric field E = ℜ[E e −iωt ] is described by its complex amplitude E. The vortex beam coming from the SLM is linearly polarized along x axis and its width is done by the extend of the phase grating on the SLM. This beam passes through a quarter wave plate rotated with its fast axis by an angle β with respect to the x axis (see Fig.1 in the main manuscript) and its influence is described by the following Jones matrix: Assuming that the field at plane A (the entrance of the objective) is a perfect image of the field in the SLM, we can describe the incident beam at plane A as: where E A0 denotes the field amplitude in plane A, w A = Mw SLM the beam waist radius with the magnification of the relaying optical system M = 0.47 between the SLM and plane A. The coordinates (x A , y A ) correspond to the transversal ones in the plane A. p and q correspond to the RH and LH circular polarization amplitudes, respectively. The RH and LH circular basis is given byê ± = (ê x ± iê y )/ √ 2, whereê x orê y denotes unit vector along x or y axis, respectively. Table A2 defines more clearly the used polarization parameters.
The amplitude of the electric field can be expressed in terms of the experimental parameters as E 0 = 2µ 0 cP/ζ . From Eq.(9), it is easy to show that the difference in energy between the RH and LH polarizations, which defines the spin angular  Table A2. Definition of polarization parameters.
momentum density, is given by: The beam passes through the microscope objective following the Richards and Wolf theory 10 (assuming Fresnel transmission coefficients t s = t p = 1) the refracted field immediately after the objective (plane B) is expressed in the circular basis as follows 11 with g(θ ) = 1 + cos θ , h(θ ) = 1 − cos θ and θ is the polar angle. The field in the beam focus (plane C) is thus a superposition of all the incident waves with angles θ and ϕ in Eq. (11), following 12,13 , where Ξ = kz cos θ + kρ sin θ cos (ϕ − ϕ). (ρ, ϕ, z) are the cylindrical coordinates measured from the focal point and θ max defines the maximum angle of the cone related to the numerical aperture of the focusing optics NA = n m sin θ max = D/(2 f ), D and f denotes the diameter of the input aperture and focal length of the focusing objective, respectively. Regrouping Eqs. (12), one gets The azimuthal-dependent exponential functions indicate that the original vortex with initial topological charge transforms to one in the focal plane composed of several vortex beams with topological charges , ± 1, and ± 2 with amplitudes defined by the circular components p and q and by the polar angle θ max . The circular components in the tightly focused beam are the result of the interference of two vortices with topological charges and − 2 forê + and and + 2 forê − , while the longitudinal component results from the interference of two vortices with topological charges + 1 and − 1. The total energy is proportional to the intensity of these three components, which can be expressed by three terms each: Estimating these terms numerically it is easy to see that the leading terms with the weight ≈ 70% − 80% are proportional to |A | 2 , interfering third terms are very small in comparison to the others, at least for the experimental conditions we have (NA = 1.2). This is very important, since the gradient force, that attracts the particle to the radial stable position, is mainly given by the intensity. We can say that the particle will be affected by the force coming from two main vortices with topological charge and four weaker vortices with topological charges + 1, − 1, + 2, − 2 owing to the spin-to-orbit conversion.

Force description -Rayleigh approximation
However such field distribution can be detected only with a local probe either using scattering 11 or velocity of a nanoparticle. Following the approach of a Rayleigh particle, i.e. the particle is much smaller that the wavelength which is treated as the induced dipole. The force components acting upon such nanoparticle in the cylindrical system of coordinates (r, ϕ, z) can be expressed as 14,15 where j = ±, z correspond to the field components in circular basis E(r, ϕ, z) = E +ê+ + E −ê− + E zêz , which is related to the field components in cylindrical system of coordinates as: and α SI is a polarizability of the particle in SI units, expressed with non-dimensional form α 0 , and α R with included radiation reaction term 16 , : where relative refractive index is denoted as m = n p /n m , ε m = n 2 m , ε 0 is permittivity of vacuum. Expressing the azimuthal force using Eq. (13) one would obtain: where the amplitudes were denoted: To express F r , F z we use of the fact, that the derivative ∂ z E acts only on the exponential factors in the integrand of A m , B m , C m , which yields extra factor ik cos θ . In similar way the derivative ∂ r E affects only the Bessel functions. In analogy, we define other two sets of integrals where X m should be substituted with A m , B m , C m : The expressions for A z ±0 are identical, with the upper index r exchanged for z in each term.
When the paraxial beam incident on the lens is circularly polarized, the expression reduces significantly: Analyzing F r we found that the radial position of the particle depends on the RH or LH circular polarization ±σ of the beam with respect to the topological charge . Thus detecting the radial particle position one should detect the spin-orbital coupling of the focused vortex beam, as we show below. Similarly the axial force F z changes its magnitude with the beam polarization, however, we can not detect these dependence in our experimental arrangement. Looking at F ± ϕ one discovers that this force is fully non-conservative for RH or LH polarization and for paraxial beam it is proportional to ± σ 17, 18 , however, force magnitude increases with stronger non-paraxiality of the beam. Thus detecting the mean time of the particle orbit T = 2πrγ/F ϕ , where γ is the hydrodynamic drag coefficient, one should be also able to detect the spin-orbit interaction, too.
Estimate of equilibrium position of orbiting particle Using Eq. (15) and assuming α SI α SI it is that seen the radial force is proportional to the radial gradient of the optical intensity I + + I − + I z expressed in Eq. (14). This can be interpreted that the particle moves in an interference field of five vortex beams with topological charges , + 1, − 1, + 2, − 2 represented by terms A , C ±1 , and B ±2 , respectively. Let us further assume that in radial direction the particle settles in the equilibrium position R eq and each vortex beam has intensity maximum at radial distance assuming for simplicity the same δ for all . This linear behavior can be seen in the data plotted in Fig. 3 for large values of . We can write for the radial force, assuming optical trap of each vortex as the Hookian spring with stiffness κ Solving for R eq , utilizing Eq. (32) and direct proportionality between trap stiffness and κ , κ ±2 , κ ±1 and corresponding terms |A | 2 , |B ±2 | 2 and |C ±1 | 2 in Eq. (14) one obtains The second term represents a sum of four terms and each of them is proportional to the ratio of the optical intensity in the corresponding vortex to the total intensity I 0 in the beam. Utilizing significant terms in Eq. (14) one ends with R eq ≈ m + 2 cos 2 β |C +1 | 2 p 2 − |C −1 | 2 q 2 + |B +2 | 2 p 2 − |B −2 | 2 q 2 + δ .

Force description -Generalized Mie approach
However the above mentioned analytical equations are not fully valid for larger particles used in the experiment. Therefore our calculations of the optical force field are based on the exact wave solution in spherical coordinates without applying any small/large particle approximations. Entering the force formula given in Ref. 19 multipole expansion coefficients for the incident field and for the scattered one, (referred to as the 'beam-shape' coefficients 20 ) are determined as projections of the (spherical) radial components of the beam field on a spherical harmonic function Y m n (θ , ϕ) where j (1) n being the spherical Bessel function and the first kind, index p stands for multipole TE (p = 1) and TM (p = 2) modes. Extending the procedure to a vortex beam 12 described by Eq. (13) and following 21 we obtained the following expressions a inc p,nm = e i( −m)(ϕ+ π 2 ) i n 4πk f e −ik f where other functions v 1 , v 2 are given also as integrals over aperture angle α: .
The shortcuts J ± m denote Pauli matrix σ 2 = 0 −i i 0 , (E x 0 , E y 0 ) is the Jones vector of the incident field (i.e. 1/ √ 2(1, ±i) for RH/LH polarization) , π(θ ) = mY m n (θ )/sin θ , τ(θ ) = ∂ θ Y m n (θ ), R(ϕ) is the standard 2D rotation matrix. Employing the equations presented above with the parameters used in the experiment (see Table A1) we calculated the forces acting upon the particle placed at plane C, (i.e. at the plane of the beam focus) and determined the radial equilibrium position R eq , radius of the maximal field energy density R ed , and orbiting frequency f 0 for different circular polarizations and topological charges of the beam presented above.