Spin-Orbit interaction

Theoretical and experimental studies of the SOI of light in Optical Tweezers

Spin-orbit interaction (SOI) has wide implications in physics - most importantly perhaps, in determining the structure of the atom. The electron's orbital angular momentum (L) and spin (S) couple to remove the accidental degeneracy that is a consequence of the solution of the Schrödinger equation without spin, where the energy levels do not depend upon the orbital angular momentum L. The LS coupling gives rise to the fine structure of the atom, and the coupling of the total angular momentum (J) and the nuclear spin (I) further splits the energy levels into what is known as the hyperfine structure. In condensed matter physics, the SOI of electrons in a semiconductor also result in a transverse spin flow or current, whose signature is measurable by polarization properties of light that interacts with the sample (basically a circular polarization component is induced in linearly polarized light). For light, SOI manifests itself through the interaction of the spin (polarization) and orbital angular momentum (both intrinsic and extrinsic) of the light, and leads to two main effects: accumulation of a geometric or Berry phase, and the Spin Hall effect (SHE) of light. The former often manifests itself in interconversion of spin to orbital angular momentum (SAM to OAM) that may lead to diattenuation, or polarization dependent intensity profiles of light, whereas the latter results in a transverse spatial separation of opposite circular polarization states of light. OT, where the trapping light is very tightly focused provides a great testing ground of SOI, where both effects are observable. Unfortunately, these were not really considered in the literature earlier, mostly due to the fact the effects (especially trajectory shifts, or intensity asymmetries) are very small - of the order of the wavelength of the trapping light. Effects of SAM to OAM conversion were observed for higher order Gaussian beams (Laguerre-Gaussian or Airy beams) having circular polarization - but no effects were ever reported for fundamental Gaussian beams. This is where we come in. We have found that even Gaussian beams show large SOI and experimentally realizable effects on particle dynamics by making a small innovation in the OT set-up - basically by using cover slips that are refractive index (RI) mismatched with the microscope objective immersion oil. Essentially, this creates a stratified medium in the trapping sample chamber (see figure) and immediately causes a host of interesting effects. 

Controlled transportation of mesoscopic particles

When we use our special cover-slips that have a higher RI (1.575) compared to the objective immersion oil (1.515), as well as higher thickness (250 um) compared to standard cover-slips (160 um), the intensity distribution near the focal region of the trap is modified diffraction effects in combination with enhanced SOI. This enhancement can be understood by calculating the electric field distribution for our system. This is no easy job, considering that the paraxial approximation no-longer works due to the tight focusing! We thus have to use the angular spectrum method where one decomposes any field into plane waves in Fourier space, propagates them taking into account the polarization characteristics of the medium by appropriate Fresnel coefficients, and performs another inverse Fourier transform to get back the field in real space. This is the famous Debye-Wolf method, and is the basis of all our calculations. In our system, the stratified medium formed causes the intensity distribution near the focal plane to change drastically from a Gaussian structure, with a ring-like structure formed along with local regions of higher intensity as well. The ring-like distribution itself is interesting, and results in a self-assembly of polystyrene beads inside it - something never seen in OT before for simple Gaussian trapping beams (see figure). The signature of SOI is the formation of the intensity lobes inside the ring - the breakdown of the azimuthal symmetry is due to a large anisotropic diattenuation (see our paper in Phys. Rev. A) that is developed as a result of SOI. We have trapped single asymmetric soft oxometalate (SOM) pea-pod particles (supplied to us by Dr. Soumyajit Roy - one of our principal collaborators) in these lobes, and then even managed to move them along the ring by just changing the polarization of the input beam (Fig. explains this - the location of the lobes depend on the angle of polarization of the input linearly polarized beam). This is rather rare example of moving particles in OT without moving the input beam at all, but by just turning a wave-plate kept at the input of the trap! Thus, the trap stiffness remains unchanged since the beam alignment does not change at all, and the distance by which we can move a particle is also rather large, almost 15 um. In fact, with multiple such ring traps, one can even envisage really long distances for moving particles - we plan to do these experiments soon! 

Controlled rotation of particles 

Rotation of particles in OT is nothing new, with plenty of literature available with circularly polarized Gaussian beams being able to spin particles, or orbital angular momentum carrying beams (LG, Airy beams, etc.) causing particles to revolve in orbits. But how about a fundamental linearly polarized Gaussian beams spinning particles, both clockwise and anti-clockwise, and most interestingly - one beam spinning particles and another being switched on to stop it or even spin it in the opposite direction! Well, this is what we can do with our stratified medium - thanks to the large SHE it leads to. The tight focusing generates a longitudinal field component, which naturally gives rise to a transverse component in the Poynting vector - implying that there now is a transverse flow of energy. This is the SHE - and it means that there will be a spatial shift or trajectory shift of opposite circular polarizations. It also means that if one comes in with a trapping beam having linear polarization - which can be considered to be a superposition of left and right handed circular polarizations (LCP and RCP) - the spatial degeneracy of these two polarization states would be lifted near the focal plane, and one would actually obtain local spatial regions of LCP or RCP around the focus (radially, and we have regions of large spin angular momentum (SAM) density - where an absorbing particle will actually spin. And indeed, we do observe spinning pea-pods and quartz particles (see all the videos). Moreover, when we put in another trapping beam close to the first one (which causes the initial spin), we can actually cancel out the circular polarization content of the field (see Figure 4(e) of our New Journal of Physics paper ), and stop the spinning particle - provided the intensity of the second beam is the same as the first, and even reverse the direction of the spin if the intensity of the second beam is higher (the torque exerted by the light - proportional to the SAM density of the field - depends on the degree of circular polarization and the total intensity). This is probably the first time in OT that such controlled spinning has been demonstrated - and perhaps the first time that probe particles have been used to measure local regions of high SAM density for non-paraxial beams. 

Spontaneous revolution of particles in OT

Till date, LG beams have been used to revolve particles in orbit - but we have now observed stable revolving motion of asymmetric particles inside our 'special' RI-mismatched trap. This is just due to scattering forces - a nice simulation using Lumerical (a nice FDTD software for modelling electromagnetic fields) reveals how our ring trap can generate a continuous tangential force along the ring so that particles are in dynamic equilibrium by revolving around the ring. The ring-trap thus potentially facilitates spontaneous micro-swimmers if the shape of the swimmers are right! 

Transverse angular momentum in OT

It has recently been discovered that a phase shifted longitudinal component of field plays a major role in the appearance of spin (polarization) dependent transverse momentum and spin (polarization) independent transverse spin angular momentum (SAM). This particular feature is well known as spin momentum locking in condensed matter physics in the context of topological insulators. In optics, this feature is manifested as the transverse component of the Poynting vector - which represents the flow of momentum - being independent of helicity (spin) of the beam. In case of evanescent fields, such non-trivial structures of spin and momentum density have been observed and reported. It has been shown that the general solution of Mie scattering from a spherical particle, which has phase-shifted longitudinal component indeed has the helicity dependent transverse component of Poynting vector (generally addressed as 'transverse (spin) momentum') and helicity independent transverse spin angular momentum density. Thus, keeping in mind that a tightly focused Gaussian beam has a longitudinal field component which is phase shifted from the transverse components, the obvious question that arises is whether a tightly focused Gaussian beam also contains these interesting and exotic properties. Debapriya and Indrani are currently involved in this work. 

Future work

This is just the tip of the iceberg. All these interesting results have been obtained with just a single RI mismatched layer - that too having RI only 5% different from the immersion oil. What happens if we use higher contrast RI? What happens with higher order Gaussian beams? Can evanescent waves causes interesting effects? We are on our way trying to answer these questions!