Direct observation of the effects of spin dependent momentum of light in optical tweezers

Light carries both orbital and spin angular momentum. The Poynting vector—considered to be the vector representative of the flow of energy—has contributions from both the canonical and spin part of the momentum. The spin contribution in the Poynting vector (P, or total momentum) was introduced by Belinfante through the equation P = P_o + P_s,^1,2 where P_o represents the canonical and P_s represents the spin momentum. The spin momentum P_s is rather enigmatic since—while it is responsible for spin angular momentum—it does not contribute to the energy flow and is therefore considered to be a virtual quantity. On the other hand, P_o is responsible for generating orbital angular momentum (OAM) l (l = r × P_o, where r is the distance from the beam axis) that is directly manifested in experiments by the rotation of mesoscopic particles about the beam axis in optical tweezers.^3,4 The question thus arises whether the manifestation of the elusive Belinfante spin momentum can be experimentally extracted by similar means.

It has recently been observed that a longitudinal component of the field—phase-shifted with respect to the transverse component—plays a major role in the appearance of spin (polarization) dependent transverse momentum and spin (polarization) independent transverse spin angular momentum (TSAM).^2,5–9 This particular feature is well known as spin momentum locking in condensed matter physics in the context of topological insulators,¹⁰ where special states exist at the outermost surface of the insulator, which falls within the bulk energy gap and permits surface metallic conduction. The carriers in these surface states are observed with their spin locked at a right-angle to their momentum (spin-momentum locking).¹¹ In optics, this feature is manifested as the transverse component of the Poynting vector—which represents the momentum density—being dependent on helicity (spin) of the beam. In the case of evanescent fields, such non-trivial structures of spin and momentum density have already been reported.^2,12 In fact, such a transversely spinning electric field arising in the case of transverse SAM of light and resembling the spinning movement of the spokes of a rolling bicycle wheel has recently been experimentally achieved.¹³ It has also been shown that the general solution of Mie scattering from a spherical particle contains a phase-shifted longitudinal component so that there exists a helicity-dependent transverse component of Poynting vector [generally addressed as “transverse (spin) momentum”] and helicity-independent transverse spin angular momentum density.^8,14,15 Thus, keeping in mind that a tightly focused Gaussian beam also has a longitudinal field component, which is phase-shifted from the transverse components, the question that naturally arises is whether such a beam also contains these interesting and exotic properties. Indeed, the presence of the longitudinal field component results in the observation of the spin-Hall effect (SHE) in tightly focused Gaussian beams,¹⁶ as well as interesting effects of spin–orbit interaction (SOI) of light.¹⁷ However, it is important to note that while a few practical applications such as nano-displacement probes¹⁸ or generation of optical vortices¹⁹ have been developed, in most cases, the manifestations of SOI have been quite small, with the magnitude of light trajectory shifts reported due to the SHE of light usually being in the sub-wavelength regime.^16,20–22 However, in recent experiments on the scattering of nano-particles reported in Refs. 18 and 23, the SHE was enhanced by using an imaging system with a small focal length or high numerical aperture (NA) microscope objectives so that the extent of the SHE covered the microscope exit pupil. In addition, our group has also demonstrated that the use of a stratified medium in the trapping beam path in optical tweezers enhances polarization-dependent intensity distributions and SHE effects, which can lead to controlled particle manipulation.^17,24 Thus, it makes sense to explore the use of optical tweezers in observing spin-momentum locking in Gaussian beams and investigate the manifestations of Belinfante spin.

In this paper, we demonstrate that a tightly focused spin-polarized Gaussian beam indeed possesses the very same properties of spin-momentum locking that have been observed for evanescent fields² and Mie scattering.¹⁴ We simulate the electric and magnetic field distribution for an optical tweezers configuration, and the results of our simulations clearly demonstrate the existence of spin momentum locking in such beams. Our simulations also reveal that the presence of a stratified medium in the path of the trapping beam can significantly enhance the magnitude of transverse spin angular momentum and transverse Poynting vector in the radial direction away from the beam center as well as the electric field intensity in the same direction. The overlap of the enhanced intensity and the transverse Poynting vector point to the possibility of observing the effects of Belinfante spin experimentally. We indeed proceed to verify the simulation results experimentally by observing the spin-dependent rotational motion of a birefringent particle around the beam axis for input circularly polarized light propagating through a stratified medium and carrying no intrinsic orbital angular momentum. The circular motion is dependent on the helicity of the input beam so that we can identify it to be the signature of the elusive Belinfante spin, the manifestations of which—to the best of our knowledge—has not been previously demonstrated for the propagating light.

In the case of tight focusing of light, the paraxial approximation fails with the generation of a large longitudinal component of the electric field, which is phase-shifted from the transverse components. Therefore, we calculate the electric field distribution of the tightly focused beam passing through different refractive index (RI) layers (considering both forward and backward propagation) employing the angular spectrum method²⁵ using Debye–Wolf diffraction integrals.²⁶ The expression of the tightly focused beam can be written in the form of a matrix equation as

where $X→$ is the Jones vector of the input field, I₀, I₁, and I₂ are the Debye integrals (see the supplementary material), and C is a constant. z denotes the beam propagation direction, whereas x and y denote the transverse plane. The diffraction integrals account for the forward and backward propagating beams and contain the Fresnel coefficients (T_s, T_p, R_s, and R_p), which are required to consider the effect of the stratified medium. We keep track of the evolution of the electric field in each refractive index layer of the stratified medium using

Here, i denotes the polarization states (i.e., s-polarized or p-polarized denoted by s and p, respectively), the positive and negative signs denote the forward and backward propagating waves, respectively, and j denotes a particular refractive index layer. The output electric fields can be obtained using the Jones vector for left and right circularly polarized input light as $X→=[1±i0]T$⁠. Note that the tightly focused electric and magnetic fields enjoy the following symmetry relations (in Gaussian units):²⁶ 

As circular polarized light propagates along the z direction, the longitudinal spin angular momentum (LSAM) is generated due to the intrinsic spin (helicity σ) of the light and may be represented as S ∝ σz, which can be transferred to absorbing birefringent particles to cause rotation about their center of mass. As noted previously, the Poynting vector or momentum density of the wave can be broken up into the canonical (or orbital, P_o) and spin (P_s) components. The latter can also be transferred locally to probe particles resulting in a torque T ∝ P₀ about the beam axis, with the direction of the torque (clockwise or anti-clockwise) given by the sign of P_s, in contrast to the SAM (S), which generates rotation around the particle center of mass. Now, the SAM density (S) and Poynting vector (P) in an isotropic medium, considering real ε and μ, in SI units are given as (ignoring dispersion effects)

Using the above relations, we calculate the SAM and momentum (or Poynting vector) density for right and left circular polarized light. The expressions for right circular polarized light are [the subscripts R and L are used to denote right circularly polarized (RCP) and left circularly polarized (LCP) light, respectively]

while those for left circular polarized light are

It is obvious that under the change of helicity from +1 to −1, the transverse (x, y) components of the SAM, henceforth referred to as TSAM, remains the same, while those of the spin momentum, henceforth referred to as TM, flips direction, which confirms that the helicity-independence of the former and helicity-dependence of the latter are inherent properties of a tightly focused Gaussian beam. Understandably, this is a clear signature of spin momentum locking. However, the longitudinal components behave in exactly the opposite manner, with S_z (LSAM) flipping sign, while P_z does not flip sign under change of input helicity as is clear from the preceding equations.

As discussed earlier, T_s, T_p, R_s, and R_p have a crucial role to play in determining the diffraction integrals, which, in turn, determine the final electric field and magnetic field distributions. Thus, a suitable choice of refractive indices in the stratified medium may provide us control over the Fresnel’s coefficients, which would determine the final distribution of the electric and magnetic field, and produce interesting effects.

We now run simulations on our experimental system (stratified medium in the path of the optical tweezers light beam) with the theoretical model developed in Sec. II and observe the occurrences of diverse phenomena dealing with spin momentum locking, transverse spin, and spin momentum effects for input circularly polarized Gaussian beam (TEM₀₀ mode) into the optical tweezers. The laser beam of wavelength 1064 nm is incident on the 100× oil-immersion objective of numerical aperture 1.3, followed by (a) an oil layer of thickness around 5 µm and refractive index (RI) 1.516, (b) a 160 μm thick coverslip having refractive indices 1.516 and 1.814 (note that the case where the RI = 1.516 is henceforth referred to as the “matched condition” since this refers to the most general condition employed in optical tweezers experiments where the coverslip and the microscope objective immersion oil have the same RI so as to minimize spherical aberration effects in the focused beam spot, increase the electric field intensity at the center, and thereby increase trap stiffness), (c) an aqueous solution chamber having a refractive index of 1.33 with a depth of 35 μm, and finally (d) a glass slide of refractive index 1.516 whose thickness we consider to be semi-infinite (around 2000 μm–3000 μm, being very large considered to other dimensions), as shown in Fig. 1. We first define a specific coordinate system for our simulation in which the z = 0 is taken at 5 µm inside the sample chamber near the coverslip. Considering this as the origin and with the dimensions mentioned above, the boundary for oil and the objective is at −170 µm, oil and the coverslip is at −165 µm, the coverslip and the sample chamber is at −5 µm, and the sample chamber and the glass slide is at +30 µm.

We use both left circular and right circular light as input and compare the results with respect to spin-momentum locking. Figures 2(a) and 2(c) display the TM and TSAM for a left circularly polarized beam, respectively, while Figs. 2(b) and 2(d) displays the same for a right circularly polarized beam at the beam focus. It is clear from the quiver plots that the TM vector flips sign, whereas the TSAM is independent of helicity of the input beam. This establishes our claim of the existence of spin momentum locking in tightly focused Gaussian beams. With the change in the RI contrast within different layers, the Fresnel coefficients (which determine the nature of the Debye–Wolf integrals) change, which would affect the intensity distribution, as well as the SAM density and Poynting vector. However, in order to probe effects of the angular momentum on probe particles, we need to trap them first—which makes the field intensity distribution at the sample region of optical tweezers crucial.

We now shift our attention to this problem and quantify the intensity profile in the radial direction inside the sample (water in our case) for coverslips of different RI and also investigate whether the off-axis intensity is sufficient enough for trapping and rotating micrometer sized particles, given the magnitude of the TM at the same location. We do not consider scattering effects from a trapped particle since the scattering (which itself is very small compared to the transmitted field due to the size of the particle) is predominantly in the forward direction and does not contribute to the transverse field anyway.

We plot Figs. 3(a) and 3(b) for the matched condition (coverslips of RI 1.516) at the focus and 2 µm axially (the Rayleigh range using Gaussian approximation is ∼175 µm) away from the focus, respectively. The intensity distribution is clearly Gaussian with negligible side lobes. However, since we are working with a fast diverging Gaussian beam, the beam intensity reduces rapidly. The side lobes also disappear as the beam expands in the radial direction when we move 2 µm beyond the focus, as shown in Fig. 3(b). It is also interesting to note that the TM—given by the red solid line—is zero at the beam center and increases slightly off-axis, where the field intensity has fallen substantially. Thus, the possibility of observation of any effect of the TM on trapped particles is rather low, which is perhaps the main reason why this has not been observed in conventional optical tweezers.

However, when we increase the RI of the coverslips to 1.814 (mismatched condition), the intensity distribution is inhomogeneous, and side lobes are formed, as shown in Figs. 3(c) and 3(d). Indeed, the intensity at the beam center is substantially lowered for the higher RI due to diffraction effects (spherical aberration). The interesting point is that the TM—depicted by red solid lines—increases in magnitude off-axis for the increased RI. The intensity is also high here (close to that at the center), and we now obtain regions where both the intensity and transverse spin momentum are high enough in magnitude so that particles may be trapped and also rotate due to the presence of sufficient TM. Thus, a particle trapped in this configuration with a stratified medium in the path of the optical tweezers light beam may be used as a probe to observe the TM.

Now, we go on to investigate the LSAM for our system, which we plot in Fig. 4 for both matched [Fig. 4(a)] and mismatched conditions [Figs. 4(b)–4(d)]. We observe that the LSAM—depicted by black-dashed lines—is always high at the beam center, where the intensity is also maximum, which explains the routine observation of the rotation of transparent birefringent particles around their center of mass for input circular polarization in optical tweezers.³ However, at the center, the TM—depicted by red solid lines—is low. For different axial planes, the matched condition yields results (displayed in detail in the supplementary material) uninteresting for experiments since the field intensity falls off rapidly in the radial direction. The mismatched case, though, needs to be considered in detail. Here, we observe that while the TM has very small radial lobes at the focus, these start becoming more significant as we move away from the focus axially. At z = 1 µm, we observe that the LSAM has several off-axis peaks, whereas the TM still falls off quickly from the center. However, at z = 2 µm, there is a region where both TM and LSAM are high off-axis in a particular region, which —along with Fig. 3(d) (which shows that the field intensity is also high in that region)—suggests that it may be possible to observe interesting effects in particle rotation for this situation. This is what we experimentally pursue, as we describe in Sec. IV.

We now attempt to observe these effects experimentally on microscopic birefringent probe particles to verify our simulations. We use a conventional optical tweezers setup comprising an inverted microscope (Olympus BX71) with an oil-immersion 100× objective (Olympus, NA 1.3) and a high power diode laser (1064 nm, 500 mW) coupled into the back port of the microscope. We control the polarization of light using a quarter wave plate (QWP) and change the helicity of the beam by rotating the QWP by 90°. For the probe particles, we use RM257 vaterite liquid crystal particles, which are optically anisotropic and birefringent so as to transfer angular momentum from the beam into the particles, and thus probe the effects of TM and LSAM. We construct the sample chamber using a glass slide and a coverslip of RI 1.814, into which we add around 20 µl of the aqueous dispersion of RM257 particles,²⁷ which are elliptical and of mean size 2 × 1 μm, with a standard deviation of 20% in both dimensions. This size enables the particle to sample the TM as well as the electric field, which are spread over a spatial range of around 4 µm in the case depicted by Fig. 3(d). We ensure that the incident power is constant for both left and right circular polarizations to avoid any difference in trapping conditions.

On coupling the laser into the microscope, we observe the formation of concentric off-axis intensity rings (see supplementary material, Fig. S1) around the beam center similar to that displayed in Fig. 3. The image of the rings is optimized by changing the z—focus of the microscope (imaging is performed with transmitted light from the microscope lamp coupled into a camera attached to the side port of the microscope). As a result, we are able to image the particles that assemble in one of the rings, as we reported in Ref. 17. However, when we trap and observe a single particle in a ring at the appropriate z-depth, we are able to observe rotation of the particle around the beam axis (see Visualizations 1 and 2 in the supplementary material)—either clockwise or anticlockwise, as displayed in time-lapsed images in Fig. 5. The trajectories of the particle are shown as red dotted circles. The first row shows rotation in an anti-clockwise direction. When we modify the orientation of the QWP by 90°, the rotation direction of the particle flips (becomes clockwise), as shown in the second row of time-lapsed images in Fig. 5. However, these are for two different particles in separate experiments. We have changed the orientation of the QWP for the same particle as well, but often lose the particle from the trap in that process. This is because the input helicity, or the Jones vector in Eq. (1), is modified so that the output electric field intensity in the sample plane is also modified—as a result of which the trap generally becomes unstable. However, in a few experiments, we do manage to keep the particle trapped by rotating the QWP very slowly. One such case we report here, which is shown in Visualization 3 and in time-lapsed images in Fig. 6(i). In this case, we observe rotation of the particle both around the axis of the beam in the clockwise direction (input RCP) demonstrated in Figs. 6(a)–6(c) and around the particle body axis in the anti-clockwise direction for input LCP, as demonstrated in Figs. 6(d)–6(f). Note that while the former is due to the enhanced TM in our system, the latter is invariably the demonstration of the LSAM, which increases over TM when the axial trapping depth is modified. We have verified on this in Fig. 4, where we demonstrate that when we change the z-distance from the focus, the TM and LSAM behave differently. Therefore, it appears that when we change the QWP slowly, the intensity at the trapping plane still changes enough to change the equilibrium position of the trapped particle in the z direction. This results in the trapped particle sampling higher LSAM than TM (note that the LSM at z = 1 µm is more than the TM, whereas at z = 2 µm, these are similar). This is also apparent in Visualization 3 [snapshots in Fig. 6—compare Figs. 6(a)–6(c) and 6(d)–6(f)], where the particle seems to undergo a change in shape as we change the QWP—this clearly indicates that the z-depth is changed. The fact that the effects of TM and LSAM are different as the trapping depth is modified is thus apparent in our experiment. We also observe that the frequency of rotation increases as we increase power, which is shown in Fig. 6(ii). This is expected as the magnitudes of both the electric and magnetic field will increase when we increase the input intensity. We demonstrate these events in Visualizations 4–7 in the supplementary material.

We would like to point out that it is not feasible that this rotation occurs due to azimuthally asymmetric scattering from the particle, which may lead to the generation of orbital angular momentum,²⁸ since the particle comes to rest as soon we remove the QWP. Now, the crucial issue is to decide whether the canonical (orbital) momentum P_o is alone responsible for the rotation of particles around the beam axis. This, however, cannot be the case since there is no dependence on input helicity in P_o, so the direction of rotation should be independent of the helicity of input light. Thus, we can conclude that the change in the direction of rotation that we observe due to the change in input helicity is solely due to the effect of the Belinfante spin, which—to the best of our knowledge—makes our experiment the first observation of this erstwhile elusive quantity for propagating light fields.

We observe that the TM or the transverse Poynting vector of a tightly focused Gaussian beam (optical tweezers) is explicitly helicity-dependent, while the TSAM remains helicity independent. This dependence demonstrates spin-momentum locking for tightly focused Gaussian beams. In addition, the TM is zero at the beam center and increases off-axis as we introduce a stratified medium in the path of the beam and increase its RI contrast. Thus, when we use a coverslip with RI 1.814 (mismatched condition) for the sample chamber of the optical tweezers, we observe in our simulations that the transverse extent of both the TM and field intensity increases with the axial distance from the beam focus. As a result, there is sufficient overlap of high intensity, which can facilitate optical confinement, and high TM, which can lead to the observation of rotational effects around the beam axis. We validate this observation experimentally with birefringent particles where we demonstrate both clockwise and anti-clockwise rotation around the beam axis for input RCP and LCP with coverslips of RI 1.814. Interestingly, when we change the helicity in the course of an experiment, we sometimes observe a transition from rotation around the beam axis (in a particular sense) to rotation about the particle axis (in the opposite sense). This implies that the particle predominantly samples TM in one case and LSAM in the other case. This observation is also expected from our simulations, which demonstrate that the relative magnitudes of TM and LSAM are modified at different axial distances from the trapping beam focus, so that as the axial position of the particle is slightly modified in the process of changing helicity using a QWP, either the TM or the LSAM dominate. The rotation around the beam axis is the clear manifestation of Belinfante spin, which is solely responsible for the spin-dependence of the Poynting vector, the canonical component being spin-independent. We believe this to be a simple but robust way to observe the effects of Belinfante spin, which have often proved to be rather elusive to experimental observation. It would also be interesting to investigate whether we could obtain spin and orbital rotation in micromotors, which often possess form birefringence due to their asymmetric structures²⁹ using a simple Gaussian beam. Indeed, the asymmetric structure of the micromotors leads to azimuthally asymmetric scattering, which renders the problem more interesting and worthwhile for future research. Other than this, we are also in the process of extending our studies to the tight focusing of higher-order Gaussian beams (Hermite–Gaussian) including those carrying intrinsic orbital angular momentum (Laguerre–Gaussian) to determine more interesting and intriguing effects of the interaction between spin and orbital angular momentum in the presence of a stratified medium.

See the supplementary material for supporting content.

The experimental data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors acknowledge the SERB, Department of Science and Technology, Government of India (Project No. EMR/2017/001456), and IISER Kolkata, Mohanpur, India, for research funds. They also acknowledge Mr. Sourav Islam of IISER Kolkata for help in the experimental measurements and Dr. Georgios Vasilakis of the Foundation for Research and Technology-Hellas, Greece, for critical comments.