# Observation of Larmor-like precession of a single birefringent particle due
to spin-dependent forces in tilted optical tweezers

Sauvik Roy<EMAIL_ADDRESS>Department of Physical Sciences, IISER-
Kolkata, Mohanpur 741246, India  Nirmalya Ghosh<EMAIL_ADDRESS>Department of Physical Sciences, IISER-Kolkata, Mohanpur 741246, India  Ayan
Banerjee<EMAIL_ADDRESS>Department of Physical Sciences, IISER-Kolkata,
Mohanpur 741246, India  Subhasish Dutta Gupta<EMAIL_ADDRESS>Department of Physical Sciences, IISER-Kolkata, Mohanpur 741246, India  Tata
Institute of Fundamental Research, Hyderabad, Telangana 500046, India
Department of Physics, Indian Institute of Technology, Jodhpur 342030, India

###### Abstract

We observe clear precessional motion of highly birefringent liquid crystal
(LC) particles trapped in a spherically aberrated optical trap which is built
around a tilted refractive index stratified medium. For input circularly
polarized light, the breaking of azimuthal symmetry induced by the tilt leads
to an asymmetric intensity distribution in the radial direction near the trap
focal plane, which - in combination with the spin-orbit conversion effects for
input circularly polarized light - results in non-uniform canonical and spin
momentum densities in those regions. In addition, while the canonical momentum
remains always oriented towards the axial direction, the spin momentum
reverses direction along spatial loops in the radial direction. As a
consequence, the total momentum precesses around the canonical momentum vector
along elliptical spatial loops - akin to a Larmor-like precession of magnetic
moment (total momentum in our case) around a magnetic field (canonical
momentum). We probe this precession experimentally using the single trapped LC
particles - with the direction of precession determined by the helicity of the
input light and the precession frequency varying linearly with the laser
power. Our experimental results are validated by numerical simulations of the
system where we employ the Debye-Wolf theory for tight focusing in the
presence of a tilted stratified media.

††preprint: APS/123-QED

Introduction.- Momentum is a crucial but curious protagonist in the realm of
electromagnetic fields [1]. Recently, the total momentum $(\bm{p})$ has been
revealed to be composed of two components: canonical momentum $(\bm{p^{o}})$
[2, 3] and spin momentum $(\bm{p^{s}})$, each with physically distinct origins
and manifestations. This characterization has proved rather efficacious in
elucidating fundamental photonic interactions in terms of the associated
linear momentum, spin angular momentum (SAM) [4, 5], orbital angular momentum
(OAM) [6, 7], and their mutual inter-conversions - commonly referred to as
spin-orbit interaction (SOI) [8, 9, 10, 11, 12, 13] of light. Typical
manifestations of SOI can be seen in simple optical elements such as planar
interfaces, lenses, and also in small particles [14, 15] as well as complex
structures e.g., photonic crystals, metamaterials, plasmonic structures [16,
17, 18], etc. The in-plane and out-of-plane spatial Goos–Hänchen (GH) and
Imbert–Fedorov (IF) [14] shifts respectively, of a paraxial beam undergoing
reflection or refraction at planar interface and the spin Hall effects in
scattering and focusing [8, 14] are well studied SOI related phenomena.
Conversion from spin to intrinsic orbital angular momentum (IOAM) in an
epsilon-near-zero (ENZ) [19] slab has been utilized in generating vortex beams
out of a beam with no intrinsic orbital angular momentum (topological charge,
$l=0$). A recent revelation of simultaneous spin-dependent directional guiding
and wave-vector–dependent spin acquisition (spin-direction-spin coupling) in
plasmonic crystals [20] facilitates the observation of the Hall effect and
inverse Hall effect in the far field.

Among these manifestations, certain events are recognized to be influenced by
their inherent symmetry. The formation of spin-dependent vortices during the
processes of focusing and scattering by small particles are found to result
from the disruption of transverse translational symmetry and the violation of
the electromagnetic duality [10] in scattering, respectively. Breaking the
cylindrical symmetry [21] of a non-paraxial field is also seen to be
accompanied by an enhancement of the spin Hall effect. In short, the SOI
effects that govern the photonic response of an optical system are generally
influenced by several factors encompassing both the field parameters and the
material properties of the optical systems involved, and symmetry is one of
the key ingredients in determining the nature of the interactions.

In the context of SOI in optical tweezers [22, 23, 24, 25, 26, 27], beams with
diverse rotationally symmetric intensity or polarization configurations are
tightly focused to modify the field at the focus, and hence the magnitude or
direction of the force experienced by micro-particles positioned in an axis-
symmetric sample chamber with perfectly planar interfaces, can be altered. On
the other hand, the fundamental Gaussian beam has been reported to exhibit
helicity-dependent enhancement of transverse spin angular momentum (TSAM) [28]
at the focal regions in the direction orthogonal to the symmetry-breaking axis
when tightly focused through a tilted refractive index mismatched stratified
medium. So, how the breaking of the axial symmetry by tilting the RI
stratified medium affects the momentum distributions and subsequently modifies
the forces acting on a microparticle is of paramount interest in the realm of
micro-manipulation. In this paper, we demonstrate that a spin-responsive
microparticle placed within a tilted sample chamber exhibits an unexpected
concurrent precessional and partial orbital motion, the direction of the
precession being determined by the helicity of the illuminating beam. The
necessary insight into the underlying cause of the motion is provided by the
spatial distributions of $\bm{p^{o}}$ and $\bm{p^{s}}$. Indeed, these
distributions unveil a distinctive relationship between the two momentum
components. While $\bm{p^{o}}$ primarily aligns along the beam propagation
direction, the total momentum density $\bm{p}$ demonstrates a spatial
precession, moving from one position to another, around the local $\bm{p^{o}}$
\- mimicking the precession of magnetic moment around a magnetic field. Our
work broadens the general understanding about the vectorial nature of
canonical and the spin momentum density in an optical field, and can be
potentially discernible in various other systems [29] as well.

Figure 1: Schematic of the Gaussian beam focusing through a tilted stratified
medium.

Experiment.- Our experimental configuration involves illuminating a four-layer
stratified medium with refractive indices (RI) of $1.516$ (microscope
objective immersion oil), $1.814$ (coverslip), $1.33$ (deionized water with
$5$CB liquid crystal (LC) particles dispersed), and $1.516$ (top glass slide),
respectively, using either a left (LCP) or right (RCP) circularly polarized
Gaussian beam. Notably, the second layer, with a refractive index mismatch
with the first, establishes an unconventional, stratified optical tweezers
setup [30, 31, 32]. A tilt is applied to the coverslip by affixing a home-
built mount below it as shown in Fig. 1. The near-flat shape of the exit
aperture of the objective lens and the necessity of focusing deep inside the
sample chamber puts an experimental constraint on how much tilt can be
applied, which comes out to a maximum possible value of $\sim 2.73^{\circ}$.

The substantial refractive index (RI) contrast within the stratified medium
leads to increased spherical aberration of the focused light. Consequently,
uniform concentric high-intensity rings form, facilitating off-axial trapping
of microparticles [30, 31, 32, 24]. When a tilt is introduced, the regularity
of the rings is disrupted, causing the intensity maxima to shift to one side
of the ring, that is decided by the applied tilt axis. Consequently, particles
are predisposed to being exclusively trapped on one side of the ring. This
phenomenon is evident in Figs. 2(a), (b), where three LC particles are
observed to be exclusively trapped on the left side of the ring. Note that any
attempt to position particles on the right side of the ring by laterally
moving the sample chamber proves unsuccessful.

Figure 2:  Concurrent precession and partial orbital movement of the
designated particle in response to (a) a right circularly polarized (RCP)
Gaussian beam and (b) a left circularly polarized (LCP) Gaussian beam. The
semitransparent yellow ring delineates the off-axis high-intensity ring and
the yellow spheres in each frame resemble the orientation of the liquid
crystal particle. (c) A typical spatial trajectory of the particle when
illuminated by a RCP Gaussian beam. (d) Linear variation of the precessional
frequency with the incident laser power.

The effect of such a distinctive trapping configuration on the dynamics of the
trapped LC particles is the principal objective of our study. Indeed, single
LC liquid crystal particles characterized by significant birefringence, are
found to exhibit clear precession around an imaginary body axis, accompanied
by partial orbital motion. Thus, they actually appear to experience a spin
force that we believe originates from the spin momentum and the corresponding
spin angular momentum components, with the direction of precession and orbital
motion being determined by the helicity of the input light (LCP/RCP beam). No
such motion is observed for incident linear polarization. The motion for input
LCP/RCP is substantiated by visually tracking the Maltese-like cross adhered
to the particle, which appears by the imaging of the particles under cross-
polarization, and is a manifestation of the orientation of the internal
directors of the liquid crystal molecules. Figs. 2(a), (b) displays time-
lapsed images of this complex motion for input LCP and RCP, with the orbital
trajectory shown by the white dashed line. The corresponding videos (Video 1
and 2, for LCP and RCP, respectively) are provided in the Online Supplementary
Information. A typical orbital trajectory appears to be elliptical measuring
$1.9\mu m~{}\times 0.7\mu m$ in the major and minor axes, respectively, for
input RCP (Fig. 2(c)) and $1.6\mu m~{}\times 0.7\mu m$ for input LCP. The
precession rate of the particle appears to be influenced by the power of the
beam, increasing with higher laser power, as shown in Fig. 2(d).

Theoretical formalism. We invoke the pioneering research conducted by Debye
and Wolf [33, 34] for tight focusing of a well-collimated beam to understand
the aforementioned complex motion. In this formalism, the refracted spherical
wave from an aplanatic lens is portrayed as a vectorial superposition of plane
waves with infinite spatial harmonics. The intricate transformations of the
complex electromagnetic field within the high numerical aperture (NA) lens are
effectively elucidated by the lens’s transfer function. This transfer function
typically incorporates Fresnel’s reflection and transmission coefficients [31,
35], providing a more realistic representation of experimental configurations
with multiple parallel interfaces perpendicular to the lens axis, delineating
different media within the lens’s image space. The efficacy of the transfer
function mechanism is evident when the stratification axis aligns with the
lens axis. In this scenario, despite multiple scattering events within the
stratified medium, ray optics dictates that all constituent plane waves
consistently stay within their respective meridional planes. However,
deliberate tilting of the stratified medium relative to the lens axis
introduces a change in the plane of incidence at the initial interface [28].
This alteration in the plane of incidence, affecting the subsequent
superposition of constituent plane waves, leads to a redistribution of
electromagnetic fields in the regions beyond the lens.

Due to the non-paraxial nature of the problem, introducing the tilt-dependent
change in the angle of incidence into a new transfer function proves to be
non-trivial. Notably, as this formalism essentially involves a vectorial sum
of complex fields, the problem can be conveniently decoupled into two distinct
components: focusing and the independent propagation of plane waves through
the stratified medium [36]. This separation allows for the continuous tracking
of the spin or polarization evolution during the propagation of light through
the medium. Furthermore, it opens avenues for leveraging the full potential of
modern computational advancements. According to this formalism, the time-
independent monochromatic electric field at a point $p$ $(\bm{r}_{p})$ in a
homogeneous medium of RI $n_{0}$ is

${\bf
E}(\bm{r}_{p})=-i\beta\iint_{\Omega}\frac{{\bm{\epsilon}}(s_{x},s_{y})}{s_{z}}e^{in_{0}k_{0}({\bf
s}.\bm{r}_{p})}\,ds_{x}\,ds_{y}$ (1)

Here, $\beta=\frac{n_{0}k_{0}f}{2\pi}$, ${\bm{\epsilon}}(s_{x},s_{y})$ is the
electric field spectra on the reference Gaussian sphere, ${\bf
s}=(s_{x},s_{y},s_{z})$ is typically the direction of an incoming ray in the
geometric description, $k_{0}$ is the free space wave number, $\Omega$ is the
solid angle formed by the geometric rays converging from the lens. For
brevity, only the transformation of the electric field components is discussed
in this paper. The temporal factor is assumed to have the form
$\text{exp}(-i\omega t)$.

As mentioned earlier, the transfer function that connects the incident
amplitude $\bm{\epsilon}^{in}$ to the refracted field amplitude
$\bm{\epsilon}(s_{x},s_{y})$ involves a systematic arrangement of three
SO($3$) rotation matrices which corresponds to the decomposition of the
incident field into TE and TM components via a coordinate system rotation. The
explicit form of the transfer function is

$A=R_{z}(-\phi)R_{y}(\theta)R_{z}(\phi).$ (2)

Where, $R_{z}(\phi)$ and $R_{y}(\theta)$ are rotation matrices along $z$ and
$y$ axis respectively. Following the Jones vector, the resultant field
amplitudes on the reference sphere read as

${\bm{\epsilon}}(s_{x},s_{y})=\sqrt{\cos{\theta}}A{\bm{\epsilon}}^{in}.$ (3)
${\bm{\epsilon}}(s_{x},s_{y})=\begin{bmatrix}a-b\cos{2\phi}&-b\sin{2\phi}\\\
-b\sin{2\phi}&a+b\cos{2\phi}\\\
-c\cos{\phi}&-c\sin{\phi}\end{bmatrix}\begin{bmatrix}{\epsilon}^{in}_{x}\\\
{\epsilon}^{in}_{y},\end{bmatrix}$ (4)

where, $\bm{\epsilon}^{in}=[{\epsilon}^{in}_{x},{\epsilon}^{in}_{y}]^{T}$
corresponds to the Jones vector of the incident polarization state.

Figure 3: (a) Irregularity in the simulated off-axial intensity ring at a
transverse plane very close to the last interface ($z=28.75\mu m$). The
stratified medium is tilted by an angle of $2.73^{\circ}$. (b) and (c) are the
2D vector plots of the transverse ($x$ and $y$) components of Belinfante’s
spin momentum $(\bm{p^{s}})$ and canonical momentum $(\bm{p^{o}})$
respectively for the incident RCP beam. (d) The helicity independent
$\bm{p^{o}}$ can be seen predominantly oriented along the beam propagation
direction with minimal transverse components due to tilting and divergence.
(e) One proposed closed trajectory of nearly elliptical shape is highlighted
in the XY view of the 3D vector plot of $\bm{p^{o}}$ and total momentum
$(\bm{p})$. (f) and (g) contain the 3D orientations of the $\bm{p^{o}}$ and
total momentum $(\bm{p})$ for the LCP and RCP illuminations respectively. The
vectors along the proposed trajectory in (e) are scaled up by a factor of 15
for enhanced visualization of the point-to-point precession of $\bm{p}$ around
$\bm{p^{o}}$.

The effect of the stratified medium can be incorporated through an independent
decomposition of each plane wave coming from the lens into TE and TM modes
[36] with the help of another transfer matrix for the stratified medium at the
observation plane transverse to the stratification axis $\bm{\hat{\xi}}$.
Owing to the linearity of the system, the total electric field at a point
$(x,y,z)$ can be found as a superposition of forward and backward propagating
fields

${\bf E}^{\pm}(x,y,z)=-i\beta\iint_{\Omega}[a^{\pm}_{TE}(z){\bf
u}^{\pm}_{TE}+a^{\pm}_{TM}(z){\bf u}^{\pm}_{TM}]\\\
e^{in_{0}k_{0}(s_{x}x+s_{y}y)}\,ds_{x}\,ds_{y}$ (5)

Here, $+$ and $-$ signs are used to denote forward and backward propagating
waves respectively, ${\bf{a}}$’s are the amplitudes, and $\bf{u}$’s are the
unit vectors along TE and TM components.

By virtue of the formulation of the Debye-Wolf integral, the electric field is
obtained in the laboratory frame, say $L_{1}$, whose origin coincides with the
geometric focal point $O_{1}$ of the lens. The transverse axes are defined by
the polarization of the incident beam. But in the case of a tilted stratified
medium, instead of solving the integral in $L_{1}$ frame, it is convenient to
solve in a frame $L_{2}$ which is rotated by an angle $\theta_{tilt}$ and
whose origin coincides with the focal point $O_{1}$. At this step, all the
vector quantities $\bf{E}$, $\bm{s}$, and $\bm{r}_{p}(x,y,z)$ are transformed
by the $3\times 3$ rotation matrix $\bf{T}$ which takes care of both the axis
and the tilting angle $\theta_{tilt}$. In the rotated frame, the incident
field strength takes the form
$\textbf{E}^{{}^{\prime}}_{L_{2}}=\textbf{T}\textbf{E}$. Next, the matrix
formalism gives us the required field $\textbf{E}^{{}^{\prime}}$ at the
required plane in the rotated frame. The electric field in the laboratory
frame $L_{1}$ is obtained by the inverse transformation
$\textbf{E}_{final}=\textbf{T}^{-1}\textbf{E}^{{}^{\prime}}$. The rotation
matrix T and its inverse $\textbf{T}^{-1}$ forms the similarity transformation
mechanism that facilitates the modelling of the tilted stratified medium via
coordinate rotation and consequently simplifies the process of finding the
field components in the laboratory frame $L_{1}$.

Having obtained the electric and magnetic fields in a nonmagnetic medium
(i.e., $\mu=\mu_{0}$) with refractive index $n$, the canonical $(\bm{p^{o}})$
and spin momentum $(\bm{p^{s}})$ densities are found according to the explicit
forms[2, 37]:

$\displaystyle\bm{p^{o}}=\frac{1}{4\omega}\text{Im}[\varepsilon_{0}n^{2}\textbf{E}^{*}.(\bm{\nabla})\textbf{E}+\mu_{0}\textbf{H}^{*}.(\bm{\nabla})\textbf{H}]$
(6) $\displaystyle\bm{p^{s}}=\frac{1}{2}\bm{\nabla}\times\textbf{s}$ (7)

Where, the spin angular momentum (SAM) density s is given by:

$\textbf{s}=\frac{1}{4\omega}\text{Im}(\varepsilon_{0}n^{2}\textbf{E}^{*}\times\textbf{E}+\mu_{0}\textbf{H}^{*}\times\textbf{H})$
(8)

where, $\varepsilon_{0}$ and $\mu_{0}$ are the permittivity and permeability
of free space. The intrinsic dispersion of the medium has been taken care of
by the permittivity $(\epsilon=\varepsilon_{0}n^{2})$ and permeability $(\mu)$
of the medium.

Numerical simulations. We now attempt to understand this complex rotational
trajectory by simulating our experimental system. We thus develop a MATLAB
code that incorporates the above-mentioned Debye-Wolf method [28] with
parameters from the real experimental setup. The interface positions are set
at $-111\mu m$, $19\mu m$, and $29\mu m$ from the geometric focus of the lens.
The same experimental tilt angle of $2.73^{\circ}$ is set in the simulation.
Because of the tilt, a nonuniform distribution of the intensity across the
rings accompanying an overall focal shift can be observed in the simulated
intensity pattern at a transverse plane close to the last interface (Fig.
3(a)). To visually depict the motion, two 2D quiver plots showcasing the
transverse components ($x$ and $y$) of the spin momentum ($\bm{p^{s}}$) also
called the Belinfante’s spin momentum (BSM) and canonical momentum
$\bm{p^{o}}$ are generated (Figs. 3(b), (c)). Unfortunately, these plots do
not provide a clear understanding of the motion. Therefore, a 3D quiver plot
of canonical momentum $\bm{p^{o}}$ (Fig. 3(d)) is generated subsequently,
revealing that $\bm{p^{o}}$ is predominantly oriented along the stratification
axis $\bm{\hat{\xi}}$ with minimal $x$ and $y$ components along the transverse
directions. However, a distinct distribution characterizes the spin momentum
$\bm{p^{s}}$, where its $z$-component undergoes no change in sign on the two
sides of a high-intensity ring, but the $x$ and $y$ components exhibit
helicity dependence, as depicted in Fig. 3(b). Clearly, the understanding of
the reason for the precessional motion is contingent upon the nature of the
distribution of $\bm{p^{o}}$ and $\bm{p^{s}}$ \- a matter that will become
apparent subsequently.

Due to the elliptical nature of the trajectory and the input power dependency
of the precession rate - in the first approximation - the rate of change of
total momentum can be equated with $\bm{p}\times\bm{p^{o}}$ through a relation
of the form $\frac{d\bm{p}}{dt}\propto\bm{p}\times\bm{p^{o}}$. Quite
interestingly, this is nothing but the well-known Bloch equation. The
canonical momentum $\bm{p^{o}}$ plays the role of the magnetic field and the
total momentum $\bm{p}$ is the vector that precesses around $\bm{p^{o}}$. The
correspondence between $\bm{p^{o}}$ and the external magnetic field is further
emphasized by the fact that the spin momentum $\bm{p^{s}}$ represents a
solenoidal current $\bm{\nabla}.\bm{p^{s}}=0,\int\bm{p^{s}}\,d\text{v}=0$ and
does not contribute to energy transport. Consequently, the linear correlation
between the precession frequency and the external magnetic field, along with
the linear relationship between precession frequency and laser power or
$\bm{p^{o}}$, substantiates the soundness of our analogy. It is to be noted
that the transverse Belinfante’s spin momentum crucially contributes to the
precession of the total $\bm{p}$ around $\bm{p^{o}}$. The point-to-point
variation of $\bm{p}$ and $\bm{p^{o}}$ for both LCP and RCP along one possible
trajectory of the particle is shown in Figs. 3(e), (f), (g). As the particle
moves point by point, it is influenced by the local $\bm{p}$ which dictates
the closed path, and the Maltese-like cross on the particle can be seen
precessing due to the total momentum $\bm{p}$ while remaining trapped in the
same ring. The proportionality constant $\alpha$ resembles the gyromagnetic
ratio of the trapped lc particle.

For an intuitive grasp of the extent of the trajectory, we present the 3D
quivers of $\bm{p^{o}}$ and $\bm{p^{s}}$ in Figs. 3(e), (f), (g). It is
readily discernible that, in a region indicated by the black arrow in Fig.
3(e), the total momentum density $\bm{p}$ undergoes spatial precession around
$\bm{p^{o}}$. Notably, the blue arrows denoting $\bm{p^{o}}$ and the red
arrows denoting $\bm{p}$ along the trajectory are magnified by a factor of
$15$ for enhanced visualization. The simulated trajectory’s dimensions are
measured to be $\sim 0.56\mu m$ in length and $\sim 1.67\mu m$ in breadth,
aligning well with the experimentally observed dimensions (length $\sim 0.7\mu
m$ and breadth $\sim 1.9\mu m$). In addition, the position of the trajectory
in the simulated plot and the experimentally observed trajectory align with
each other. The point-to-point variation of $\bm{p}$ around $\bm{p^{o}}$, the
trajectory extent, and the input power dependency of the precession rate
collectively demonstrate the precession of $\bm{p}$ around $\bm{p^{o}}$ in a
tightly focused tilted stratified medium in optical tweezers.

Conclusion.- In conclusion, we observe an intriguing precessional motion of a
highly birefringent particle in optical tweezers developed around a tilted
stratified medium. Due to the breaking of the azimuthal symmetry in the
problem, the spin momentum becomes radially inhomogeneous. This, coupled with
the fact that it reverses direction along spatial lobes in the radial
direction while the canonical momentum is always aligned axially, leads to a
resultant precession of the total momentum about the canonical momentum. The
direction of precession is controlled by the helicity of the input light beam,
with the rate depending on the laser power. Indeed, the association of
canonical momentum with the magnetic field and the helicity-dependent total
momentum with magnetic moment leads to an optical analogy of a spinning
particle in an external uniform magnetic field, represented by the ubiquitous
Bloch equation. Here, however, the equivalents of both the magnetic moment and
the external magnetic field are integral components of light itself, thus
revealing another exotic manner in which it manifests itself in the mesoscopic
world. Indeed, the use of more complex input beam profiles in such tilted
optical tweezers could yield even more interesting results, and generate
diverse and unexpected avenues in particle manipulation.

###### Acknowledgements.

Sauvik Roy is thankful to the Department of Science and Technology (DST),
Government of India for the INSPIRE fellowship.

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