# Precise in situ radius measurement of individual optically trapped
microspheres using negative optical torque exerted by focused vortex beams

Kainã Diniz<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
do Rio de Janeiro
Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil  CENABIO
- Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 21941-902, Brazil  Tanja
Schoger  Institut für Physik, Universität Augsburg
86135 Augsburg, Germany  Guilherme T. Moura  Arthur L. Fonseca  Instituto de
Física, Universidade Federal do Rio de Janeiro
Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil  CENABIO
- Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 21941-902, Brazil  Diney S.
Ether Jr  Instituto de Física, Universidade Federal do Rio de Janeiro
Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil  CENABIO
- Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 21941-902, Brazil  Rafael S.
Dutra  LISComp-IFRJ, Instituto Federal de Educação, Ciência e Tecnologia, Rua
Sebastião de Lacerda, Paracambi, Rio de Janeiro, 26600-000, Brasil  Gert-
Ludwig Ingold  Institut für Physik, Universität Augsburg
86135 Augsburg, Germany  Nathan B. Viana<EMAIL_ADDRESS>Instituto de
Física, Universidade Federal do Rio de Janeiro
Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil  CENABIO
- Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 21941-902, Brazil  Paulo A.
Maia Neto<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro
Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil  CENABIO
- Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 21941-902, Brazil

###### Abstract

We demonstrate a new method for determining the radius of micron-sized
particles trapped by a vortex laser beam. The technique is based on measuring
the rotation experienced by the center of mass of a microsphere that is
laterally displaced by a Stokes drag force to an off-axis equilibrium
position. The rotation results from an optical torque pointing along the
direction opposite to the vortex beam angular momentum. We fit the rotation
angle data for different Laguerre-Gaussian modes taking the radius as a
fitting parameter in the Mie-Debye theory of optical tweezers. We also discuss
how micron-sized beads can be used as probes for optical aberrations
introduced by the experimental setup.

optical tweezers, vortex beam, negative optical torque, radius measurements,
vorticity

††preprint: APS/123-QED

##  I Introduction

The negative optical torque is a remarkable example of a nontrivial exchange
of angular momentum between light and matter. It arises from the generation of
scattered light carrying an excess angular momentum, thus leading to a recoil
torque along the direction opposite to the angular momentum of the incident
light. Recent proposals [1, 2] and experimental demonstrations [3, 4, 5, 6, 7,
8, 9] cover a wide spectrum of systems.

Negative torque experiments usually employ the spin angular momentum
associated with circular polarization. In this paper, we use instead the
orbital angular momentum of vortex beams which are used to trap a dielectric
microsphere in our otherwise typical optical tweezers setup, shown in Fig. 1.
Our experimental conditions are such that the laser beam annular focal spot is
comparable to or smaller than the microsphere radius, thus leading to a stable
trap along the beam axis, in contrast to the non-equilibrium steady state of
orbital motion measured with smaller beads [10, 11]. We then apply a lateral
Stokes force and displace the microsphere to a new off-axis equilibrium
position as illustrated by Fig. 1(a). The optical torque on the microsphere
center-of-mass leads to a rotation of the equilibrium position with respect to
the direction of the applied force, which we measure for several values of the
vortex beam topological charge $\ell.$ In most cases, the rotation is opposite
to the handedness defined by the sign of $\ell$.

For small values of the Stokes force, the rotation angle is a measure of the
vorticity of the optical force field near the beam axis. The vorticity is very
sensitive to parameters describing the trapped microsphere as well as the
focused laser beam. For instance, the chirality of a spherical bead can, in
principle, be characterized by measuring the rotation angle [12, 13]. Here, we
determine the microsphere radius with nanometric precision as we benefit from
the enhanced torque exerted by vortex beams. Indeed, these modes carry an
orbital angular momentum $\ell\hbar$ per photon that originates from the
field’s spatial variation [14], while the spin angular momentum is limited to
$\pm\hbar$ per photon.

The radius of an airborne optically trapped microsphere near a surface was
measured by analyzing its Brownian fluctuations [15]. Precise measurements of
airborne microspheres are usually based on Mie resonances, taking advantage of
the high quality factor of whispering gallery modes [16, 17]. However,
colloidal particles in a water suspension typically correspond to low
refractive index contrasts, thus leading to broader resonances. Dynamic light
scattering is a common approach for such systems [18]. Nevertheless, since it
relies on measuring intensity fluctuations of light scattered by a sample
containing many Brownian particles, this technique does not measure the radius
of single individual particles. Videomicroscopy is a simple alternative, but
inferring the size from the light intensity pattern is usually ambiguous,
particularly for an optically trapped particle. Indeed, the microsphere image
depends strongly on its axial position with respect to the focal plane [19],
which typically fluctuates by Brownian motion.

Our approach provides a precise in situ radius measurement of a specific
microsphere which is optically trapped. A comparable precision would be
obtained by employing electron microscopy. However, in this case one would
have the inconveniences and risks of isolating the chosen bead, transporting
and drying the solution, and coating the sample.

In addition to measuring properties of the trapped particle, our method also
allows us to characterize the astigmatism of the trapping beam. Indeed, there
exists a second mechanism contributing to the rotation of the equilibrium
position. When the trapping beam is such as to break rotational symmetry
around the propagation axis, the gradient of the electric energy density is
not radial and develops an azimuthal component. For example, if the paraxial
beam’s polarization is linear, the nonparaxial focal spot will be elongated
along the polarization axis [20], resulting in an azimuthal component of the
optical force [6]. Optical aberrations that define a preferential direction,
such as astigmatism, can also be responsible for producing a non-symmetric
focal spot [21]. For that reason, the comparison with the experimental data
for circularly polarized focused vortex beams allows us to measure the
astigmatism parameters of our setup. In order to implement such comparison, we
develop a theoretical model for the optical force exerted by an astigmatic
focused vortex beam.

The paper is organized as follows: Sec. II presents the experimental setup and
procedure, as well as how the rotation angle is extracted from the raw data.
Sec. III presents the theoretical model developed to describe the experiments.
The results are discussed in Sec. IV and concluding remarks are presented in
Sec. V. Technical details are presented in two appendices.

Figure 1: (a) A trapped particle is subject to a constant drag force
$\mathbf{F}_{S}$ and finds a new off-axis equilibrium position rotated with
respect to the direction of this force by an angle $\alpha$.
$\mathbf{F}_{\rho}$ and $\mathbf{F}_{\phi}$ denote cylindrical optical force
components, and $\mathbf{v}_{S}$ represents the fluid velocity. (b) Lateral
view of the sample region showing the density plot (red) of the electric
energy density for the nonparaxial focused beam. $L$ denotes the height of the
paraxial focal plane with respect to the coverslip. PI represents the
piezoelectric nano-positioning system used to move the microscope stage. (c)
Schematic representation of the experimental setup: The laser beam propagates
through a polarizing beam splitter (PBS) towards the spatial light modulator
(SLM). The resulting first order of diffraction crosses a quarter-wave plate
(QWP) before entering the microscope (dotted frame). The beam is then focused
by an oil-immersion objective into the sample chamber.

##  II Experimental setup and procedure

Our method for radius measurements goes as follows. We apply a constant Stokes
drag force $\mathbf{F}_{S}$ on a sphere trapped by optical tweezers with a
given Laguerre-Gaussian mode LG${}_{p\ell},$ with radial order $p=0$ and
topological charge $\ell,$ at the objective entrance port. The particle is
displaced until it finds an off-axis equilibrium position rotated by an angle
$\alpha$ with respect to the direction of the drag force, as depicted in Figs.
1(a) and (b). Then, we vary the topological charge, measuring rotation angles
as a discrete function of $\ell$. Finally, we fit this data to the theoretical
model presented in the next section, leaving the radius as a free parameter
and, in some cases, also the astigmatism parameters and the paraxial focal
height.

The experimental setup for the measurements of $\alpha$ is illustrated in Fig.
1(c). A $\rm TEM_{00}$ laser beam (IPG photonics, model YLR-5-1064LP) with
vacuum wavelength $\lambda_{0}=1064\,\mathrm{nm}$ illuminates a spatial light
modulator (SLM). This device enables the modulation of both the field’s
amplitude and phase [22], which allows us to convert the incident beam into
chosen $\text{LG}_{0\ell}$ modes with appropriate waist $w_{0}$ via a complex
phase modulation. Notice that we choose a different beam waist for each mode
so as to ensure an appropriate filling at the objective’s entrance. The
criteria for choosing the waists and the method to measure them are described
in Appendix A, while the waist values are given in Table 5. Before entering an
inverted microscope, the beam is left-polarized by a quarter-wave plate. It is
then focused by a $100\times$, oil-immersion objective with numerical aperture
$\rm NA=1.4$ and back aperture radius $R_{\rm obj}=2.8\,\rm mm$ through the
glass-water interface into an aqueous dispersion of polystyrene beads
(Polysciences), where the optical trapping occurs (see Fig. 1(b)). This
solution is illuminated by a 470 nm blue LED whose scattered light, after
being collected by the objective, is recollimated by a tube lens ($L_{t}$) to
form an image on a CMOS camera (Hamamatsu Orca-Flash 2.8 C11440-10C).

After trapping a microsphere with a focused Gaussian beam ($\ell=0$), we first
lower the objective until the microsphere touches the bottom of the sample
chamber. Then, we move up the objective by a height of $d=(2\pm
1)\,\upmu\mathrm{m}$ so as to define a new trapping position a few microns
above the coverslip. To apply a constant drag force, we use a piezoelectric
nano-positioning stage (Digital Piezo Controller E-710, Physik Instrumente) to
move the microscope stage with the velocity ${\bf v}_{S}=\pm
v_{S}\,\mathbf{\hat{x}},$ $v_{S}=20\,\upmu{\rm m/s},$ for 0.5 s, alternating
between the positive and negative $x$ directions. At each cycle, the Stokes
drag force displaces the particle from its on-axis equilibrium position. In a
given run, the bead’s movement is recorded for 10 s. The center-of-mass
position on the $xy$-plane at each frame is later determined by video analysis
with Fiji [23], as depicted by way of example for the $x$ coordinate in the
upper panel of Fig. 2.

The camera itself is rotated by an angle close to $45^{\circ}$ to ensure that
the displacements along the $x$ and $y$ directions are roughly of the same
magnitude. The central panel in Fig. 2 depicts all particle positions obtained
from the measured $x$ coordinates shown in the upper panel as well as from the
corresponding $y$ coordinates (not shown) for a typical run. All recorded
positions are categorized into three separate clusters by means of a Scikit-
learn routine [24] and marked as blue, green, and orange points in Fig. 2. In
this way, the orange points that lie in between the two main clusters are
identified and no longer considered for the analysis.

The marginal distributions for the $x$ and $y$ directions can be fairly
described by Gaussian distributions so that we fit the two main clusters to
two-dimensional Gaussians in order to determine the coordinates of the two
equilibrium positions. We then obtain the angle depicted in Fig. 2 between the
line connecting the two equilibrium positions and the $x$ direction. Its error
is computed by propagating the error of the coordinates. Finally, to determine
the rotation angle $\alpha$ depicted in Fig. 1(a), we subtract the camera
offset angle, which is determined by recording the position of a fixed
reference spot in the coverslip as the microscope stage is driven.

Apart from the rotation angle, we also extract the radial displacement
$\rho_{\rm eq}$ of the sphere from the focal point. By assuming that the
movement of the bead is symmetric, we can compute the displacement from the
distance between the two equilibrium positions.

For each beam mode $\ell$ the measurement was repeated several times.

Figure 2: The $x$ and $y$ coordinates of the bead are measured over a time of
$T=10\,\mathrm{s}$. The upper panel shows a typical distribution of the
measured $x$ coordinates. Green and blue clusters correspond to opposite
velocities of the microscope stage. The central panel shows the measured
positions on the $xy$ plane. The histograms for the $x$ and $y$ coordinates
are fitted by Gaussian functions in order to determine the equilibrium
positions under the Stokes force (red points). The rotation angle $\alpha$ is
then obtained from the slope of the red line connecting the two equilibrium
positions.

##  III MDSA+ theory of the optical force by a focused vortex beam

The optical force acting on an illuminated particle is generally computed by
integrating the time-averaged Maxwell stress tensor over a surface which
encloses the particle. The stress tensor includes the incident and scattered
field contributions. Within the Mie-Debye theory [25, 26], the former is
described by a Debye-type nonparaxial model [20] for a tightly focused beam,
while the latter follows from standard Mie scattering by a spherical particle.
An extension to include the spherical aberration introduced by focusing
through the glass-water planar interface led to the MDSA (Mie-Debye with
spherical aberration) theory [27, 28]. In [29], the model was further extended
(MDSA+ theory) to allow for the presence of any primary aberration on the
paraxial Gaussian beam before focusing.

To mimic the experimental setup discussed in Sec. II, here we consider that
the laser beam at the objective entrance port is a Laguerre-Gaussian mode
LG${}_{0\ell}.$ Expressions for the aplanatic focusing of a Laguerre-Gaussian
beam were developed in [30]. Building on these earlier results, we find the
following angular spectrum decomposition for the electric field in the aqueous
solution obtained from focusing a circularly polarized LG0ℓ beam at the
objective entrance port:

$\displaystyle\mathbf{E}^{(\sigma)}_{\ell}(\mathbf{r})$
$\displaystyle=-\frac{ikfE_{0}e^{-ikf}}{2\pi}\left(\sqrt{2}\gamma\right)^{|\ell|}\int_{0}^{2\pi}d\varphi\,e^{i\ell\varphi}$
(1)
$\displaystyle\quad\times\int_{0}^{\theta_{m}}d\theta\,\sin\theta\sqrt{\cos\theta}\sin^{|\ell|}(\theta)e^{-\gamma^{2}\sin^{2}\theta}$
$\displaystyle\hskip 40.00006pt\times
T(\theta)e^{i(\Psi_{\text{g-w}}+\Psi_{\text{ast}})}e^{i\mathbf{k}_{\text{w}}\cdot\mathbf{r}}\hat{\epsilon}_{\sigma}(\theta_{\text{w}},\varphi_{\text{w}})\,.$

The spherical components $(k,\theta,\varphi)$ describe the wave vectors in the
glass slide. The parameter $\gamma=f/w_{0}$ defines the ratio of the objective
focal length $f$ and the beam waist $w_{0}$ at the objective entrance port.
The polar angle $\theta_{\text{w}}$ in the host medium is defined through
Snell’s law: $\sin\theta_{\text{w}}=\sin\theta/N_{\text{s}}$ where
$N_{\text{s}}=n_{\text{w}}/n_{\text{s}}$ is the relative refractive index
between the fluid and the glass slide. The integration is performed up to a
maximum angle $\theta_{\text{m}}$ given by
$\sin\theta_{\text{m}}=\text{min}\left(N_{\text{s}},\sin\theta_{0}\right)$
with $\sin\theta_{0}=\mathrm{NA}/n_{\text{w}}$, where NA denotes the numerical
aperture of the objective. The wave vector $\mathbf{k}_{\text{w}}$ in the
sample is characterized by its modulus $k_{\text{w}}=N_{\text{s}}k$ and the
spherical angles $(\theta_{\text{w}},\varphi_{\text{w}})$ where
$\varphi_{\text{w}}=\varphi$. The unit vector
$\hat{\epsilon}_{\sigma}(\theta_{\text{w}},\varphi_{\text{w}})=e^{i\sigma\varphi_{\text{w}}}(\hat{\theta}_{\text{w}}+i\sigma\hat{\varphi}_{\text{w}})/\sqrt{2}$
accounts for a right- ($\sigma=-1$) or left-handed ($\sigma=1$) circularly
polarized Fourier component along the propagation direction
$(\theta_{\text{w}},\varphi_{\text{w}}).$ It is obtained by rotating the unit
vector $(\hat{x}+i\sigma\hat{y})/\sqrt{2}$ at the entrance of the objective by
the Euler angles $(\varphi_{\text{w}},\theta_{\text{w}},-\varphi_{\text{w}}).$

The astigmatism (ast) introduced by the optical elements of the experimental
setup is accounted for by the Zernike phase [31, 32]

$\Psi_{\text{ast}}=2\pi
A_{\text{ast}}\left(\frac{\sin\theta}{\sin\theta_{0}}\right)^{2}\cos\left[2(\varphi-\phi_{\text{ast}})\right]$
(2)

given in terms of the amplitude $A_{\text{ast}}$ and of the angle
$\phi_{\text{ast}}$ defining the astigmatism axis and thus breaking rotational
symmetry.

We also take into account the spherical aberration introduced by refraction at
the interface between the coverslip and the aqueous suspension. Neglecting the
dependence of the Fresnel refraction coefficients on the polarization, the
transmission amplitude is [27]

$T(\theta)=\frac{2\cos\theta}{\cos\theta+N_{\text{s}}\cos\theta_{\text{w}}}$
(3)

whereas the spherical aberration phase reads [33]

$\Psi_{\text{g-w}}=kL\left(N_{\text{s}}\cos\theta_{\text{w}}-\frac{\cos\theta}{N_{\text{s}}}\right)$
(4)

where $L$ is the distance between the glass-water (g-w) interface and the
paraxial focus as illustrated in Fig. 1(b).

The incident field (1) is inserted into the Maxwell stress tensor together
with the scattered field, which is obtained by solving the scattering problem
using Mie theory for spherical particles combined with the appropriate Wigner
rotation matrix elements. The integration over the stress tensor can be worked
out analytically. Explicit expressions for the cylindrical force components
($F_{\rho},F_{\phi},F_{z}$) as functions of the position of the sphere center
$\mathbf{R}(\rho,\phi,z)$ with respect to the focus can be found in Appendix
B.

##  IV Results and discussion

###  IV.1 Numerical analysis

The sphere radius $R$ as well as the parameters for the optical aberrations
can be obtained by fitting the experimental data with the rotation angles
obtained from the MDSA+ theory. Two optical aberrations are considered in our
model. The first one is the spherical aberration produced at the glass-water
interface, which is characterized by the phase (4) proportional to the
distance $L$ of the paraxial focus from the coverslip. The second is
astigmatism, defined by (2) in terms of the amplitude $A_{\text{ast}}$ and the
angle of the principal meridian $\phi_{\text{ast}}$. The distance $L$ can, in
principle, be determined by emulating the experimental procedure described in
Sec. II [27]. Considering a Gaussian mode at the entrance of the objective, we
first compute the initial focal height $L_{0}$ for which the on-axis
equilibrium position is such that the sphere is touching the coverslip. The
equilibrium position is obtained from the requirement of a vanishing axial
force component

$F_{z}^{(\ell=0)}\left(\rho=0,\phi=0,z=R-L_{0},L=L_{0}\right)=0\,.$ (5)

Afterwards, the focal plane is displaced upwards by $d$ leading to the final
height $L=L_{0}+N_{s}d$. The rotation angle for each mode $\ell$ can then be
found by solving the following system of equations for the equilibrium
position
$\mathbf{R}_{\text{eq}}=\mathbf{R}_{\text{eq}}(\rho_{\text{eq}},\alpha,z_{\text{eq}})$
under the applied Stokes drag force:

$\displaystyle F_{z}^{(\ell)}(\mathbf{R}_{\text{eq}})$ $\displaystyle=0$ (6)
$\displaystyle\arctan\left(\frac{F_{\phi}^{(\ell)}(\mathbf{R}_{\text{eq}})}{F_{\rho}^{(\ell)}(\mathbf{R}_{\text{eq}})}\right)$
$\displaystyle=\alpha$

with the force expressions given in Appendix B. The rotation angle $\alpha$
corresponds to the azimuth angle $\phi$ of the bead at its equilibrium
position. The radial component $\rho_{\text{eq}}$ is extracted from the
experimental data as described in the previous section. We found that the
rotation angle only weakly depends on $\rho_{\text{eq}}$, meaning that the
changes in the rotation angle with the radial distance are negligible. For
each $\text{LG}_{0\ell}$ mode, we thus average $\rho_{\text{eq}}$ over all
measurement runs for a given bead and use the resulting average value as input
for the radial equilibrium coordinate.

Note that without astigmatism ($A_{\text{ast}}=0$) the force components do not
explicitly depend on the azimuthal angle $\phi$. Hence, the set of equations
(6) no longer needs to be solved simultaneously. Instead, after determining
the axial equilibrium position, we can directly calculate the rotation angle.
The computation time using only the MDSA theory is thus significantly reduced
compared to the full calculation within the MDSA+ theory.

The fitting is done by minimizing a weighted sum of squared errors between the
theoretical and experimental results for the rotation angle

$\chi^{2}=\frac{1}{N}\sum_{\ell}\frac{1}{M_{\ell}}\sum_{r}\left(\frac{\alpha(\ell)-\alpha_{\text{exp}}(\ell,r)}{\Delta\alpha_{\text{exp}}(\ell,r)}\right)^{2}$
(7)

where $\alpha_{\text{exp}}(\ell,r)\pm\Delta\alpha_{\text{exp}}(\ell,r)$ is the
experimentally obtained rotation angle and its error for the beam mode $\ell$
and run $r$, $N$ is the number of modes used for fitting and $M_{\ell}$ is the
number of measurement rounds performed for each beam mode. The maximum number
of runs $M_{\ell}$ was ten, while in a few cases, due to experimental reasons,
only data from one run could be used.

The force calculation was implemented in Python using the scientific libraries
NumPy [34] and SciPy [35]. For the minimization of $\chi^{2}$ we used the
iminuit package [36].

Relevant parameters for the simulations not yet mentioned include the
refractive index of the immersion oil $n_{\text{s}}=1.518$ and the refractive
index of the polystyrene beads $n=1.5694$, which we obtained by linearly
interpolating the data given in [37] to our wavelength of
$\lambda_{0}=1064\,$nm. The laser power, below $120\,{\rm mW}$ at the
objective entrance port, is too low to heat the sample. Therefore, we assume
it to be in thermal equilibrium with its environment and use the refractive
index for water $n_{\text{w}}=1.3246$ at $T=19\,^{\circ}$C in our
calculations. For comparison, we also performed the calculation for
$T=24\,^{\circ}$C with $n_{\text{w}}=1.3242$ and found no significant change
in the fitted parameters. These refractive indices for $\lambda_{0}$ were
obtained from a linear interpolation of the data given in [38].

###  IV.2 Beads of nominal radius 1.5 microns

We begin by discussing the results for two beads whose nominal radius is
$(1.50\pm 0.04)\,\upmu\mathrm{m}$. In the following, we refer to them as beads
A and B. Fig. 3 shows the experimental results for the rotation angles of each
bead for beam modes ranging from $\ell=-4$ to $\ell=4$. The error bars are the
standard deviations of $\alpha$ computed from the multiple rounds of
measurement.

Figure 3: Microsphere rotation angle $\alpha$ versus beam mode $\ell$ for
beads A and B. The circles depict the experimental results and the squares and
triangles are the result of fitting with MDSA and MDSA+ theories,
respectively, with the latter including the effect of astigmatism. In all
plots, we have added lines connecting the data points as a guide to the eye.

For each sphere, we performed a fit, where we excluded the $\ell=4$ case for
which we found no reasonable value for $\alpha$. We believe that the reason
for this is that $\ell=4$ is close to the bistable trapping regime
demonstrated in Ref. [11]. Although the bead does have stability around the
axis, it also probes a region of positive torque near the annular focal spot,
due to its thermal fluctuations. This causes the experimental value of
$\alpha$ to be much less negative than the value of about $-15.7^{\circ}$
predicted by theory.

First, we fitted the experimental curve by just using the MDSA theory with $L$
determined as described above. The fitted radii can be found in Tab. 1 and
$\alpha$ as a function of $\ell$ is depicted in Fig. 3. Overall, the fitted
data points show good qualitative agreement with the experimental result,
except for $\ell=0$. In this case, theory predicts a negative rotation angle,
while the experimentally observed one is positive. We will discuss the reason
for this discrepancy below.

Table 1: Optimal radii for beads of nominal radius $(1.50\pm 0.04)\,\upmu\mathrm{m}$ from a fit within the MDSA theory (zero astigmatism). bead | $R(\upmu\mathrm{m})$ | $\chi^{2}$
---|---|---
A | 1.500 $\pm$ 0.004 | 21.7
B | 1.494 $\pm$ 0.003 | 26.4

Next, we used the MDSA+ theory for fitting with $A_{\text{ast}}$,
$\phi_{\text{ast}}$ and $L$ as additional parameters. The parameters obtained
from the fit can be found in Tab. 2. The fitted data points depicted in Fig. 3
show almost perfect agreement with the experimental results. Notice that the
$\chi^{2}$ value does not indicate overfitting. In addition, in spite of the
large number of parameters, the two independent fits for beads A and B found
astigmatism parameters $A_{\text{ast}}$ and $\phi_{\text{ast}}$ that agree
within error bars. This is to be expected since the astigmatism is a
characteristic of the experimental setup, thus remaining the same for
measurements with both beads. On the other hand, the difference of about 1
$\upmu\mathrm{m}$ between the values of $L$ for the two beads is still
consistent with the large error of adjusting the height of the objective as
mentioned in Sec. II. Following this reasoning, we also performed a joint fit
for beads A and B with shared parameters for the astigmatism. The results can
be found in Tab. 3.

As for the fitted radii, notice that they are close to the ones found by
fitting the data to the MDSA theory. This shows that if one wishes solely to
characterize a bead’s radius, the method can be performed with this simpler
version of the theory, drastically reducing the necessary computation time.
Furthermore, we have performed the MDSA-fit with each individual round and
found values for the radius compatible with the ones in Tab. 1, suggesting
that the method could be greatly simplified.

Table 2: Fit parameters for beads of nominal radius $(1.50\pm 0.04)\,\upmu\mathrm{m}$. bead | $R(\upmu\mathrm{m})$ | $L(\upmu\mathrm{m})$ | $A_{\text{ast}}$ | $\phi_{\text{ast}}\text{(rad)}$ | $\chi^{2}$
---|---|---|---|---|---
A | 1.503 $\pm$ 0.007 | 5.8 $\pm$ 0.6 | 0.245 $\pm$ 0.026 | 0.37 $\pm$ 0.14 | 2.2
B | 1.492 $\pm$ 0.005 | 4.7 $\pm$ 0.6 | 0.262 $\pm$ 0.024 | 0.44 $\pm$ 0.12 | 2.3

It should be noted that only the presence of higher-order modes allows the
distinction between spheres A and B. The measured rotation angles for a
Gaussian mode are $\alpha_{\mathrm{A}}(\ell=0)=(0.7\pm 0.2)^{\circ}$ and
$\alpha_{\mathrm{B}}(\ell=0)=(1.1\pm 0.4)^{\circ}$ respectively for beads A
and B. They are practically indistinguishable within error bars. However, if
we now look at the case $\ell=2$, the corresponding results are
$\alpha_{\mathrm{A}}(\ell=2)=(-6.3\pm 0.1)^{\circ}$ and
$\alpha_{\mathrm{B}}(\ell=2)=(-7.3\pm 0.1)^{\circ}$. Note that in that case
not only are the values further apart, but they are distinguishable beyond
error bars.

Finally, we emphasize that for all modes except $\ell=0$ the angular momentum
gained by the particle is opposite to that carried by the paraxial beam. This
exception happens because for this mode the total angular momentum per photon
is relatively small, and the symmetry-breaking effect of astigmatism dominates
over the angular momentum exchange. This is corroborated by $\ell=-1$, where
the total angular momentum per photon is zero, and the torque, caused
exclusively by the spot-asymmetry, is positive. Therefore, the negative
optical torque reported in [6] extends to $\rm LG_{0\ell}$ modes and seems to
be a characteristic of near-focus interactions, rather than a particularity of
that setup.

Table 3: Parameters from the joint fit for two beads with nominal radius $(1.50\pm 0.04)\,\upmu\mathrm{m}$ using shared values for the astigmatism.  $R_{A}(\upmu\mathrm{m})$ | $R_{B}(\upmu\mathrm{m})$ | $L_{A}(\upmu\mathrm{m})$ | $L_{B}(\upmu\mathrm{m})$ | $A_{\text{ast}}$ | $\phi_{\text{ast}}\text{(rad)}$ | $\chi^{2}_{A,B}$
---|---|---|---|---|---|---
1.502 $\pm$ 0.010 | 1.492 $\pm$ 0.007 | 5.56 $\pm$ 0.83 | 4.85 $\pm$ 0.78 | 0.254 $\pm$ 0.026 | 0.41 $\pm$ 0.14 | 2.4, 2.4

###  IV.3 Beads of nominal radius 2.260 microns

We now discuss the results for two beads whose radii have a nominal value of
$R_{\mathrm{N}}=(2.260\pm 0.075)\,\upmu\mathrm{m}$ and are referred to as
beads C and D in the following. Due to their larger size, these beads were
stably trapped on the optical axis for a broader range of topological charges
[11] when compared to the smaller beads discussed above. The experiments were
performed with $\ell$ ranging from $-7$ to $7$ yielding the data shown in Fig.
4. Beads in the geometrical optics regime $R\gg\lambda$ are less sensitive to
optical aberrations because the asymmetries in the field distribution are
averaged out over the sphere. We thus neglect the astigmatism and use fixed
values for the distance $L$ as determined from the known displacement $d$ of
the objective when fitting the experimental data.

The experimental data for $\ell$ ranging from $-6$ to $6$ were fitted. In the
cases $\ell=\pm 7$, we could not find equilibrium positions for all radii in
the fitting interval ranging from 2.11 to 2.41 $\upmu$m. We believe that the
explanation lies again in the size of the annular focal spot, which is too
large to allow for stable trapping on the beam axis for the highest values of
$|\ell|$. The radii obtained from the fit are given in Tab. 4 and the fitted
data for $\alpha$ are depicted in Fig. 4. For comparison, we also performed
the fit using the MDSA+ theory with the astigmatism parameters obtained from
the joint fit of beads A and B. The resulting radii lay within the error bars
of the radii obtained within the MDSA theory and the quality of the fit shows
no significant improvement. Due to the high relative error of measuring the
displacement $d$, we also performed the fit with $d=3\,\mu\mathrm{m}$ which
again leads to the same optimal radii. Our findings thus confirm that both
astigmatism and spherical aberration are less critical for larger spheres,
which implies that here the fitting process is even more robust than for
smaller spheres.

The sensitivity of our method with respect to changes in the radii can be
inferred from Fig. 4, where the rotation angles are also depicted for the
nominal radius $R_{\mathrm{N}}$. The agreement with the experimental data
becomes much worse in this case and the qualitative aspects of the curve
change, with even an opposite sign for the rotation angles for modes with
topological charges in the range $-3\leq\ell\leq 2$.

Table 4: Optimal radii for beads of nominal radius $(2.260\pm 0.075)\,\upmu\mathrm{m}$. bead | $R(\upmu\mathrm{m})$ | $\chi^{2}$
---|---|---
C | 2.339 $\pm$ 0.003 | 40.1
D | 2.331 $\pm$ 0.003 | 67.7

Fig. 5 displays $\chi^{2}$ as a function of the sphere radius for bead A
(upper panel) and beads C and D (lower panel). Notice that the curves for
beads C and D do not only exhibit a minimum at the optimal fit but also
possess local minima at other values of the radius. The height of the dashed
and dotted line serves as a reference for comparison with $\chi^{2}$ for the
optimal fit. The grey area indicates the error interval around the nominal
radius. The observed pseudo-oscillations originate in the semi-classical
scattering limit from the interference between direct reflection and radial
round-trip propagation inside the sphere over a distance $4R$ [25], and their
period is given by $\lambda_{0}/4n\approx 0.169\,\upmu\mathrm{m}$. The
distance between the first two and the last two minima is found for bead C as
$0.170\,\upmu\mathrm{m}$ and $0.171\,\upmu\mathrm{m}$, respectively, and for
bead D as $0.164\,\upmu\mathrm{m}$ and $0.172\,\upmu\mathrm{m}$, respectively.
These values are close to the theoretically expected period considering the
error bars of the fitted radii.

The ambiguity arising from multiple minima does not have an impact on the
results for bead A, as indicated in the upper panel of Fig. 5. Since its
radius is closer to the wavelength of the trapping light, the interference
effect described above is not as pronounced as in the case of the larger
spheres C and D. Consequently, their fitted radii values correspond to more
clearly defined global minima.

As for beads C and D, the radii given in Table 4 correspond to the global
minima of $\chi^{2}$ and they lie closer to the nominal value than radii
associated with additional local minima of $\chi^{2}$. The minima displayed in
the lower panel of Fig. 5 at radii beyond $2.5\,\upmu\mathrm{m}$ have values
of $\chi^{2}$ comparable to the global minima as can be seen by means of the
horizontal lines. However, these radii differ from the nominal value by more
than $3\sigma$ and therefore should be excluded. Even though the local minima
of $\chi^{2}$ at a radius around $2.17\,\upmu\mathrm{m}$ clearly lie above the
respective global minimum, we need to assess whether the difference between
the two values of $\chi^{2}$ is significant. To this end, we consider the
maximum of $\chi^{2}$ within the error interval of the fitted radii according
to Table 4. For bead C, we find a maximal value of $41.2$ as compared to the
value of $48.1$ at a radius of $2.169\,\upmu\mathrm{m}$. Therefore, the global
minimum is significantly better than the local minimum. A similar analysis can
be carried out for bead D confirming that also in this case the global minimum
is indeed significantly better.

Figure 4: Rotation angles for beads C and D as functions of the beam mode
$\ell$. The circles depict the experimental results. The fitted data from the
MDSA theory is shown in blue. The values at $\ell=\pm 7$ were excluded from
the fit. As in Fig. 3, the lines connecting the data points serve as a visual
aid.  Figure 5: In the bottom panel, $\chi^{2}$ as a function of the radius
$R$ for beads C (circles) and D (squares) is shown within the MDSA theory. The
grey area depicts the error interval for the nominal radius $(2.260\pm
0.075)\,\upmu\mathrm{m}$ provided by the bead manufacturer. The dotted
(dashed) line shows the value of $\chi^{2}$ for bead C (D) found from the fit
and given in Tab. 4. The top panel shows, for comparison, $\chi^{2}$ for bead
A. The grey area represents the error interval $(1.50\pm
0.04)\,\upmu\mathrm{m}$.

##  V Conclusions

In conclusion, we have presented a method for determining the radii of
optically trapped microspheres with errors on the order of a nanometer, well
below the usual manufacturer’s standard error. One of its main advantages is
that it allows to measure a specific spherical particle in situ during
trapping experiments.

Furthermore, we showed that micro-sized particles with $R\approx\lambda$ can
be used to probe the optical aberrations of the experimental setup as well as
to get an estimate for the height of the sphere above the coverslip, which is
an experimental parameter hard to determine with precision.

## Acknowledgments

We are grateful to Bruno Pontes, Cássia S. Nascimento, Cyriaque Genet, Paula
Monteiro and Suzana Frases for fruitful discussions. P. A. M. N. and N. B. V.
acknowledge funding from the Brazilian agencies Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq–Brazil), Coordenação de
Aperfeiçamento de Pessoal de Nível Superior (CAPES–Brazil), Instituto Nacional
de Ciência e Tecnologia Fluidos Complexos (INCT-FCx), and the Research
Foundations of the States of Rio de Janeiro (FAPERJ) and São Paulo (FAPESP).

##  Appendix A Waist and Objective Filling

To ensure good trapping conditions in optical tweezers, one must guarantee
that the waist of the paraxial beam is such that it illuminates the objective
entrance with a significant fraction of its power distributed near the border
of the lenses. In this way the marginal rays are strongly deflected,
optimizing the focused beam’s intensity gradient. For a Gaussian beam, this
means simply overfilling the entrance. However, Laguerre-Gaussian beams
require a more careful choice of waist, since most of its power is localized
far from the optical axis and may leak outside the objective if the beam waist
is too large. In the experiments presented here, we have used two criteria for
the choice of each mode’s waist. One is the theoretical criterion presented in
[11] to control the objective filling by the ratio $r_{\ell}/R_{\rm obj}$,
where $r_{\ell}$ is the radius of maximum intensity of the paraxial $\rm
LG_{0\ell}$ beam. The second criterion was empirical, based on the limitations
of the SLM. As the topological charge increases, the modulation masks
necessary to reduce the beam’s waist become larger, to the point that they can
no longer fit inside the SLM’s display. Hence, one must choose waist values
that sufficiently optimize the optical tweezers’ intensity gradient and, at
the same time, are produced by masks that fit inside the SLM’s display.

To experimentally measure the waist values of each Laguerre-Gaussian mode, we
have employed a variation of the method described in [27]. It consists of
measuring the power through a diaphragm of radius $a$ centered at the beam’s
axis as a function of this aperture, and then fitting the power to the
integrated intensity in the diaphragm’s area. In the case of a Laguerre-
Gaussian mode, we have:

$P_{\ell}(R)=P_{t}\left[\Gamma(|\ell|+1)-\Gamma\left(|\ell|+1,\frac{2a^{2}}{w_{0}^{2}}\right)\right]$
(8)

where $P_{t}=(\pi/2)w_{0}^{2}I_{0}$ is the total beam power in the $\ell=0$
(Gaussian) case. Instead of an actual diaphragm, we have simulated its effect
by using the spatial light modulator to divert all light outside a circle with
controllable radius $R$ centered at the beam axis. The measured waist values
and the ratio $r_{\ell}/R_{\text{obj}}$ for each mode are presented in Table
5.

Table 5: Measured waist for each $\text{LG}_{0\ell}$ mode $\ell$ | $w_{0}\,(\mathrm{mm})$ | $r_{\ell}/R_{\rm obj}$
---|---|---
0 | $2.150\pm 0.004$ | -
$\pm 1$ | $1.732\pm 0.008$ | $0.44$
$\pm 2$ | $1.204\pm 0.004$ | $0.43$
$\pm 3$ | $1.171\pm 0.005$ | $0.51$
$\pm 4$ | $1.039\pm 0.007$ | $0.52$
$\pm 5$ | $0.948\pm 0.003$ | $0.54$
$\pm 6$ | $0.867\pm 0.002$ | $0.54$
$\pm 7$ | $0.806\pm 0.002$ | $0.54$

##  Appendix B Multipole expansion of the optical force

Here, we present the series expansion of the optical force components for a
Laguerre-Gaussian beam within the MDSA+ theory [29].

We use a dimensionless force $\mathbf{Q}=\mathbf{F}/(n_{\text{w}}P/c)$, with
the laser beam power $P$, the speed of light $c$ and the refractive index in
the sample region $n_{\text{w}}$. The force is given by subtracting the loss
at the sphere (-$\mathbf{Q}_{s}$) from the total extinction ($\mathbf{Q}_{e}$)

$\mathbf{Q}(\mathbf{R})=\mathbf{Q}_{s}+\mathbf{Q}_{e}$ (9)

and it is calculated at the position $\mathbf{R}=\mathbf{R}(\rho,\phi,z)$ of
the spherical particle with respect to the focal spot.

First, we present the components of the scattering force. The axial part of
the scattering force is given by

$\displaystyle
Q_{sz}^{(\sigma,\ell)}=-\frac{8\gamma^{2}}{A_{\ell}N_{s}}\mathrm{Re}\sum\limits_{j=1}^{\infty}\sum_{m=-j}^{j}$
$\displaystyle\left[\frac{\sqrt{j(j+2)(j-m+1)(j+m+1)}}{j+1}(a_{j}a_{j+1}^{*}+b_{j}b_{j+1}^{*})G_{j,m}^{(\sigma,\ell)}G_{j+1,m}^{(\sigma,\ell)*}\right.$
(10) $\displaystyle\quad\left.+\sigma
m\frac{2j+1}{j(j+1)}a_{j}b_{j}^{*}\left|G_{j,m}^{(\sigma,\ell)}\right|^{2}\right]$

where $a_{j}$ and $b_{j}$ are the Mie coefficients and $A_{\ell}$ defines the
so-called filling factor given by

$A_{\ell}=8(2\gamma^{2})^{|\ell|+1}\int_{0}^{\sin\theta_{\text{m}}}t^{2|\ell|+1}e^{-2\gamma^{2}t^{2}}\frac{\sqrt{(1-t^{2})(N_{\text{s}}^{2}-t^{2})}}{\left(\sqrt{1-t^{2}}+\sqrt{N_{\text{s}}^{2}-t^{2}}\right)^{2}}\,\mathrm{d}t\,.$
(11)

The transverse components of the scattering force are given by

$\displaystyle\left\\{\begin{array}[]{c}Q_{s\rho}^{(\sigma,\ell)}\\\
Q_{s\phi}^{(\sigma,\ell)}\end{array}\right\\}=\frac{4\gamma^{2}}{A_{\ell}N_{s}}\left\\{\begin{array}[]{c}\mathrm{Im}\\\
-\mathrm{Re}\end{array}\right\\}\sum\limits_{j=1}^{\infty}\sum_{m=-j}^{j}$
$\displaystyle\left[\frac{\sqrt{j(j+2)(j+m+1)(j+m+2)}}{j+1}(a_{j}a_{j+1}^{*}+b_{j}b_{j+1}^{*})\right.$
(12) $\displaystyle\hskip
50.00008pt\times\left(G_{j,m}^{(\sigma,\ell)}G_{j+1,m+1}^{(\sigma,\ell)*}\pm
G_{j,-m}^{(\sigma,\ell)}G_{j+1,-m-1}^{(\sigma,\ell)*}\right)$
$\displaystyle\quad-\left.2\sigma\frac{2j+1}{j(j+1)}\sqrt{(j-m)(j+m+1)}\mathrm{Re}(a_{j}b_{j}^{*})G_{j,m}^{(\sigma,\ell)}G_{j,m+1}^{(\sigma,\ell)*}\right]$

where the upper sign corresponds to the radial force component, while the
lower sign is for the azimuthal part. The multipole coefficients of the
circularly polarized Laguerre-Gaussian beam, including optical aberrations,
are given by

$\displaystyle
G_{j,m}^{(\sigma,\ell)}(\mathbf{R})=(\sqrt{2}\gamma)^{|\ell|}\int_{0}^{\theta_{m}}\mathrm{d}\theta\sqrt{\cos(\theta)}\sin^{|\ell|+1}(\theta)e^{-\gamma^{2}\sin^{2}(\theta)}d_{m,\sigma}^{j}(\theta_{w})f^{(\sigma,\ell)}_{m}(\mathbf{R})e^{ik_{w}\cos(\theta)z+i\Psi_{\text{g-w}}}\,.$
(13)

The coefficient $f_{m}^{(\sigma,\ell)}$ accounts for the astigmatism. The
explicit expression for a Gaussian beam can be found in Eq. (8) of [29] and
for a Laguerre-Gaussian beam we obtained

$\displaystyle f^{(\sigma,\ell)}_{m}(\mathbf{R})=$
$\displaystyle\sum_{s=-\infty}^{\infty}(-i)^{s}J_{s}\left(2\pi
A_{\text{ast}}\frac{\sin^{2}\theta}{\sin^{2}\theta_{0}}\right)$ (14)
$\displaystyle\times
J_{2s+m-\sigma-\ell}(k\rho\sin\theta)e^{2is(\phi_{\text{ast}}-\phi)}\,.$

In the absence of astigmatism $A_{\text{ast}}=0$, only the $s=0$ term
contributes to the sum and the coefficient reduces to

$f^{(\sigma,\ell)}_{m}(\mathbf{R})=J_{m-\sigma-\ell}(k\rho\sin\theta)\,.$ (15)

Next, we present the expressions for the extinction force. The axial part is
given by

$Q_{ez}^{(\sigma,\ell)}=\frac{4\gamma^{2}}{A_{\ell}N_{s}}\mathrm{Re}\sum\limits_{j,m}(2j+1)(a_{j}+b_{j})G_{j,m}^{(\sigma,\ell)}\left(G^{(\sigma,\ell)^{\prime}}_{j,m}\right)^{*}.$
(16)

The coefficient $G^{(\sigma,\ell)^{\prime}}_{j,m}$ can be expressed in terms
of the multipole coefficients (13) by applying the recursion relation for
Wigner-$d$ matrix elements [39, p. 90]

$G^{(\sigma,\ell)^{\prime}}_{j,m}=\frac{\sqrt{j(j+2)\left[(j+1)^{2}-m^{2}\right]}}{(2j+1)(j+1)}G_{j+1,m}^{(\sigma,\ell)}+\frac{\sqrt{(j^{2}-m^{2})(j^{2}-1)}}{j(2j+1)}G_{j-1,m}^{(\sigma,\ell)}+\sigma\frac{m}{j(j+1)}G_{l,m}^{(\sigma,\ell)}\,.$
(17)

The transverse components of $\mathbf{Q}_{\mathrm{e}}$ are given by

$\left\\{\begin{array}[]{c}Q_{e\rho}^{(\sigma,\ell)}\\\
Q_{e\phi}^{(\sigma,\ell)}\end{array}\right\\}=\frac{2\gamma^{2}}{A_{\ell}N_{s}}\left\\{\begin{array}[]{c}\mathrm{Im}\\\
-\mathrm{Re}\end{array}\right\\}\sum\limits_{j,m}(2j+1)(a_{j}+b_{j})G_{j,m}^{(\sigma,\ell)}\left(G^{-,(\sigma,\ell)}_{j,m+1}\mp
G^{+,(\sigma,\ell)}_{j,m-1}\right)^{*}\,.$ (18)

The negative (positive) sign corresponds to the radial (azimuthal) component.
The multipole coefficients $G^{\pm,(\sigma,\ell)}_{j,m-1}$ can also be
expressed in terms of the coefficients $G^{(\sigma,\ell)}_{j,m-1}$ by again
applying recursion relations

$\displaystyle G^{\pm,(\sigma,\ell)}_{j,m}$
$\displaystyle=\mp\frac{\sqrt{(j\pm m)(j\pm
m+1)(j^{2}-1)}}{j(2j+1)}G_{j-1,m}^{(\sigma,\ell)}+\sigma\frac{\sqrt{(j\mp
m)(j\pm m+1)}}{j(j+1)}G_{j,m}^{(\sigma,\ell)}$ (19)
$\displaystyle\qquad\pm\frac{\sqrt{(j\mp m)(j\mp
m+1)((j+1)^{2}-1)}}{(j+1)(2j+1)}G_{j+1,m}^{(\sigma,\ell)}\,.$

Making use of the recursion relations (17) and (19) reduces drastically the
need to explicitly evaluate integrals of the form (13) and thus significantly
improves the runtime of the numerical calculations.

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Hadron structure from basis light-front quantization

Chandan Mondal1,2,$\star$, Jiangshan Lan1,2,3, Kaiyu Fu1,2, Siqi Xu1,2, Zhi
Hu1,2, Xingbo Zhao1,2, James P. Vary4 ^i_a

& +g_s¯ψγ_μT^aA_a^μψ+12g_s^2¯ψγ^+T^aψ1(i∂+)2¯ψγ^+T^aψ},

where $\psi$ and $A^{\mu}$ are the quark and gluon fields, respectively.
Meanwhile, the confinement in the leading Fock sector is given by [3]

$\displaystyle P_{\rm
C}^{-}P^{+}=\kappa^{4}\left\\{x(1-x)\vec{r}_{\perp}^{2}-\frac{\partial_{x}[x(1-x)\partial_{x}]}{(m_{q}+m_{\bar{q}})^{2}}\right\\},$
(1)

with $\kappa$ being the strength of the confinement. With the framework of
BLFQ [1], we solve the Hamiltonian in the leading two Fock sectors. We
truncate the infinite basis space by introducing limits $K$ and $N_{\rm max}$
in longitudinal and transverse directions, respectively. By fitting the
constituent parton masses and coupling constants as the model parameters, we
obtain a good quality description of light meson mass spectroscopy [4]. We
evaluate the pion electromagnetic form factor (EMFF) and the parton
distribution functions (PDFs) from our resulting LFWFs obtained as
eigenfunctions of the Hamiltonian.

Figure 1: The EMFF of the pion. The experimental data can be found in Ref.
[4].

Figure 2: Left: the PDFs of the pion at $\mu^{2}=16$ GeV2. Right: the pion’s
gluon PDF at $\mu^{2}=4$ GeV2 is compared with the global fits, JAM [7] and
xFitter [8].

Figure 3: The quark unpolarized GPDs as functions of $x$ and $b_{\perp}(\equiv
b)$ for the up (left panel) and down quark (right panel) in the proton.

The EMFF of the charged pion is compared to the experimental data in Fig. 1.
We find an impressive agreement between our results and the precise low
$Q^{2}$ EMFF data. We show the pion PDFs after QCD evolution in Fig. 2. Our
prediction for the pion valence quark distribution is found to be consistent
with the reanalyzed FNAL-E-615 data. The gluon distribution significantly
increases in our approach compared to that in the BLFQ-NJL model [5, 6] as
well as to the global fits [7, 8] as can be seen from Fig. 2. We notice that
gluons carry $\\{39.5,\,42.1,\,43.9,\,44.6,\,45.1\\}\%$ of pion momentum at
the scale $\mu^{2}=\\{1.69,\,4,\,10,\,16,\,27\\}$ GeV2, respectively.

##  1 Nucleon

We solve for the mass eigenstates for the nucleon from a light-front effective
Hamiltonian in the leading Fock sector representation, consisting of a 3D
confinement (pairwise), Eq. (1), and a one-gluon exchange interaction with
fixed coupling. [9, 10]. The parameters in the effective interaction are fixed
by fitting the nucleon mass and the flavor Dirac form factors. We then employ
the resulting LFWFs to calculate the EMFFs, PDFs, GPDs, TMDs and many other
observables of the nucleon [9, 10, 11]. The unpolarized GPDs for zero
skewness, $H(x,b_{\perp})$ and $E(x,b_{\perp})$, as functions of $x$ and the
transverse impact parameter $b_{\perp}$ for up and down quarks are shown in
Fig. 3. The GPDs evaluated in our BLFQ approach are qualitatively similar to
the GPDs calculated in phenomenological models [12, 13]. The quark TMDs:
$f_{1}(x,p_{\perp}^{2})$, $g_{1L}(x,p_{\perp}^{2})$, and
$h_{1}(x,p_{\perp}^{2})$ in the proton are presented in Fig. 4. The
qualitative behavior of our results is also consistent with the TMDs
calculated in phenomenological models [14, 15].

Figure 4: The quark TMDs: $f_{1}(x,p_{\perp}^{2})$, $g_{1L}(x,p_{\perp}^{2})$,
and $h_{1}(x,p_{\perp}^{2})$ (related to leading twist PDFs) in the proton.
Left: vs $x$ for fixed $p_{\perp}^{2}$; right: vs $p_{\perp}^{2}$ for fixed
$x$ [11].

## Acknowledgements

C. M. thanks the Chinese Academy of Sciences Presidents International
Fellowship Initiative for the support via Grants No. 2021PM0023. X. Z. is
supported by new faculty startup funding by the Institute of Modern Physics,
Chinese Academy of Sciences, by Key Research Program of Frontier Sciences,
Chinese Academy of Sciences, Grant No. ZDB-SLY-7020, by the Natural Science
Foundation of Gansu Province, China, Grant No. 20JR10RA067 and by the
Strategic Priority Research Program of the Chinese Academy of Sciences, Grant
No. XDB34000000. J. P. V. is supported in part by the Department of Energy
under Grants No. DE-FG02-87ER40371, and No. DE-SC0018223 (SciDAC4/NUCLEI).

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