On-Axis Optical Trapping with Vortex Beams: The Role of the Multipolar Decomposition

Optical trapping is a well-established, decades old technology with applications in several fields of research. The most common scenario deals with particles that tend to be centered on the brightest part of the optical trap. Consequently, the optical forces keep the particle away from the dark zones of the beam. However, this is not the case when a focused doughnut-shaped beam generates on-axis trapping. In this system, the particle is centered on the intensity minima of the laser beam and the bright annular part lies on the periphery of the particle. Researchers have shown great interest in this phenomenon due to its advantage of reducing light interaction with trapped particles and the intriguing increase in the trapping strength. This work presents experimental and theoretical results that extend the analysis of on-axis trapping with light vortex beams. Specifically, in our experiments, we trap micron-sized spherical silica (SiO2) particles in water and we measure, through the power spectrum density method, the trap stiffness constant κ generated by vortex beams with different topological charge orders. The optical forces are calculated from the exact solutions of the electromagnetic fields provided by the generalized Lorentz–Mie theory. We show a remarkable agreement between the theoretical prediction and the experimental measurements of κ. Moreover, our numerical model gives us information about the electromagnetic fields inside the particle, offering valuable insights into the influence of the electromagnetic fields present in the vortex beam trapping scenario.

Supporting Note 1. Theoretical description of electromagnetic fields and forces

S1.1 Multipolar decomposition of electromagnetic fields
As a first step, we consider an arbitrary monochromatic electromagnetic field with welldefined helicity.The incident electric field is written in terms of the well-defined helicity multipoles [1][2][3] as where j is the total angular momentum, m z is the angular momentum in z direction and p is the helicity of the incoming light beam (for a non-focused light beam helicity is equal to the polarization).We also have that D j = i j (2j + 1) 1 2 and C j,mz,p are the Beam Shape Coefficients (BSC).The BSCs contain the transversal mode amplitude and the focusing information following the Aplanatic Lens Model, 4 and are defined as in reference 5 for a focus placed in the center of coordinates and in references 6,7 for the displaced case.A p j,mz are the well-defined helicity multipoles, formed combining the electric and magnetic multipoles [1][2][3] as which are eigenstates of the helicity operator Λ = ∇× k , respecting the relation ΛA p j,mz = pA p j,mz .
The scattered electric field has the contribution of both positive and negative helicities, and are expressed as where 10][11] The magnetic field can be derived from the electric field using the relation Finally, for our calculation of the forces the total electric and magnetic fields must be taken, which are derived as (5)

S1.2 Multipolar expansion of cylindrically symmetric optical systems
In this subsection, we are going to analyze the consequences of the on-axis configuration in a optical system with cylindrical symmetry.This is also the case when a spherical particle is optically trapped centred along the longitudinal axis (z) of a focused Gaussian or Laguerre-Gaussian beam, which is the main system under study in this work.We refer to this kind of trapping as "on-axis" configuration and it has deep implications for the multipolar decomposition of the electromagnetic field described in this work.The cylindrical symmetry implies that the incident E i and scattered E sc fields are eigenstates of the operator of rotations around z axis R z (θ) = exp(−iJ z θ), what, at the same time, means that the m z is well-defined in E i and E sc , leading to the eigenstate relations J z E i = m z E i and In consequence, we have that each individual multipole forming these fields has the same fixed value of m * z = L + p, allowing to simplify their expressions as and Note that the notations of both electric fields and the BSCs include an "on" superscript, meaning that they are describing the on-axis configuration.The lower limit of the j summation has been also set to |m * z | due to the properties of the spherical harmonics basis.This last effect suppresses a certain number of the lowest multipolar orders depending on the values of L (topological charge) and p (helicity) of the trapping beam, simplifying the multipolar analysis of this kind of systems.
On the other hand, the cylindrical symmetry along z would be broken if any relative displacement between the Gaussian or Laguerre-Gaussian beam and the spherical scatterer is performed along the transversal directions (x or y).This would require again the general description of the electric field with the complete summation of multipoles, described in Subsection S1.1.

S1.3 Theoretical calculation of optical forces
We employed the Maxwell Stress Tensor (T ij ) 12 to calculate the optical force acting on a spherical scatterer.In particular, we have integrated T ij over a sphere in the far-field surrounding the illuminated spherical particle.The i-component of the forces is given by where Here E and H denote the total electric and magnetic field, n j is a unitary vector that is normal to the differential of the area of a sphere surrounding the sample dA.Since in farfield the electromagnetic fields only contain transverse components, the first two terms of the left-hand side of Eq. ( 9) do not contribute to the optical force, yielding where Ω denotes the solid angle.Finally, we analytically calculated these integrals as in 13 to obtain the optical forces under the illumination of LGs with well-defined helicity and total angular momentum.
resolution LCOS-SLM (element 4) operating in an off-axis configuration.This device modulates the phase of the wavefront of the incoming light pixel by pixel, guided by a set of phase shift masks.The fundamental pattern for generating vortex beams superposes an azimuthal phase onto a blazed grating, so that the vortex beam forms in the first diffraction order, and is easily separated from stray light.Elements 2 and 3 are mainly devoted to adapting the laser beam to the requirements of the LCOS-SLM.First, its polarization is set with a half-waveplate (element 2).Then, the beam size is adjusted with a beam expander (element 3) formed by two plane-convex achromatic doublets with focal distances 25.4mm and 75mm.
Note that the beam expander includes a spatial filter using a pinhole of 30µm of diameter to eliminate higher frequencies of the beam.
After the phase modulation is applied, the beam wavefront at the SLM's screen plane is reconstructed onto the back focal plane of the focusing objective (element 8) using a 4f system (element 5).This ensures that the back aperture of the objective is filled with a constant amount of light when the different beam modes are employed and allows us to achieve a tight diffraction limited vortex beam with little to no distortion.For the trapping experiments shown in this work the back aperture with a diameter D ≃ 6mm was filled with Gaussian-shaped beam of 1/e 2 halfwidth w 0 ≃ 2mm.The value of w 0 was chosen to be smaller than the radius of the back aperture of the microscope objective to focus as much beam power as possible, 14 but specially to reduce the interference between the beam and the edges of microscope entrance.Note that, along the 4f line, between the SLM and the focusing elements, the polarization of the beam is adjusted to pure circular using a quarter-wave plate and an additional half-wave plate (elements 6).Ideally, a single quarterwave plate applied to a linearly polarized beam would be able to generate perfect circular polarization, but experimentally, only the combination of both elements provides the finetuning of the polarization state required for this work.Along the 4f system, we can also find a dichroic mirror (element 7) that reflects the 976 nm trapping laser towards the objective lens (element 8) and transmits the visible illumination light going to the camera (element

14).
The next part of our setup is formed by the optical trap (element 9) and the elements conforming to it.Here, the laser beam is tightly focused by the previously mentioned 100X oil immersed objective lens of NA=1.25 and WD=0023mm (element 8), which is capable of producing a beam waist with sizes < 1µm.
In our case, the trapped samples are silica (SiO 2 ) spherical particles with a diameter of 2µm.These spherical particles are part of a monodisperse suspension in water, which is confined in a micro-channel slide of 0.4mm of height made of a transparent polymer (see Supporting Fig. 2).Due to gravitational forces, these silica beads fall right to the bottom of the channel.Hence, this plane must be finely aligned with the focal plane of our objective lens to make the particles interact with the highest intensity part of our laser beam to get trapped.For this purpose, we adjust the position of the slide with a micrometric precision positioner, aided, at the same time, by the imaging system that will be explained shortly.
Supporting Fig. 3 shows two pictures obtained through the imaging system of the trapping After the laser light interacts with the trapped particle, the forward scattering is collected and collimated by a second 10X objective lens (element 10 of Supporting Fig. 1) with NA=0.25 and WD=7mm.Then, the laser light is reflected on a dichroic mirror (element 11) and guided to the four quadrant photo-detector (element 12), placed at the end of the optical path.At the same time, the second objective focuses the white light (element 13) traveling in the opposite direction over the slide, playing the role of a condenser in the imaging system of our setup simultaneously.Finally, the image of the trapping plane will be formed on the screen of a CCD camera (element 14).
Supporting the four-quadrant photo-detector is processed in order to obtain the stiffness parameter of the optical trap.

Supporting Figure 2 :
Experimental trapping environment.a) Picture of the experimental trapping elements.It is possible to differentiate the micrometric precision positioner holding the micro-channel slide, the trapping laser focusing objective lens (bottom) and the condenser objective lens (top).b) Sketch of the experimental trapping elements.In this drawing it is possible to see the configuration of the micro-channel slide and the way it contains the monodisperse suspension of spherical silica particles in water.

Supporting Figure 3 :
Imaging of the X/Y trapping plane a) SiO 2 particles in the bottom of the micro-channel slide in the absence of trapping light.b) SiO 2 particles in the bottom of the micro-channel slide, but in this case one of them (the one in the centre) is optically trapped.Note that the trapped particle is slightly out of focus with respect to the imaging plane.plane of the experimental setup.Note that the monodisperse suspension of silica particles in water must be carefully diluted in order to reach the proper concentration of particles in the bottom of the micro-channel slide.If the number of particles is too high in this plane, trap stiffness measurements could easily be spoiled by the interference of other attracted particles.

Figure 5 :
Example of PSD procedure for stiffness constant calculation.Examples of 6 different κ x measurements: 3 optical powers with incident light beam with topological charge L = 0 and another 3 with L = 1.It can be observed the processing of the particle oscillation frequencies in order to extract the κ x parameter.First by applying the Lorentzian fitting and then by determining the f c .