Aaron Schäpers

Gaute Hellesø

Jochen Fick
email: jochen.fick@neel.cnrs.fr

Optical trapping in air on a single interference fringe

Keywords: optical tweezers, trapping in air, interference fringes, micro-particles

Stable and reproducible trapping in air of 1 µm and 500 nm dielectric particles has been realized using a dual beam optical fiber tweezers with cleaved commercial single mode fibers. The influence of the interference fringes of the two coherent and counter-propagating trapping beam is investigated by controlling the fringe visibility. Optical trapping on a series of up to 10 fringes or trapping on only one to two fringes has been observed in distinct experiments. High axial trapping efficiencies of up to 1 nN•µm -1 W -1 is observed. The experimental results are supported by numerical simulations.

Introduction

Since its development in 1986 by Ashkin [START_REF] Ashkin | Observation of a single-beam gradient force optical trap for dielectric particles[END_REF], optical tweezers have found many applications in different research domains such as biology, chemistry or physics. Most optical trapping experiments are done with particles in suspension. Optical trapping in air [START_REF] Gong | Optical trapping and manipulation of single particles in air: Principles, technical details, and applications[END_REF], and more recently in vacuum [START_REF] Millen | Optomechanics with levitated particles[END_REF], is, however, of great interest.

Optical traps can, in most cases, be described by the harmonic oscillator model. Because of the high water viscosity, trapping in suspension corresponds to a heavily over-damped oscillator. Trapping in vacuum drastically reduce damping and allows to study light-matter interactions in more detail. The viscosity of air is about one order of magnitude lower than for water, but trapping in air still corresponds to an over-damped oscillator. However, the higher viscosity makes trapping in air more stable than in vacuum. In addition, the problem of particle heating is reduced.

In 1997 R. Omori et. al. trapped micrometer-sized glass spheres in air using a single beam tweezers [START_REF] Omori | Observation of a single-beam gradient-force optical trap for dielectric particles in air[END_REF]. A comparison between suspension and air tweezers showed that trapping in air requires significantly less optical power [START_REF] Summers | Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps[END_REF]. This fact is mainly due to the higher refractive index difference of, for example, polystyrene particles giving ∆n ≈ 0.53 in air instead of ∆n ≈ 0.20 in water.

Previous works on optical trapping in air concerned mainly trapping of quite large dielectric particle in the 8 to 10 µm range [START_REF] Michihata | New technique for single-beam gradient-force laser trapping in air[END_REF][START_REF] Kampmann | Optical system for trapping particles in air[END_REF]. A micro-displacement sensor has been developed, using an optically trapped microprobe based on the interference scale [START_REF] Michihata | Microdisplacement sensor using an optically trapped microprobe based on the interference scale[END_REF]. In a different work, large, absorbing particles were trapped and transported in free space using standard diffraction lenses and using vortex or bottle beams [START_REF] Shvedov | Optical vortex beams for trapping and transport of particles in air[END_REF][START_REF] Shvedov | Robust trapping and manipulation of airborne particles with a bottle beam[END_REF][START_REF] Zhang | Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques[END_REF]. Single dielectric and metallic micro and nanoparticles were trapped in air by L. Jauffred and S. Taheri using a trapping chamber designed to minimize any turbulence [START_REF] -R. Taheri | Tube length-assisted optimized aerosol trapping[END_REF][START_REF] Jauffred | Optical Trapping of Gold Nanoparticles in Air[END_REF]. In this context, the paramount influence of the surface of the trapped particle was revealed for gold nanoparticles [START_REF] Arita | Invited article: Optical trapping of ultrasmooth gold nanoparticles in liquid and air[END_REF].

In the field of meteorology, trapping in air was motivated by the study of droplet behavior inside clouds. Single and dual beam geometries were applied to micron-sized droplet trapping in air with focus on optical spectroscopic or micro-rheology studies to investigate droplet surface tension or particle viscosity [START_REF] Mcgloin | Optical manipulation of airborne particles: techniques and applications[END_REF][START_REF] Guillon | Optical trapping and binding in air: Imaging and spectroscopic analysis[END_REF][START_REF] Brzobohatý | Optical manipulation of aerosol droplets using a holographic dual and single beam trap[END_REF][START_REF] Power | Probing the micro-rheological properties of aerosol particles using optical tweezers[END_REF]. Furthermore, the spectroscopic characterization of fluorescent particles in air presents an interesting approach as it allows to eliminate all environmental influence of the surrounding medium [START_REF] Wang | Optical trapping and laser-spectroscopy measurements of single particles in air: a review[END_REF][START_REF] Wu | Stable optical trapping and sensitive characterization of nanostructures using standing-wave raman tweezers[END_REF].

The first optical fiber tweezers was already realized in 1993 by Constable et. al. [START_REF] Constable | Demonstration of a fiber-optical light-force trap[END_REF]. Their work represent perhaps the most straightforward optical trapping experiment as they used just two pigtailed lasers and realized fiber alignment by pushing them towards a capillary between two glass slides. Today, the use of nano-structured optical fibers allow very efficient trapping in single-fiber or dual-fiber geometries. In fact, one of the advantages of using optical fibers is that they facilitate the realization of dual beam tweezers. In this configuration, the constraints on beam focusing are significantly reduced as the scattering forces from the two beams cancel each other. Finally, as will be shown in the present paper, the possibility to trap on one single interference fringe results in high trapping efficiencies. Optical fiber based dual optical tweezers were used by different groups to trap water droplets or 3 -5 µm dielectric particles [START_REF] Rudd | Fiber based optical trapping of aerosols[END_REF][START_REF] Xiao | A pzt-assisted single particle loading method for dualfiber optical trap in air[END_REF] In this article, we investigate optical trapping in air by a dual fiber optical tweezers. The set-up is an adapted version of our suspension tweezers used for trapping and spectroscopic characterization of different kinds of nano-particles [START_REF] Decombe | Single and dual fiber nano-tip optical tweezers: trapping and analysis[END_REF][START_REF] Decombe | Luminescent nanoparticle trapping with far-field optical fiber-tip tweezers[END_REF][START_REF] Kumar | Emission spectroscopy of nayf4:eu nanorods optically trapped by Fresnel lens fibers[END_REF]. In contrast to former work, the best results are obtained with cleaved fibers instead of fiber tips or Fresnel lens fibers [START_REF] Asadollahbaik | Highly efficient dual-fibre optical trapping with 3d printed diffractive fresnel lenses[END_REF]. The influence of optical interference fringes will be studied in detail, both experimentally and using numerical simulations.

Methods

Experimental setup

A very compact and entirely fibered optical tweezers, fitting on a 750 x 450 mm breadboard, was developed (Fig. 1). A pigtailed diode-laser (λ = 808 nm, P max = 250 mW) is coupled to a 2 × 2 fused fiber coupler. Each of its two outputs is connected to one of the trapping fibers. A variable attenuator in one light path allows to compensate for differences in transmission between the light paths. In the second light path, a three-paddle fiber polarization controller is introduced to adjust the relative polarization between the two trapping beams. For trapping, two cleaved, standard single mode fibers are used (Nufern, S630-HP) with a mean mode diameter of 4.2 µm and a numerical aperture of 0.12. Each fiber is fixed on a set of xyz piezoelectric translation stages for efficient fiber alignment.

Inspired by the work in Ref. [START_REF] -R. Taheri | Tube length-assisted optimized aerosol trapping[END_REF][START_REF] Jauffred | Optical Trapping of Gold Nanoparticles in Air[END_REF], an original two-compartment trapping chamber was designed and manufactured by 3D printing. It ensures effective particle feeding and shielding of trapped particles against any air draft (Fig. 1.b). A portable commercial nebulizer (Omron U22) is used to produce an aerosol from an ethanol suspension with microparticles. The aerosol is blown by means of a nozzle into the upper compartment of the trapping chamber (10 × 30 × 7.5 mm 3 ). Some droplets are then diffusing through a small hole (∅ ≈ 1 mm) into the lower compartment which contains the optical trap (h = 2 mm). Once a droplet is trapped, the ethanol is quickly evaporating and the particle will remain trapped. The two trapping fibers are passing through opposite openings in the trapping chamber, with sufficiently large openings to allow alignment of the fibers. With this setup, trapping of a new particle is realized within less than ten minutes and particles remain trapped for a couple of hours.

Data acquisition and analysis

Videos of trapped particles are recorded at framerates of 1000 fps through a 50x long working distance microscope objective coupled to a CMOS camera. A homemade particle tracking algorithm in python is used to extract the time-dependent particle position in the x -y observation plane. The video resolution is about 130 nm/pixel. By fitting over a number of pixels, the tracking algorithm has, however, an enhanced resolution of some picometers. The position records are then used to calculate the position probability P independently in x and y directions for application of Boltzmann statistics in the framework of the equipartition theorem [START_REF] Decombe | Single and dual fiber nano-tip optical tweezers: trapping and analysis[END_REF]. Supposing a harmonic trapping potential U = 1/2κx 2 , P can be written separately in x and y direction as:

P (ξ) = 1 Z • e -κ ξ •(ξ-ξ 0 ) 2 k b T , (1)
with ξ ∈ {x, y}, Z the partition function, k B the Boltzmann constant, T the absolute temperature, ξ 0 the mean position, and κ ξ the trap stiffness. In the presence of interference fringes in the probability distribution in axial direction, P (x) shows a series of n equidistant peaks and Eq. 1 can be modified to:

P n (x) = A 0 • e -κ e x •(x-x e 0 ) 2 2k B T •    1 -V *   1 - n-1 j=0 e -κ p x •(x-x p 0 -j•δx) 2 2k B T      (2)
with V the peak visibility factor, δx the peak (fringe) spacing, x p 0 the position of the first peak, and x e 0 the position of the Gaussian envelope function maximum. Two specific trap stiffnesses are defined: κ e x corresponding to trapping without interference fringes and κ p

x corresponding to trapping on a single fringe. In our model κ p x is supposed to be the same for all observed peaks.

Numerical simulations

For nanoparticles, the Rayleigh approximation can be used to analytically find the optical force on a particle trapped by two counter-propagating Gaussian beams [START_REF] Zemánek | Optical trapping of rayleigh particles using a gaussian standing wave[END_REF]. However, the particles used in this study are rather large for applying the Rayleigh approximation. Furthermore, the fibers have a mode radius ω 0 = 2.1 µm, giving a Rayleigh length z 0 = πω 2 0 /λ = 17 µm. The distances between the fibers considered here are 20 -50 µm. As this is just one or a few times the Rayleigh length, the Gaussian approximation for the beams cannot readily be justified. The finite element method is commonly used to simulate field distributions, but requires 5 -10 mesh-points per wavelength. This implies that a full 3D simulation of the problem becomes very memory-consuming. There are thus several obstacles for simulating the problem at hand, and also several paths that can be followed to simplify the problem. Initially, we tested a full finite element 3D simulation for a small separation between the fibers and also a quasi-3D model taking advantage of vertical and horizontal symmetries, thus simulating a quarter of the space. This is the same approach as has previously been used for trapping with counter-propagating beams in a gap between two planar optical waveguides [START_REF] Hellesø | Surface transport and stable trapping of particles and cells by an optical waveguide loop[END_REF]. Results from these accurate models were used to verify our simplified model.

By considering only particles on the optical axis, and only calculating forces along the optical axis, the problem has rotational symmetry around the optical axis. It can thus be simulated with the finite element method in 2D, taking into account the rotational symmetry. This is our choice for a simplified model as the memory requirements are low, it is fast to solve and it gave the same results as a full 3D-model for small separations between the fibers.

Mathematically, the force F with a full 3D-model is found by integrating the Maxwell stress tensor T and the surface normal n over the surface S of the sphere [START_REF] Gaugiran | Optical manipulation of microparticles and cells on silicon nitride waveguides[END_REF][START_REF] Dienerowitz | Optical manipulation of nanoparticles: a review[END_REF]:

F = S T • nds. ( 3
)
When simplifying to rotational symmetry, the expression becomes:

F = 2π L T • nrdl, (4)
with L the line describing the sphere in 2D and r the radius to the line. For simplicity, the same symbols are used for the vectors and tensor T for 3D and 2D. As mentioned, applying rotational symmetry limits the study to particles on the axis and finding the force F z along the axis. The 3D-model can handle any polarization through T, while for the 2D model, the polarization must be out of the 2D plane. This corresponds to beams with parallel (∆ϕ = 0 • ) polarization. For crossed polarization (∆ϕ = 90 • ), the beams don't interfere and the force from one beam can be found independently of the force from the other beam. This makes it possible to find the force also for crossed polarization with a 2D model, but this is not considered in the following. The finite element method (Comsol Multiphysics 6.0, RF-module) is used to find the optical mode of the trapping fibers. Using rotational symmetry, the field-distribution between the fibers is found for various positions of the particle. Both fibers are taken as sources with the fundamental mode and the same polarization (out of the plane, ∆ϕ = 0 • ). The polarization is thus parallel and the beams generate an interference pattern between the fibers. The axial force F z on the particle is found with Equation 4 and using Comsol's definition of the Maxwell stress tensor. The integration is performed in post-processing, using built-in functions.

Experimental Results

In the present work, 500 nm and 1 µm commercial polystyrene particles are trapped at laser powers of P = 24 -66 mW and fiber-to-fiber distances of d = 20 -50 µm. In general, optical trapping of a single particle is realized in less than 10 minutes once the fibers are mounted and aligned. In the case of dual-beam optical tweezers with one single coherent light source, the two trapping beams are interfering and the resulting interference fringes have a significant influence on the behavior of the trapped particle. The visibility of the interference fringes can be controlled by rotating the polarization axis of one trapping beam. If the polarization axes of both beams are parallel, i.e. for zero polarization mismatch ∆ϕ = 0, the fringe visibility is maximal. On the other had, for crossed polarizations (∆ϕ = 90 • ) no interference fringes are observed. In this case, the trapping efficiency is linearly increasing with optical power. For 1 µm particles and a fiber-to-fiber distance of d = 50 µm we measured a normalized trapping efficiency of κx = 0.15 pN•µm -1 •W -1 and κy = 0.23 pN•µm -1 •W -1 in axial (x) and transverse (y) directions, respectively. The transverse trapping efficiency shows a 1/d 2 dependency from the fiber-to-fiber distance, whereas the axial efficiency was found to be constant.

The influence of the interference fringes on the optical trapping is investigated by recording series of trapping videos of the same particle but with different polarization mismatches. The polarization of one beam is rotated in 30 • steps over a 360 • range. After the experiment, the zero polarization mismatch ∆ϕ = 0 • is found from the fringe visibility. The presented results correspond to the forward series (0 • → 360 • ). The backward series (360 • → 0 • ) shows similar results, but are not included here. The nominal fringe spacing, corresponding to λ/2 = 404 nm was used for fine-tuning the image resolution of ≈ 130 nm/pixel for each individual measurement series. For the following experimental series the laser power was fixed to P = 66 mW, with a fiber-to-fiber distance of d = 50 µm.

Optical trapping on a series of interference fringes

Trapping of 1 µm polystyrene particles is found to be significantly influenced by the presence of the interference fringes (Fig. 2). For zero polarization mismatch, the positions corresponding to the interference fringes are well distinguishable in the particle position plots (Video 01 and Video 2). With increasing mismatch, the fringe visibility is decreasing and totally disappears for crossed polarizations (∆ϕ = 90 • ). The fringes reappear with a second, but less prominent maxima at ∆ϕ = 180 • . The lower fringe visibility at ∆ϕ = 180 • originates from the limits of our polarization controller. This device also slightly decreases the right beam intensity. Consequently, the mean trapping position and the interference fringe positions are shifted to the right for ∆ϕ ≳ 150 • . For ∆ϕ = -30 • to 60 • the fringe positions are stable, with some variation in the mean particle position.

The axial position probability records are fitted to Equation 2 by applying 10 peaks. The peak visibility V , shows a clear 180 • periodicity with a maximum of V = 99% (Fig. 2

.b). The peak trapping efficiency shows a maximum of κ p

x (0 • ) = 8.3 pN•µm -1 (Tab. 1). The second maximum at 180 • is lower with κ p x (180 • ) = 6.6 pN•µm -1 (Fig. 2.c). The envelope trapping efficiency is about 600 times smaller and shows a similar dependence with two maxima of κ e

x (0 • ) = 16 fN•µm -1 and κ e x (180 • ) = 9 fN•µm -1 . Trapping in transverse direction is not affected by the polarization mismatch change and is quasi-stable with a mean efficiency of κy = 30 fN•µm -1 (Fig. 2.e). The mean time the particle is trapped on the main peak shows a sharp maximum of tr = 11.3 ms at ∆ϕ = 0 • . Besides this peak, tr is in the range 0.2 to 0.3 ms.

We now turn to the analysis of the trapping dynamics, i.e. the time dependent position and the particle speed. The axial position x(t) for four polarizations is shown on Fig. 3. For ∆ϕ = 0 • the particle is "hopping" between discrete positions. With increasing polarization mismatch, this effect is disappearing, resulting in a random motion for orthogonal beams. This effect is more pronounced when looking at the position-dependent particle speed. The most interesting feature concerns the axial distribution of the axial speed v x (x). The plot, shaped as an eye-diagram, suggests that the particle is hopping directly from one fringe to the neighbor during a single video frame interval, resulting in an transfer speed of v t

x ≈ 404 µm•s -1 (dotted horizontal lines in Figure 3). For larger polarization mismatch and less efficient trapping on the fringes, the eye-diagram is getting less pronounced. The influence of the interference fringes remains, however, visible for ∆ϕ = 90 • . The influence of the polarization mismatch on the transverse speed distribution v y (x) is less pronounced. For zero polarization mismatch, the maximum transverse speed v y is slightly higher than v x and decreases with increasing ∆ϕ.

Optical trapping on few interference fringes

A different effect is observed in a subsequent optical trapping experiment with identical conditions, but improved fiber alignment. In this series, the particle is actually trapped on a single, or up to three interference fringes (Fig. 4). For the different videos the particle is trapped on five uniformly spaced interference fringes. Even in the case of ∆ϕ = 90 • , the imperfect polarization of the beams results in trapping on only two interference fringes.

Depending on the number of peaks, the trapping efficiency was obtained by fitting the data to Equation 1 or 2. Here, the envelope trap stiffness has no physical meaning and becomes a free fitting parameter. The maximum axial peak trapping stiffness of κ p x = 65 pN•µm -1 is about eight time larger than for the first series. The polarization mismatch dependency is, however, less pronounced with no clear minimum or 180 • periodicity (Fig. 4.b). In the transverse direction, the trapping efficiency is quite constant with a mean value of κ y = 0.037 pN•µm -1 , about ten times higher than in the first series.

The highest trapping efficiency is observed for a sequence with trapping on two interference fringes, and not for trapping on a single fringe as expected. During the 10 s of the sequence, the particle is, however, hopping only once between two fringes (Fig. 4.d, ∆ϕ = 0 • ). For ∆ϕ = 30 • , the particle is already moving between the fringes four times and at ∆ϕ = 60 • , the particle is hopping more than 30 times. Whether the particle is trapped on one or two fringes can therefore be regarded as a statistical coincidence. Moreover, it is random which specific fringe the particle is trapped on.

Optical trapping of 500 nm particles

A similar trapping behavior is observed for optical trapping of 500 nm particles, including experiments with trapping on several fringes and experiments with trapping on only one to three fringes. These results will not be reported in detail as they do not provide fundamentally new insight into the underlying physics. The main results are summarized in Table 1. Optical trapping is observed on 9 interference fringes with a maximum axial on-fringe trap stiffness of κ p x (∆ϕ = 0 • ) = 5.5 pN•µm -1 , which is of the same order as for the 1 µm particles. In the transverse direction, the mean trap stiffness is κy = 0.066 pN•µm -1 , which is twice the value for 1 µm particles. A significant difference is observed for the mean time on the main fringe tr = 1.7 ms, which is ten times shorter and reflects the general observation of less stable trapping for 500 nm particles.

Numerical simulations

Numerical simulations were performed to mimic the experimental trapping of 1 µm polystyrene particles, two cleaved fibers at a distance of ≈ 50 µm and an optical power of P = 66 mW in each fiber. The high index difference between the polystyrene particle and air leads to strong light scattering with three to four interference fringes inside the particle volume (Fig. 5a). The calculated optical force for parallel optical polarization (∆ϕ = 0 • ) is shown in Figure 5b. The periodic oscillations are attributed to the interference fringes. The three chosen fiber distances are (i + δ) • λ with λ = 808 nm, i = 62, and δ ∈ {0, 1 4 , 1 2 }, close to the experimental value of d = 50 µm. The higher force for δ = 1 2 is due to higher light intensity inside the Fabry-Pérot type cavity which is formed between the two cleaved fibers.

The trapping potential U is calculated by integrating the optical force (inset Fig. 5.b). It shows the superposition of the parabolic (harmonic) potential of a two incoherent beam tweezers with the periodic modulation caused by the interference fringes. The potential well between two adjacent minima is very high, above ≈ 100 • k B T . For better comparison with the experimental results, the position probability was calculated using P = 1 Z exp(U/2k B T ) (Fig. 5.c). Under these conditions, optical trapping is limited to trapping on the two central interference fringes. The comparison with the experimental curve for ∆ϕ = 0 • is very good. Experimentally, the occupation rate is not equal. This is because the particle is initially trapped on one fringe and only moves to the second one at the end of the observation time.

The axial peak trap stiffness is obtained from the calculated position distribution applying the same numerical fitting approach as for the experimental data (Fig. 5.d). The constant decrease of κ p

x with increasing fiber distance can be approximated by a 1/d 2 dependence. Moreover, the trapping stiffness is enhanced by 30% for δ = 1 2 with respect to δ = 0. To observe this enhancement experimentally, a very precise angular alignment of the two fibers is required, with perfectly cleaved end-faces. These conditions are quite difficult to realize experimentally.

Discussion

Optical trapping with two counter-propagating, weakly diverging laser beams results in general in an anisotropic trapping potential. Trapping in the transverse direction has been observed to be about three times more efficient than in the axial direction [START_REF] Decombe | Single and dual fiber nano-tip optical tweezers: trapping and analysis[END_REF][START_REF] Decombe | Luminescent nanoparticle trapping with far-field optical fiber-tip tweezers[END_REF]. In the case of converging beams, this anisotropy is even more pronounced [START_REF] Asadollahbaik | Highly efficient dual-fibre optical trapping with 3d printed diffractive fresnel lenses[END_REF]. Trapping in the transverse direction is dominated by the gradient force, which pulls the particle towards the beam center, and the forces created by the two beams are adding up. The highest trapping efficiency is obtained for beam waists close to the particle size. In the axial direction, particles are pushed by the scattering force. The forces created by the two beams are opposing each other. Diverging or focused beams, namely with varying intensity, are required to obtain a stable axial trapping position.

The single mode optical fibers used here have a mode diameter of ω = 4.2 µm and a numerical aperture of NA = 0.12. Thus, the large beam diameter explains the relatively low transverse trapping efficiency, whereas the weak beam divergence results in a low axial envelope trapping efficiency. In previous experiments in air, we have used fiber tips with beam parameters better adapted to the particles, with ω = 0.9 µm and NA = 0.26. It was, however, impossible to trap particles with these fiber tips. In fact, the stronger beam divergence resulted in larger transverse scattering forces, prohibiting the particles from reaching the stable optical trapping region.

In the present case of weakly diverging and coherent beams, the sum of the opposing scattering forces is relatively small and the optical gradient force related to the interference fringes dominates. Thus, trapping on one single fringe can be realized. This on-fringe trapping is of great fundamental and experimental interest as the light intensity gradient of an interference fringe is the steepest possible gradient in the optical far field.

For the highest observed trapping efficiency of 65 pN•µm -1 , the axial position probability peak width (FWHM) is 19 nm, about seven times smaller than the resolution given by the camera pixel size, and 50 times smaller than the particle size. We estimate that this value is near the experimental resolution of our set-up. On the other hand, a very good agreement of the experimental results with the numerical simulations is observed. The possibility of trapping on one or two interference fringes was predicted by the simulations and shown experimentally. Also, the simulated and measured trapping efficiencies correspond well.

For particle trapping in suspension, we have previously observed similar on-fringe trapping, but only for 300 nm particles and with considerably lower visibility [START_REF] Decombe | Luminescent nanoparticle trapping with far-field optical fiber-tip tweezers[END_REF]. In fact, several conditions are required for on-fringe trapping. First, the particle diameter should be similar to, or smaller than, the fringe spacing to have only a few fringes "inside" the particle. For example at 808 nm wavelength, three or four fringes are inside a 1 µm particle. Secondly, in order to have a significant gradient force, a high index difference between the particle and the surroundings is essential. For a polystyrene particle in air, ∆n air = 0.57, which is more than twice that in water (∆n water = 0.25). Finally, for small particles, Brownian motion becomes dominant and prevents single fringe trapping.

Using a focused laser beam, a normalized trapping stiffness of κ = 16 pN•µm -1 •W -1 was reported for 1 µm silica particles [START_REF] Jauffred | Optical Trapping of Gold Nanoparticles in Air[END_REF]. This value is about eight times higher than the transverse stiffness observed here, but 60 times lower than the normalized peak trapping stiffness of κp x = 985 pN•µm -1 •W -1 .

Conclusion

Stable and reproducible optical trapping of micron-sized dielectric particles in air was realized using our dual beam fiber optical tweezers. The geometry of this tweezers set-up is straightforward, with two commercial cleaved single-mode optical fibers, a commercial nebulizer, a 3D-printed trapping chamber and a homemade microscope. The optics path from the pigtailed laser to the trapping fibers is entirely fibered, thus resulting in a very low footprint.

Depending on the alignment conditions, trapping on a series of interference fringes or on a single interference fringe is observed. A very high trapping efficiency of up to κ p x = 65 pN•µm -1 is observed for a low light power of 66 mW. The paramount influence of the interference fringes is elucidated by controlling the polarization mismatch of the two trapping beams and thus gradually suppressing the fringe visibility.

The results presented in this article show that dual fiber optical tweezers allow us highly efficient trapping of micron-sized objects in a compact set-up using the extremely steep intensity gradient that occurs from the interference fringes of two coherent laser beams. Our set-up could be easily modified to integrate spectroscopic characterization of the trapped particles.

Figure 1 :

 1 Figure 1: Optical fiber tweezers setup: (a) Schematic of the all-fibered tweezers, (b) CAD image of the trapping chamber: the upper compartment (brown), the lower chamber (blue), the two optical fibers (orange), the plastic films to seal and separate the compartments (yellow), and the nozzle for aerosol feeding (red). (c) Photo of the trapping chamber with the nozzle and the two optical fiber holders. (d) 1 µm polystyrene particle trapped between two cleaved fibers at 50 µm distance. (e) Same particle in a trapping video frame.

Figure 2 :

 2 Figure 2: Optical trapping of a 1 µm particle on a series of interference fringes. (a) From left to right: 2D position tracking plots and position probabilities in axial (x) and transverse (y) directions for different polarization mismatches ∆ϕ. The black circle is an on scale sketch of the trapped particle. (b) -(f) Polarization mismatch dependence of indicated trapping parameters. The continuous lines in (a) and (b) are numerical fits, whereas the dotted lines in the other plots are guides to the eye.

Figure 3 :

 3 Figure 3: Axial particle dynamics for four polarization mismatches of the trapping series shown on Fig. 2. From left to right column: time-position records, axial speed (v x ) and transverse speed (v y ) as a function of the axial position (x). Horizontal dashed lines correspond to a speed of v x = ±404 µm•s -1 .

Figure 4 :

 4 Figure 4: Optical trapping of a 1 µm particle on one to three fringes. (a) 2D position tracking plots and position probabilities in axial (x) and transverse (y) directions for different polarization mismatches ∆ϕ. The black circle is an on scale sketch of the trapped particle. Axial peak (b) and transverse trap stiffness (c). The dotted lines are guides to the eye. (d) Axial and transverse time position records for three polarization mismatches.

Figure 5 :

 5 Figure 5: (a) 2D map of the electrical light field norm showing the strong light scattering of the trapped particle. (b) Calculated axial optical force at P = 66 mW on a 1 µm particle between two cleaved fibers for three distances d = (62 + δ) • λ as labeled in (c) (∆ϕ = 0, P = 50 mW). Inset: Corresponding potential U . (c) Calculated position probability (dots) and numerical fits as used for the experimental data. (d) Calculated axial peak trap stiffness as a function of the fiber distance (gray circles all δ = 0, color dots as labeled in (c)). The experimental result for ∆ϕ = 0 (Fig. 4) is added as pink lines in (c) and (d).

Table 1 :

 1 Main optical trapping efficiency parameters: κ e x , κ p x axial envelope and peak stiffnesses, κ y transverse stiffness, and tr mean remaining time on main peak (P = 66 mW, total recording time: 10 s).

		1 µm particle	500 nm particle
		1. series 2. series
	κ e x κ p x κy	0.016 a 8.3 a 0.030 b	-65 a 0.37 b	0.010 b 5.5 a 0.066 b	[pN•µm -1 ] [pN•µm -1 ] [pN•µm -1 ]
	tr	11.3 a	10 000 a	1.7 a	[ms]
	nbr peaks	10	1 -3	9
	a maximum value, b mean value

Acknowledgments

We thank M. Fick for his contribution in designing and fabricating the optical trapping chamber. J.Fick acknowledges funding from Agence Nationale de la Recherche (project no. ANR-16-CE24-0014-01) whereas O.G. Hellesø acknowledges support from the Research Council of Norway (grant no. 302333).


domain: phys spi

text:
Martin Kjell

Marie Mølster
email: moelster@kth.se

Patrick Guionie

Jean-Baptiste Mutter

Jean-Michel Dherbecourt

Xavier Melkonian

Andrius Delen

Carlota Zukauskas

Fredrik Canalias

Patrick Laurell

Georges

Kjell Martin Mølster

Marie Guionie

Patrick Mutter

Jean-Baptiste Dherbecourt

Jean-Michel Melkonian

Xavier Délen

Andrius Zukauskas

Carlota Canalias

Fredrik Laurell

Patrick Georges

Myriam Raybaut

Antoine Godard

Valdas Pasiskevicius

Precisely tunable narrowband mid-infrared mJ pulses from a backward wave optical parametric oscillator

Introduction

Rotational-vibrational spectroscopy using mid-infrared (MIR) absorption lines in small molecules is at the basis of diagnostic and analytical tools benefiting medical diagnostics, environmental monitoring, and providing crucial input to climate models. For the latter applications, long-distance differential absorption MIR LIDARs (DIAL) operating on mobile platforms require high MIR photon flux delivered in Fourier transform-limited nanosecond pulses generated using mechanically robust coherent source designs. The detection protocols typically require rapid and precise switching between on-and off-wavelengths of the Doppler broadened absorption line. This switching range encompasses a few GHz for small molecules in the atmosphere. Tunable optical parametric oscillators (OPOs) employing type-II quasi-phase-matching (QPM) in PPLN can provide sufficient output energy for seeding power amplifier chains in MIR LIDAR [START_REF] Melkonian | Long-wave infrared multiwavelength optical source for standoff detection of chemical warfare agents[END_REF]. However, a typical type-II PPLN OPO acceptance bandwidth of about 300 GHzcm in the 2 µm spectral range mandates doubly-resonant OPO designs, single-longitudinal mode pumping, and active mechanical stabilization of the OPO cavity length. In this work, we investigate a robust arrangement for the generation of narrowband, mJ-level radiation around 2 µm, which offers rapid and precise tuning without the need for optomechanical adjustments or cavity alignment. The highly efficient source relies on a precise and rapid tuning of a single-frequency fiber laser seeded master oscillator power amplifier (MOPA) pump source at 1030 nm [START_REF] Hamoudi | Hybrid master oscillator power amplifier single-frequency, nanosecond, multi-mJ, 5 kHz at 1030 nm[END_REF] and the frequency translation properties of a backward wave optical parametric oscillator (BWOPO) [START_REF] Harris | Proposed backward wave oscillation in the infrared[END_REF][START_REF] Canalias | Mirrorless optical parametric oscillator[END_REF][START_REF] Strömqvist | Coherent phase-modulation transfer in counter-propagating parametric down-conversion[END_REF]. Concept and experimental setup BWOPO operation relies on a three-wave mixing (TWM) process with counter-propagating signal and idler waves. The three-wave mixing process involved in BWOPO needs to satisfy energy conservation (𝜔 = 𝜔 + 𝜔 ) and momentum conservation (Δk = k -k + k -K = 0). Here, the subscripts p, b, and f represent the pump, backward, and forward propagating fields, respectively, and K = 2𝜋 Λ ⁄ is the wavevector of the QPM grating with the period . The grating vector has to be similar in size to that of the pump wave vector inside the crystal to achieve phase matching, implying sub-µm periodicity of the QPM structure. Counter-propagating TWM geometry ensures that the QPM structure starts oscillating once the pump threshold intensity is exceeded. The material of choice is periodically poled KTiOPO4 (PPKTP), the only bulk crystal to produce BWOPO oscillation so far. The parametric output wavelengths in the BWOPO are determined by dispersion and QPM grating period. For a changing pump frequency, the changes in the parametric frequency are given by: Δ𝜔 = (1 + 𝜀) Δ𝜔 and Δ𝜔 = -𝜀 Δ𝜔 ,

where 𝜀 = 𝑣 𝑣 -𝑣 𝑣 𝑣 + 𝑣 is of the order of 10 -3 and is given by the group velocities of the interacting waves [START_REF] Canalias | Mirrorless optical parametric oscillator[END_REF]. Then Eq. ( 1) implies that the frequency of the forward-generated (codirected with the pump) wave tunes approximately at the same rate as the pump, while the frequency changes in the backward wave are small [START_REF] Strömqvist | Coherent phase-modulation transfer in counter-propagating parametric down-conversion[END_REF]. Group velocity dispersion (GVD) introduces only a minor correction which can be neglected in the 2 µm region where GVD for z-polarized waves is close to zero in KTP. For a given pump wavelength, either signal or idler can be generated in the backward direction, depending on the QPM period. Although these configurations might appear equivalent at the threshold, they are not at higher pumping rates when the device is far from degeneracy. When the BWOPO has the period which enforces forward signal generation, the efficiency is fundamentally limited due to cascaded BWOPO processes, akin to the cascades observed in stimulated Raman scattering [START_REF] Zukauskas | Cascaded counter-propagating nonlinear interactions in highlyefficient sub-µm periodically poled crystals[END_REF]. The same signal wavelength can be generated backward for a shorter QPM period. In such BWOPO, the cascaded processes are prohibited by momentum conservation, and higher efficiencies can be obtained at target wavelengths. Closer to degeneracy, however, when signal and idler are less than an octave apart, the cascading will not be possible even when signal is generated in the forward direction. Therefore in this work, we chose PPKTP crystals with a period of 580 nm, resulting in a forward-traveling signal at 1981.1 nm and a backward-traveling idler of 2145.6 nm. A stable and narrowband ytterbium based pump at 1030 nm with a maximum tuning (Δ𝜔 ) exceeding 10 GHz, specially designed for DIAL systems [START_REF] Hamoudi | Hybrid master oscillator power amplifier single-frequency, nanosecond, multi-mJ, 5 kHz at 1030 nm[END_REF], was used to pump BWOPO PPKTP crystals. The pump system consisted of a single-frequency fiber MOPA system followed by two freespace Yb:YAG based amplifiers. It produced 11 ns (FWHM) long millijoule pulses at a repetition rate of 5 kHz. The typical frequency bandwidth was 100 MHz. The pump wavelength tuning was obtained by modulating the central wavelength of the seed fiber laser. The pump beam 1/e 2 radius in the BWOPO crystal was 140 µm. The wavelengths of the pump and the BWOPO output were simultaneously measured using an optical spectrum analyzer and a wavemeter. We also measured the waves' powers, bandwidths (by Fabry-Perot interferometer), and temporal traces. The PPKTP crystals had two different QPM grating lengths, one of 7 mm and one of 17 mm. The latter is by far the longest grating length of sub-µm gratings to date. The crystals were uncoated.

Results and Discussion

Fig. 1(a) shows the measured and calculated frequency tuning for different pump frequency modulation amplitudes. The experiment results agree well with the dependencies calculated using Eq.1. Tuning of the forward BWOPO wave closely follows pump modulation, while the modulation of the backward wave was below the resolution of the wavemeter. Moreover, the BWOPO tuning is free of discrete jumps and tunes continuously with the pump. Thus, using this simple and mechanically robust arrangement, precise tuning over several GHz, suitable for targeting the on-and off-absorption line resonances, has been realized. The available fiber seed laser limited the tuning range of the pump. The pump can be tuned within the repetition rate of the laser, allowing pulse-topulse frequency targeting. In this measurement, the modulation rate was restricted by the integration time of the MIR wavemeter. The above-discussed characteristics do not come at the expense of efficiencies, as shown in Fig. 1 (b-c). The conversion efficiencies (Signal+Idler) are above 70% and 50% for the long and short QPM grating, respectively. The efficiency corresponds to the combined parametric energy of 1.13 mJ (1.92 mJ pump) and 0.68 mJ (0.96 mJ pump), respectively. The threshold of the 7-mm BWOPO was 0.37 mJ, while it dropped to 70 µJ in a 17-mm long BWOPO crystal. Pulse durations of the parametric waves were about 12 ns at the average pump power of 10W. As a consequence of a high conversion efficiency, the temporal profile of the depleted pump pulse was a flat-top of 19 ns. In summary, we have demonstrated a narrowband mJ source with continuous and precise tuning over several GHz at 2 µm, with energy conversion efficiency reaching 70%. Average powers of up to 3W and 2.65W were generated at the signal and idler wavelengths, respectively. The mechanically robust, the cavity-free arrangement would be beneficial for stable performance in challenging environments, e.g., space and airborne applications. This work has been done under the framework of the LEMON project, EC grant agreement No. 821868.

Figure 1 (

 1 Figure 1 (a) Tuning of the backward and forward wavelengths resulting from a detuned pump. (b-c) The efficiency and parametric output energies for the two different QPM grating lengths used in the experiment. The efficiency is the ratio between the parametric energy and the pump.

© 2 0 2 2 T h e Au t h o r ( s )


Sonia Gueguen

Serge Amselem

Annick Clement

Paul Landais

Radico

RaDiCo (Rare Disease Cohorts): a platform for rare disease (RD) e-cohorts


Gabriela Rezende

Lorna Le Stanc

Iris Menu

Mathieu Cassotti

Ania Aite

Emilie Salvia

Olivier Houde

Grégoire Borst

Arnaud Cachia

Differential effects of mindfulness meditation and cognitive training on cool and hot inhibitory control in children and adolescents

Ces résultats soutiennent le modèle de développement spécifique du traitement émotionnel à l'adolescence. Les capacités de CI chaud à chez les adolescents étant souvent caractérisées par des difficultés de régulation émotionnelle, l' efficacité d'un entraînement au CI basé sur la PC chez les adolescents peut dériver positivement dans de nombreux aspects, tels que la prise de décision et les comportements à risque, souvent présents à cet âge.

• CT overall improved cool IC (p<0.05) but not hot IC (p=0.47), while MM practice improved hot IC (p<0.01) but not cool IC (p=0.61). Children's hot IC improved after MM (p<0.01) and cool IC after CT (p<0.001), while adolescents improved only hot IC after MM (p<0.05).

• Children benefited more (p<0.001) than adolescents (p=0.20) from the interventions.

Conclusion

The present study supports the benefits of meditation interventions in children and adolescents, and open educational perspectives focused on improving emotion regulation with more ecological practices. This is particularly important for adolescence since this developmental period is characterized by impaired difficulties in emotion regulation.

Results

• We recruited 129 healthy participants from public schools: 66 children (36 females, 9 -10 years old) and 63 adolescents (37 females, 16 -17 years old).

• Participants performed cognitive practice on a tactile tablet, either CT or MM, for 15 minutes per day, 5 days per week, for 5 weeks. All participants had normal intelligence without sensory, psychiatric, or neurological problems.

• To evaluate the cool and hot dimensions of the IC, we chose the Stroop task in its two versions (cool and hot).

Introduction

  Differential effects of mindfulness meditation and cognitive training on cool and hot inhibitory control in children and adolescents Inhibitory Control (IC) is an Executive Function that plays an essential role in cognitive and social-emotional development (Borst et al., 2015). The IC occurs either in a neutral context (cool IC) or in social contexts involving emotions (hot IC) (Aïte et al., 2018). Cool IC improves linearly from childhood to adulthood while hot IC follows a non-linear, a U-shaped trajectory curve (Luna & Wright, 2016). Beyond development based on brain maturation, IC can also progress as a result of certain activities, such as cognitive training (CT) and mindfulness meditation (MM) (Takacs and Kassai, 2019). The purpose of our study is to compare the effects of two 5-week tactile tablet-based interventions, MM practice versus IC-based CT training, in children and adolescents by specifically analyzing cold and hot dimensions of IC. o Aïte, A., Cassotti, M., Linzarini, A., Osmont, A., Houdé, O., & Borst, G. (2018). Adolescents' inhibitory control : Keep it cool or lose control. Developmental Science, 21(1), e12491. https://doi.org/10.1111/desc.12491 o Borst, G., Aïte, A., & Houdé, O. (2015). Inhibition of misleading heuristics as a core mechanism for typical cognitive development: Evidence from behavioral and brain-imaging studies. Developmental Medicine & Child Neurology, 57(s2), 21-25. o Luna, B., & Wright, C. (2016). Adolescent brain development : Implications for the juvenile criminal justice system. In APA handbook of psychology and juvenile justice (p. 91-116). American Psychological Association. https://doi.org/10.1037/14643-005 o Takacs, Z. K., & Kassai, R. (2019). The efficacy of different interventions to foster children's executive function skills : A series of meta-analyses. Psychological Bulletin, 145(7), 653-697. https://doi.org/10.1037/bul0000195 • We fitted a linear mixed-effect model on the Stroop scorewith Time (Pre-test vs Post-test) and Type of conflict (Cool vs Hot) as within-subject factors, Intervention group (CT vs MM) and Age group (Child vs. Adolescent) as betweensubject factors and participants as a random factor.