On-chip optical tweezers based on freeform optics

which comprise optical reﬂectors or refractive lenses integrated on waveguide end facets via two-photon polymerization. The freeform optical design offers unprec-edenteddegreesoffreedomtodesignopticalﬁeldswithstrongthree-dimensionalintensitygradients,usefulfortrappingandmanipulatingsuspendedparticlesinanintegratedchip-scaleplatform.Wedemonstratethedesign,fabrication,andmeasurementofbothreﬂectiveandrefractivemicro-opticaltweezers.Thereﬂectivetweezersfeaturearemarkablylowtrappingthresholdpower,andtherefractivetweezersareparticularlyusefulformultiparticletrappingandinterparticleinteractionanalysis.Ourintegratedmicro-opticaltweezersuniquelycombineacompactfootprint,broadbandopera-tion,hightrappingefﬁciency,andscalableintegrationwithplanarphotoniccircuits.Thisclassoftweezersispromisingforon-chipsensing,cellassembly,particledynamicsanalysis,andiontrapping.

In this paper, we demonstrate a new class of on-chip optical tweezers whose functionality is based on waveguide-integrated, freeform micro-optical elements. Freeform optics opens up a large design space for optical wavefront control and has been regarded as a disruptive optical systems technology. However, the development of freeform optics in their bulk incarnation has been hampered by the fabrication challenges of free surface shaping with subwavelength accuracy. Recent developments in two-photon polymerization (TPP) [43], a process capable of on-demand printing of three-dimensional (3D) structures with critical dimensions well below the classical diffraction limit, have opened up a promising route for scalable fabrication of freeform optics-with a miniature form factor amenable to on-chip integration [44][45][46]. We show that freeform micro-optical elements are ideally suited for on-chip optical tweezers benefitting from several unique features. First, this design enables precise positioning of the trap location in the 3D region above the chip and facile adaptations of the trapping potential. Second, the micro-optics are seamlessly integrated with on-chip waveguides and can be readily interfaced with photonic integrated circuits to impart new functionalities such as dynamic tuning and switching of optical traps. In addition, compared to other planar devices that can also be used for free-space outcoupling such as diffractive gratings, plasmonic scatterers, and metasurfaces, freeform reflective or refractive micro-optics offer excellent optical efficiency (>95% efficiency from the waveguide to the focal spot) and broadband operation [46]. The high efficiency facilitates a low trapping threshold power, whereas the large spectral bandwidth is useful for quantum state preparation and fluorescence or Raman excitation. Finally, the waveguides can further function as integrated optical monitors, providing real-time information about the motion of trapped particles through the measurement of the backscattered light intensity [20]. In the following, we will present the design, fabrication, and characterization of two embodiments of the on-chip tweezers based on reflective and refractive micro-optics.

A. Design and Modeling
In this section, we describe the design and modeling of on-chip optical tweezers based on freeform reflectors. The basic concept is to use a pair of freeform reflectors attached to the end facets of waveguides to redirect and focus light to a predefined spot above the chip through total internal reflection (TIR) as shown in Fig. 1(a). Specifically, we use two ellipsoidal reflectors to focus the counterpropagating light output from two waveguides to the same point. The focused beams intersect with an optimized angle to generate a 3D gradient field for particle trapping in a way similar to a dark-field objective [ Fig. 1(b)]. The ellipsoidal reflectors are designed such that one of the foci coincides with the waveguide output and the other with the trap location as indicated in Fig. 1(c). The structure was further optimized by slightly offsetting the elliptical reflector along the longitudinal direction of the waveguide, where the amount of offset is determined by full-wave simulations [47].
In this demonstration, we used single-mode polymer waveguides with a core size of 2.5 µm × 2.5 µm, with the mode profile shown in Fig. S1. The core and cladding materials are EpoCore and OrmoCore (Micro Resist Technology GmbH) with refractive indices n core = 1.575 and n clad = 1.537, respectively. To eliminate Fresnel reflection at the waveguide facet, EpoCore is also used as the reflector material. The waveguide design yields a divergence angle of the beam output of approximately 10 • . To ensure TIR at the reflector-water interface, the angles θ 1 and θ 3 must meet the conditions 56. Fig. 1(c)], assuming that the refractive index of water at the 1550 nm trapping wavelength is n water = 1.315. In our design, the chief ray direction is given by θ 1 = 65 • and θ 3 = 50 • , corresponding to an equivalent NA of 1.0 (= n water · sin θ 3 ) in water for the crossing beams. Figure 1(d) depicts the optical intensity distribution on the center cross-sectional plane (x − z) of the waveguide for the transverse-electric (TE) polarized mode, simulated using 3D finite-difference time-domain (FDTD). In this example, the beam's focal spot is set to be 8 µm above the chip surface. The x − y -plane intensity distribution is shown in the inset of Fig. 1(d). The focal spot full widths at half-maximum (FWHM) in the x , y and z directions are 1.9, 2.1, and 2.5 µm, respectively.
A microparticle in this focal spot region experiences a conservative optical gradient force and a nonconservative optical scattering force. In the y direction, the optical scattering force is negligible since Poynting vectors of the incident beams lie within the x − z plane. In the x direction, a standing wave is formed [inset of Fig. 1(d)], and the scattering forces from the two beams cancel out. Therefore, the optical forces in the x and y directions are dominated by the conservative gradient force, which form the trapping potential for a microparticle. However, in the z direction, the scattering force cannot be neglected and tends to propel the microparticle away from the trap. Considering the fact that the gradient force originating from the strong light intensity gradient is dominant in the z direction for our case, we can define an effective potential depth by restricting our analysis to the z direction [48]. The calculated optical forces and trapping potentials of a 4.5 µm diameter polystyrene sphere (n = 1.56) in water are presented in Fig. 2. The modes in both waveguides are TE polarized and in phase. Interference between the two beams occurs at the focal spot where they cross, resulting in spatially varying intensity fringes, shown in Fig. 2(a). In Figs. 2(b)-2(d) we plot the optical force and trapping potential versus displacement from the equilibrium position, both normalized to the total power sent into the two waveguides. The optical forces are calculated with the Maxwell stress tensor (MST) method. Maximum forces of 1.0, 1.0, and 1.2 pN/mW are obtained along the x , y , and z directions, corresponding to trapping potential depths of 519, 574, and 330 k B T/mW, respectively, where k B denotes the Boltzmann constant and T represents temperature. By fitting the simulated trapping potential with a harmonic well (Fig. S2), theoretically expected trapping stiffnesses can be obtained along the x , y , and z directions as 0.23, 0.67, and 0.39 pN/µm/mW, respectively. In water, Brownian motion, with an average kinetic energy of 1.5 k B T, will randomly drive the particle to diffuse out of the trap. To achieve stable trapping, a minimum potential depth of 10 k B T is usually defined as a trapping threshold [2]. The 10 k B T stability threshold corresponds to trapping powers down to 19 µW (x ), 17 µW (y ), and 30 µW (z) in our design. These threshold power values are much lower than previous reports of on-chip optical tweezers [27,49], benefitting from the high optical coupling efficiency and optimized 3D field profile produced by the micro-optics. We also note that the two beams' shallow incidence contributes to the reduction of the force due to radiation pressure along the z direction and therefore stabilizes trapping compared to a conventional high-NA objective. To further demonstrate this point, we simulated the same particle trapped by a single Gaussian beam with NA = 1.0 (Fig. S3) and found that the potential depth along the z direction is 137 k B T/mW, 60% less than our design.
In the simulations, we assume that the output beams from the two waveguides have identical polarization, phase, and power. To investigate the robustness of the device operation to structure variations, we consider the extreme scenarios where the two beams have orthogonal polarizations [Figs. 2(e)-2(h)], 10% of power fluctuation (Figs. S4a-S4e), and a π phase difference (Figs. S4f-S4j). The nearly identical optical forces and potential depths along all three axes suggest insensitivity of the device performance to polarization, power, and phase changes. The result shows that the optical tweezer design is inherently tolerant against fabrication variations.

B. Experimental Characterizations
The fabrication process is schematically illustrated in Fig. S5, with details and parameters described in Supplement 1. The waveguide was patterned with an i-line UV stepper (GCA AutoStep 200) on top of a 10 µm thick bottom cladding layer, followed by spin coating of the top cladding layer with 10 µm thickness. A trench with a lateral dimension of 120 µm × 40 µm and a depth of 18 µm was then etched into the bottom cladding layers via reactive-ion etching (RIE) to define the waveguide output facets to which the reflectors are attached. The freeform reflectors were then sculpted using TPP in the photosensitive polymer EpoCore with a Photonic Professional GT 3D printer station (Nanoscribe GmbH). The 3D structures for TPP process of the focusing reflectors can be found in Visualization 1 and Fig. S6. The fabricated device is displayed in Fig. 4(a) and Fig. S7. Light coupled into the chip was first split into two equal parts by a Y-branch and then directed to the two waveguide ports of the tweezers.
The setup used to measure the focal spot's light intensity profile and characterize the trapping performance is depicted in Fig. 3(a). A liquid chamber was assembled on the chip using a coverslip and was filled with an aqueous microparticle solution. A 1550 nm wavelength laser was coupled into the chip by a lensed fiber via edge coupling. To map the intensity profile near the focal spot, the position of the 63× objective was swept along the out-of-plane (z) direction using a piezoelectric motor, and a series of images, each representing the in-plane intensity distribution at one z position, Fig. 3. Schematic of the experimental setup used to demonstrate on-chip trapping, map the focal spot profile, and measure the trapping strength of the on-chip optical tweezers. (a) A 660 nm laser beam, focused through an oil-immersion 63× objective, brings a particle to the trapping location. The particle's position is visualized using light collected through the oil-immersion lens and focused onto the VIS-camera and IR-camera. (b) An illustration of the procedure used to transfer a particle from a conventional microscope optical trap (660 nm laser) to the on-chip trap (1550 nm laser). was captured by an IR camera (Visualization 2). The 3D intensity distribution was then reconstructed, as shown in Fig. 4(b) (x − z plane cut) and Fig. 4(c) (x − y plane cut across the focal spot center). The measured focal spot FWHM is approximately 5.7 µm (x ) × 2.8 µm (y ) × 4.3 µm (z). The increased spot size compared with the simulated results is attributed to deviation of the reflector shape from the design caused by volume change during polymer cross-linking.
A conventional microscope optical tweezer with a waterimmersed objective and 660 nm wavelength laser was used to move a particle to the on-chip trap [ Fig. 3(b)]. After turning off the 660 nm laser and turning on the 1550 nm laser, the particle can be trapped stably near the focal spot of the two reflectors as shown in Fig. 4(d). To confirm successful trapping, the 1550 nm laser was switched off and on multiple times. When the laser was off, the sphere drifted away from the trap due to Brownian motion. The particle was recaptured by the trap after the laser was switched on (Visualization 3). Its spatial positions were extracted from the recorded video by fitting the position and diameter of the captured particle as shown in Fig. S8. The displacement of the trapped particle from its equilibrium position during a 4 min measurement is plotted in Fig. 4(e) (Visualization 4). The data imply that the particle is better confined along the y and z directions than in the x direction, consistent with the field distribution along the three directions. Traping stiffness is estimated by fitting the displacement distribution with a Gaussian function under different laser powers, which yields 0.09 ± 0.01, 0.16 ± 0.02, and 0.13 ± 0.02 pN/µm/mW along the x , y , and z directions, respectively [ Fig. 4(f )].

TRAPPING WITH REFRACTIVE MICROLENSES
In this section, we demonstrate that refractive micro-optics can similarly be applied to on-chip optical trapping. The refractive optical tweezers consist of two microlenses with a Cartesian oval shape (Visualization 5, Fig. S9). Considering the waveguide beam as a pointlike source, the shape of the lens can be determined by seeking a constant optical path length between the pointlike source and the target focal spot [50]. The lenses are attached to the output facets of two waveguides facing each other [ Fig. 5(a)]. The counterpropagating beams exiting from the waveguides are focused to the same spot by the microlenses to form a trap. The shape of the trap can be readily controlled by changing the lenses' focusing power: tightly focused beams lead to strong confinement along the longitudinal direction, whereas lower-NA microlenses lead to extended traps along the x direction. In our experiment, we designed an elongated trap shape suitable for multiparticle trapping: the beam is tightly localized in the transverse y − z plane and only loosely confined along the longitudinal x direction. The optical intensity distribution corresponds to a FWHM spot size of 12.2 µm (x ) × 1.6 µm (y ) × 1.6 µm (z). This intensity profile creates a single stable trap site in the y − z plane while permitting multiple particles to be simultaneously trapped along the longitudinal x direction (Fig. S10).
We experimentally validated dual particle trapping using the microlens-based tweezers [ Fig. 5(c), Visualization 6], which can serve as a simple platform for investigating the interactions between the two particles. Figure 5(d) plots the statistical distribution of the relative displacement between the two trapped particles measured over a 5 min period, with the corresponding probability distribution functions (PDFs) of D x and D y (relative displacement along the x and y directions, respectively), shown in Figs. 5(e) and 5(f ). The histograms reveal that D y follows a Gaussian distribution due to the single trapping site defined by the transverse optical confinement. In contrast, the dispersion of D x cannot be accurately described with a Gaussian distribution. To elucidate this observation, we consider two types of interactions acting along the longitudinal x direction: the optical attractive force coming from both the radiation pressure force and the gradient force, which attracts the particles toward each other, and the counteracting double-layer repulsive force [51]. The probability distributions of D x and D y are given by p( where U (D i ) is the potential energy. Here the index i takes value x or y . The potential energy (per unit optical power) along the y direction is a harmonic potential U (D y ) = 1 2 κ y D 2 y , where κ y is the trap stiffness. The potential energy in the x direc- , where κ is the Debye parameter and D x ,m gives the interparticle spacing when the potential energy is minimum. By fitting the model with the experiment data, we obtain κ y = 8.27 k B T/µm 2 = 0.033 pN/µm, κ = 3.26 µm −1 , D x ,m = 5.11 µm, and F attr = 32.29 k B T/µm = 0.13 pN. This model fit is in agreement with the measured results as illustrated in Figs. 5(e)-5(h).
An added benefit of the on-chip micro-optical tweezers is that light backscattered into the feeding waveguides can also be employed to monitor the movement of the trapped particles in real time [20]. Taking the refractive optical tweezers as an example, displacement of the particle from its equilibrium position along the x direction breaks the mirror symmetry and induces a large difference in reflected light intensity received by the two waveguides. Our FDTD simulation results suggest that the particle displacement can lead to a reflection difference as large as 25% (Fig. S11), which offers a convenient technique for monitoring the motion of the trapped particle with high precision. The measured center-to-center displacement between the two trapped particles in the x and y directions. Three thousand experimental data points were collected over a 5 min measurement duration. (e), (f ) Extracted probability distribution histograms of the x and y spacings between the particles. The optical attractive force between the particles F attr , double-layer repulsion force F DL , and optical gradient force F grad are schematically illustrated as an inset. (g), (h) Potential energy profiles along the x and y directions computed from the data in (e), (f ). The red curves in (e)-(h) are theoretical predictions based on our particle interaction model.

CONCLUSION
In summary, we proposed and experimentally validated freeform micro-optics-based optical tweezers as a new platform for on-chip trapping and manipulation. Leveraging the ability to shape the free-space wavefront on demand by freeform micro-optics, we show that 3D gradient optical fields can be created and engineered to trap one or multiple suspended particles using a chip-scale platform. Reflective and refractive micro-optical designs enabling low trapping power and interparticle interaction analysis were demonstrated. The micro-optical tweezers uniquely combine versatile configuration, high optical efficiency, compact footprint, broadband operation, and integration scalability, qualifying it as a promising platform for a plethora of applications such as on-chip sensing, cell analysis and assembly, microscale dynamics study, and atom/ion trapping and cooling.