Light-momentum-driven soft optical waveguide micro-actuators

Abstract

Radiation pressure of light allows us to exert forces and trap microscopic objects by highly focused laser beams in the form of optical tweezers. In principle, light also exerts mechanical forces in curved optical waveguides as photons alter their direction and momentum. While these optical forces are typically negligible due to the rigidity of conventional materials, we demonstrate that soft photopolymer optical waveguides can undergo significant mechanical deformation driven by light momentum. Using two-photon polymerization direct laser writing, we fabricate micron-scale curved nanowire waveguides that exhibit large-scale movements when guiding light. An analytical optomechanical model accurately predicts the observed shape deformations and their power-dependent behavior. Our findings pave the way for the development of light-driven micro-actuators, offering fresh opportunities for photonic and microelectromechanical systems.

Optical force emerges when photons change momentum while altering their propagation direction inside curved optical waveguides. The authors demonstrate significant deformation of flexible waveguides induced by the forces exerted by guided light.

Introduction

Optical waveguides are increasingly utilized in a wide range of applications across various fields, including photonic integrated circuits or photonic sensors^1–3. Like optical fibers, which can be bent as needed while transferring light to long distances, the mechanical flexibility of smaller-sized optical waveguides is critical in some scenarios. Flexibility on the millimeter and centimeter scale is essential in the case of implantable/wearable waveguide sensors^4–11, biointerfaces¹² or soft robotics applications^13,14, to prevent damage and ensure proper functionality during operation. Aside from that, there are optomechanical waveguide applications that rely on micrometer or nanometer-scale size changes and shape distortions. Flexible waveguide-based systems were employed for ultrasound detection¹⁵, force and strain¹⁶ or pressure¹⁷ sensing. Besides the passive systems mentioned so far, there are flexible waveguide applications with active shape control. Micro-actuating platforms have the capability to deliberately cause controlled deformation in light-guiding structures, thereby actively modifying waveguide performance. This phenomenon was employed for constructing electro-optical switches¹⁸ or light attenuators¹⁹. Light can induce mechanical deformations in organic crystal waveguides composed of photo-switchable molecules^20,21. Thermally^22–24 and chemically²⁵ induced bending of flexible waveguides and actuators was reported. From a material point of view, deformable waveguides of different stiffness can be prepared from silica or glass^13,18,19, organic crystals^{20–23,25,26}, silicon rubber¹⁴, hydrogels⁸ or polymers^7,24, the later one being important in our present work.

Two-photon polymerization direct laser writing (TPP-DLW) is a convenient and powerful technique to fabricate microstructures with an extensive variety of shapes^27,28. The typical spatial resolution achievable with TPP-DLW is approximately 100 nm. As the optical properties of many photoresist materials allow for light propagation, TPP-DLW was utilized to prepare wave guiding systems, such as optical interconnects²⁹, graded index waveguides³⁰ or fiber to chip couplers³¹. Small optical waveguides connected to a set of spheroidal handles were employed in optically trappable micro-illuminator tools³².

At this point, it is important to highlight the mechanical properties of photopolymerized microstructures. A noticeable decrease in the Young’s modulus was observed when the size of objects (immersed in different liquids) was lowered to the micrometer/nanometer range, as compared to the modulus value in bulk^33–36. Various non-waveguiding applications take advantage of the low microstructure stiffness, including micro-mechanical logic gates³⁷, single cell manipulators³⁸ or flow sensors³⁹. The viscoelastic properties of used polymer materials affect the dynamic behavior of flexible microstructures^40–42. The remarkable flexibility of photopolymer nanowires was utilized for viscosity measurements⁴³ through analyzing the Brownian fluctuation dynamics⁴⁴ of spherical beads anchored to them.

Photopolymer nanowire-based geometries (helical springs^33,45, straight rods^38,41,42,45 or torsional strings³⁸) are readily deformed by pico-Newton-range optical forces reachable in optical tweezers. In previous experiments, the nanowires were deformed through optically trapped microbeads attached to them. The trapping force in optical tweezers emerges when photons diffracted from micro-beads change their propagation direction, with the overall change in photon momentum resulting in the desired force action. In contrast to light scattering, optical waveguides can change the direction of photon propagation in a highly controlled manner, allowing much higher utilizations of the available momentum flow for force generation^46,47. While reversed propagation is the maximum change in photon momentum that can be achieved in a single back-reflection event, curved waveguides may go beyond this limit when aiming to create distributed force action along a specific path. In helical waveguides, for example, the direction of photon propagation can change continuously in the whole length of the waveguide, without any theoretical limit on the number of turns and the total internal optical force created. Therefore, such systems are attractive from the momentum-usage point of view, raising the question: Can optical forces induced by light propagating inside of curved waveguides cause significant deformations in these structures?

In this paper, we address the above question by fabricating curved nanowire optical waveguides (using TPP-DLW of OrmoComp photoresist) and investigating light propagation and mechanical light–matter interactions in them. Our primary goal is to optimize the waveguides and find conditions for the highest deformation effects induced by optical forces. An analytical waveguide bending model is presented to predict and explain experimental observations.

Results

Theoretical considerations

Consider a semi-circular optical waveguide of a radius R₀, fixed at its input side to a static support and free at the opposite end, as illustrated in Fig. 1a. The momenta of guided photons vary as light travels through the waveguide, resulting in a distributed optical force directed outward from the half-circle. This force action, under low-stiffness waveguide conditions, distorts the waveguide; the half-circle gradually straightens as the guided light power is increased. A theoretical model is presented in the “Methods” section to explain the soft waveguide deformations under these conditions. The deformed nanowire shape is obtained by solving three equilibrium equations that relate the bending moment and the forces acting inside the curved waveguide to each other. The optical force is perpendicular to the waveguide axis. Based on the model results, this section identifies the mechanical requirements for achieving significant waveguide deformations.

Fig. 1. Theoretical aspects of deformable optical waveguides.

a The scheme of an undeformed half-circular waveguide attached to a static support on the input side. b Theoretical flexural rigidity values for which a half-circle waveguide opens to α = 90° (see the inset), plotted against the initial waveguide radius R₀ for three different guided light powers (5, 10, and 20 mW).

The stiffness of the waveguiding structure is dependent upon its flexural rigidity, defined as the product of the material’s Young’s modulus E and the nanowire’s second moment of inertia I. The flexural rigidity, together with the waveguide radius and the guided light power, dictates the magnitude of waveguide deformations. Achieving extremely low flexural rigidities while maintaining wave guiding capability may pose challenges. Consequently, it is essential to determine the range of E *I values suitable for substantial waveguide deformations. The practical radii of half-circular waveguides range from 5 to 50 µm, with the attainable guided light powers in our experiments being approximately several tens of milliwatts. To quantify the extent of the semi-circular waveguide deformation, we define the deflection angle α, which is the angle between the line segments that connect the static nanowire base with the free end of the nanowire before and after deformation (see the inset in Fig. 1b). We target opto-mechanical conditions that permit half-circles to open to a minimum of α = 90°, as illustrated in the inset of Fig. 1b. Our curved cantilever bending model was used to find the flexural rigidities for which the waveguides deform to this extent; the results are presented in Fig. 1b for various initial (undeformed) half-circle radii and guided light power values. The threshold flexural rigidity increases with the square of the radius. Consequently, larger radius half-circles are more advantageous for deformation applications, as they permit greater flexural rigidity. A linear relationship exists between the guided light power and the required flexural rigidity.

To fabricate self-deforming optical waveguides via TPP-DLW, it is essential to identify the appropriate photoresist material and establish the polymerization conditions that produce the flexural rigidity range indicated in Fig. 1b. In addition to possessing suitable mechanical properties, the fabricated structures must enable efficient light propagation. The subsequent two sections illustrate that soft curved nanowires fabricated from OrmoComp can be optimized to satisfy both the mechanical and optical criteria for self-deforming waveguide applications.

Optimized mechanical properties of soft nanowires

The mechanical properties of OrmoComp nanowires, produced by TPP-DLW (see Fig. 2a), were examined in a separate experiment to identify optimal polymerization conditions that yield suitable candidates for deformable waveguide applications. The flexural rigidity of the studied water-immersed straight nanowires prepared with different polymerization laser powers was determined by two different techniques.

Fig. 2. Material properties of OrmoComp nanowires prepared by two-photon polymerization direct laser writing (TPP-DLW).

a The TPP-DLW process. b Brightfield image of a 30 µm long cantilever used to measure the flexural rigidity of photopolymer nanowires. The scheme depicts the force of the calibrated optical trap F_T exerted on the end-sphere attached to the nanowire, where x is the deflection of the end-sphere. c The bent cantilever prepared at 7 mW laser power as modeled in COMSOL Multiphysics for F_T = 30 pN force applied to the sphere. The color bar indicates the von Mises stress in N/m² units. The laser trap is shown schematically. d The optical trap force (F_T) plotted against the nanowire deflection (x) as measured for nanowires prepared with different polymerization laser powers. The curves were fitted with linear functions (indicated by dashed lines) to evaluate the cantilever stiffness. e Probability distributions of the fluctuating end-bead positions measured for cantilevers prepared at 6, 7, and 8 mW polymerization laser powers. The solid lines represent Gaussian curve fits to the data. f The flexural rigidity (E*I) of the studied cantilevers was evaluated from the bending and the fluctuation measurements. g The thickness and the height of OrmoComp nanowires prepared at 12 µm/s speed. Interpolated data from the Supplementary information of ref. ³⁸ are shown. h The Young’s modulus as a function of the polymerization laser power. i Representative model of the nanowire bundle used for Raman measurements. j Typical Raman spectrum of OrmoComp nanowires. The insets show the 1412, 1451 , 1635, and 1726 cm⁻¹ Raman peaks after background subtraction. The data measured for different polymerization laser powers were normalized on the left and right insets by the 1412 and 1726 cm⁻¹ peak intensities, respectively. k The intensity ratios of the (1451 cm⁻¹)/(1412 cm⁻¹) and (1635 cm⁻¹)/(1726 cm⁻¹) Raman lines plotted against the polymerization laser power. The error bars represent the standard error of the mean value calculated from four independent data sets.

First, well-controlled forces were applied to the end of straight cantilevered beams (Fig. 2b, c) by means of calibrated laser tweezers. A spherical bead polymerized to the end of the nanowires was pulled by a laser trap in a direction perpendicular to the beam axis. In the small deformation limit, the beam end deflection (x) is a linear function of the applied trapping force (F_T): x = F_T/K (Fig. 2d). The obtained beam stiffness K can be used to calculate the flexural rigidity (E *I): K = 3EI/l³, where l is the cantilever length. We assume here that the nanowire interior is homogeneous and is characterized by a single Young’s modulus (E). Given the possible differences between the different regions on the nanowire cross-section, denser and stiffer center part vs. weaker side region, the used Young’s modulus represents an effective value. The second moment of inertia I is calculated for a nanowire with elliptical cross-section: I = πt³h/64, where t and h are the thickness and the height of the ellipse, respectively. The dimensions of the nanowires prepared from OrmoComp at different polymerization laser powers and line drawing speeds were determined in our previous work (see the Supplementary data of ref. ³⁸). Interpolated dimension values are used in the present calculations.

The second method used to determine the flexural rigidity relies on the fluctuations of the microbeads attached to the cantilevers. The probability density distributions of the freely fluctuating structures were fitted by Gaussian distributions as shown in Fig. 2e. The equipartition theorem relates the distribution variance (σ²) to the structure stiffness, σ² = k_BT/K⁴⁸; the camera blur effects are neglected.

The experimental results are shown in Fig. 2d and e for the bending and fluctuation experiments, respectively. The flexural rigidity values determined by the two measurement methods are plotted together in Fig. 2f as a function of the polymerization laser power. As these results show, flexural rigidities as low as 0.25 × 10⁻¹⁹ N m² are achievable, which aligns with the upper segment of the desired rigidity range established through calculations (Fig. 1b).

The thickness and the height of the nanowires (Fig. 2g) are used in the next step to calculate the effective Young’s modulus that characterizes the elastic material properties of the nanowires prepared at different laser powers. The results are plotted in Fig. 2h. The effective Young’s modulus was found to vary in the range of E = 4-11 MPa, depending on the polymerization laser power, which is in good agreement with the values measured for OrmoComp nanowires by other methods^38,42. The discrepancies between the fluctuation and bending techniques may stem from inaccuracies in optical trap calibration due to the cantilever’s influence on the bead trapping force.

The obtained Young’s modulus is about a hundred times lower than reported for bulk OrmoComp (manufacturer data, and ref. ⁴⁹), paving the way towards self-deforming light-guiding structures. Previously, the lowered elastic moduli of nanosized polymer objects immersed in different solvents were explained by the limited degree of polymer cross-linking and the enhanced permeability of the polymer network^50,51. Raman spectroscopy was used to measure the degree of conversion in two-photon polymerized structures, which correlates well with the Young’s modulus of the material^52–54. The Raman spectra published for OrmoComp structures prepared under different conditions and by different groups are not identical^53,55,56, showing minor deviations from each other. It is important to note that in our present work, we focus on OrmoComp nanowires prepared at near-threshold polymerization conditions. The polymerization threshold power is around 3.5 mW (under this power, no nanowires were found in the sample after washing off the uncured Ormocomp). Raman spectroscopy is applied to get a quantitative measure that reflects the power-related changes of the effective Young’s modulus. For this purpose, a bundle of straight nanowires was polymerized between two supports as shown in Fig. 2i. The obtained Raman spectra reveal a correlation between the polymerization laser power and the intensity ratios of the 1635 and 1726 cm⁻¹ as well as the 1451 and 1412 cm⁻¹ Raman lines (Fig. 2j,k). The analyzed intensity ratios exhibit a similar saturation pattern at higher laser powers as observed in the Young’s modulus. Based on the proposed structural formula of the OrmoComp precursor^55,57, the Raman bands at 1635 and 1726 cm⁻¹ are assigned to the vibrational modes of the C = C (vinyl) and C = O (carbonyl) groups of the monomer units, respectively. With enhanced polymerization, as power rises, the concentration and consequently the signal intensity of the CC double bond decreases as expected^52–55. Here we present a second intensity ratio, which involves the Raman lines at 1451 and 1412 cm⁻¹. The covalent cross-linking of the base polymer affects the entire molecular structure, establishing a clear relationship between the CH₃/CH₂ (1451 cm⁻¹) and S–CH₂ (1412 cm⁻¹) vibrations and the degree of polymerization (Fig. 2j,k).

Detailed analysis of the presented data reveals essential information for optimizing the used nanowires for self-deforming waveguides. Importantly, the Young’s modulus measured on nanowires is orders of magnitude lower than reported for bulk conditions. In the studied polymerization laser power range (from 6 mW to 9 mW), the Young’s modulus changes by about a factor of two (Fig. 2h). At the same time, higher polymerization laser powers cause the height and thickness of the nanowires to increase (Fig. 2g), which has a significant impact on the second moment of inertia of the nanowire cross-section. These two effects combine to produce a dramatic change in flexural rigidity, which increased by more than tenfold when the polymerization laser power was increased from 6 mW to 9 mW (Fig. 2f). Flexural rigidity is critical when curved nanowire waveguides are bent by guided light; therefore, the polymerization laser power must be carefully chosen to ensure self-supporting nanowires while avoiding excessive stiffness. Our results show that the 6.5–7.5 mW polymerization power range is best suited for the polymerization scanning speed of 12 µm/s, resulting in self-supporting nanowires with flexural rigidities between 0.5 and 1.2 × 10⁻¹⁹ N m².

Optimized light propagation in curved nanowires

As previously discussed, curved waveguides with larger radii are advantageous as they can be more rigid while still undergoing significant shape deformation with the same guided optical power. Still, making waveguide actuators as small as possible may have practical benefits, particularly when considering lab-on-a-chip devices. However, smaller radii may also limit the light-guiding efficiency of nanowires. Small curvature radii cause light coupling from the core modes to the surroundings, which leads to unwanted power loss. To optimize the nanowires for self-deforming applications, it is important to understand the details of light propagation inside curved structures. Two types of nanowire waveguides were prepared for transmission measurements. In the first case, a set of semi-circular nanowire arches with different radii was polymerized on a glass substrate (Fig. 3a). These half-circles, assigned as type-I structures, are polymerized in the plane perpendicular to the glass surface. The transmission of the light passing through the arches was measured (see the “Methods” section for details). At the same time, the electric field distribution inside the arches was simulated numerically using the Wave-optics module in COMSOL Multiphysics. The computation results obtained for three different curvature radii are shown in Fig. 3b. Excessive losses are demonstrated for the smaller structures. A large portion of light does not couple into the 7.5 μm radius arch and propagates as a diverging beam in water. Even the light that enters the small-radius waveguide loses its intensity continuously along the beam. By contrast, light readily couples into and propagates through the larger arches. Figure 3c shows a comparison between the experimental and simulated transmission data. Light guiding ability diminishes for arches with radii below 5 μm. For waveguide radii above 15–20 μm the transmission is mostly limited by the coupling efficiency at the entrance side and saturates at about 50–60%. The good agreement between the measured and simulated transmission values indicates a low level of absorption and scattering losses, as these are not included in the numerical model. We assume that the slightly lower simulated values are mostly caused by differences in experimental and modeling beam characteristics at the arch entrance. The effects of non-homogeneous refractive index distribution within the waveguide and the deviation from the ellipsoidal voxel shape could also be the cause. As described in the Methods part, due to the ellipsoidal TPP-DLW voxel shape, the cross-section of the type-I nanowire arches changes along the beam. This fact, however, seems to have a minor effect on the light-guiding ability of the large radius waveguides.

Fig. 3. Light propagation in curved nanowires.

a The scheme of half-circled waveguides polymerized in a plane perpendicular to the glass surface (type-I). b The electric field instantaneous absolute value calculated in COMSOL Multiphysics for the central plane of type-I nanowire waveguides with radii of 7.5, 12.5, and 17.5 μm (λ₀ = 594 nm). The outer shell represents the perfectly matched absorbing boundary layer. The color bar is in arb. units. c Comparison of the measured and simulated transmission data for type-I waveguides. The error bars represent the standard deviation of the measured data. d The schematic view and a SEM image of a half-circled waveguide polymerized in a plane parallel to the glass surface (type-II). e The electric field absolute value calculated in COMSOL Multiphysics for a type-II half-circled nanowire waveguide with 30 μm radius (λ₀ = 1070 nm). Inset: The distribution of the electric field along the type-II waveguide center (R = 30 μm). f The scheme of the cantilevered semi-circular waveguide used for deformation experiments. g Theoretical flexural rigidity values plotted from Fig. 1b. The colored regions represent the parameter range achievable in our experiments. The open white circle displays the fabrication conditions of the used self-deforming waveguides.

In the second set of experiments, the transmission was measured in a 30 μm radius half-circle nanowire oriented in parallel to the support glass surface (type-II structures) as shown in Fig. 3d. In this case, the nanowire cross-section is identical at all points. Two solid coupling parts (plotted in red in Fig. 3d) were used to guide the laser beam entering the sample “from below” into- and out from- the horizontal nanowire. The total transmitted signal was 4 %, which is much lower than for the type-I structures. The electric field data calculated for light passing through the nanowire part of type-II structures (see Fig. 3e) indicate that the light intensity stays nearly constant within this arc. Consequently, the low transmission value measured for the whole structure, including the coupling parts (about 4%), is due to inefficient coupling. Two extreme possibilities are distinguished: (i) light losses arise predominantly on the entering side; only 4% of light is guided in the nanowire, (ii) identical losses arise in the input and output coupling elements; 20% of light is coupled into the nanowire in this latter case. The real percentage of light coupled into the nanowire, which is assumed to fall between these two extremes, determines the highest possible guided power that can be reached in type-II structures. As the self-deforming waveguides developed in this work derive from the type-II structure, all this will be of high importance in the next section.

Self-deforming curved optical waveguides

Based on the findings from the preceding sections, we propose self-deforming waveguides capable of extensive deformation. Microstructures featuring a semi-circular cantilever design (Fig. 3f) were fabricated by excluding one of the two coupling elements from the type-II structures. Figure 3g integrates and summarizes the findings from the previous sections. The theoretical results for opening the semi-circular waveguide to 90^o (taken from Fig. 1b) are overlapped by the parameter regions achievable in our experiments: (i) the lowest possible flexural rigidity is about 0.25 × 10⁻¹⁹ N m² (see Fig. 2f); and (ii) the smallest radius for efficient light guiding is approximately R₀ = 20 µm (see Fig. 3c). We chose to use self-deforming waveguides with a 30 µm radius, fabricated at a polymerization laser power of 7.4 mW and a scan speed of 12 µm/s. The estimated flexural rigidity of these nanowires (1.1 × 10⁻¹⁹N m²) is represented by the white open circle in Fig. 3g. Obviously, the guided laser power required to “open” the half-circles must be around 20 mW. Given the maximum laser power at the sample (300 mW in our experiments) and the input coupling efficiency (4-20% estimated for type-II waveguides), we can expect 12–60 mW of guided power in the waveguide.

The experimental findings obtained with the fabricated waveguides are presented in Fig. 4a and in Supplementary Movie #1. Remarkable self-deformation of the semi-circular structures was observed, which intensified with the increase in input laser power. At the highest excitation power, the waveguide end was displaced to a distance twice the diameter of the initial semicircle arc. For quantitative measurements, we optimized light coupling into waveguides at the highest laser power. The deformation response was maximized by moving the in-coupled laser focus in three dimensions. After optimized coupling, the optical power-dependent deformations were measured by lowering the laser power in discrete steps and evaluating the deflection angle α, as shown in Fig. 4a. The outcomes are plotted in Fig. 4b as a function of the input laser power. The highest deflection angle attained was 105°. A thorough examination of the experimental data (refer to Fig. 4b) indicates a slight permanent deformation as the light intensity is gradually reduced, and the waveguide reverts to its relaxed position.

Fig. 4. Self-deformation of curved nanowire waveguides.

a Overlapping images of the deformed half-circle waveguide at different input powers. b The waveguide deflection angle α, plotted against the input power. c Comparison of selected experimental nanowire shapes (thick gray lines) with the results of the basic model (blue lines). The gray lines represent the average shape of six deflection measurements. d The moment-curvature dependences used in the basic (blue squares) and non-linear (red circles) models. e Comparison of selected experimental nanowire shapes (thick gray lines) with the calculated results of the non-linear model (red lines). f The calculated light power within the waveguide is plotted as a function of the experimental input laser power. The blue squares and the red circles belong to the basic and the nonlinear model. The slope of the fitted linear dependence (dashed line) estimates the light coupling efficiency at low-power conditions to be 4.7%. g The distribution of the bending moment M, axial force F, and shear force V along the waveguide. L is the waveguide length; s is the curvilinear abscissa; the waveguide free end is positioned at s/L = 0. h Computed deformation of a three-quarter circle waveguide. The input power increases as the color transitions from dark blue to light blue, orange, and red. The inset depicts the optical forces generated in the “orange” waveguide configuration. i Overlapping images of the deformed three-quarter circle waveguide at different input powers.

The curved cantilever bending model presented in the Methods section was used to describe the experimental waveguide deformations. The model accounted for minor deviations from the ideal half-circular profile by using the waveguide’s actual initial shape derived from experiments. Figure 4c depicts the calculated waveguide shapes (blue lines) for various light powers assumed, which are superimposed on selected experimental curves derived from experimental images (gray lines). The only free parameter in the basic model was guided laser power, which was adjusted to match experimental deflection angles. In general, the nanowire shape changes are well reproduced by the proposed model; differences occur only at higher input powers, where the model predicts a slightly different s-shape for the deformed waveguide. This later observation points to the limits of the proposed model, which is based on a linear moment-curvature relation, as expressed by Eq. (4) in the Methods section. For the “basic” model version, the results of which are plotted in Fig. 4c, the flexural rigidity interpolated from the bending experiments (EI = 1.1 × 10⁻¹⁹ N m², see Fig. 2f) was introduced into Eq. (4). The blue squares in Fig. 4d represent the corresponding linear moment-curvature dependence. We hypothesize that deviations from this linearity may account for the differences between measured and calculated waveguide shapes. Indeed, nonlinearities emerging from geometrical and constitutive effects have been reported to result in nonlinear moment-curvature dependence of hyper-elastic slender beams⁵⁸. In this study, the moment-curvature dependence was empirically modified to improve agreement with our experiments. We assumed that the waveguide stiffens under extreme deflection conditions. The red circles in Fig. 4d were fine-tuned step by step until the non-linear model with the modified moment-curvature dependence accurately reproduced the experimental waveguide shapes (Fig. 4e).

Figure 4f shows the light powers used in the model plotted against the experimental input power values. Both the basic and nonlinear models show that the efficiency of light coupling into the waveguide changes when the nanowire shape is modified. Higher power values result in more efficient coupling, which can be attributed to the fact that the position of the focused laser beam entering the coupling element was optimized for the highest deflection at the highest power. The linear fit of the calculated vs. input power data in the low-power limit gives an estimate of 4.7% for light coupling efficiency (dashed line in Fig. 4f). This value falls within the limits established in the previous section, supporting the validity of our model assumptions.

The modeling results provide information about the distribution of bending moment, shear force, and axial force along the waveguide. Figure 4g shows the calculated distributions for the data in Fig. 4e. The waveguide free end is positioned at s/L = 0. The bending moment absolute value increases monotonically towards the anchored end at all the studied light powers. The axial force has a pronounced maximum in the nanowire center part, when the power is the highest and the nanowire is nearly symmetrically s-shaped. The highest axial and shear force values are roughly 70 and 110 pico-Newtons, respectively.

The guided light intensity was repeatedly toggled on and off in Supplementary Movie #1, while maintaining a consistent on-power level. Obviously, the waveguide accommodates abrupt power changes on the order of several seconds, which can be attributed to two effects. First, the viscosity of the surrounding water counteracts the overdamped relaxation motion. Second, the waveguide itself has viscoelastic properties, which also have a significant impact on motion dynamics⁴². A detailed description of the system dynamics exceeds the scope of this work.

The curved cantilever bending model was also used to predict light-induced deformation of other waveguide configurations, besides the half-circle. Perhaps most interestingly, when the circular shape was extended to three-quarters of a circle while keeping the radius constant, an interesting phenomenon appeared. Above a certain light power value, the straightened waveguide started to turn back, as shown in Fig. 4h by the gradual orange-to-red shape transition. This behaviour is specific to guided-light-induced deformations. It can be explained by considering the optical forces arising in the s-shaped nanowire, as schematically depicted in the inset of Fig. 4h. In the waveguide part closer to the anchored end, the curvature of the nanowire is reversed (becomes negative) when compared to the original curvature. In this region, the optical forces (blue arrows pointing left) act against the forces driving the free end of the waveguide (red arrows pointing right). Above a certain power level, the reversed forces become dominant, and the waveguide begins to move backwards.

Figure 4i and the Supplementary Movie #2 show the experimental proof for the reversed waveguide deformation at high powers. The observed deformations of a three-quarter circle waveguide closely match the model predictions. Due to imperfections in manufacturing, the undeformed shape of the waveguide is not exactly circular, which was accounted for in the model. A precise overlap between the calculated and experimental waveguide shapes was not achieved, which may be related to the minor out-of-plane deformations observed in the experiments; see the slightly defocused images in Fig. 4i.

Discussion

Curved nanowire waveguide structures fabricated via TPP-DLW from OrmoComp demonstrate significant deformation when light propagates through them. The light power required for deformation is on the order of a few tens of milli-Watts. The primary parameter facilitating deformations, the flexural rigidity, is ~1 × 10⁻¹⁹ N m². This low stiffness is achievable only because the Young’s modulus of nanowires is one hundred times lower than that of bulk OrmoComp.

Self-deforming waveguide structures offer an efficient method for converting photon momentum changes into force action. In the case of optical tweezers, the efficiency of photon momentum usage can be expressed by the dimensionless factor Q defined as: F_opt = QnP/c₀, where F_opt is the force, P is the light power, c₀ is the speed of light in vacuum, and n is the medium refraction index⁵⁹. The Q values accessible by optical tweezers are approximately on the level of Q = 0.5, while Q = 2 is the highest possible factor that can be achieved by simple (back)scattering events. In the case of curved waveguides, the force action is distributed along the waveguide arc, allowing the momentum change to be utilized more efficiently. For example, for light passing through a half-circle-shaped nanowire section, the photon momentum vector is reversed, resulting in a net force determined by Q = 2. At the same time, the apparently competing forces from the two opposite ends of the half-circle, which cancel out from the net force, jointly contribute to the arc opening. One can integrate the force density p = dF_opt/ds along the curved waveguide to define an effective factor, Q_eff, as follows: $\int p\mathrm{d}\mathrm{s}=Q_{\mathrm{eff}}nP/c_0$. This way, for the half-circle waveguide section, we get Q_eff = π. It is important to note that there is no theoretical upper limit for Q_eff in helical waveguides.

We conclude that the strongly deforming waveguides described in this paper are promising for light momentum applications and could serve as the foundation for future opto-mechanical micro-actuator systems. For example, the presented micro-actuators could be used in surface-mounted cell trapping microstructures³⁸, which rely on their elasticity to grip cells. The current version of these devices was designed to be operated by optical tweezers, but this could be avoided by incorporating waveguide actuators into them, making their operation simpler and more suitable for use in lab-on-a-chip devices (since only a low NA objective would be required).

Alternative materials, distinct from Ormocomp, may be investigated to further reduce the Young’s modulus. This would allow for a larger waveguide cross-section, which would significantly increase light propagation efficiency and allow for higher forces. The low thickness of the current waveguides limits them to operate in single-mode or possibly few-mode regimes. The presence of higher-order guided modes within larger cross-section waveguides would have no effect on the conclusions drawn from the curved cantilever bending model, which only considers total guided power.

Methods

Waveguide fabrication and testing

The TPP-DLW apparatus was described in detail earlier³⁸. Shortly, the beam of a 785 nm pulsed laser (pulse length = 150 fs, C-Fiber A, Menlo Systems, Germany) was focused into the photoresist drop-casted on a standard microscope slide using a ×40 oil immersion objective (NA 1.3). OrmoComp photoresist was purchased from Micro Resist Technology GmbH. The sample was placed on a 3-axis translator stage (P-563.3CL PIMars XYZ, Physik Instrumente GmbH, Germany). The curved nanowires were polymerized using continuous translator movement in a single line, resulting in a smooth nanowire surface. After polymerization, the sample was rinsed twice in Ormocomp’s developer solution (Ormodev, Microresist GmbH, Germany) for 10 min each, followed by another two 10-min rinses in ethanol. During the second developer bath, the entire sample was illuminated with a microscope mercury lamp (HBO50) to aid in post-polymerization.

The mechanical properties of nanowire waveguides were tested by measuring the deflection of straight cantilevers equipped with a spherical bead at their free end (see Fig. 2b and c). The 30 µm long cantilever nanowires were polymerized with 6–9 mW laser powers at a constant 12 μm/s scan speed. The nanowires’ thickness and height were obtained by interpolating the data published in our previous work³⁸. The deflection force was exerted by trapping the bead in a calibrated optical tweezer setup built on a wide-field fluorescence microscope (Nikon Eclipse Ti) equipped with a 1070 nm trapping laser (YLM-10-1064-LP, IPG) and moving it perpendicular to the cantilever. The microstructures were observed in brightfield. Correlation-based image analysis routines were used to determine the position of the trapping sphere and its displacement from the trap center. The trap stiffness was calibrated by measuring the fluctuations of non-tethered spheres. In addition to the optical-trap-based deflection experiments, the stiffness of the cantilevers was assessed by analyzing the fluctuations of the attached microbeads, which were monitored through video tracking at 400 frames per second.

The Raman spectra of OrmoComp nanowires, fabricated under near-threshold conditions, were recorded in a bundle of ~50 wires, horizontally separated by 1 μm and vertically by 2 μm (see Fig. 2i). The bundle design improved the signal detected by a Renishaw inVia Raman microscope excited at 532 nm. The nanowires were polymerized at 12 μm/s scan speed using laser powers ranging from 5 to 9 mW.

The optical quality of curved waveguides was determined by measuring light transmission through a series of half-circle-shaped nanowires with varying radii. Two types of nanowires were used. The micro-arches polymerized in the plane normal to the support glass surface (type-I) are depicted in Fig. 3a. These structures were polymerized with 6 mW laser power at 25 μm/s scan speed. The half-circles situated in the plane parallel to the glass surface (type-II) are depicted in Fig. 3d. In this case, the waveguides were attached to the substrate via a pair of solid coupling components that directed the light into the nanowire. The type-II nanowires used for transmission measurements were prepared with 7.4 mW laser power at 12 μm/s scan speed. It is noticed that the vertically elongated ellipsoidal voxel shape (the minimal polymerized volume) results in differences between type-I and type-II half-circles. The cross-section of the type-I structures varies along the beam, exhibiting a circular shape at the two bases and an elliptical form at the arch top. In contrast, the cross-section of the type-II waveguide is consistently elliptical throughout the entire nanowire. Type-I structures are more suitable for transmission measurements due to their uncomplicated configuration. Nonetheless, type-II structures imply the deforming actuators created in this study (see below). The dimensions (thickness and height) of the nanowires prepared at different polymerization conditions can be estimated using the data presented in the Supplementary Materials of our previous work³⁸.

The transmission of the type-I waveguides was measured with a 594 nm laser beam (Cobolt Mambo) focused into one of the arch arms using a 60x objective (NA = 0.8). To improve light coupling into the waveguide, the effective numerical aperture was reduced by working with a narrow laser beam entering the objective back aperture. The light that passed through the arc was detected in a back-reflected arrangement using a silicon photodiode. The system was calibrated for relative transmission measurements by detecting the signal reflected by a mirror placed above the glass slide. Typically, the average transmission of three arches was determined for each half-circle radius.

The 30 µm radius type-II waveguide transmission was measured by focusing a 1070 nm laser beam into one of the coupling parts with a ×20 objective (NA = 0.4). To achieve the best transmission signal, the laser focus position within the coupling part base was carefully optimized. The transmission signal was evaluated from the light intensity at the microstructure output arm detected by the microscope camera (ORCA Flash 4.0v3 CMOS camera, Hamamatsu, Japan).

Numerical simulations

Numerical simulations of mechanical stress inside bent nanowires and electric field distribution in curved waveguides were performed in COMSOL Multiphysics using Structural Mechanics and Wave Optics modules. The material properties within the nanowires were assumed to be homogeneous. The Young’s modulus was set to the value determined in the deflection experiment. The COMSOL Electromagnetic Waves, Frequency Domain interface was used for electromagnetic field calculations. When light propagation through type-I and type-II nanowires was simulated, the investigated volume consisted of three major parts: the nanowire, the surrounding water layer, and an outer “perfectly matched” boundary layer in which the light was absorbed without back-reflection. A Gaussian light intensity distribution with a width adapted to the experimental conditions was set on the type-I waveguide input side. The simulations of the type-II structures began with a uniform light distribution at the nanowire entrance.

Self-deforming waveguide experiments

The curved waveguides used for self-deforming experiments (Fig. 3f) were derived from the type-II structures (Fig. 3d). The nanowire half-circles were aligned parallel to the glass surface and connected to the substrate through a single coupling part, leaving the other nanowire end free. The 1070 nm laser beam was focused into the coupling part using a 20x objective (NA = 0.4). The laser intensity entering the sample was set from 0 to 300 mW. The nanowire deformation was observed in brightfield. Waveguides with three-quarter-circle geometries (Fig. 4h,i) were also fabricated and tested under identical conditions.

Curved cantilever bending model

A simple model is used to characterize the deformation of light-transmitting nanowires. Our model describes a thin waveguide that is nearly circular in shape (corresponding to the experimental structure’s shape) and deforms only into the plane of its original curvature as light passes through it. Light leakage through the waveguide walls is not considered. Also, the possible force arising at the waveguide end, where the light exits the nanowire, is ignored. This latter assumption includes the symmetry of the spread light with respect to the nanowire end axis and the neglect of the minor force caused by the refractive index difference between the polymer and the surrounding liquid⁶⁰. Finally, the minor stretching of the deformed cantilever is neglected. The model is built using a curvilinear coordinate system. The curvilinear abscissa s represents the distance measured along the waveguide arc (see Fig. 5a). At each point s, the waveguide is characterized by the tangential angle θ(s). The local curvature κ(s), and the radius of curvature R(s), are given by the derivative of θ:

$$\kappa =\frac{1}{R}=\frac{d\theta }{ds}.$$ 1

Fig. 5. Geometric and physical parameters are considered in the thin waveguide model.

a The curvilinear abscissae s, measured along the waveguide. θ(s) is the tangential angle. b A scheme showing a short section of the waveguide. M—bending moment, F— axial force, V—transverse (shear) force, R—radius of curvature, p—optical force density.

Figure 5b shows a short section of the curved waveguide with the axial force F acting along it, the transverse shear force V, and the bending moment M. The light that propagates inside the curved waveguide acts perpendicular to the waveguide axis, producing the force density p.

Three equilibrium equations describe the stationary state. Given that the sum of tangential and radial forces, as well as the sum of bending moments acting on a curved waveguide section, must be zero, we obtain:

$$\frac{dF}{ds}=-\kappa V\frac{dV}{ds}=\kappa F-p\frac{dM}{ds}=V$$ 2

Let us denote the curvature of the waveguide before deformation as κ₀(s) = 1/R₀(s) and the deviation from the original curvature as Δκ(s). Our experiments used nearly circular waveguides, where κ₀(s) is almost constant. The waveguide curvature after deformation is expressed as

$$\kappa \left(s\right)={\kappa }_0\left(s\right)+\mathrm{\Delta }\kappa \left(s\right).$$ 3

In the first linear approximation, the bending moment M is proportional to Δκ. The moment–curvature relation is given by⁶¹

$$M\left(s\right)=EI\mathrm{\Delta }\kappa \left(s\right),$$ 4

where E and I are the Young’s modulus and area moment of inertia, respectively. In a broader context, the bending moment is a nonlinear function of Δκ and must be determined through alternative considerations.

The last quantity needed to describe waveguide deformation is the light-induced force density p. The greater the curvature of the waveguide, the stronger the light will push in the direction perpendicular to it:

$$p=\alpha \left({\kappa }_0+\mathrm{\Delta }\kappa \right),$$ 5

where α is a proportionality factor. By considering Minkowski photon momentum⁶² and its change in a curved waveguide, we get:

$$\alpha =n\frac{P}{c_0}.$$ 6

P is the total light power that propagates in the waveguide, and c₀ is the speed of light in vacuum. For simplicity, we use the refractive index n of the cured photopolymer in our calculations.

After substituting Eqs. (3) and (5) into Eq. (2) and applying Eqs. (4) and (6), we numerically solved the equilibrium equations by integrating from the free cantilever end. The boundary conditions at the free end were set as: $F=V=M=0$. The obtained M(s) dependence was used to calculate the deformed waveguide curvature (see Eq. (4)). The deformed shape of the curved beam was determined by assuming θ = 0 at the fixed cantilever end. The calculations were repeated with various light powers P.

Author contributions

G.B., G.V., and L.K. conceptualized and designed the work. Z.T. and L.K. provided laboratory access, coordinated the work of partners. G.T.I. and J. K. designed and fabricated the micro-structures. G.T.I., C.S., J.K., A.S., G.B., and G.V. performed the experiments, acquired and analyzed the data. C.S. and A.H. performed COMSOL modeling. Z.J. acquired, analyzed, and interpreted Raman data. G.B. developed the theoretical model. G.B. and G.V. drafted the manuscript. All authors reviewed the paper.

Peer review

Peer review information

Nature Communications thanks anonymous reviewer(s) for their contribution to the peer review of this work. [A peer review file is available].

Data availability

The data presented in this study are provided in the Source Data file. Source data are provided with this paper.

Code availability

The algorithms applied in this work can be built following the instructions in the “Methods” section. The codes that support the findings of this study are also available from the corresponding authors on request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-64826-9.