1. Introduction

Micro-objects can easily be trapped, and displaced by optical forces that are generated by highly focused laser beams [1]. In addition, particles with anisotropic shapes can rotate around the laser axis due to unsymmetrical forces that are generated at their surfaces. Such remotely photo-driven micro/nanorotors are unique tools to elaborate mechanical and sensing functions at the micron and nano scales. They require no mechanical contact and no electrical wire.

A first class of synthetic microrotors is fabricated in silica or resins using conventional lithography processes. They have planar geometries. Their optical torques originates from the change of momentum of the light that escapes from their side surfaces [2–4] or from the net forces that are produced onto their surfaces [5]. In addition, based on an actuating technique called scanning laser optical trapping (SLOT), Terray et al. [6] realized pumps and valves. In this technique, micrometer-sized spheres are trapped in a laser beam. By moving the laser beam, it is possible to rearrange micro-spheres into functional structures and to actuate them. Micro-pumps have been realized and flow speeds up to 4 μm/s in a 6μm wide channel have been reached [6]. Ladavac et al. [7] developed another technique to assemble and actuate a collection of dielectric spheres of micrometer size. These particles are trapped by optical gradient forces along an array of light rings which are created by a computer generated hologram. The particles are moved by photon orbital angular transfer of momentum, and a flow speed of 3 μm/s is reached for 3W optical power. Leach et al. [8] used the birefringence of vaterite particles to transfer the momentum of a circularly polarized light. When placed in a channel, the rotation of these circular particles is able to induce a drag force on the fluid, and a fluid speed of 1 μm/s in a 15×15 μm channel is demonstrated. Other optically actuated rotating structures have also been published. Rotation rates up to 0.042 Hz/mW have been recently demonstrated with a 3-wing shuttlecock structure [9]. However there are fewer possibilities to optimize the geometry of these planar micro-motors, although they have the advantage to use the high reproducibility and throughput of microelectronics fabrication techniques. In this paper, we propose another way to actuate micro-structures. We are able to fabricate non-symmetric structures by means of two-photon based fabrication technique. Thanks to the special non-symmetric geometry, we will show that it is possible to directly transfer photon momentum from light to micro-structures.

Our structures belong to a second class of synthetic microrotors. They are fabricated by the two-photon induced polymerization technique that allows the production of polymer micro-objects with arbitrary tri-dimensional shapes [10]. The polymerization occurs only at the focal point of the laser following the two-photon absorption step of photo-initiators. Rotors with complex shapes are directly obtained by scanning the focus along predetermined trajectories. Photo-driven rotations can be induced either from the net optical torque resulting from the complex shape [11, 12] or by driving a movable part with an optical tweezers trap [13]. Recent works have focused on the realization of more complex devices that integrate the optical motor with its laser power [14], and on the demonstration of optical driven micropumps in their microfluidic channels [15–17].

Previously, we have demonstrated that two-photon microfabrication technique can be efficiently implemented with Q-switched Nd:YAG microlasers in replacement to femtosecond laser sources [18]. We have applied this technique to fabricate and study the properties of photo-driven microsensors for microfluidic applications [19–21]. Here, we report on the fabrication of Archimedes micro-screws. Their rotations at the focal point of optical tweezers are observed and analyzed. We show that it can be explained by the photon momentum transfer, and described by a simple model of light reflection on their helical surfaces. Finally, we present a demonstration of a photo-driven micropump in a microfluidic channel.

2. Experimental details

2.1 Fabrication of Archimedes micro-screws using two-photon polymerization

Archimedes micro-screws were fabricated by two-photon photopolymerization using a passively Q-switched Nd:YAG microchip laser (Teem Photonics Inc.) with 532 nm wavelength, 550 ps pulse-width, and 6.5 kHz repetition rate. The laser beam, expanded with an X3 telescope, was coupled into an inverted microscope (Zeiss Axiovert 200), and focused with a microscope objective lens (Zeiss, A-plan x100, NA=1.25). A commercial resin (Photomer 3015 from Henkel) is used with 1% photoinitiator (N ⁴,N ^4'-bis-(4-methoxyphenyl)-N ⁴,N ^4'-diphenyl-4,4'-diaminobiphenyl) specially designed for two-photon absorption. Polymerization occurs in the resin at the focal point with about 1 mW laser power, 1ms exposure time, and 0.12 μm displacement step. A 3-axis piezoelectric stage (Nanocube, Physik Instrumente) translates the sample through the laser focus to produce the required structure. Several hundreds of micro-objects are fabricated on each sample. A few drops of acetone are added to dissolve the unexposed resin and to free the polymer micro-objects in a sealed chamber (5 mm diameter) which is made with two cover slips spaced by a double-faced tape.

Micron-sized Archimedes screw was fabricated by scanning the laser focus along special designed trajectory as shown in Fig. 1 . The voxel-based ladder-like elements are used to fabricate Archimedes micro-screws. A voxel corresponds to a single laser spot illumination and has the shape shown on Fig. 1(a): its radius is approximately Vx_R=300nm and its length Vx_L=800nm. Each ladder-like element has a width Vx_R (azimuthal), a thickness Vx_L (axial, Z), and a length R (radial) which corresponds to the micro-screw radius (R=2μm). Therefore each ladder-like element is approximately constituted of R/Vx_R voxels. This corresponds to 10 laser shots in our case for the presently studied micro-screws. Consecutive ladder-like elements are shifted by an axial increment (ΔZ) and a radial increment (Δθ) such that 65, 95, and 126 of them are respectively required for screw number 1, 1.5 and 2 (Fig. 1(b)). As a result, fabrication of these screws requires total durations of laser illumination of respectively 0.65s, 0.95s and 1.26s. In practice, effective total fabrication time will depend on the electronic control and computer speed. Figure 1(c) shows the shape of the final screws. All three screws have the same dimensions, radius R=2μm and height H=5μm. The only varying parameter is the screw pitch and consequently the number of screws N such that H=Na is kept constant. Figure 1(d) shows a photograph of three fabricated Archimedes screws (screw number is N=1.5) which are floating freely in acetone solution.

 

Fig. 1 (a) Voxel-based ladder-like element used to fabricate Archimedes micro-screws; (b) Trajectory of the laser focal point: each ladder-like element is scanned after an axial increment (ΔZ) and a radial increment (Δθ); (c) Screw composed of elementary ladder-likes; (d) The photograph of three Archimedes screws (screw number=1.5) floating freely in acetone solution.

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2.1 Optical tweezers set-up

The same microscope and objective lens are used for the optical tweezers. A CW Nd-YAG laser at λ=1064 nm provides the trapping beam. The incident laser power on the sample varies up to 200 mW by adjusting the current of the pumping laser diode. During experiments, the rotation of the trapped object is ascertained by direct observation through the microscope ocular or using a video camera. In the microscope image plane, an optical fiber, connected to a photodiode, samples a spot of the image near the center of the rotating object. The light detected by the photodiode is periodically modulated by the image of the object in rotation. An oscilloscope is used to measure the photodiode signal.

3. Experimental results on optically driven rotation of Archimedes micro-screws

A typical behavior of an Archimedes micro-screw being trapped by an optical tweezers is shown in Fig. 2 . When a free-floating screw (Fig. 2(a)) is approaching the laser focus (Fig. 5(b) ), it gets trapped, and aligns its axis in the laser direction. Then, it starts to rotate spontaneously around its axis (Fig. 2(c)). The rotation frequency ranges from a few to tens of Hz, depending on the laser power and screw geometry. The laser-induced rotation is due to the optical torque which is transferred to the object by the laser scattering. After the laser power is switched off, the screw stops rotating and freely leaves away (Fig. 2(d)).

 

Fig. 2 Demonstration of optical trapping and rotation of an Archimedes micro-screw in an optical tweezers. (Media 1)

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Fig. 5 Ω/P_laser versus 1/N (N: screw number). Theoretical curves are plotted for three different values of the laser beam radius, R₀. We notice that experimental data fits well with $R_0=0.5\mu m$.

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The rotation frequency is measured by detecting the modulated light that is transmitted through the micro-screw. Measurements have been performed with micro-screws having three different screw numbers, i.e. N=1, 1.5, and 2. Figure 3 shows that the rotation frequency is linearly proportional to the laser power after a power threshold of about 18mW. In addition, we notice that micro-screws with shorter pitch (higher N) rotate slower than the one with longer one. These experimental results are in good qualitative and quantitative agreement with theoretical predictions that are presented in the next section.

 

Fig. 3 The dependence of rotational frequency when increasing the laser power for micro-screws with screw numbers: 1.0, 1.5, and 2.0.

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4. Theoretical calculation

4.1 Model for the light-induced rotation of Archimedes screw

The light-induced rotation results from the conservation of momentum that occurs when laser photons are reflected on the helical micro-screw surface as shown in Fig. 4(a) . The momentum of the incident photons is ${\stackrel{\rightharpoonup }{P}}_{in}=-hk\stackrel{\rightharpoonup }{z}=-h\upsilon n_s/c\stackrel{\rightharpoonup }{z}$ where $n_s$ is the refractive index of the solution, c the speed of light in vacuum, h the Planck constant, and υ the light frequency, associated with the laser wavelength λ=1064nm. The force $\stackrel{\rightharpoonup }{F}$ which is exerted on the screw surface results from the change in photon momentum from ${\stackrel{\rightharpoonup }{P}}_{in}$ to ${\stackrel{\rightharpoonup }{P}}_{re}$ under the angle of incidence α. Partial reflection at the interface between solution and polymer is taken into account, and photons are reflected with a probability which is given by the Fresnel reflection coefficients on the solution/polymer interface (s/p). Since the screw is rotating and has all orientations, average reflectivity between TE(s) and TM(p) polarizations has to be considered:

 

Fig. 4 (a) laser photon reflected on helical micro-screw surface; (b) the screw decomposed to individual wedges; (c) an elementary unit of wedge; (d) schematic representation of the main geometrical parameters: After a turn elevation changes by the pitch, a whatever the value of r. It is clear that α is a function of r and takes the value α_ο for r=R, i.e. at the border. The total height of the screw is H which corresponds to a rotation of an angle Δθ. This angle can take any value and is here shown as if it were smaller than 2π. For actual screws in this paper, Δθ=Nπ where N=1, 1.5, 2 is the number of turns of the screw.

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The Archimedes screw can be decomposed into individual wedges as shown on Fig. 4(b). As it is clear from this picture the angle of incidence α depends on the position of current point M, more precisely on the distance r between M(r, θ) and the screw axis. An elementary unit of wedge (Fig. 4(c)) is considered to obtain a motion model for the light-induced rotation of screw. α is the local angle of the screw with the horizontal plane. It is also the angle of incidence at this point and is a function of the distance r to the screw axis. The Fig. 4(d) shows the layout of screw length, radius, and height. Let us denote ${\alpha }_R$ the value taken by the angle of incidence $\alpha (r)$at the border of the screw, i.e. for $r=R$. Thus we have $a=2\pi R\tan ({\alpha }_R)$ where a is the screw pitch. $H=Na$ is the total height of the screw and N is the number of screw periods which has already been addressed as screw number. The consequence is that the angle of incidence increases as M becomes closer to the screw axis since we have: $a=2\pi \tan [\alpha (r)]$.

Taking into account the Gaussian profile of the laser beam as well as the α-dependence of the reflection coefficient $R_{s/p}(\alpha )$ is easy but requires a numerical approach. However the aim of this paper is to demonstrate that the screw rotation is actually actuated by the photon momentum of laser and to give orders of magnitude of forces in presence rather that giving an accurate model. Therefore we will make a few assumptions in order to be able to derive simple analytical formulas: we consider the case of a non-polarized collimated laser beam with a constant intensity I, on a disc of radius $R_0$ which may be smaller that R. The total laser beam power is thus $P=\pi R_0{}^{2}I$. The optical force (f_opt) and optical torque (M_opt) produced on the elementary surface drdl by the change of photon momentum Δp of a photon flux (I/hv) with an incidence angle α are respectively given by:

The total torque is proportional to the screw length H. According to the last expression of Eq. (6), H finally disappears and the main geometrical parameters are the beam radius R_o and the number of screw periods, N. The screw pitch also plays an important role through the angle α in the factor of Eq. (5), i.e. $f(t_0)$. If light was shielded by the first screw period, the effective number N would be limited to unity. However, in our case, polymer transmission is high and we assumed that all periods are equally effective. This results in the fact that total optical torque is relatively independent on the screw number N when H is maintained constant. Indeed, the increase of N is almost totally compensated by the decrease of α and thus the decrease of optical torque efficiency on each screw period.

Now, we need to consider the hydrodynamic torque. In the case of photo-driven screw rotating in solution, the standard equation of rotation around an axis is $I\dot{\Omega }=M_{opt}-M_{hyd}$ where $M_{opt}$ is the optical torque given by Eq. (5), and $M_{hyd}$ the hydrodynamic torque. According to ref [22], the symmetry of the screw allows us to derive a general expression for this hydrodynamic torque:

A precise determination of the G parameter which appears in Eq. (7) would require a numerical resolution of Navier-Stokes equations for the actual screw as has been done for instance in ref [23]. for a different although similar problem. However it is beyond the scope of this paper to solve this hydrodynamic problem. According to ref [22], $G=4\pi$ is the rigorous value for a rotating sphere. In our numerical simulations we chose the value $G=2\pi$ which is reasonable since the factor $H/\sin ({\alpha }_R)$ in Eq. (7) already takes into account for the fact that the contact area between the screw and the fluid is longer than for a sphere. In the steady-state regime, the angular rotation speed Ω is given by $M_{opt}=M_{hyd}$. From previous equations it is easy to derive that Ω is proportional to the total laser beam power P and is given by:

4.2 Numerical results of simulations for the three micro-screws

In the case of our experiment$R=2\mu m$and$H=5\mu m$. In agreement with theoretical predictions, the rotation speed has been experimentally found to be proportional to the laser power. The measured ratio Ω/P are respectively 200, 130, and 65 s⁻¹W⁻¹ for screw numbers 1, 1.5, and 2. We plotted on Fig. 5 the theoretical result of Eq. (9) for three different values of the laser beam radius R_o. Experimental data are also shown on the same figure and are found in excellent agreement with the curve for$R_0=0.5\mu m$. Let us insist on the fact that there is no really adjustable parameter in the model, the only point is that geometrical and optical parameters are not known with a very high precision, due to the very small size of the system. Such inherent uncertainties justify that a more accurate model is not relevant at this stage. However, we can derive two interesting conclusions from this work. First of all, the order of magnitude of the predicted rotation speed is indeed in excellent agreement with what is experimentally observed and this proves that, despite the photon momentum being small; it is actually at the origin of the screw rotation. Second, the N-dependence of the rotation speed is also found in good agreement with the theoretical predictions and this also proves that our interpretation is correct.

We notice from Fig. 5 that the N-dependence is expected to be very sensitive to various parameters, for instance R₀. This feature is quite surprising and requires additional investigations and comments, as will be done with the help of Fig. 6 . We will show that several parameters have a strong influence which may compensate or not, resulting in a quite complicated and interesting behavior. First of all, the angle of incidence at the border of the beam is plotted on Fig. 6(a) and we see that variations are very important. As a result reflectivity on the screw surface, plotted on Fig. 6(b) varies quite significant and in addition strongly depends on R₀. However the smaller R₀, the higher the reflectivity, but the smaller the resulting torque because the distance to rotation axis is reduced. These two effects largely compensate such that the total resulting optical torque shows variations of limited amplitude (a factor 2) as shown on Fig. 6(c). Finally the hydrodynamic torque is the main parameter which governs the N-dependence of the rotation speed has shown on Fig. 6(d). The main reason is that it depends on the total curvilinear lengths of the coil screw.

 

Fig. 6 The influence of the different parameters and physical effects is shown here: (a) incidence angle of screw surfaces, (b) averaged reflectivity on these surfaces, (c) total optical torque per unit power, (d) total hydrodynamic torque per unit rotation speed.

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5. Demonstration of photo-driven micropump in a microcannel

The micropump action of a photo-driven Archimedes micro-screw has been demonstrated by inserting it in the open side of a microfluidic microchannel as shown in Fig. 7 . The microchannel has also been fabricated during the same two-photon polymerization fabrication process. It has been made by two vertical walls and an upper cover to obtain a 10×10µm ² cross-section on a cover glass. The upper side of the channel was kept open to insert, by the optical tweezers action, a vertical micro-screw inside the channel.

 

Fig. 7 Schematic of the photo-driven micropump action in a microfluidic microchannel.

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The micropump action is shown in the sequence of photos in Fig. 8 . The time interval between photos is about 3 seconds. The micro-screw (N=1) is rotating at 40 Hz with a driving power of 200 mW, i.e. 0.2 Hz/mW. A micron-size bead is pulled by the micro-pump flow at a speed 1$\mu m/\sec$. Thus, one can estimate the flow rate at 6$pL/\min$for this not optimized geometry. The similar performance is obtained as for pumps [6–8] based on different principles and described in the introduction. However we believe that the present screw is a first realization, and is far from being optimized. Our very versatile fabrication technique will permit other and more efficient structures, although based on the same principle of photon momentum transfer. One advantage could be that this approach does not require light polarization control.

 

Fig. 8 The optofluidic demonstration of Archimedes micro-screw in a micro-channel (sequence: from left to right). (Media 2)

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6. Conclusion

The two-photon polymerization technique has been used to fabricate Archimedes micro-screws that have been designed for photo-driven micro-pump applications. Due to their chiral geometry, they can rotate spontaneously under laser irradiation. This light-induced rotation results from the conservation of momentum that occurs when laser photons are reflected on their helical surface. The optical torque increase with the incident laser power, and with the total reflectivity of the irradiated surface, that has a complex dependency with the micro-screw geometry. An analytical expression is derived to evaluate the angular speed in the steady-state regime. The predicted rotation speeds and its dependency with the screw-number N are in very good agreement with experimental measurements. This photo-driven rotation generates a hydrodynamic torque that has been used to demonstrate a micro-pump action in a microfluidic microchannel. We believe that this work is promising and opens the way to the design and the realization of very efficient pumps.