1. Introduction

One of the fundamental phenomena in optics is refraction, wherein naturally occurring materials obey Snell’s law as a result of having positive refractive indices. However, in the 1960s, Veselago considered a notional material that had a negative refraction and proposed its use as a flat lens.[1] Within the last several years, work on metamaterials[2,3,4,5] and ‘perfect lenses’[6] revived Veselago’s ideas and trigged intense discussions. Meanwhile, negative refraction was also investigated in photonic crystals (PhCs) by engineering their dispersion properties.[7] Along these lines, experiments have demonstrated negative refraction and imaging based on negative refraction by two-dimensional PhC flat lenses.[8,9] More recently, we demonstrated experimentally subwavelength imaging at microwave frequencies with a three-dimensional (3D) PhC flat lens that exhibited a full 3D negative refraction.[10,11] As negative-index materials are rewriting the laws of optics, they may allow for the realization of a lens having flat surfaces, no optical axis, and a resolution independent of working wavelength. These merits will find significant applications in a wide variety of disciplines. Herein, we demonstrate one potential application of the negative-refraction flat lens in a novel technique, optical tweezers, [12,13] where super-resolution and field-curvature-free systems are especially beneficial. Using a full-3D negative-refraction lens, we have been able to demonstrate, for the first time, electromagnetic gradient force trapping, similar to that of optical tweezers, but at microwave frequencies. The negative-refraction flat lens provides a unique mechanism to create a highly focused, strongly convergent beam independent of a single optical axis. This technique allowed us to stably trap and accurately manipulate a variety of neutral dielectric particles by simple movement of an imaged microwave source over a field-of-view not limited by imaging aberrations. The far-reaching implications of this work include a novel mechanism for using gradient forces as arrays of manipulations [14] through a single lens—for example in fluidic systems particles can be manipulated by arrays of sources.

The belief that light carries momentum and therefore can exert force on electrically neutral objects by momentum transfer dates back to Kepler, Newton and Maxwell. However, the radiation force had not attracted much interest until the invention of lasers, which can generate light of extremely high intensity and thus exert a significant force on small neutral particles. This capability enables an unprecedented tool to trap and manipulate small particles ranging in size from the micrometer-scale down to molecules and atoms,[15, 16] as well as to drive specially designed particles as sensitive nano-probes.[17,18] The techniques based on radiation force have found applications in a wide range of fields including biomedical science, atomic physics, quantum optics, isotope separation, and planetary physics. One of the most successful applications is the use of optical tweezers,[12,13] which relies on a single-beam gradient-force trap. In biology, optical tweezers are widely used for their ability to nondestructively manipulate small particles ranging in size from tens of nanometers to tens of micrometers.[19, 20, 21] In atomic physics, optical tweezers have found applications in cooling atoms to record low temperatures and trapping atoms at high densities.[13, 22, 23] To implement the optical tweezers for achieving a stable trap, one requires a highly focused and strongly convergent laser beam, [13] which is often realized through a microscope system and is limited by the working wavelength and numerical aperture (N.A.). To manipulate or “tweeze” particles in a large field of view, the system is required to be devoid of field curvature. However, high N.A. and small field curvature are often incompatible in a conventional optical system. In practice, optical tweezers are very expensive, custom-built instruments that require a working knowledge of microscopy, optics, and laser techniques. These requirements limit the application of optical tweezers. However, the invention of the negative-refraction flat lens could potentially overcome the shortcomings of conventional optical systems and further advance this technique. The combination of super-resolution (high N.A.) and lack of an optical axis provide a superbly focused beam in a system with translation symmetry (absence of field curvature) and enable an optical tweezers array using a single lens. In this article, we validate this concept in the microwave regime by demonstrating the application of negative-refraction flat lens based on 3D PhC to manipulate small objects.

2. Three-dimensional subwavelength imaging by full 3D negative refraction

 

Fig. 1. (a) Three-dimensional PhC fabricated layer by layer (20 layers in total). The inset shows a conventional cubic unit cell of the bcc structure. (b) Band structure of the bcc lattice PhC.

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The PhC flat lens employed in this work, as shown in Fig. 1(a), has been reported in our recent publications.[10, 11] The flat lens is made of a body-centered cubic (bcc) PhC with the unit cell as shown in the inset of Fig. 1(a). Low loss microwave material with dielectric constant 25 was used to fabricate the PhC in a layer-by-layer process (there are 20 layers in total, and each layer has a thickness of 6.35mm). Negative refraction is obtained by properly engineering the dispersion properties of the PhC, which are best shown using a photonic band diagram, see Fig. 1(b). In the photonic band diagram, group velocity is found by calculating the gradient of the frequency in k-space (wavevector space), i.e. v _g = 2π∇_k f. Dispersion curves of regular materials have a group velocity with a positive radial component, resulting in k ∙ v _g > 0. However, the dispersion curve at the top (15.6GHz~17.0GHz) of the third band of our PhC shows that frequency decreases with ∣k∣ increasing, resulting in k ∙ v _g < 0. In other words, phase velocity is opposite to group velocity for a given electromagnetic wave as it propagates in the 3D PhC within this frequency range. The result is negative refraction. The constant-frequency surface is nearly spherical for a frequency in this range, which makes full 3D negative refraction possible.[24] In our previous work,[10]we have demonstrated that the flat lens is fully functional and produces images with subwavelength feature size in all three dimensions.

 

Fig. 2. Image of the optimized monopole source achieved through the 3D PhC flat lens at 39mm away from the source (the lens is 25mm thick). The outlined region shows the area of the image where the intensity was>0.5 max.

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To create a highly efficient point source, we constructed a monopole using a section of a coaxial cable (50 ohm, 1.8mm in diameter) by first exposing a long section of core at the end, and then progressively cutting it until optimal radiation efficiency was achieved in the frequency range of interest. We placed the monopole source at the lens center, 1 mm away from its surface. A monopole detector was employed for characterizing the image formed by the flat lens. Both the source and the detector were connected to a vector network analyzer (VNA). By spatially scanning the detector, we obtained good images in the 15.9GHz~17.1GHz frequency range. Figure 2 shows the image of the source when the working frequency is set to f=16.3GHz. The image size is found to be only 6.5 mm in diameter or 0.35λ (wavelength λ=18.4 mm) using the full width at half maximum (FWHM) criterion. The image is located 39 mm away from the source (the flat lens is 25 mm thick). By adjusting the image distance ±2 mm, good images with diameter 6.0mm~7.0mm for other frequencies in this range (15.9GHz~17.1GHz) are also observed.

3. Trapping and manipulation of particles

In the Rayleigh scattering regime (λ >> r, where r is the radius of the particle.), the radiation force acting on a dielectric particle can be explained as the interaction between the polarized particle and the applied electric field. The radiation force produced by a focused beam has two components: scattering force and gradient force. Optical tweezers rely on the gradient force, which is proportional to the dipole moment of the particle and the gradient of power density.[25] For a spherical particle in a dielectric liquid medium, the total dipole moment can be shown to take the form $\mathbf{p}=4\pi r^3{\epsilon }_b\left(\frac{{\epsilon }_a-{\epsilon }_b}{{\epsilon }_a+2{\epsilon }_b}\right)\mathbf{E}$,[26] where ε_a and ε_b are the dielectric constants of the particle and the medium, respectively, and E is the applied electric field. For simplicity, we approximate the beam focused by the flat lens as a Gaussian beam. In this case, the maximum gradient force is $F_{\mathit{grad}}\propto r^3\sqrt{{\epsilon }_b}\left(\frac{{\epsilon }_a-{\epsilon }_b}{{\epsilon }_a+2{\epsilon }_b}\right)\frac{P}{W_0^3}$, [27] and the resulting gradient acceleration is $a\propto \sqrt{{\epsilon }_b}\left(\frac{{\epsilon }_a-{\epsilon }_b}{{\epsilon }_a+2{\epsilon }_b}\right)\frac{P}{W_0^3}$, where P is the power and W ₀ is the diameter of the beam waist. Since the acceleration is inversely proportional to the cube of the beam width, squeezing the beam size is a very efficient way to increase the acceleration, and thus improve the particle trapping.

 

Fig. 3. The schematic of the basic apparatus used for the microwave tweezers. The inset shows the polarization of the electric field with regard to the PhC.

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To this end, we established our experimental setup as illustrated in Fig.3. A 10-watt amplifier is employed to amplify the electromagnetic waves from a local oscillator, which, in our case, is a vector network analyzer. The source monopole is connected to the output port of the amplifier through a coaxial cable and an isolator to prevent back-reflection. The flat lens is placed 1 mm above the monopole with the orientation as shown in the inset. A 10-mm air gap is formed using a thin petri dish and the sample is contained in another petri dish. Both petri dishes are optically transparent, so we can see the sample and the flat lens at the same time. By tuning the frequency, the focused image of the monopole source can be located directly at the bottom of the sample dish. A stereomicroscope with a digital video camera was employed to record our experimental results.

The sample used in our experiment consists polystyrene particles dispersed in a liquid medium, dioxane (1, 4-dioxane: C4H8O2). Dioxane has dielectric constant ε_b=2.1, compared to polystyrene ε_a=2.6; the inequality ε_a>ε_b ensures the presence of a trapping force. More importantly, dioxane molecules are nonpolar and the material is transparent at microwave frequencies and therefore exhibits very low absorption—we measured its loss tangent to be 2×10^-3 in the 16.0 GHz~17.0 GHz frequency range. In addition, the density of dioxane is 1.035g/cm³, which is very close to that of polystyrene, 1.04g/cm³. This helps in decreasing the effect of gravity and reduces the friction of particles that have sunk to the bottom of the container.

 

Fig. 4. (a,b) Microwave electromagnetic trapping of neutral particles through a negative-refraction flat lens [Media 1]. (c, d) Microwave electromagnetic dragging of neutral particles along the vertical axis [Media 2]. Based on the result shown in Fig. 4(a,b), the monopole source has moved 10mm along the vertical axis. (e,f) Microwave electromagnetic dragging of neutral particles along the horizontal axis. Based on the result shown in Fig. 4(c,d), the monopole source has moved 6mm along the horizontal axis [Media 3].

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To further demonstrate the trapping and the ability to manipulate particles, we translated our monopole source 10.0 mm along the vertical axis without moving the lens itself. Figure 4 (c, d) and Movie 2 show that the cluster first splits into small groups, which are then pulled to a new position where the cluster is reconstructed. In this process, the particles are dragged 9.2 mm along the vertical axis. When we translated our monopole source 6.0 mm along the horizontal axis, the particles were again following the monopole image, as shown in Fig. 4 (e, f) and Movie 3. Notice that the cluster size changes little as it moves over long distances. This is expected—because the flat lens has translation symmetry for imaging and is devoid of a unique optical axis, as a result there is no field curvature and the quality of the monopole image is independent on its lateral position. This is a very advantageous property for trapping as it also allows for effective manipulation without lens movement. In addition, the flat lens provides a novel mechanism for using gradient forces not only as traps but also as arrays of manipulations (optical tweezers) through a single lens— for example in fluidic systems particles can be manipulated by arrays of sources turned on and off sequentially to affect their movement. In contrast, in a conventional system arrays of optical tweezers are realized by arrays of lenses, which have the limitations of a fixed array pattern and element spacing restricted by the lens size. [14]

 

Fig. 5. Microwave electromagnetic trapping of neutral particles with different sizes. The average diameter of the large particles and that of the small particles are 100μm and 600μm, respectively. Two circles are used to track the motion of particles. [Media 4]

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4. Discussion

The power fed to the monopole is measured to be 7.0 watts in the 16.0GHz~17.0GHz frequency range. Since about 50% of the radiated power is focused to form the image (the other half radiates away from the lens), the total power on the image side is roughly 3.5 watts. Using W ₀=6.5 mm in the Gaussian beam approximation, we calculated the maximum acceleration due to the gradient force to be 40 μm/s², while that due to the scattering force is less than 1 nm/s², which is negligible. When a particle is accelerated, a retarding force to the motion will occur due to the viscosity of the liquid—the higher the speed, the stronger the retarding force. As a result, the particle reaches terminal velocity, which can be estimated by Stokes’ law, and in our case it is expected to be less than 1 μm/s. However, the maximum speed we observed is considerably higher than this estimate. This discrepancy may be attributed to the interaction among a large number of particles, i.e., a collective dragging effect similar to drafting known and exploited by drivers/riders in competitive car/bike racing.

To investigate the reason for the unexpected velocity increase more closely, we tested another sample consisting of polystyrene particles with two different sizes, Ø100μm and Ø600μm, mixed together and dispersed in 4-mm-deep dioxane. We repeated the experiment, and recorded the motion of the particles as shown in Fig. 5 and Movie 4 where it is clear that both types of particles are attracted to the center. However, it is also clear that large particles move faster than small particles. This result supports the hypothesis of electromagnetic trapping: if the motion of the particles were driven, for example, by the thermal convection of the liquid, then their speed would be independent of the particle size. In contrast, the electromagnetic trapping force on a particle is proportional to r ³ while the retarding force is proportional to vr (according to Stokes’ law), where v is the speed of the particle and r is its radius. The balance of the trapping force and the retarding force results in the terminal speed v_t ∝ r ². In other words larger particles will move faster through viscous fluid when driven by electromagnetic tweezers force. In the case of small particles, as several of them cluster together, they behave as a single larger particle with the corresponding increase of terminal speed. Movie 2 reveals this effect clearly: larger groups move faster than smaller groups during the migration process.

5. Conclusion

In conclusion, we experimentally demonstrated microwave electromagnetic trapping and manipulation of neutral particles using a negative-refraction flat lens. The advantages of the flat lens, including super-resolution and the absence of field-curvature, play an essential role in improving the overall performance of such electromagnetic tweezers. Although this experiment was carried out in the microwave regime, the technique is suitable for optical wavelengths, where the gradient acceleration is expected to increase by twelve orders of magnitude according to the formula $a\propto \sqrt{{\epsilon }_b}\left(\frac{{\epsilon }_a-{\epsilon }_b}{{\epsilon }_a+2{\epsilon }_b}\right)\frac{P}{W_0^3}$ when the same level of power is applied. However, in optical regime the challenge lies in the fabrication of such a full-3D negative-refraction flat lens. Recent progress in 3D micro-fabrication is promising to overcome this challenge.[28, 29, 30] For optical wavelengths, the flat lens would become a thin patterned film, which would provide a focused beam with size limited only by the size of the source. It is expected that the use of a flat lens for optical tweezers will give rise to more widespread application of the technique.

Further work will demonstrate that the migration route of particles can be controlled by a source array. Neither physical motion on the sources nor on the lens is required to manipulate the particles. We will report this in the future.