We present a novel optical element that efficiently generates orbital angular momentum (OAM) of light and transforms light between OAM modes based on a polarization grating with a fork-shaped singularity. This forked polarization grating (FPG) is composed of liquid crystalline materials, and can be made either static or switchable with high diffraction efficiency (i.e., 100% theoretically) into a single order. By spatially varying the Pancharatnam–Berry phase, FPGs shape the wavefront and thus control the OAM mode. We demonstrate theoretically and empirically that a charge FPG creates helical modes with OAM charge when a Gaussian beam is input, and more generally, transforms the incident helical mode with OAM charge into output modes with OAM charge . We also show for the first time that this conversion into a single mode can be very efficient (i.e., experimentally) at visible wavelengths, and the relative power between the two possible output modes is polarization-controllable from 0% to . We developed a fabrication method that substantially improves FPG quality and efficiency over prior work. We also successfully fabricated switchable FPGs, which can be electrically switched between an OAM generating/transforming state and a transmissive state. Our experimental results showed conversion efficiency for both configurations at 633 nm. These holographically fabricated elements are compact (i.e., thin glass plates), lightweight, and easily optimized for nearly any wavelength from ultraviolet to infrared, for a wide range of OAM charge, and for large or small clear apertures. They are ideal elements for enhanced control of OAM, e.g., in optical trapping and high-capacity information.
© 2012 Optical Society of America
It has long been recognized that photons carry spin angular momentum that corresponds to polarization. During the last two decades, research has revealed that lightwaves can carry orbital angular momentum (OAM) as well, which is associated with the helical characteristic of the phase front . It was first suggested and experimentally demonstrated by Allen’s group that lightwaves with an azimuthal angle-dependent phase term carry OAM of per photon, where can take any integer value. Since this discovery, OAM has attracted intense attention and has been studied in various applications, including particle manipulation , microfabrication , and quantum information sciences [4,5]. However, the generation, manipulation, and detection of OAM is still challenging; in particular, very few methods have been found that can generate and transform OAM states efficiently, compactly, and with generous tolerance to input polarization, incidence, and OAM purity.
Here we present a highly efficient OAM generator and controller based on polarization gratings (PGs). PGs are a category of diffraction gratings that are formed in anisotropic materials, and function by affecting the polarization of incident light in such a way as to control the Pancharatnam–Berry phase [6,7]. This in-plane wavefront shaping occurs within a thin anisotropic layer and leads to unique behavior: 100% diffraction into a single order for wide angular acceptance and a wide range of periods. In one subclass, researchers suggested to embody the birefringence profile by spatially aligning liquid crystal (LC) materials , which was later achieved with efficiency by others [9–11]. These PGs are thin and lightweight, operate with extremely high efficiency, and can be made either static or switchable. These attractive properties of PGs have been used in several applications including laser-beam steering , optical filters , attenuators , polarization imaging , and displays [16,17]. Traditional PGs have a one-dimensional spatially varying optical axis that follows . Our new OAM element introduced here is a two-dimensional variation that deftly manipulates OAM while otherwise preserving the advantageous features of traditional PGs.
Seo et al. have investigated two possible static PG structures as OAM beam generators and have shown mixed experiment results . The first is a reflective, reverse twisted nematic PG that diffracts always into multiple orders, wherein which they noticed the first-order diffraction manifests OAM. Because of the undesired zero and high-order diffraction, this structure has very limited efficiency and utility. The second structure is a transmissive PG made by two adjacent diffraction orders from a forked computer-generated hologram (CGH). They demonstrate that their samples can convert a Gaussian beam into an OAM beam, and that the OAM mode is sensitive to the polarization handedness. However, their preliminary report suggests very low diffraction efficiency and high scattering losses. Furthermore, they only discuss OAM generation from a Gaussian wave, and ignore the potential for conversion between OAM modes in general. All these issues make their structures insufficient for a practical mode converter.
Here we report on forked polarization gratings (FPGs), which can not only generate OAM from a Gaussian wave (i.e., with ) input, but can also convert incoming OAM modes into higher or lower modes (Fig. 1) with high efficiency. Some of our preliminary results have been presented on these two conferences [19,20]. In this work, we go substantially beyond our initial reports [19,20] of the basic concept, by providing a more clear and concise theoretical analysis, more complete experimental results, and much more comprehensive characterization. In particular, high-quality polymer FPGs for visible wavelength are demonstrated for the first time. We also discuss the issues with the current fabrication process and directions of the future improvement.
2. Forked Polarization Grating
FPGs are a two-dimensional variation of the traditional one-dimensional PG, with a spatially varying birefringence profile whose optical axis follows
An FPG is essentially an inhomogenous waveplate, with its optical axis spatially varying in the plane; it reshapes the wavefront by locally modifying the Pancharatnam–Berry phase of the incoming light. This modulation can be achieved by spatially varying birefringence and/or dichroism and is polarization sensitive, as shown in Fig. 1. Among many possible types and creation methods, in this work, we utilize the “circular PGs”  created by polarization holography and recorded on LC materials with positive birefringence. Figure 2 shows FPG structures and two examples of FPGs. Here we only show the center areas since they are of the most interest. The arrows indicate the local optical axis, which is also the local nematic director in LC-based FPGs. Colored background is also used as an indication of the optical axis orientation, where purple (red) corresponds to the optical axis parallel to the axis. It clearly shows the “forks” at the FPG center. It also predicts the throughput distribution of FPGs when viewed under a polarizing microscope.
A. Theoretical Behavior of FPGs
The well-known Jones matrix of a waveplate can be directly written for the FPG using Eq. (1), which describes the spatially varying orientation of the local optical axis:
Note that is the net phase change (via Pancharatnam–Berry effect) caused by the FPG, for the LCP and RCP components [the second and the third terms in Eq. (3)], respectively. Therefore, an FPG with charge should add or subtract an azimuthal phase variation of about the singularity, and thus transform light OAM to a state higher or lower.
To further observe this transformation, consider a circularly polarized incident beam with electric field , which has OAM charge , RCP/LCP polarization, and unity magnitude. The near-field electric field immediately after the FPG is
Three waves can be identified in this output, by inspection. The linear dependence on in the second term results in a diffracted wave (symmetric directions for the two circular polarizations), whereas the first term is the directly transmitted wave. The azimuthal dependence in both terms suggests helical wavefronts, which persist in each wave under paraxial conditions. The far-field electric field in each diffraction order , can be summarized as5). However, if the input is polarized in any other way, the diffraction will be the superposition of the circular cases, containing all the three orders, , , and .
The net power transfer to any of these diffraction orders is most easily described by a ratio of output to input intensity , called the diffraction efficiency:
- 1. The output of an FPG has only three possible diffraction orders: and . The relative power balance between the zero and the first orders depends only on the retardation . The first orders reach 100% efficiency when the retardation is halfwave (i.e., ), and all of this light is directed into a single first order for circularly polarized input. This behavior is the same as traditional one-dimensional PGs.
- 2. The first-order outputs () have their OAM charge changed by , such that and , as can be observed in Eq. (5). The linear momentum is also affected, as these propagate along first-order diffraction directions set by the classic grating equation (), and will include the normal grating dispersion. The first orders are always circularly polarized, orthogonal with each other, and orthogonal to the circular polarization present in the input.
- 3. The zero-order output () has the same OAM, polarization, and direction as the input beam. Note that both the output OAM and polarization behavior of the FPG output are independent of the retardation.
It is worthwhile to note that the first-order output of the FPG transforms all three types of momentum simultaneously: OAM, spin angular momentum (i.e., polarization), and linear momentum (i.e., wave vector). This is very rare to occur in a single element, and can be preferable in some applications.
B. Numerical Simulation of FPGs
As a first step to empirically validate the theoretical behavior of FPGs, we used numerical simulation of the Jones calculus above. For various inputs and FPG characteristics, we calculated the near-field output electric field via Eq. (4) on a two-dimensional grid. We then used Fourier transforms to simulate beam propagation and examined the resulting far-field electric field. Because the three possible diffraction orders of the FPG spatially separate in the far field, we can observe amplitude and phase information of each separately. The results from several simulations where the FPG had halfwave retardation are shown in Fig. 3, with both linear and circular polarization inputs. The doughnut-shaped intensity is a characteristic of an OAM beam, with the dark center radius increasing with its charge. Phase plots in the bottom rows show the OAM charge of each output beam. Each light-to-dark transition presents a phase difference. In the phase plots of the cross sections normal to the diffraction wave vectors, the direction of this transition going about the center decides the sign of phase change and therefore the sign of OAM charge. In the bottom phase plots, which are views at an oblique cross section, OAM charge can be observed through the number of the fork-shaped branches. The results confirm the theory above—most notably, the OAM charge of the first orders is changed by the charge of the FPG, and 100% of the input is diffracted into a single first order when the input is circularly polarized.
3. Fabrication Methods
The fabrication of FPGs involves polarization holography and, in our approach, axial waveplates called -plates [21,22] acting as helical wave generators. We will discuss these both in turn, and then discuss the detailed fabrication of the FPG itself.
A. Polarization Holography
We fabricated the FPGs by writing a polarization hologram into photo-alignment materials. Polarization holography records the polarization standing wave of the interference of two beams, rather than the intensity (bright/dark) fringes. Two coherent beams with orthogonal circular polarization are superimposed with a small angle between them, which creates a spatially varying standing wave in the polarization pattern, with constant intensity everywhere. For two equal-power plane waves, the resulting field is linearly polarized everywhere, with a one-dimensional periodic linear modulation of its polarization orientation [6,8].
B. Liquid-Crystal and Photo-Alignment Principles
While polarization holograms can be recorded and developed in many polarization-sensitive materials, we use a thin layer of photo-sensitive polymer called linear photo-polymerizable polymer (LPP)  on a glass substrate. This establishes a spatially varying alignment condition in the LPP that follows the local linear polarization direction of the standing wave. When LC encounters this LPP profile, the latter aligns the former to achieve the desired optical axis orientation. For a static (i.e., passive) configuration, we spin-coat a reactive mesogen mixture, and after it aligns to the LPP we photo-polymerize it into a solid cross-linked film. This processing is based on the procedure detailed in . For a switchable (i.e., active) configuration, the fabrication process is as follows: first, ITO-coated substrates are spin-coated with the LPP and then assembled into a cell with a fixed thickness set by glass bead spacers; second, the cell is exposed to the polarization hologram; third, this cell is heated and filled with nematic LC in its isotropic phase and then cooled down to room temperature.
C. -Plate Fabrication
To record FPGs, we need one or two of the writing beams to carry helical wavefront(s). In this work, we realized this by inserting inhomogeneous waveplates called -plate(s) in the optical path(s) . A -plate is essentially a radially symmetric halfwave plate. It has azimuthal varying optical axis in the transverse plane that follows . In our project, we realized the azimuthal-varying photo-alignment by relative rotation between the sample and the light polarization orientation, as a revised setup in . Specifically, we focused the linearly polarized light from an HeCd laser using a cylindrical lens into a slim strip on the sample and simultaneously rotated the cylindrical lens and the substrate at respective speeds during the exposure. Since we will use them in the UV polarization holography, we optimized their retardation for halfwave condition at 325 nm wavelength, with the anisotropy recorded by the same materials that we used for the FPGs.
D. FPG Fabrication
We adapted the conventional PG fabrication setup for FPG fabrication by inserting a -plate into the holography optical path [see Fig. 4(a)]. To avoid the low-intensity center caused by the OAM singularity during light propagation, we minimized the distance between the -plate and the sample. This resulted in a uniform intensity distribution to align the photo-sensitive polymer, leading to good alignment over the whole sample. This is crucial to FPG quality, since the center area is where the fork-shaped singularity will be formed. In order to keep the -plate close to the sample, it has to be in the path of both writing beams, as shown in Fig. 4(a). As a result, two singularities will be recorded on the sample. That is, for every singular point we record, there will be a doubled singularity side by side on the same sample [Fig. 4(b)]. While this may be a concern for some contexts, for this work it was not, since the -plates we used were of low charges and the noncenter part of them resembled a normal halfwave plate. We therefore considered the result as two FPGs on one sample, with singularities located several hundred micrometers or several millimeters apart. Since our area of interest is confined to the region around one singularity, it is equivalent to polarization holography with a -plate and a halfwave plate in the two writing arms, respectively, as shown in Fig. 4(c).
The fabrication recipe was tailored as following: we exposed LPP (DIC: LIA-C001) under polarization holography by an HeCd (325 nm) laser with . The LC layer was always arranged into a halfwave thickness (i.e., here for 633 nm), depending on the configuration as follows. For the polymer FPGs, we used the reactive mesogen mixture from Merck (RMS10-025) with birefringence. We coated a first layer with diluted LC (25%) at 1500 rpm for 30 s, then a second layer with 100% LC at 630 rpm for 45 s. For the switchable FPGs, we filled LC material MDA-06-177 ( at 589 nm, EMD Chemicals) into a gap cell.
4. Experimental Results
A. LC Alignment Fidelity
We first verified that our samples have the desired optical axis alignment. Figure 5 shows the polarized optical microscope pictures of the FPGs fabricated as described in the previous section. The two polymer FPGs are of charge [Fig. 5(a)] and [Fig. 5(b)]. In both cases, the pictures are centered at the singularity to show the variation. In sample areas other than the singularities, the alignment is uniform in one dimension with a period of 16 μm in the other dimension, just like a PG [shown in Fig. 5(c)]. Comparing to Figs. 2(c) and 2(d), we see the alignment is a good match with both theory and simulation. Our fabrication method is valid as discussed in Section 3.D. For the charge FPG, we also show its location relative to the doubled singularity in Fig. 5(d). The two singularities are measured apart on this sample. A switchable sample is shown in Figs. 5(e) and 5(f). The same overall periodic pattern and a bifurcation are present. Some of the point defects are the spacers in the cell gap. With a voltage of 15 V () applied, the grating structure fades out and disclination lines appear over large length scales [Fig. 5(f)]. This phenomenon suggests the LC molecules are mostly aligned perpendicular to the substrates as expected. This state of a switchable FPG is defined as the “on” state and the FPG at this state is addressed to be on its transmissive mode. Correspondingly, the state with zero applied voltage is defined as the “off” state and the FPG is addressed to be on its generating/transforming mode.
B. Far-Field Spectral and Diffraction Behavior
We then investigated the far-field spectral and diffractive behavior. To measure the spectra, we used a collimated, unpolarized broadband light source with small beam size along with an integrating sphere connected to a fiber spectrometer (Ocean Optics) to isolate individual diffraction orders. The ratio of the power measured within a diffraction order to the input power is the transmittance, and corresponds to the total net throughput into that order. In all our samples, this varied from 95% to 99% in the visible range. However, as with many diffraction grating analyses, it is more useful to consider the ratio of the power measured within a diffraction order to the total power transmitted into the output hemisphere, a quantity called the diffraction efficiency. This isolates the diffractive behavior of the PG itself in the experimental measurements and allows direct comparison to Eq. (6) by normalizing out the minor effect of the substrate, interface reflections, and absorption, but preserving the effect of any scattering.
The diffraction efficiency spectra of zero order, first order, and total first orders of a polymer FPG are shown in Fig. 6. Two aspects are particularly important. First, as predicted in PG theory [Eq. (6)], the zero-order and first-order efficiency vary with wavelength. Their valley and peak locations verify that the birefringence of our sample gives halfwave retardation at wavelength of 633 nm, which was our target. Second, where the normalized retardation at 633 nm, most light is diffracted into the two first orders. Very little power is leaked into zero order or high orders. In this case, the input light is unpolarized; each of the two first orders equally obtain half of the total power. This is very close to the theoretical values, and . The small difference is due to scattering and very dim higher orders, which may be caused by material point defects or slight alignment mismatch.
Switchable FPGs on generating/transforming mode were measured to have the same zero-order and first-order efficiency spectra as the polymer FPGs. The same switchable FPGs on transmissive mode become nearly isotropic and nondiffractive; thus a zero-order efficiency across the measured wavelength range was obtained as expected .
C. Polarization Response
Here we describe the polarization behavior of FPGs at their optimized wavelength. We used a collimated beam from an HeNe laser (633 nm) as the normally incident light, and received the diffraction from the FPG by a photodetector with integrating sphere in the far-field region. The laser beam has a waist of 0.5 mm, so that we can shine it at one FPG on our sample without involving the other singularity. As shown in Table. 1, most of the transmitted light () is diffracted to the lowest three orders, among which the zero order was much weaker than the first orders. Around 2% of the light goes to higher orders; the remaining power () is lost by scattering. The power distribution between the two first orders is dependent on the polarization of the incident light as predicted by Eq. (6b). Left circularly polarized light (, ) is mostly diffracted to the first order (, ), whereas right circularly polarized light (, ) is diffracted to the negative first order (, ). When the incident light is linearly polarized, which is equivalent to equally composed right and left circular polarization (), the total power is equally distributed into the two first orders (). Unpolarized input works the same way, as the case in Fig. 6 showed. Thus, the FPGs are demonstrated to have the polarization sensitivity predicted by theory and simulation.
We also graphed these data points in Fig. 6 (black squares) to compare them with the data obtained by the spectrometer. The first-order diffraction efficiencies appear slightly lower and the zero-order leakage higher for the HeNe laser than those from the broadband light source. We think the reason for the difference is beam-size related. The laser beam was much narrower than the white light and covered a smaller area of the FPG. Thus the singularity on the FPG took a higher percentage of the whole effective area. Since the singularity is essentially a kind of defect, we expect more loss from the laser.
D. OAM Mode Transformation Behavior
To examine the OAM charge of each individual diffracted beam, we set up a Mach–Zehnder interferometer with the HeNe laser (633 nm) and captured the far-field interference pattern with a CCD camera. The objective beam is each order of the diffraction from FPG; the reference beam is an expanded Gaussian wave. We used the direct output mode from the laser, which was Gaussian with waist of 0.5 mm as input for the FPG. This beam size ensured that only one singularity on each sample was lit.
Figures 7(a)–7(d) show the results for a linearly polarized Gaussian input beam and a charge FPG. The interference patterns show that the three orders of diffraction have different OAM charges, which are indicated by the different fork-shape topological charges. Thus, OAM modes of are generated at the diffraction. Meanwhile, the very weak order leakage remains Gaussian. Figures 7(e)–7(h) show the results for an OAM charge input light and the same FPG. The diffracted light transforms to OAM mode of charge , which is the input mode lowered by 1; diffracted light transforms to OAM mode of charge , which is the input mode raised by 1; and leakage remains the same mode, . Note the topological resemblance between these patterns and Fig. 3(e). Easily from the interference principle, the result matches our FPG formula and simulation, aside from some distortion that we will discuss in a later section.
E. Voltage Response of Switchable FPGs
For the switchable configuration, the samples work the same way when no external voltage is applied. To verify the diffraction properties when switching, we did the same characterization with several applied voltages using an HeNe laser that has the targeted wavelength 633 nm. Table 2 and Fig. 8 show the diffraction efficiency and OAM analysis of a switchable FPG for three states: “off”, “on” and an intermediate state (details in ). The result agrees very well with our theory in Section 2.A. First of all, the OAM of all three diffraction orders is polarization insensitive, just as the polymer FPGs. Secondly and the most significantly, we testify that the output OAM change is unrelated to applied voltage either. The proof is that the interference patterns at different applied voltages are topologically identical (Fig. 8, bottom row).
We then did comparative measurements of the voltage response of the diffraction efficiency at two wavelengths: the optimized wavelength of the sample, 633 nm, and a control wavelength, 532 nm. The zero-order and total first-order diffraction efficiencies at 633 nm are shown in Fig. 9, as the solid curves with blue and red, respectively. The squares and circles are actual measured data points. In the same manner, cyan and magenta dashed curves with diamond and circle data points are the zero-order and total first-order diffraction efficiencies at 532 nm. As expected for most LC materials, all these diffraction efficiencies start to flip at a threshold voltage and reach saturation at some high voltage. An interesting observation is that for the nonoptimal wavelength 532 nm, its diffraction efficiencies reach maximum and minimum at a nonzero applied voltage . The values are and for zero order and total first orders, which are very close to the values of 633 nm with zero applied voltage. This result suggests a useful feature of switchable FPGs: a sample that is optimized for a certain wavelength at its “off” state can be tuned for other wavelengths by simply applying a small external voltage. In other words, switchable FPGs can achieve broadband high efficiencies through electrical tuning.
The raise of the fall switching times of the switchable FPGs was measured with an HeNe laser and an oscillating square voltage. With detailed measurements shown in , here we highlight the fast total switching time of all samples, which is on the order of 3 ms. This is about the average value of a conventional homogeneous halfwave plate configured with the same type LC materials. Therefore, our FPGs preserve the good electro-optical switching fashion of LC elements.
FPGs with good OAM mode transformation and efficiency have been fabricated. The key enabling our fabrication success is utilizing the classic polarization holography method, which resulted in excellent LC alignment without defects and minimal scattering. The resulting FPGs exhibit all the properties predicted by theory and simulation. Power measurements of the diffraction orders show our current FPGs have very good diffraction efficiencies (92%–95%). We believe by adjusting the recording setup and fine-tuning the processing, it is possible to achieve diffraction efficiency . A layer of antireflection coating on the substrate will help improve the absolute transmission as well. This means that these FPGs could modulate the OAM state of light with very little power loss.
For switchable FPGs, the applied voltage controls the efficiency ratio between zero-order and the total first-order diffraction, which is identical to the power ratio between the unchanged and changed OAM. Thus, by adjusting applied voltage, we can control the percentage of light being transformed, from entirely transformed (100%) to not transformed at all (0%). And for every voltage (percentage of transformation), the incident polarization determines the efficiency ratio between and diffraction, that is, how much of the transformed light goes to a upper OAM state and how much goes to a lower OAM state. As a result, by combining voltage and polarization controls, a switchable FPG can amazingly control these three output OAM states: the original OAM, the OAM state lowered by , and the OAM state raised by , with arbitrary power ratio from 0% to . This is very unique among all the current OAM controlling techniques.
The dynamic response of a switchable FPG is fast, with a total switching time of , which is on the order to a conventional nematic LC halfwave switch. For certain applications, switchable FPGs can make low-cost, flexible, and compact substitution for SLM systems.
The special diffractive property of FPG makes it a significantly better OAM mode transformer in some circumstances, compared to its close kin -plate. Unlike -plates, which only work well with circularly polarized light, FPGs work well with arbitrary polarization. The fork-shaped birefringence pattern modulates the wavefront through Pancharatnam–Berry phase and results in both linear momentum and OAM change. A pure helical input will always be transformed to pure helical beams. For instance, if an OAM eigenstate is needed from a Gaussian source, we need to make sure the input light is perfectly circularly polarized before sending it through a -plate. Otherwise, the output will be a superposition of and modes. Additionally, the -plate should be exactly halfwave, or the leakage wave (with unchanged OAM) will add to the output as well. Although supplementary polarization filters could be used, this adds to the complexity of the system. Alternatively, with FPGs, even if the input is not perfectly circular or is not the optimal wavelength, we will always get a pure OAM eigenstate at the first-order diffraction. The undesired modes are automatically filtered (i.e., diffracted into other directions). This feature makes them superior in applications where purity of a single eigenstate is preferred.
Challenges remain as well. One might notice the little distortion in some of the interference patterns of the high-charge beams, such as in Fig. 7(g). The distortion indicates that the charge singularity tends to split into two unit charge ones. We know this can happen when the beam is not perfectly symmetric , due to perturbation or ellipticity of the beam itself. In our case, the imperfect alignment of the FPG may have caused this problem. A close look at Fig. 5(b) suggests that the anisotropy is slightly different from the vector plot in Fig. 2. One possible reason relates to the optical axis patterning process. For a high-charge OAM beam, the center area carries very high topological strength, which means the photo-alignment material we exposed to the beam will experience more difficulties in anchoring accordingly. This happens for both -plates and FPGs. This could be improved by increasing exposure fluence or adjusting the processing methods. We will leave the comprehensive diagnosis and improvement for our further work.
With high-quality FPGs now realized, many applications of FPGs can be considered; instead of spiral phase plates or CGHs, one can use FPGs to easily generate helical light modes, which can be used in particle manipulation or microfabrication. We can expect easier fabrication and simpler control in these fields with the use of FPGs.
We have proposed a polarization-controlled OAM mode generator and transformer called a forked polarization grating and have successful realized it. We demonstrated that with our polarization hologram setup and photo-alignment technique we can make high-quality FPGs that modulate light’s OAM highly efficiently. By adjusting the input polarization and FPG retardation (by applied voltage on switchable FPGs), we are able to achieve a OAM conversion of one beam, or generate three beams with different OAM at the same time and freely control their ratio. Moreover, FPGs modulate the wavefront through Pancharatnam–Berry phase and can spatially separate the outgoing beams with different OAM states. With these competitive advantages, including simple processing, compact size, and light weight, FPGs will greatly promote the development of applications that utilize the OAM of light, especially in high-capacity information sciences.
The authors gratefully acknowledge the support of the National Science Foundation (NSF grant ECCS-0955127).
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